EQUADIFF 2003

Preface.

In this note we discuss the motion of a particle near the Lagrangian points of the real Earth-Moon system. We use, as real system, the one provided by the JPL ephemeris: the ephemeris give the positions of the main bodies of the solar system (Earth, Moon, Sun and planets) so it is not difficult to write the vectorfield for the motion of a small particle under the attraction of those bodies. Numerical integrations show that trajectories with initial conditions in a vicinity of the equilateral points escape after a short time. On the other hand, the Restricted Three Body Problem is not a good model for this problem, since it predicts a quite large region of practical stability. Therefore, we introduce two analytic models that can be written as a quasi-periodic perturbation of the Restricted Three-Body Problem, to try to account for the effect on the Sun and the eccentricity of the Moon. Then, we compute some families of normally elliptic invariant tori at some distance from the triangular points, that give rise to regions of effective stability for the considered models. By means of numerical simulations, we show that these regions seem to persist in the real system, at least for time spans of 1000 years.

A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems are represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. Symmetry of coupled cell systems can lead to synchronized cells. We show that symmetry is not the only mechanism that can create such states in a coupled cell system. The first main result shows that robust synchrony is equivalent to the condition that an equivalence relation on cells is 'balanced'. The second main result shows that admissible vector fields restricted to synchrony subspaces are themselves admissible vector fields for a new coupled cell network, the 'quotient network'.

The existence of solitary-wave solutions to the three-dimensional water-wave problem with strong surface-tension effects is predicted by the KP-I model equation. The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and the KP-I equation admits explicit solutions for three different types of solitary wave. A line solitary wave is spatially homogeneous in the direction transverse to its direction of propagation, while a periodically modulated solitary wave is periodic in the transverse direction. A fully localised solitary wave on the other hand decays to zero in all spatial directions. In this article we outline mathematical results which confirm the existence of all three types of solitary wave for the full gravity-capillary water-wave problem in its usual formulation as a free-boundary problem for the Euler equations.

We review Mather theory, weak KAM theory and viscosity solutions of Hamilton-Jacobi PDE's in an interconnected way.

Singular and non-smooth constrained evolution problems in mechanics often lead to evolution (quasi-)variational inequalities for regulated functions with values in a Hilbert space X. The goal of this contribution is to show that conversely, the technique of variational inequalities and hysteresis operators in the Kurzweil integral setting can be used for showing the rich topological structure of the space of regulated functions including two independent weak convergence concepts.

Examples of various types of chaos are described in some physical systems:

(1) Subshifts of second species orbits in the circular restricted three body problem (with S.V. Bolotin);

(2) Anosov energy levels for the frictionless dynamics of a mechanical linkage, and a normally hyperbolic Anosov submanifold for weak friction and suitable feedback control (with T.J. Hunt);

(3) "Ergodic pumping": a proposed mechanism for the power stroke of myosin, and a design principle for nanobiotechnology (with D.J.C. MacKay).

In this note we outline recent developments in the study of mixed-type, or forward-backward, functional differential equations. Such equations arise naturally in various contexts, for example, in the study of traveling waves in discrete spatial media such as lattices. Since initial value problems for mixed-type equations are not in general well-posed, it is our goal to decompose solutions of such equations as sums of "forward" solutions and "backward" solutions. We show that for autonomous equations, the set of all forward solutions defines a semigroup which can be realized by a retarded functional differential equation except for possibly finitely many modes, and similarly for the set of backward solutions as an advanced functional differential equation. Holomorphic factorizations play a crucial role in our results. We also study the boundary value problem associated to an interval [-τ, τ] of finite length, with emphasis on the asymptotic case of large τ.

A general model for the dynamics of three competitors is considered. The equations can have periodic dependence in time. The conditions for the extinction of one or two species are discussed and they lead to a semi-exclusion principle. The problem can be reformulated in topological language and it is linked to the dynamics of certain homeomorphisms of a two-dimensional disk.

This note is primarily concerned with symmetry properties of solutions of parabolic equations. To put the questions in broader perspective, we first recall several symmetry results on elliptic equations (on bounded and unbounded domains). We then proceed by summarizing earlier theorems on asymptotic symmetry for parabolic equations on bounded domains. Finally, we announce a new theorem on nonautonomous equations on ℝ^N which asserts that positive solutions decaying at spatial infinity are asymptotically radially symmetric about some center. As our discussion of the symmetry problem suggests, dealing with parabolic equations on ℝ^N one is faced with interesting extra difficulties not present in elliptic equations or in parabolic equations on bounded domains.

As it is well-known, the principal difficulty in the study of a planar vector field family is to understand how in the family, the polycycles bifurcate into limit cycles. The reason is that such a bifurcation is in general a highly non-analytic phenomenon. Here, we only consider limit cycles created in unfolding a Hamiltonian vector field. Limit cycles bifurcating from a regular Hamiltonian cycle, are in general directly controlled by the zeros of the associated Abelian integral unfolding. Then, a general question is to understand in what measure the Abelian integral unfolding also controls the limit cycles bifurcating from a Hamiltonian polycycle. First one gives examples of singular cycles containing just a hyperbolic saddle as a unique singular point. In this cases, the bifurcating limit cycles are all related to zeros of the associated Abelian integrals. Next, investigating the case of the 2-saddle cycle unfoldings breaking just one connection, one has observed that the number of bifurcating limit cycles is related to the number of bifurcating zeros, but in general strictly greater than it. Finally if an unfolding breaks more than one connection in an k-saddle loop with k ≥ 2, generically just one zero bifurcates in the integral unfolding but k limit cycles may bifurcate from the loop. This text is essentially based on two recent papers written in collaboration with Freddy Dumortier ¹ and Daniel Panazzolo ² respectively.

No abstract received.

In this paper we relate notions from linear time series analyses, like autocovariances and power spectra, with notions from nonlinear times series analysis, like (smoothed) correlation integrals and the corresponding dimensions and entropies. The complete proofs of the results announced in this paper will appear in the Chemical Engineering Journal.¹

We study the persistence of Poincaré - Treshchev tori on a resonant surface of a nearly integrable Hamiltonian system in which the unperturbed Hamiltonian needs not satisfy the Kolmogorov non-degenerate condition. The persistence of the majority of invariant tori associated to g-non-degenerate relative equilibria on the resonant surface will be shown under a Rüssmann like condition.

We describe a set of algorithms (and corresponding codes) that serve as a basis for an automated proof of existence of a certain dynamical behaviour in a given dynamical system. In particular it is in principle possible to automatically verify the presence of complicated dynamics using these tools. The Hénon map is used to illustrate these techniques.

We illustrate how numerical boundary value techniques can be used to obtain a rather complete classification of certain types of periodic solutions of the Circular Restricted 3-Body Problem (CR3BP), for all values of the mass-ratio parameter.

The presence of viscosity normally has a stabilizing effect on the flow of a fluid. However, experiments show that the flow of a fluid in which viscosity decreases as temperature increases tends to form shear layers, narrow regions in which the velocity of the fluid changes sharply. In this paper, we present evidence that the formation of shear layers is due to an inherent instability in the fundamental conservation laws governing fluid flow when the viscosity decreases sufficiently quickly with temperature.

A saddle-periodic orbit in a delay differential equation can be computed together with its unstable eigenfunctions using the latest version of the continuation package DDE-BIFTOOL. For the case of a single unstable Floquet multiplier we show how this information can be used to compute the one-dimensional unstable manifold of the associated fixed point of the Poincaré map.

Consider a discrete dynamical system, having a reducible invariant curve. Here we present an algorithm for the computation of the Fourier coefficients of the curve and of its reducing (Floquet) transformation, provided we have an approximation to the curve and to the Floquet transformation. The method presents a high degree of parallelism: The computing time is essentially divided by the number of processors, and the computational effort grows linearly with the number of Fourier modes needed to represents the curve. For this reason the method seems a good option to compute quasi-periodic solutions with several basic frequencies.

We discuss a method to compute periodic solutions of delay differential equations that combines the advantages of collocation and single shooting. We present results for a test case to demonstrate the efficiency of the method.

In this paper we study the numerical accuracy of computing one-dimensional manifolds of (non-hyperbolic) equilibria in a planar vector field for which the manifolds are known explicitly. We consider the (strong) stable manifolds of a saddle, a sink, and a centre-stable equilibrium.

The simulation of periodically forced processes in packed bed reactors leads to the development of partial differential equations that are normally solved in time using dynamical simulation. Depending on the convergence properties of the system at hand, the number of cycles that needs to be computed up to a cyclic steady state is reached, can be large. Hence, direct iterative methods are essential. However, to overcome severe memory constraints many authors have reverted to pseudo-homogeneous one-dimensional models and to coarse grid discretization, which renders such models inadequate or inaccurate. Presently, we propose a limited memory method, called the Broyden Rank Reduction method, to simulate a full two-dimensional model with radial gradients taken into account.

Connecting orbits in delay differential equations (DDEs) are approximated using projection boundary conditions, which involve the stable and unstable manifolds of a steady state solution. The stable manifold of a steady state of a DDE is infinite-dimensional. We circumvent this problem by reformulating the end conditions using a special bilinear form. The solution of the resulting boundary value problem using a collocation method is implemented in the Matlab package DDE-BIFTOOL. In a model system for intracellular signalling, we show how the method can be used to obtain travelling wave solutions of a delay partial differential equation.

The purpose of this survey is to briefly summarize some recent advances in the study of water waves with vorticity.

The equations for the resonant four wave interaction describe the approximate dynamics of four resonant wave packets. In this paper we give estimates between the associated formal approximation and true solutions of the original system. We explain that the ideas for the justification of the Nonlinear Schrödinger equation can be transfered to the justification of the equations for the resonant four wave interaction.

In existence theories for water waves usually the inviscid irrotational Euler equations are studied. We make a connection with a theory for viscous waves driven by wind as it was developed by the author in two previous publications. We formulate open questions concerning possible new existence results and a possible classification of inviscid surface waves as viscosity solutions of the Euler equations.

We deal with the existence of Carathéodory solutions for initial value problems for first order ordinary differential equations. Our approach consists in passing from the differential equation to a solvable differential inclusion, and then we look for Carathéodory solutions of the former equation among those of the inclusion.

We study the existence of heteroclinics for a model of fourth order equations related to the extended Fisher-Kolmogorov equation and the Swift-Hohenberg equation. We prove the existence of multitransition heteroclinics. These solutions are obtained as local minimizers of the associated functional.

No abstract received.

This paper deals with boundary value problems for systems of equations whose nonlinear part involves periodic functions and such that the linear part has a one-dimensional solution space. This is the case, for example, of linearly coupled pendulums and systems of equations arising in the theory of Josephson multipoint junctions. We shall study the existence and multiplicity of solutions using various methods of Nonlinear Analysis such as the alternative method (or Liapunov-Schmidt reduction) and methods of critical point theory.

Equations x″ + g(x) = 0, where g(x) is an odd type superlinear function, exhibit large amplitude rapid oscillations. We are looking for conditions, which ensure that solutions of perturbed equation x″ + g(x) = f(t, x, x′) behave similarly.

We present estimates for the number of unbounded branches of periodic solutions in a problem of bifurcation from infinity with degeneration of the linear part of order two and suggest sufficient conditions for the existence of many branches.

We prove existence and non-existence results for two point higher order equation

for f : [0, 1] × ℝⁿ → ℝ and p : [0,1] → ℝ⁺ continuous functions, s a real parameter with the boundary conditions with a, c, A, B ∈ ℝ and b, d ≥ 0 such that a² + b > 0 and c² + d > 0. The proof makes use of a priori estimates, lower and upper solutions technique and coincidence degree theory.

We present new existence results for the nonlinear second order impulsive periodic boundary value problem. They rely on the presence of a pair of associated lower/upper functions. In contrast to the results known up to now, we need not assume that they are well-ordered.

The forced sine-Gordon equation can be considered as a natural extension to partial differential equations of the forced pendulum equation. It is known that, if f is almost periodic and not too large, the pendulum equation has almost periodic solutions. Our aim is to extend this result to the sine-Gordon equation. A crucial tool in the proofs is a recent maximum principle for the telegraph equation. This maximum principle holds up to space dimension three.

No abstract received.

In this work we study the existence of positive solutions for a nonlinear Dirichlet problem involving the m-laplacian. The nonlinearity considered depends on the first derivatives; in such case, variational methods cannot be applied. So, we make use of topological methods to prove the existence of solutions. We combine a blow-up argument and a Liouville type theorem to obtain a priori estimates. Some Harnack type inequalities which are needed in our reasonings are also proved.

No abstract received.

We consider a fourth order quasilinear Sturm-Liouville problem with unilateral constraints. Bifurcation of smooth solution branches as well as smooth dependence of the contact sets of the bifurcating solutions on parameters are shown.

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow-up time and if it blows up again, it can only do so finitely many times.

The second order uniformly elliptic equations with lower-order terms is studied in unbounded domains having a various structure at infinity. The homogeneous Neumann boundary conditions are posed on a non-compact part of the boundary. We investigate the asymptotic behaviour of solutions at infinity. In particular, the localization effect for the support of solutions is studied. The conditions on the growth of solutions at infinity, which cannot be realized, are found.

We analyze the limit of solutions of an elliptic problem, with zero flux boundary conditions when the reaction terms are concentrated in a neighborhood of the boundary and that shrinks to the boundary as a parameter goes to zero. We prove that this family of solutions converges, in the sup norm, to the solution of an elliptic problem with no reaction terms and with a nonhomogeneous flux condition. This non zero flux condition for the limiting problem comes from the concentrated reactions around the boundary of the domain.

No abstract received.

Imagine a carpet that is woven out of nonlinearly elastic strings whose deformation energy is given by with 1 < p < ∞. The corresponding differential operator is reminiscent of the p-Laplace operator , but obviously different from it and it has no rotation invariance. Therefore some standard symmetry results like the radial symmetry of the first eigenfunction if Ω is a ball or the Faber Krahn inequality have to be modified for this operator. In this lecture I explain to what extent such symmetry properties can be generalized and investigate also the limit p → ∞. Most of the results were obtained in cooperation with M. Belloni from Parma and V. Ferone from Napoli.

A quasilinear elliptic equation with unilateral nonlocal boundary conditions is used for explanation of our recent results concerning smooth bifurcation branches for variational inequalities, their direction and stability.

In this paper we obtain existence and uniqueness results for the Dirichlet boundary value problem -Δu = f_α(u + c) in a bounded domain Ω ⊂ ℝ^d, with the nonlocal condition ∫_Ω f_α(u + c) = M. The solutions of the above BVP can be regarded as steady states for some evolution system describing self-gravitating Fermi-Dirac particles.

No abstract received.

It is well known that the cyclicity in one parameter families of planar vector fields can be obtained by computation of Melnikov functions. In this note, the question is addressed if and how one can transfer results obtained in one parameter subfamilies to multi parameter families of planar vector fields.

We consider the families of quadratic systems in the projective plane with algebraic limit cycles of degree 2 or 4. There are no algebraic limit cycles of degree 3 for a quadratic system. Until the moment, no other families of quadratic systems with an algebraic limit cycle, not birrationally equivalent to the ones that we study, have been found. We prove that none of these systems has a Liouvillian first integral. Our main tool is the characterization of the form of the cofactor of an irreducible invariant algebraic curve, when this curve exists, by means of the study of the singular points of the system. For obtaining this characterization of the form of the cofactor we consider the behavior of the solutions of the system in a neighborhood of a critical point.

LaSalle, in his book of 1976 (see Ref. 8), proposes to study several conditions which might imply global attraction of a fix point for a discrete dynamical system x_n+1 = F(x_n), defined in IR^m. Assuming that the fix point is the origin, one of his proposals appears after writing F(x) = A(x)x, and then imposing conditions on the eigenvalues of the matrices A(x), for all x ∈ IR^m. Cima et al. (see Ref. 3) have given an adaptation to ordinary differential equations of these conditions. In that paper the authors study the effect of imposing both conditions, either in the case of discrete dynamical systems or for ordinary differential equations. They also observe that such a decomposition of F(x) is in general not unique. In this note we consider the extension of LaSalle's Condition to ordinary differential equations, when the choice of A(x) is somehow canonical. Concretely, the choice of A(x) that we consider in this work is , where the integration of the matrix DF(x) is made term by term. Unfortunately, our conclusion is that the condition obtained for ordinary differential equations, when the decomposition of F(x) is given by F(x) = A_c(x)x, is just useful to give global attraction in dimension one.

We study the possible existence of first integrals of the form I (x, y) = (y - g₁(x))^α₁(y-g₂(x))^α₂⋯(y-g_ℓ(x))^α_ℓ h(x), where g₁(x), …, g_ℓ(x) are unknown particular solutions of dy/dx = Q(x, y)/P(x, y), α_i are unknown constants and h(x) is an unknown function. For certain systems some of the particular solutions remain arbitrary and the other ones are explicitly determined or are functionally related to the arbitrary particular solutions. We obtain in this way a nonlinear superposition principle that generalizes the classical nonlinear superposition principle of the Lie theory. In general, the first integral contains some arbitrary solutions of the system but also quadratures of these solutions and an explicit dependence on the independent variable, see¹.

We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact α–or ω–limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then K lies in the zero set of V.

In this note we get several results on the number of limit cycles of a subfamily of the so called rigid or uniformly isochronous systems. Recall that in polar coordinates these systems are such that the derivative of the angular variable is constant.

We discuss the first persistence result in the theory of polynomial dynamical systems: persistence of heteroclinic points of periodic orbits in a family of Henon mappings, and more generally, polynomial automorphisms of the complex plane. Traditional results on analytic continuation of geometrical objects related to polynomial dynamical systems, like periodic orbits, deal with algebraic functions. Global analytic continuation of these functions is always possible. On the other hand, coordinates of heteroclinic points are in general transcendental functions, and the possibility of global analytic continuation of these functions is not at all obvious. The ideas of the proof are presented. As a corollary we claim genericity of the Kupka Smale property for polynomial automorphisms of the complex plane. Several related results are formulated and numerous problems are stated.

We show that polynomial Liénard systems may have limit cycles enclosed by period annuli, or enclosing period annuli. In both cases one can construct polynomial systems having an arbitrarily high number of limit cycles with such properties. As a limit case, we prove the existence of an analytic Liénard system with infinitely many limit cycles surrounding a central region. We also show that for every n there exists a Liénard system of degree n with n - 2 limit cycles.

This is part of the effort, the program launched by Dumortier, Roussarie and Rousseau, in proving the finiteness part of Hilbert's 16th problem. In this paper, we highlight the ideas of proving the finite cyclicity of PP-graphics in quadratic systems.

We consider the reduced Stokes equations, which can be derived from the (generalized) Stokes equations by expressing the pressure in terms of the data and the velocity. The new system is equivalent to the generalized Stokes equations. It has the advantage that the pressure and the divergence equation are no longer part of the system. But a non-local operator, which is the product of a Poisson and a trace operator, enters the system, instead of the pressure. Using the structure of the reduced Stokes equations, the corresponding solution operators can be represented with the aid of the solution operators to the Laplace resolvent equation and the inverse of an operator acting on the boundary of the domain. This representation can be used to prove the existence of a bounded H_∞-Calculus of the associated reduced Stokes operator.

In classical L^q(ℝⁿ)–spaces, n = 2 or n = 3, 1 < q < ∞, we investigate a singular integral operator arising from the linearization of a hydrodynamical problem with a rotating (nonsymmetric) obstacle. Since the integral operator is not a classical Calderón-Zygmund operator, we use Littlewood-Paley theory and a decomposition of the singular kernel in Fourier space to prove L^q–estimates of solutions.

In this note we would like to demonstrate a problem that occurs when proving regularity of time derivative of velocity vector field describing a flow of a generalized Newtonian fluid under the Dirichlet boundary condition.

We state a partial regularity result for steady flows of a class of incompressible fluids, their viscosity varies with the shear-rate and the pressure. Apart from partial regularity in three dimension we obtain, as a consequence of the method, full regularity of solution in two dimensions. The analysis leads to the study of regularity of a weak solution to a system that generalizes the Stokes system in two directions: the Laplace operator is replaced by a general linear second order elliptic operator and the pressure gradient is replaced by a linear first order operator, acting on the pressure.

We study the long-time behavior of solutions of the Navier-Stokes equation in ℝ² × (0, 1). After introducing self-similar variables, we compute the asymptotics of the vorticity up to second order and prove that they are governed by the two-dimensional Navier-Stokes equation. In particular, we show that the solutions converge towards Oseen vortices.

We consider the system modelling the density and the temperature of a cloud of self-interacting particles. Besides the potential generated by self-gravitating particles we assume also the existence of the external potential. In the paper the existence and nonexistence of radially symmetric steady states in a ball is studied.

We study the initial-boundary value problem for the Stokes equations with Robin boundary conditions in the half-space . It is proved that the associated Stokes operator is the generator of a bounded holomorphic semigroup on , which is even strongly continuous on . By a counterexample we will show that this assertion is wrong on , except for the special case of Neumann boundary conditions.

The purpose in this paper is to derive the regularizing rate estimates for the strong solutions including higer derivatives to the Navier-Stokes flows in the whole space. The solutions need not be small. Similar as in the case of the heat equation, these estimates also imply the analyticity in space varables as well as decaying with respect to time. Moreover, we can obtain its analyticity rate, that is, the estimate for the size of the radius of the convergence of Taylor's expansion.

We present a result for an incompressible ideal fluid with a free surface subject to surface tension; it is not assumed that the fluid is irrotational. A priori estimates are shown by combining energy and vorticity estimates. We indicate how these results can be used in order to derive a local existence result with a Navier-Stokes approximation.

No abstract received.

We consider dynamical systems on compact manifolds, which are local diffeomorphisms outside an exceptional set (a compact submanifold). We are interested in analyzing the relation between the integrability (with respect to Lebesgue measure) of the first hyperbolic time map and the existence of positive frequency of hyperbolic times.

In this paper we discuss different nonequivalent mathematical formalizations of the concept of an attractor. Relations between different definitions of attractors are presented. Some results about attractors of generic dynamical systems are formulated, and numerous problems are proposed.

We demonstrate the use of an algorithm to compute a two-dimensional global invariant manifold as a sequence of approximate geodesic level sets. The resulting information of the parametrization by geodesic distance can be used to visualize and even crochet the manifold. This is illustrated with the example of the stable manifold of the origin in the Lorenz system, which is also shown on the Equadiff 2003 poster.

We explore the complicated dynamics arising in a neighbourhood of a homoclinic point associated to a homoclinic bifurcation of a two parameter family of three dimensional diffeomorphisms. Besides stating the existence of strange attractors with two positive Lyapounov exponents for the associated return map, we also select a curve in the space of parameters in order to numerically detect the presence of possible new families of one-dimensional and two-dimensional strange attractors. The end of this curve of parameters corresponds to a return map which is conjugated to a "two dimensional tent map".

The metric entropy of a C²–diffeomorphism with respect to an invariant smooth measure μ is equal to the average of the sum of the positive Lyapunov exponents of μ. This is the celebrated Pesin's entropy formula, h_μ(f) = ∫_MΣ_{λi> 0} λ_i. The C² regularity (or C^1+α) of diffeomorphism is essential to the proof of this equality. We show that at least in the two dimensional case this equality is satisfied for a C¹–generic diffeomorphism and in particular we obtain a set of volume preserving diffeomorphisms strictly larger than those which are C^1+α where Pesin's formula holds.

We study an equivariant vector field possessing a heteroclinic network. We focus on the complexity that arises from the existence of switching at the nodes of the network.

We use symmetry methods to study coupled cell networks, which are models for central pattern generators (CPGs). The cells in these networks obey identical systems of differential equations. In particular, we study a CPG model for legs rhythms in bipedal locomotion, proposed by Golubitsky, Stewart, Buono, and Collins. It consists of four identical cells and admits periodic solutions corresponding to the standard bipedal gaits of slow, fast, Tun, walk, skip, and gallop, among other gaits. We present numerical results on primary and secondary gaits in the CPG model using Hodgkin-Huxley equations to model cell dynamics. We use a partial linear synaptic coupling.

We show how intrinsic complete transversals simplify both classification and recognition of equivariant bifurcations.

The stability properties of a recently discovered solution of the general three body problem with equal masses and the shape of an eight are analyzed as the masses are varied. It is shown by numerical continuation and the evaluation of the characteristic multipliers that the solution is stable only in narrow mass interval. Other less symmetrical and unstable solutions with equal masses in the same homotopy class as the figure-8 orbit have been found. The branching behavior is also analyzed.

In the framework of the SUBMODELS program¹ an exact solution of gas dynamics equations is constructed. The solution describes a flow in the strip between a source and a sink of ideal gas. In case of the matched gas rates the flow is continuous. Otherwise the solution gives an explicit description of a two-dimensional non-stationary gas motion with a shock wave.

We consider the vibrations of a thin elastic plate that contains a small region where the density is much higher than elsewhere. The thickness, the density and the size of the region depend on a parameter ε that tends to zero.

We give a short review of available methods to determine the non–degeneracy of Hamiltonian Hopf bifurcations in three–degree–of–freedom systems. We illustrate the geometric method to more detail, using the example of the Lagrange top.

We announce two topological results that may be used to estimate the number of relative periodic orbits of different homotopy classes that are possessed by a symmetric Lagrangian system. The results are illustrated by applications to systems on tori and to strong force N-centre problems.

We derive conditions for complete synchronization of two symmetrically coupled identical systems of ordinary differential equations and differential-delay equations.

We deal with the proof of the hydrodynamical limit from kinetic equations (including B.G.K. like equations) to multidimensional systems of conservation laws (including isentropic gas dynamics). It is based on a relative entropy method, hence the derivation is valid only before shocks appear on the limit system solution. However, no a priori knowledge on high velocities distributions for kinetic functions is needed. The case of the Saint-Venant system with topography is included.

We give some stability results on vortex sheets solutions to the two-dimensional compressible Euler equations. The linearized problem is shown to obey an energy estimate with loss of one derivative.

There are discussed formation of singularities in finite time for infinite-dimensional semilinear Smoluchowski system, describing coagulation process, and an interesting heirarchy of Hopf-type equations arising in the study of these infinite-dimensional linear systems. The singularities exist for smooth initial data of the Cauchy problem and it is necessary to formulate correctness theorems for global generalized (Sobolev) solutions of infinite-dimensional Cauchy problem for semilinear systems.

In the present paper we consider a shallow water equation with Chezy friction term. It is well-known that this hyperbolic PDE admits a one-parameter family of discontinuous periodic roll wave solutions parametrized by their wavelength as well as a discontinuous homoclinic wave.

We show that analogous traveling waves exist when small viscous terms of size ε are added to the equation and determine how the velocity of the viscous homoclinic waves differs from the velocity of the inviscid waves. The correponding traveling wave equation leads to a singularly perturbed problem involving points on the slow manifold which are not normally hyperbolic. The periodic roll waves follow stable and unstable parts of the slow manifold and are therefore of "canard" type.

We survey some recent and ongoing research using Evans function techniques to study the stability problem for detonation waves. Much of the work in this area falls into two categories. It is either (i) numerical in nature and focused on the standard (inviscid) ZND model, or (ii) restricted to the Majda or Majda-Rosales models which have the simplifying feature of scalar kinetics. In contrast the Evans function approach allows the treatment of the reacting Navier-Stokes equations.

We present results about the nonlinear stability of large Ekman-Hartmann boundary layers which appear in rotating magnetohydrodynamics models. This model is interesting for the understanding of the dynamics in the Earth core. The proofs can be found in⁶.

We study existence of solutions of H-systems on the two-dimensional disc with prescribed boundary datum γ ∈ H^1/2 (∂𝔻,ℝ³) and H: ℝ³ → ℝ is bounded, continuous and belongs to L^p(ℝ³) for some p ∈ [3, + ∞].

In this work we obtain the following biological principle of two competing species: no species can be driven to extinction by a competitor if it possesses a refuge and its birth rate in the overall habitat is large enough. Moreover, it will be shown how the species concentrates within the refuge when the suffered aggression rises huge values.

We present some perturbation results for a class of quasilinear elliptic equations involving the p -Laplace operator, contained in a recent paper by Cingolani and Vannella ⁸. The existence of a nontrivial solution for quasilinear problems is proved, using Morse tecniques.

This work deals with the existence of an eigenvalue sequence to mixed Dirichlet-Neumann problem given below, and to show the properties of the associated principal eigenvalue. When moving the boundary conditions in a suitable way, we are able to analyze the behaviour of the first eigenvalue and associated normalized eigenfunctions.

We examine the bifurcations to positive and sign-changing solutions of degenerate elliptic equations. We show that there is a countable infinity of bifurcations at the origin. Moreover, a bifurcation occurs at each point in some unbounded interval in parameter space. We apply our results to non-monotone eigenvalue problems, degenerate semi-linear elliptic equations, boundary value differential-algebraic equations and fully non-linear elliptic equations.

We study the existence and nonexistence of positive (super) solutions to a semilinear elliptic equation in cone-like domains of ℝ^N. On the plane ℝ² we determine the set of (p, σ) such that the equation has no positive (super) solutions, depending on the parameters A, B ∈ ℝ and the geometry of the domain.

We investigate the existence of self-similar solutions for semilinear heat equations by making use of the methods for semilinear elliptic equations. In particular, we apply variational methods to show the existence of the second solutions, which implies the non-uniqueness of solutions to the Cauchy problem for semilinear heat equations with singular initial data.

We look for two positive solutions of the Brézis-Nirenberg problem -Δu - λ u^{q - 1} = u^2*-1 in bounded domains on the sphere , with 1 < q < 2 and 2* the critical Sobolev exponent. The first solution will be found as a minimizer of a cutoff functional and the second as a critical point with the mountain pass theorem.

Minimax method has been a powerful tool in constructing multiple solutions in nonlinear elliptic problems. When invariant sets of the gradient flow are present, minimax method can be used to give locations of the critical points in relation to the invariant sets. We present applications of this idea to several examples in studying nodal solutions of nonlinear elliptic problems.

We consider optimal control systems governed by the second-order differential equations with periodic and Dirichlet boundary conditions, formulate sufficient conditions for the continuous dependence of solutions on controls, and prove the existence of optimal processes.

Parametrization aspects of the Riemannian manifold structure of several classes of linear systems are investigated by three case studies. The aim is to design finite atlases of bounded charts exhibiting finite Milnor distortion. Key questions are the existence of a uniform upper bound on distances between systems, and finiteness of the Riemannian volume of the manifold. Model reduction issues and a new isometry result between stable AR and all-pass systems are briefly discussed.

We consider the local state isomorphism approach towards identifiability analysis of nonlinear state-space systems. It is shown that, under certain conditions, the local state isomorphism is linear for homogeneous polynomial systems. The ORC at the origin is shown to play a key role. Worked examples are given to illustrate the results.

We derive algorithms that compute a balanced state representation from the differential equation describing a finite-dimensional linear system.

The goal of this paper is to show some properties of real analytic Hamiltonian systems of the form H = ½y² + V(x, t), where V is periodic in x and t. We prove that this system has invariant tori of rotational type for sufficiently large values of |y|. The measure of the set of such tori is shown to be exponentially close to full measure as y → ±∞.

A class of strange attractors is described, occurring in a low-dimensional model of general atmospheric circulation. The differential equations of the system are subject to periodic forcing, where the period is one year – as suggested by Lorenz in 1984. The dynamics of the system is described in terms of a Poincaré map, computed by numerical means. It is conjectured that certain strange attractors observed in the Poincaré map are of quasi-periodic Hénon-like type, i.e., they co-incide with the closure of the unstable manifold of a quasi-periodic invariant circle of saddle type. A route leading to the formation of such strange attractors is presented. It involves a finite number of quasi-periodic period doubling bifurcations, followed by the destruction of an invariant circle due to homoclinic tangency.

Work is presented of how a new method to compute 1D unstable manifolds of saddle periodic orbits of delay equations can be used to identify transitions to chaos in a physical system that is subject to delayed feedback. Specifically, we study an interior crisis bifurcation and an intermittent transition in a semiconductor laser subject to phase-conjugate feedback.

Empirical time series of financial market data, like day-to-day stock returns, exhibit the phenomenon that although usually tomorrow's price is unpredictable, the absolute value of the price change is correlated with the magnitude of past price changes; though the corresponding correlation coefficients are not very large, they are significantly different from zero. This phenomenon is known as 'volatility clustering' in the financial liturature. In this note a micro-economic model of volatility clustering, introduced by Gaunersdorfer and Hommes⁷, will be analysed. The deterministic skeleton of the model has a Chenciner bifurcation, and hence periodic points and invariant quasi-periodic circles coexisting with the 'fundamental' equilibrium. Adding noise in form of stochastic supply shocks, volatility clustering is generated by the system jumping between the bases of attraction of the fundamental equilibrium (low volatility), and that of the non-fundamental attractor (high volatility).

Let us consider a smooth discrete dynamical system such that one of the variables is a one-dimensional angle whose dynamics is a rigid Diophantine rotation. We assume that the system has an invariant curve whose rotation number is the rotation angle of the angular variable of the system. In this note we study the fractalization of the invariant curve in the context of bifurcation theory.

We present a numerical algorithm for the continuation of periodic orbits of high-dimensional dissipative dynamical systems. It is based on single shooting and Newton-Krylov methods. A non-trivial fluid dynamics problem, which after a pseudo-spectral discretization gives rise to a system of dimension O(10⁴), has been used as test. The efficiency of the algorithm, which allows the unfolding of a complex bifurcation diagram of periodic orbits, shows the suitability of the method for large-scale nonlinear dissipative partial differential equations.

Different methods have been proposed for the numerical computation of Lyapunov exponents. After a quick review we shall describe a new powerful method which is specially suitable to detect domains of ordered motions. Then several examples will be presented, applied to conservative and non-conservative systems, discrete and continuous. They show how one can obtain relevant information on the global dynamics in a fast way. Finally, an application to compute the metric entropy of a family of diffeomorphisms will be done. In that case to have an accurate value of the entropy is important to support different conjectures.

No abstract received.

We consider a semi-linear parabolic equation on a union of thin bounded tube domains Ω_{1, ε} = Γ × (0, ε) and Ω_{2, ε} = Γ × (-ε, 0) joined at the common base Γ ⊂ ℝ^d, d ≤ 3. Unknown functions are coupled by some interface condition on Γ. This problem can model a reaction-diffusion system of two components reacting at the interface. The reaction intensity k(x, ε) depends on ε (i.e. on domain's cross size). We study limiting properties of the global attractor of the corresponding evolution semigroup as ε → 0 (i.e. as the initial domain is getting thinner). These properties depend crucially on the behaviour of the intensity k(x, ε) as ε → 0 and completely different scenarios are possible.

We consider spiral waves in reaction-diffusion systems. One example is the light sensitive Belousov-Zhabotinsky reaction. Due to the Euclidean SE(2)-symmetry, in the spatially homogeneous case spiral wave patterns appear, which rotate rigidly around the fixed tip position of the spiral. We are interested in the persistence of such solutions under a symmetry breaking perturbation, which still keeps a discrete lattice symmetry. In particular we study the resulting tip motions via a reduction of the original PDE-systems to vectorfields on the two-dimensional torus. Our methods include global center manifold reductions.

In this work we describe how to prove with computer assistance the existence of fixed points and periodic orbits for infinite dimensional discrete dynamical systems. The method is based on Krawczyk operator. As an example we prove the existence of three fixed points, one period–2 and one period–4 orbit for the Kot-Schaffer growth-dispersal model.

In this note we show that any solution of a nonlocal phase-field system with temporal memory converges to a unique stationary state. We make use of a non-smooth version of Lojasiewicz inequality.

We study the large time behavior of the solutions of a hyperbolic systems of conservation laws with relaxation. In particular, we study the case where the behavior is as the heat kernel. We show a weakly nonlinear behavior to obtain the optimal convergence rate in the L^P-sense.

We review some transversality results for the semiflow generated by a scalar semilinear parabolic equation with periodic boundary conditions. In particular, we consider the tranversality results for the spectral filtrations of the stable and unstable manifolds of equilibria and periodic orbits.

In two recent papers the concept of adjacency has been used to formulate a criterion for the existence of a heteroclinic orbit, connecting a pair of given equilibria or rotating waves for scalar semilinear parabolic equations. Here, we want to introduce the concept of adjacency on a general level, then summarize the above mentioned results, and in addition show a further application of this concept as a criterion for possible limiting states for a given initial condition.

In this paper we announce we found representation in Lax form and θ-functional solutions of an integrable real billiard system whose billiard map is given by a shift on a stratum of the Jacobi variety of a hyperelliptic curve. To our knowledge this is the first example of integrable discrete system on a stratum solved explicitly. The proof of the main theorems will be published elsewhere.

The distribution of digits and mantissae in dynamical systems (both in continuous and discrete time) is discussed in light of two simple yet fundamental results. By utilizing shadowing and uniform distribution techniques, it is shown that systems with regular long-term behavior are surprisingly likely to exhibit Benford's logarithmic mantissa distribution — much in contrast to systems with stationary statistical properties. The results complement and extend recent work.

The classical KAM theorem deals with Lagrangean invariant tori in nearly integrable Hamiltonian systems. The stability formulation of the KAM theorem states that, when restricting to a large measure Diophantine "Cantor set" of such tori, the integrable approximation is smoothly conjugate to the nearly integrable perturbation. Action-angle variables are used in this setting and therefore the theorem usually is confined to local trivialisations of the whole bundle of Lagrangean tori. This is of special importance when this bundle is non-trivial. The present paper asserts that the conjugacies can be extended globally in a consistent way, thereby preserving the geometry of the global torus bundle.

In the conservative dynamics of certain quasi–periodically forced oscillators, normal–internal resonances are considered in a bifurcational setting. The unforced system is a one degree of freedom oscillator, under forcing the system becomes a skew–product flow with a quasi–periodic motion on an n–dimensional torus as driving system. In this work, we investigate the persistence and the bifurcations of quasi–periodic n–dimensional tori (so–called "response solutions") in the averaged system, filling normal–internal resonance 'gaps' that had been excluded in previous analyses.

We consider the persistence problem of quasi-periodic, Floquet, Diophantine invariant tori in Hamiltonian vector fields. The standard KAM theory of [1, 2] deals with the case where the Floquet matrix has simple eigenvalues. In the present paper this result is generalized to the case of multiple eigenvalues, in particular to the case of normal 1 : -1 resonance.

We consider a perturbation of a Hamiltonian planar vector field. The bifurcation set of limit cycles is studied. If the vector field is defined in an annulus, limit cycles are in one to one correspondence to the zeros of a polynomial. Catastrophe theory is relevant for the study of the ensuing bifurcations. A similar conclusion is obtained if the Hamiltonian vector field has a center. In these two cases the geometry of the bifurcation set is polynomial. We focus on the case where the Hamiltonian is defined near a homoclinic loop of a hyperbolic saddle. The study now reduces to the zeros of a displacement function that involves perturbations of Dulac unfoldings. The latter expand on a logarithmic scale. In this note, we show that, after a blow up in the parameter space, the geometry of the bifurcation set of limit cycles is again polynomial in an exponentially small domain.

Persistence of invariant tori at a 1:l resonance in reversible systems is investigated. Both the generic and the semisimple case are considered. As parameters are needed for persistence, a general unfolding theory for the reversible context is developed. This allows to generalize Broer, Huitema² and prove the persistence of a suitable union of quasi periodic (diophantine) tori.

We outline averaging principles for systems of differential equations whose fast variables evolve quasiperiodically. This special class of systems is important, as it naturally arises in modeling the dynamics of particle beams in accelerators. We give results that are tailored to such systems (proofs will appear elsewhere), and we briefly describe related work and applications to beam dynamics.

Local bifurcations of invariant tori are induced by the normal behaviour and occur where the latter changes from elliptic to hyperbolic behaviour. For invariant tori in Floquet form this is described by the Floquet exponents. When one of the Floquet exponents vanishes even the persistence of the bifurcating tori themselves is in question. With the actions conjugate to the toral angles serving as unfolding parameters, one can look for persistence of the pertinent bifurcation scenario instead. Such a study lies at the intersection of KAM theory and singularity theory. This paper presents the results of current research and ends with a speculation on how far the borders may be pushed.

Parabolic resonance orbits (PROs) are orbits of n ≥ 3 d.o.f. near integrable Hamiltonian systems with initial conditions near a resonant normally parabolic (n - 1)-torus, which is part of a family of bifurcating (n - 1)-tori. Vast numerical evidence indicates that PROs exhibit large instabilities in the slow variables ("actions"). Here the structure and the driving mechanism of PROs are realised, using a specific example as an intuition. Explicit calculations, numerical simulations and some heuristic arguments are enrolled to outline this mechanism and to construct the building blocks for a rigorous proof that PROs exhibit maximal possible instabilities w.r.t. the boundaries of their energy surface.

We establish some sufficient conditions ensuring existence, uniqueness and exponential stability of nontrivial stationary solutions for a class of delay stochastic partial differential equations.

The paper discusses the asymptotic behaviuor of the solutions of nonautonomous delay equation

where a(t) ∈ C^∞. Assuming that a(t) ≠ 0 for all t ∈ ℝ₊ (though a(t) possibly tends to 0, as t → ∞), we prove that under some additional conditions on a(t), every solution y(t) of (1) decreasing fast enough as t → ∞, vanishes identically. We consider the case, when a(t) is an 1-periodic function, as well. Due to Verduyn Lunel, in this case (under some mild assumptions on a(t)) equation (1) has small (super-exponentially decreasing ) solution if and only if a(t) has a sign change. How small can such a small solution be? Under some additional conditions on a(t), we prove, that if for some C > 0 and γ > γ₀, then y(t) vanishes identically. Examples demonstrate that our results are sharp up to the constants.

Motivated by experiments in nonlinear optics and laser physics, mathematical models using delay differential equations have appeared in the recent literature. Singular perturbation techniques that take advantage of the natural values of the physical parameters are used to determine simpler equations and/or analytical solutions. These approximations are tested by comparing bifurcation diagrams obtained analytically and numerically using continuation techniques.

We discretize a scalar linear delay integral equation (DIE) by a quadrature method [1]. Conditions are proposed under which the discretized equation retains certain stability properties of the DIE. This allows to approximate the rightmost characteristic roots which determine the stability of the zero solution of the DIE.

We investigate an inverted pendulum on a cart subject to a delayed feedback control force which tries to balance the pendulum. This is modeled by a two-dimensional system of delay-differential equations and can be considered as a prototype system for control problems arising in mechanical engineering. The linear stability analysis shows that there is only a bounded region of linear stability of the origin (corresponding to successful balancing), and identifies a singularity of codimension three as the organizing center for all dynamics of small amplitude.

Here we present the numerical bifurcation analysis of the ordinary differential equation governing the dynamics on the three-dimensional center manifold. This is compared directly with a bifurcation study of the full delay system in the vicinity of the singularity.

We investigate two instabilities of spiral waves in oscillatory media subject to different types of forcing using the complex Ginzburg-Landau equation. First, the transition of spiral waves via so-called superspirals to spatio-temporal chaos is related to a coexistence of the Eckhaus instability of the wave field and the intrinsic oscillatory meandering instability of the spiral core. Second, resonantly forced oscillatory media are shown to possess a novel scenario of spiral breakup. Bifurcation analysis and linear stability analysis yield explanations for the phenomenology observed by direct simulations.

In this note we study the appearance and the relevance of edge bifurcations in the stability problem associated to a front pattern in a certain class of (singularly perturbed) bi-stable reaction-diffusion equations.

This paper is devoted to some nonlinear propagation phenomena for reaction-diffusion-advection equations with Kolmogorov-Petrovsky-Piskunov (KPP) type nonlinearities. Special emphasis is put on pulsating travelling fronts in periodic unbounded domains and on the dependance of their speeds on the geometry of the domain and on the given reaction, diffusion and advection coefficients. More general domains without periodicity are considered as well.

We present a brief survey on our recent results concerning the existence and properties of travelling wave solutions to reaction-convection-diffusion equations

with different types of reaction terms G. In particular, we show that in certain cases the presence of the additional transport term H causes the disappearance of wavefronts.

The generalized Swift-Hohenberg equation with a quadratic term is studied. Temporally stable localized stationary solutions have been found, stationary fronts as the source of traveling fronts have been shown to exist. The study is a combination of the mathematical theory using the theory of homo- and heteroclinic solutions and numerical simulation.

We consider the sine-Gordon equation describing the phase difference of a Joseph-son junction with a π-discontinuity point. It is known that the time-independent equation can have non-monotone semifluxons. A stability analysis to the semifluxons is performed. It is shown numerically that the presence of defects can stabilize the semifluxons.

We study the Gray-Scott model in a bounded two dimensional domain and establish the existence and stability of symmetric and asymmetric multiple spotty patterns. The Green's function and its derivatives together with two nonlocal eigenvalue problems both play a major role in the analysis. For symmetric spots, we establish a threshold behavior for stability: If a certain inequality for the parameters holds then we get stability, otherwise we get instability of multiple spot solutions. For asymmetric spots, we show that they can be stable within a narrow parameter range.

An applicable version of the Sharkovskii cycle coexisting theorem to differential equations is presented. Then it is applied to ordinary differential equations without uniqueness.

The existence of an attractor for a 2D-Navier-Stokes system with delay is proved. The theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous. The formulation allows to treat, in a unified way, terms containing various classes of delay features (constant, variable, distributed delays, etc.).

The talk is devoted to the study of quasi-linear non-autonomous difference equations: invariant manifolds, compact global attractors, almost periodic and recurrent solutions, invariant manifolds and chaotic sets. We prove that such equations admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions of the existence of a compact global attractor and characterize its structure. We give a criterion for the existence of almost periodic and recurrent solutions of the quasi-linear non-autonomous difference equations. Finally, we prove that quasi-linear difference equations with chaotic base admits a chaotic compact invariant set. The obtained results are applied while studying triangular maps: invariant manifolds, compact global attractors, almost periodic and recurrent solutions and chaotic sets.

We use methods of non-autonomous differential equations to study a quasi-periodic saddle node bifurcation scenario which is present in a differential equation of Duffing type.

A technical property based on comparison with ODEs is added to the definition of uniform complete guiding set in order to assure the existence of an a priori bounded solution for a class of finite-delay equations. Some examples constitute a sample of the applicability of this result. In particular, Landesman-Lazer type conditions guaranteing a bounded solution for an n-order escalar equation are refined.

Some results on the spectrum of Schrödinger operators with quasi-periodic real analytic potential and Diophantine frequencies are presented. The eigenvalue equation associated to this operator is a quasi-periodic generalization of Hill's equation with periodic forcing. A dynamical study of this equation allows an accurate description of the structure of stability zones close to constant coefficients. This study has direct applications to the structure of the spectrum of Schrödinger operators with small quasi-periodic forcing. In particular we discuss the occurrence of Cantor spectrum.

We consider one-parameter families of maps close to a linear rotation in ℝ²⁺ⁿ with conditions that imply that they are weakly attracting in the rotation plane and weakly repelling in the 'rotation axis'. We get that the unstable manifolds converge to the 'rotation axis' and that, in the case of time one maps of vector fields, the stable manifolds converge to an independent of the parameter rotationally symmetric surface when the maps approach the rotation.

An optically-injected semiconductor laser exhibits chaotic behaviour for certain values of the parameters. The underlying model is an example of a general three-dimensional system of ordinary differential equations, and existing analyses agree very well with experiments. We outline a method for computing an approximate but powerful description of the behaviour in the chaotic regime in terms of symbolic dynamics. The method is based on computing the stable and unstable manifolds of the system, which can then be used to give a natural description of the orbits.

Let {x_i}_i∈ℕ be a regular orbit of a C² dynamical system f. Let S be a subset of its Lyapunov exponents. Assume that all the Lyapunov exponents in S are negative and that the sums of Lyapunov exponents in S do not agree with any Lyapunov exponent in the complement of S. Denote by the linear spaces spanned by the spaces associated to the Lyapunov exponents in S. We show that there are smooth manifolds such that and .

In this work are studied periodic perturbations, depending on two parameters, of planar polynomial vector fields having an annulus of large amplitude periodic orbits, which accumulate on a symmetric infinite heteroclinic cycle. Such periodic orbits and the heteroclinic trajectory can be seen only by the global consideration of the polynomial vector fields on the whole plane, and not by their restriction to any compact set. The global study involving infinity is performed via the Poincaré Compactification. It is shown that, for certain types of periodic perturbations, one can seek, in a neighborhood of the origin in the parameter plane, curves C^(m) of subharmonic bifurcations, for which the periodically perturbed system has subharmonics of order m, for any integer m.

Assume that a Hamiltonian system looses some of its degrees of freedom in the following way: the motion in the slow component stops in the limit ε → 0. In this case, the small parameter enters the dynamics through the corresponding symplectic form instead of the Hamiltonian function. The slow manifold can be defined in the usual way, but unlike the general dissipative case the slow manifold may be normally elliptic even for a generic Hamiltonian. We study a mechanism, which destroys the normally elliptic slow manifold and use a specially developed averaging technique to study the dynamics nearby.

The basic dynamics of the Yamada model consists of a slow drift along a slow manifold which changes its stability along a not normally hyperbolic line and a fast jump from the stable part of the slow manifold to the unstable part. Depending on parameters, a homoclinic orbit and a transcritical bifurcation occur. We present the main ideas of the proof that in the parameter space a smooth curve of homoclinic orbits exists exponentially close to a curve of transcritical bifurcations. In the singular limit, where the two curves meet, a singular homoclinic orbit exists. The proof is based on the blow-up method in combination with the geometric singular perturbation theory.

We treat degenerate planar turning points from a dynamical systems approach, following a method introduced in ³. We show the existence of solutions to boundary value problems and examine the behaviour of the transition time of the orbits as they pass near the turning point.

A system which differs from an integrable Hamiltonian system with two degrees of freedom by a small Hamiltonian perturbation and a much smaller non-Hamiltonian perturbation is considered. The unperturbed system is isoenergetically nondegenerate. The averaging method is used for approximate description of solutions of the exact system on time interval inversely proportional to the amplitude of the non-Hamiltonian perturbation. The averaged over initial conditions error of this description is estimated from above by a value proportional to the square root of the amplitude of the Hamiltonian perturbation.

A method for proving the existence of periodic and heteroclinic orbits in a singularly perturbed ODE system called a slow-fast system is given by using the Conley index theory. This is a continuation of the authors' earlier work¹ which is now extended to systems with multi-dimensional slow variables. As an application, we show, in a system of reaction-diffusion equations studied by Gardner and Smoller², the existence of periodic traveling waves solutions as well as the set of traveling wave solutions that are encoded by symbolic sequences of two symbols. This is based on joint works^3,5 with M. Gameiro, T. Gedeon, H. Kokubu, and K. Mischaikow.

We give a geometric singular perturbation analysis of a classical problem proposed by Lagerstrom to illustrate the ideas involved in the rather intricate asymptotic treatment of low Reynolds number flow. We present a geometric proof based on the blow-up method for the existence and uniqueness of solutions. Moreover, we show how asymptotic expansions for these solutions can be obtained in this framework, thereby establishing a connection to the method of matched asymptotic expansions.

We consider a class of random differential equations with two time scales and prove the existence of a slow random inertial manifold. To establish this result we introduce an appropriate random graph transform and prove the existence of a random fixed point. The graph of this fixed point yields the invariant manifold.

No abstract received.

We give a geometric analysis of relaxation oscillations described by singularly perturbed systems with two slow and one fast variable. It is shown that near a singular periodic orbit a Poincaré map can be defined which under certain assumptions possesses a one-dimensional invariant slow manifold. This allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter. Additionally we show how this invariant structure of the Poincaré map is destructed by canards and turning points which give rise to more complex dynamics and chaotic behaviour. The analysis is based on methods from geometric singular perturbation theory with emphasis on the blow-up method.

Cartwright and Littlewood discovered "chaotic" solutions in the periodically forced van der Pol equation in the 1940's. Subsequent work by Levinson, Levi, and others has made this singularly perturbed system one of the archetypical dissipative systems with chaotic dynamics. Despite the extensive history of this system, many questions concerning its bifurcations and chaotic dynamics remain. We use a combination of analysis of the singular limit and numerical simulation to describe a horseshoe map that arises in the three-dimensional phase space. The canards that form at a "folded saddle" play a crucial role in this analysis.

Here are studied qualitative properties of the families of curves –foliations– on a surface immersed in ℝ⁴, along which it bends extremally in the direction of the mean normal curvature vector. Typical singularities and cycles are described, which provide sufficient conditions, likely to be also necessary, for the structural stability of the configuration of such foliations and their singularities, under small C³ perturbations of the immersion. The conditions are expressed in terms of Darbouxian type of the normal and umbilic singularities, the hyperbolicity of cycles, and the asymptotic behavior of singularity separatrices and other typical curves of the foliations. They extend those given by Gutierrez and Sotomayor in 1982 for principal foliations and umbilic points of surfaces immersed in ℝ³. Expressions for the Darbouxian conditions and for the hyperbolicity, calculable in terms of the derivatives of the immersion at singularities and cycles, are provided. The connection of the present extension from ℝ³ to ℝ⁴ to other pertinent ones as well as some problems left open in this paper are proposed at the end.

In this paper we show that a necessary and sufficient condition for the differential equation of lines of axial curvature factors into differential equations of mean directionally curved lines and asymptotic lines is the vanishing of the normal curvature.

The unsolved Carathéodory conjecture for a smooth surface in ℝ³ would follow from Bendixson's formula, were it known that the principal foliations on such a surface never admit an elliptic sector at an isolated umbilic. Evidence in favour of the nonexistence of elliptic sectors in principal foliations is the main result and it offers a first geometric explanation of why the conjecture might be true for smooth surfaces.

We study the dynamics near degenerate homoclinic orbits in conservative systems. Typically bellows bifurcate from those orbits. We show the existence of one parameter families of suspended horseshoes, parameterized by the level of the first integral, near the bellows configuration. By analysing the nearby 1- and 2-periodic orbits we give an idea how the horseshoes dissolve while the bellows merge into the primary degenerate homoclinic orbit.

We give the topological classification of real analytic vector fields around configurations of multiplo saddle-connections. We consider connections provided by the skeleton of a normal crossings divisor that have a single "face to face" connection.

We are concerned with the dynamics arising in generic unfoldings of the nilpotent singularity of codimension three on ℝ³. Any of such unfoldings can be written as a perturbation of the family:

where λ² + μ² + ν² = 1. When λ < 0, μ < 0 and ν = 0 the two equilibrium points are hyperbolic of saddle-focus type with different stability indexes. For specific parameter values (λ_K, μ_K, 0) it is known that the one-dimensional invariant manifolds intersect. We prove that in such case there also exists a topologically transversal intersection between the two-dimensional invariant manifolds. The breaking of the heteroclinic cycle gives rise to Shil'nikov homoclinic orbits.

We consider reversible and ℤ₂-symmetric systems of ordinary differential equations (ODEs) that possess a symmetric homoclinic orbit to a degenerate equilibrium which itself undergoes a reversible pitchfork bifurcation. The homoclinic bifurcation has been analysed in Ref. [3] where it has been shown that a reversible homoclinic pitchfork bifurcation occurs. In this note we review results about this homoclinic bifurcation.

Recently, Melnikov-type techniques have been developed to analyze homoclinic and heteroclinic behavior for periodic orbits and invariant tori in Hamiltonian systems having an invariant plane on which there is a homoclinic orbit to a saddle-center. We extend the techniques for studying the creation of such homoclinic and heteroclinic connections by perturbations in the class of Hamiltonian systems. We illustrate the theory for a multi-degree-of-freedom model of the Euler elastica. Finally we give a comment on another extension of the techniques which enables us to answer a remaining question on the generalized Hénon-Heiles system.

In this paper we answer a question Alain Chenciner posed at the AIM ARCC Workshop on Variational Methods in Celestial Mechanics: What are the central configurations of the four-body problem which lie on a common circle and which have their center of mass at the origin of the circle? It is shown that only four equal masses in a square satisfy these conditions.

We prove the existence of symmetric choreographies - solutions of the planar N-body problem on which all bodies travel on the same curve. We describe general method of computer assisted proofs using reflectional symmetry of the orbit to isolate a solution. Using this method we proved, as an example, the existence of many choreographies of 4 and 5 bodies.

Collinear and triangular homographic solutions of the planar three-body problem with potentials of the form r^-α, α ∈ (0,2) are considered. Given the masses, m_i, i = 1, 2, 3 normalised by and a value of the energy, these solutions depend on an additional parameter, either the angular momentum c or a (generalised) eccentricity e. The purpose is to describe the stability properties and the bifurcations for any values of m_i, e and α. After several reductions, it is enough to study a three-parameter family of four-dimensional linear differential equations depending periodically on time. We are interested in the domains where the linear stability is of type EE (elliptic-elliptic), EH (elliptic-hyperbolic), HH (hyperbolic-hyperbolic) or CS (complex-saddle), as well as in the boundaries of these domains. The bifurcations emerging from e = 0 are studied by using normal forms. These provide also approximations for the boundaries. For e close to 1 (the value e = 1 corresponding to homographic solutions with a triple collision) a blow up method, applied to the variational equations, is the key tool used to study the bifurcations. For the intermediate region the bifurcation diagrams are completed numerically.

Using global variational methods we establish a family of symmetric periodic orbits of the Isosceles three body problem with arbitrary masses. Stability type may be determined for families of symmetric periodic orbits on the energy-momentum levels by counting Lagrangian singularities (Maslov index) along the orbits. We also present numerical simulations which afford an illustration of the variational-stability method for periodic orbits, as well as providing an overview of the complicated dynamics. The symmetric periodic orbit families admit some surprising regularity (reminiscent of Kepler's third law for scaling of elliptical orbits) amongst a chaotic background.

New PDE of composite type, which have the concrete physical contents, are discussed in this work. The application of method of dynamic potentials is demonstrated for example of initial-boundary value problems for the equation of such type. One initial-boundary value problem with moving boundaries are considered also. Specific properties of fundamental solutions are represented.

Quasi-uniform grid offered for numerical solution of boundary-value problems in infinite domain. The offered approach was successfully applied for calculation of spectra for differential operator, parabolic, elliptic, hyperbolic and non-classical linear and non-linear problems.

The stability of coupled identical systems is considered where the individual units are governed by ordinary or delay differential equations and interact with other units through processes possibly involving further delays. The problem arises in the study of the so-called oscillator death, a phenomenon that has significance in several physical and biological systems, where oscillatory activity of isolated units is suppressed once they are coupled. In this work, the oscillations are assumed to result from a supercritical Hopf bifurcation, and necessary and sufficient conditions for stability are obtained using averaging theory.

New L^∞ estimate of solutions of the 2D Navier-Stokes problem in an infinite strip is obtained.

No abstract received.

This paper deals with a generalization to the Lie group SO(3) of the classical cubic polynomials in the Euclidean space based on the study of a second order variational problem. The corresponding Euler-Lagrange equation gives rise to a interesting system of nonlinear differential equations. In this paper the reduction of the essential size and the separation of the variables of the system are obtained by means of invariants along the cubic polynomials.

The equation

is viewed as a perturbation of the equation which does not oscillate at infinity. The sequences , are assumed real, r_n > 0 for all n ≥ 0, the sequences may be complex-valued. We study the Hartman-Wintner problem on asymptotic "integration" of (1) for large n in terms of solutions of (2) and the perturbation .

The asymptotic behaviour of some types of retarded differential equations is analyzed. The existence of global attractors is established for autonomous equations without uniqueness using the notion of attractor for multivalued semiflows. Several applications to biological and physical models are given.

No abstract received.

We consider a Darboux problem associated to a hyperbolic differential inclusion and we provide the Lipschitz dependence on the initial data of the solution set and a version of Filippov's theorem concerning the existence of solutions of our problem.

In this note we present some observations of bifurcation in nonlinear elliptic boundary value problem -Δu = f(λ, u), in Ω, u = 0 on ∂Ω. In particular, we are interested in the effect of concave and convex combination of Ambrosetti, Brezis and Cerami type and get new class of concave and convex nonlinearity.

We consider the mathematical treatment of a two dimensional climate model (latitude - deep) which models the coupling of the mean surface temperature of the Earth with the ocean temperature. The model incorporates a dynamic and diffusive boundary condition. Our results concern the existence of a bounded weak solution. In the case of multivalued coalbedo functions the uniqueness of solutions may fail for initial datum flat enough.

New results on stability of integro-differential equations are proposed. The technique is based on a reduction of integro-differential equations to corresponding systems of ordinary or delay differential equations.

In this paper we explain some primary results from the forthcoming work⁵ concerning quasiconvexity and the stability of classes of solutions to differential relations.

No abstract received.

We costruct a canonical Kukles system and carry out the global qualitative analysis of its special case corresponding to a generalized Liénard equation.

For parameter-dependent Stochastic Delay Differential Equations (SDDE) and numerical methods depending on specific control parameters one question of interest is for which SDDE- and algorithmic parameters the numerical solution generated by the method applied to the respective SDDE is asymptotically stable in the p-th absolute moment (p ∈ 1 + ℕ). A second question of interest is how the disturbances in the initial data propagate with time.

For so-called test-equations from a class of linear SDDE and a class of numerical methods both questions are related to the asymptotic behaviour of p-th absolute moments (p ∈ 1 + ℕ) of parameter-dependent stochastic linear recurrence relations. We give criteria for such recurrence relations to be asymptotically stable in the p-th absolute moment for fixed SDDE- and algorithmic parameters. If p ∈ 2 + 2ℕ the criterion is exact. In the case p ∈ 1 + 2ℕ we provide a criterion for stability and a criterion for instability of a parameter.

Unfortunately, the key criterion suffers from high-dimensionality. However, a transformation is proposed which reduces the dimensionality at computational costs.

We study the behavior of global solutions of the scalar Zakharov system (ZS)

in dimension d = 2. We obtain that for any 2 < p ≤ 4, the L^p norm of the global solutions satisfy the following property, There exists a sequence {t_n|n ∈ ℕ} with t_n → +∞ such that lim_n→+∞||ψ(t_n)||_p = 0.

We give a quick review of mathematical models used to elucidate the biological mechanisms behind the collective behaviour of the amoeba Dictyostelium discoideum. Moreover, we identify still-open questions of biophysical nature for which mathematical modelling, analysis and simulation may help in their answering.

A new numerical algorithm for solving semilinear elliptic problems is presented. A variational formulation is used and critical points of a C¹-functional subject to a constraint given by an intersection of level sets of finitely many C¹-functionals are sought. First, constrained local minima are looked for, then constrained mountain pass points. The approach is based on the deformation lemma and the mountain pass theorem in a constrained setting.

The present note "compares" some forward-backward parabolic equations in population dynamics with negative cross-diffusion models. The cross-diffusion systems can be interpreted as time delay approximations of the forward-backward parabolic equations.

A local existence theorem for a class of Volterra integrodifferential equations will be presented. In the proof we use compactness methods based on the properties of M-accretive operators and Schauder fixed point theorem. We apply the results to the study of a general n species competitive system in population dynamics.

Sufficient Lyapunov-type and frequency-domain conditions for the finite-time stability of a class of evolutionary variational inequalities (resp. equations) are derived. These inequalities (resp. equations) are considered in a scale of Hilbert spaces. The use of finite-time stability results for the parameter prediction of bifurcations is demonstrated.

Systems of equations which approximate two-dimensional Bingham flows are considered. The solvability of initial boundary value problems for these equations is proved. The convergence of the solutions of approximating systems to the solutions of nonperturbed systems is demonstrated.

No abstract received.

In this paper the quenching of aeroelastic oscillations of a seesaw oscillator by means of a dynamic absorber is considered. The optimal absorber tuning is obtained. For a specific case, a comparison between the original and the quenched aeroelastic behaviour is shown.

No abstract received.

We present a reflectionally symmetric planar vector with an additional 2π-periodicity that features a reduced saddle-node Hopf (SNH) bifurcation with global reinjection.

The aim of the paper is the substantiation of a constructive method for verification of hyperbolicity and structural stability of discrete dynamical systems. The main tool to do so is a symbolic image which is a directed graph constructed by a finite covering of the projective bundle. Hyperbolicity is tested by the calculation of the Morse spectrum (the limit set of Lyapunov exponents of pseudo trajectories) which can be found for a given accuracy by the construction of a symbolic image ³. If the Morse spectrum does not contain 0, then the chain recurrent set is hyperbolic and the system is Ω-stable. Thus, the symbolic image gives us an opportunity to verify these properties. A diffeomorphism f is shown to be structurally stable if and only if the Morse spectrum does not contain 0 and for the complementary differential there is no connection CR⁺ → CR^- on the projective bundle. These conditions are verified by an algorithm based on the symbolic image of the complementary differential.

We study the asymptotic behaviour of an equation of Kirchhoff's type on ℝ^N. We prove the existence of an absorbing set and the existence of a strong global attractor for this equation. We also give stability results for the generalized equation of Kirchhoff's type on all of ℝ^N.

Almost sure asymptotic and generalized mean square stability of a trivial solution to the system of stochastic differential equations with delay and with non-autonomous nonlinear main part have been proved. The Liapunov-Krasovkii functional technique and some approaches from the Theory of Stochastic Processes have been applied.

We present a model describing heat flow through solids at temperatures in the neighbourhood of a "lambda" point where the specific heats of certain materials develop a singularity and their heat conductivities attain a maximum. The model is mathematically interesting due to a change of type which occurs here, with aspects of hyperbolicity initially being prominent close to the cold side of the phase transition, the hot side always being parabolic. At later times the differences between hyperbolic (thermal, wavelike) solutions and solutions to a corresponding equation describing steady state conductivity diminish, which eventually leads to steady state behaviour characterized by a degenerate, parabolic ("slow diffusion") equation on both sides of the phase transition.

In ([2]), with n = 3 and p = 3/2, solutions of the above problems are obtained as minimizers of some Thomas-Fermi functionals for an ionized (c > 0) atom; they are non negative and radial elements of C¹(ℝⁿ) ∩ H(ℝⁿ) with compact supports. Here We obtain those solutions and establish some of thier properties.We consider the maximal domain for p and n ≥ 3 i.e. with a := n - 1, p ∈ (1, ( a + 1)/(a - 1)). We recall that the "finite mass condition" in ([1]) requires that and this demands that the admissible solutions have the estimate |x|^{1 - a} at 0 ( see [5, 6] ).

When the bounded domain Ω is regular enough ( ∂Ω ∈ C²) and contains a ball centered at the origin with a radius R₀ > 0, under the hypotheses displayed below, we find a lower bound of large solutions of the problems (1) below.

We consider improved Hardy inequalities involving the biharmonic operator. We give necessary conditions under which these inequalities cannot be further improved. In establishing these results we derive various sharp Improved Hardy-Sobolev inequalities.

No abstract received.

No abstract received.

No abstract received.

The aim of this paper is to show that 'standard' second order Hardy inequality considered in the new book by Kufner and Persson can be also obtained from the second order inequality derived by Bronisław Florkiewicz and Katarzyna Wojteczek in previous papers. The conditions under which the inequality is valid will be also shown.

No abstract received.