Equadiff 99

The following sections are included:

• Preface

• Contents

Using Melnikov techniques we get analytically two different families of periodic orbits in the Sitnikov problem [5]. The first family is obtained proving the existence of transverse heteroclinic orbits and the second one is gotten from the unperturbed periodic orbits using a generalization of the Hartman-Grobman theorem that we state here.

A new analytical method is developed to generate 2nπ-periodic solutions of the Sitnikov problem. The periodic solutions of Sitnikov’s system are expanded around the periodic solutions of an auxiliary integrable differential equation. This leaves us with a forced ODE having polynomial nonlinearities for which approximate analytical solutions can be derived.

Collisions appear as singularities in the N-body problem. A binary (resp. total) collision occurs when only two (resp. all) masses collide. Between these cases other types of collisions appear, like triple (TC), simultaneous binary (SBC), etc. We consider a geometrical approach, searching for continuity with respect to initial conditions for orbits passing close to a collision. After surveying known results, some recent progress, dealing with finite order of smoothness in the SBC case and with the classification of passages near TC, is presented.

The integral manifolds of the N-body problem are the level sets of center of mass, linear and angular momentum, and energy. Understanding the topology of these manifolds, and determining how that topology depends on the parameters, is a basic problem in celestial mechanics. We have been studying the integral manifolds, by computing their homology groups , and using changes in the homology to detect changes in the topology and dynamics of the manifold.

This brief note describes a new point of view about diffusion of orbits along a chain of invariant tori in Hamiltonian systems. Under certain conditions the problem reduces to the study of the dynamics on an invariant Cantor set of annuli analogous to the invariant Cantor set of points of the horseshoe map. Within such an invariant set, diffusing orbits can often be found quite easily.

Some results are provided in support of Moeckel’s dominant mass conjecture—that any linearly stable relative equilibrium contains a mass much larger than the others. One result is that for n ≥ 24,306, any equal mass relative equilibrium is spectrally unstable. By adding masses of smaller order in the appropriate places, we also show that it is possible to thicken the ring around the 1 + n-gon relative equilibrium and maintain its linear stability.

We construct the first examples of flows with robust multidimensional Lorenz-like attractors: the singularity contained in the attractor may have any number of expanding eigenvalues, and the attractor remains transitive in a whole neighbourhood of the initial flow. These attractors support an SRB (Sinai-Ruelle-Bowen) measure and, contrary to the usual (low-dimensional) Lorenz models, they have infinite modulus of structural stability.

The author’s results on the moduli of Ω-conjugacy are reviewed.

We consider dynamics of infinite-modal maps on an interval, as a model for differential equations with saddle-focus homoclinic orbits. We study the existence of periodic and strange attractors for these infinite-modal maps.

We report on a numerical study of codimension-three resonant inclination flip homoclinic orbits. Two cases of unfoldings are presented: one with a single homoclinic doubling, and the other containing a homoclinic-doubling cascade. Our results confirm the unfoldings given in Ref. [3].

An analytical approach to predict a critical parameter value of homoclinic bifurcation in a three-dimensional system is reported. A perturbation method is performed to construct approximation of the periodic solution. The analytical period doubling value is also given. To support our analytical predictions and to describe the dynamical behaviour of the system, a numerical study is provided.

Hyperbolic subsets and some bifurcations under varying the system parameter and value of a Hamiltonian are studied, in particular, all homo- and heteroclinic orbits to saddle-foci are enumerated. The symbolic description of hyperbolic subsets are given. A countable set of parameter values is distinguished for which each saddle-focus possesses a nontransverse homoclinic orbit. Applications to the stationary and travelling wave solutions to the generalised Swift-Hohenberg equation are discussed.

The following sections are included:

• Introduction

• Outline of the proofs

• References

A class of codimension two dynamical systems with very simple dynamics is considered. In spite of simplicity of dynamical behavior, the structure of the bifurcation set is rather complicated. It is shown that there is no versal two-parameter family of dynamical systems which could describe completely the structure of the bifurcational set.

We present an algorithm for computing rigorous solutions to a large class of ordinary differential equations. The main algorithm is based on a partitioning process and the use of interval arithmetic. We illustrate the presented method by computing solution sets for two explicit systems.

Relations between the solutions of singularly perturbed differential equations and bifurcations in the associated one-parameter families of differential equations are examined. We discuss bifurcations with single or double zero eigenvalue, as well as the Hopf bifurcation.

We consider a singularly perturbed boundary value problem with Dirichlet conditions and study the sensitivity of the internal layers solutions with respect to small changes in the boundary data. Our approach exploits the existence of smooth invariant manifolds and their asymptotic expansions in the small parameter of perturbation. We show that the phenomenon is extremely sensitive since the shock layers are only obtained by exponentially small perturbations of the boundary data.

The following sections are included:

• INTRODUCTION

• EQUATION OF LIENARD TYPE

• EXISTENCE OF PERIODIC SOLUTIONS OF (1_ε)

• REFERENCES

This text summarizes the talk presented during the conference and is based on joint work with Robert Roussarie. It deals with singular perturbation problems for C^∞ ordinary differential equations. It is shown how the use of centre manifolds and blow up permits to obtain results where a high differentiability is needed. It can be applied for instance to the study of multiple limit cycle bifurcations in the plane.

It is known that for each ε > 0 the optimal controller for the infinite horizon singularly perturbed problem can be designed by finding a Lagrangian invariant manifold of the corresponding Hamiltonian system. We construct an asymptotic approximation to the optimal solution on the basis of the slow-fast decomposition of the Hamiltonian system and of the invariant manifold into the slow and the fast ones.

We investigate the boundary value problem for the second order parabolic equation with a small parameter h in a bounded domain. We construct the global exponential asymptotics of the solution of this problem for a finite time interval uniformly with respect to h.

Two-dimensional polynomial dynamical systems are considered. We develop Erugin’s two-isocline method for the global analysis of such systems, construct canonical systems with field-rotation parameters and study limit cycle bifurcations. Using the canonical systems, cyclicity results and Wintner-Perko termination principle, we outline a global approach to the solution of Hilbert’s 16th Problem.

Stiff problems model vibrations of elastic systems consisting of two materials, one of them is very stiff with respect to the other. We study the asymptotic behavior of eigenvalues and eigenfunctions of a spectral stiff problem as the stiffness constant of only one of materials tends to 0. Using the WKB technique the complete asymptotic expansions of so-called high frequency vibrations are constructed.

In this paper a perturbation method based on integrating vectors and multiple scales will be presented for initial value problems for regularly or singularly perturbed ordinary differential equations. Asymptotic approximations of first integrals can be constructed which are valid on long time-scales.

We consider the Abraham model of a rigid charge distribution coupled to the electromagnetic field and subject to slowly varying external potentials. It is rigorously shown that the true dynamics is well-approximated by an effective dynamics, induced by the Lorentz-Dirac equation. We characterize all “physically relevant” solutions of the latter as lying on a center-like manifold of a singular perturbation problem . In particular, we characterize the famous “run-away”-solutions.

Recent results for different classes of nonlinear singularly perturbed system which are based on the development of asymptotic method of differential inequalities are considered. This approach is demonstrated for some porblems in the case of exchange of stabities, some classes of elliptic and parabolic equations, including systems with fast and slow equations, and integro-differential equations with contrast structures.

For analytical singularly perturbed differential equations, I present two very general theorems on the existence of solutions having Gevrey asymptotic expansions and an idea of their proof (work with M. Canalis-Durand, J.-P. Ramis and Y. Sibuya). Three applications are given, e.g. the proof of Wasow’s conjecture on the nature of turning points (thesis of C. Stenger).

We give a short introduction to chaotic behaviour of random dynamical systems in discrete time and in particular we describe via a random version of the Smale-Birkhoff Theorem how random homoclinic points lead to chaos. For stochastic systems in continuous time we discuss the appearance of random homoclinic points and present a version of Melnikov’s method applicable to randomly perturbed differential equations.

Under the uniformly elliptic condition or the hypoelliptic condition as in the diffusion case, see [3] and [1], the linear stochastic jump diffusion process projected on the unite sphere is a strong Fellor process and has an unique invariant measure using the relation of the transition probability between diffusions and jump diffusions in [4]. The almost sure exponential stability of the linear stochastic jump systems depends on the sign of the deterministic Lyapunov exponent.

This paper reviews the uses of Lyapunov’s second method in the study of various properties of stochastic differential equations. The paper emphasises the new ideas and approaches developed recently.

We consider the behaviour of generic special Euclidean (SE(n)) group extensions of dynamical systems that are chaotic or quasiperiodic. Results of Nicol et al (1999) [13] show that for a generic extension of a chaotic base dynamics, one will see a Brownian-like random walk if n > 1 is odd or if n = 2. For SE(2)-extensions of quasiperiodic dynamics, there is bounded motion for almost all smooth enough extensions.

The paper deals with the problem of reduction of a smooth reversible vector field near a hyperbolic singularity to a resonant polynomial normal form via a smooth symmetric transformation. The authors show that the Sternberg-Chen theory remain true in that specific context.

The pattern forming problem corresponding to a Hopf bifurcation with the spatial periodicity of the face centered cubic (FCC) lattice is considered, and a large number of primary solution branches found. This analysis is by extending the symmetry group and first solving the Hopf bifurcation on the simple hypercubic lattice in four spatial dimensions.

The notion of (Γ, Γ′) co-equivariance is defined and used to analyze the most general effects of vector anisotropy on steady pattern formation on the hexagonal lattice.

Systems with continuous symmetry typically have equilibrium sets that are n-manifolds with n > 0. Under symmetry-breaking perturbations such a manifold breaks up into lower-dimensional pieces or to a discrete set of points. We describe the generic geometry in the case n = 2, and remark on the case when some residual symmetry persists.

Using the results of [1], [2], and [3] we develop a stability theory for bifurcating periodic points from a fixed point in parametrized families of reversible or symplectic diffeomorphisms. As an application we provide the analysis of the stability of periodic orbits appearing in the simplest bifurcation scenario for both Hamiltonian and reversible systems.

We describe recent results on the existence of stably ergodic symmetric attractors for diffeomorphisms equivariant by a compact Lie group. We also discuss the question of existence of canonical ergodic measures for symmetric attractors.

We review recent developments in the theory for generic bifurcation from periodic and relative periodic solutions in equivariant dynamical systems.

We present new results on the validity, universality and structure of the Ginzburg-Landau equation on the line.

The point symmetries of Navier equations for linear elastostatics are found. The system of equations is reduced into the system ordinary differential equations. The exact fundamental solution corresponding to a singular force has been found by the integration of the system subject to the appropriate subsidiary conditionsnon-homogeneous system, a system of ordinary differential equations is obtained.

We formulate a result of the type of the Implicit Function Theorem for abstract equivariant equations, and we demonstrate by two examples (problems for ordinary and partial differential equations) how the assumptions can be verified and how the assertions can be interpreted.

Second order ordinary differential equations admit r-dimensional, r=1, 2, 3 or 8 Lie groups of point symmetries. On the other hand Painlevé classification produces 50 classes of equations from the set of second order equations. In this paper the overlap between the two classification is studied.

We present a symmetry reduction for the flow near relative equilibria in S¹-equivariant Hamiltonian systems; our approach is based on the idea of Krupa [1] to decompose the vectorfield into a part which is tangential and one which is normal to the group orbits. We show for the case under consideration that this decomposition can be done in a way which is compatible with the symplectic structure.

We use equivariant bifurcation theory to reduce an optimisation problem on R²ⁿ to a more numerically efficient problem of several optimisations over R and R². The problem concerns the placement of hydrophones in relation to each other to improve the signal to noise ratio where the performance measures which are derived will have an inherent O(2) × S_n, symmetry.

In the Conley index theory, transition matrices are used to detect bifurcations of codimension one connecting orbits in a Morse decomposition of an isolated invariant set. Here we shall give a new axiomatic definition of the transition matrix in order to treat several existing formulations of transition matrices in a unified manner.

We present two algorithmic constructions of the chain homomorphism induced by a multivalued representable map, introduced in [A1Ka1]. We also give an example of computation of chain homomorphism induced by Rössler equations, using these methods.

We prove the existence of a periodic solution to a well known system of differential equations in IR³. In the proof a topological approach using Conley index theory both for flows and for discrete dynamical systems is combined with a computer assisted construction of a guaranteed enclosure of a time-one map and computation of its homology.

We present a recent improvement of the homotopy Conley index, the so-called Conley index over a base. We apply it to endow the homology and cohomology Conley indices with the structure of module over the cohomology ring of the phase space and indicate how that structure provides information on continuation of cohomology classes of isolated invariant sets.

The following sections are included:

• INTRODUCTION

• SECOND-ORDER LAGRANGIANS

• EXISTENCE OF SIMPLE CLOSED TRAJECTORIES

In this note we present a method combining topological data with local hyperbolic behavior to obtain homoclinic trajectories for maps and ODE’s.

In this paper it is obtained that, in contrast with ordinary differential equations of second order, corresponding delay equations can have unbounded solutions. Nesessary and sufficient conditions of boundedness of all solutions of delay equations are obtained. Tests of existence of unbounded solutions are proposed.

The equation

is studied. Along with (1) we consider a linear equation We suggest an approach for investigation of the nonlinear equation (1) via the linear equation (2).

Feedback systems are important in applications, for example, optical feedback lasers, phase-locked frequency synthesizers and wave equations with feedback stabilization at the boundary, and the problem regarding sensitivity and robustness of the feedback system with respect to time delays in the loop has attracted a lot of attention. In this paper we shall discuss some recent developments.

A network of two identical neurons with delayed feedback is modeled by a system of delay differential equations. Reported in this note are recent results about the detailed information of the topology and smoothness of the global attractor of the system and the global dynamics. A 5-dimensional invariant set is considered, which belongs to the global attractor and contains a phase-locked periodic orbit and a synchronous periodic orbit and their connecting orbits. Limiting profiles of these periodic orbits as square waves are described.

We investigate the method of collocation based on piecewise polynomials to compute periodic solutions of DDEs. Periodic solutions are found by solving a periodic two-point boundary value problem which is an infinite-dimensional problem for DDEs. Although our results are purely numerical they give clear insight into the phenomena that occur and demonstrate efficiency of the collocation procedure for calculating periodic solutions of DDEs.

We consider the Hamiltonian reduction of Einstein’s equations on a (3 + 1) vacuum spacetime and its relation to the geometrization of three-manifolds. We introduce a dimensionless non-local time-dependent reduced Hamiltonian H_reduced which has a unique strict local minimum and is a strictly monotonically decreasing function along the Einstein flow away from this minimum. We show that the infimum of H_reduced is related to the topological invariant σ(M), the σ-constant of M, by

Further applications and developments relating the Einstein flow to σ(M) and to the geometrization conjecture for 3-manifolds are discussed.

We study the intrinsic heat equation governing the motion of plane curves. The normal velocity υ of the motion is assumed to be a nonlinear function of the curvature and tangential angle of a plane curve Γ. By contrast to the usual approach, the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional a. Short time existence of a regular family of evolving curves is shown in the case when υ = γ(ν)[k]^{m − 1} k, 0 < m ≤ 2 and the governing system of equations includes a nontrivial tangential velocity functional.

Relying on Dafermos’ concept of generalized characteristics, we study nonclassical solutions to scalar conservation laws. Our analysis clarifies the geometrical structure of nonclassical solutions and provides an alternative proof of their existence. This approach should be useful in studying regularity and large time behavior of these solutions.

We study traveling wave solutions of scalar hyperbolic balance laws

and their viscous counterpart where the viscosity ε is assumed to be small. We assume the following about f and g:

(F) f is convex: f ∈ C³, f″(u) > 0

(G) g ∈ C² with simple zeroes

and look for entropy traveling waves of (1).

In this paper viscous shock profiles of the Riemann problem of systems of viscous hyperbolic balance laws will be considered. Even for strictly hyperbolic flux terms and gradient like source terms with real eigenvalues, oscillating viscous shock profiles can be found. They appear near a Hopf-like bifurcation point of the traveling wave o.d.e. The structure of this bifurcation and its implications for the shock solutions will be studied in detail.

We present recent results concerning the convergence rates for a relaxation scheme approximating scalar conservation laws. The initial layer effects on the convergence rates are quantified and sharp global error estimates are obtained by combining the relaxation error and the discretization error. When the flux is strictly convex, we further obtain the lip⁺ stability under an additional scale condition, which yields the local and global convergence rates.

We discuss some of the recent progress for hyperbolic and viscous conservation laws. The following topics are discussed: well-posedness, viscous waves, relaxation, vacuum, discretization, and multi-dimensional gas flows. Key results and references are briefed and listed. We will also mention open problems.

Shock wave study has become a very rich subject. To make the presentation concise, we have concentrated on some of the topics of direct interest to the author and have listed few references, mostly only the latest ones directly related to the topics under discussion.

We consider the Cauchy problem for the system of 1-D compressible Euler equations with damping and the related diffusive problem. By combining the usual weighted energy methods with special L¹-estimate, we proved the global existence and large time behavior for the smooth solutions and the time-asymptotically equivalence with convergence rates for both problems under some mild initial conditions.

In this note, our aim is the proof of local wellposedness for quasilinear wave equations for initial data less regular than what is required by energy method. This implies to prove Strichartz type estimates for wave operators whose coefficients are only lipschitz. We improve the index of regularity obtained in [3].

The subject of this lecture is the notion of hyperbolicity for systems of Euler-Lagrange equations and the associated domain of dependence property. We shall introduce a new notion of hyperbolicity which represents a significant departure from the notions which have hitherto been proposed. To place the present work in proper context, we shall begin with a brief exposition of pre-existing notions of hyperbolicity and discuss what are, from our point of view, their shortcomings…

We give necessary and sufficient conditions for boundedness, blow-up and attractivity to the zero and to the nonzero equilibria for a nonlinear Klein-Gordon equation. We use the the stable and unstable sets, introduced by Payne and Sattinger [1].

This paper concerns the large time asymptotic behavior of weak solutions of Maxwell’s equations with possibly nonlinear conductivity in an arbitrary spatial domain. Weak and local L^q-convergence (q ≤ 2) to stationary states is shown.

The following sections are included:

• Strichartz inequalities for magnetic potential wave operators

• Idea of the proof of the main theorem

• References

A modulational approach to stability of non-topological solitons ϕ(t, x) = e^iωt f_ω(x) in □ϕ + m²ϕ = |ϕͼ^{p − 1} is given. This yields a proof of stability and a proof that on a pseudo-Riemannian manifold solitons move along geodesics (in an appropriate scaling). The stable range of frequencies ω can be determined from the Gagliardo-Nirenberg inequality.

Some method of constructing intrinsically defined discrete models for differential operators on the space with the Lorentz metric is described. The difference wave equation and a discrete model of the mixed problem are constructed.

Global dynamics of nonlinear wave equations with cubic nonlinear and non-monotone damping are studied. By the asymptotical bootstrap method, it is shown that for any initial status with finite energy, a unique solution exists globally and is bounded. Moreover, there exists an absorbing set for this solution semiflow.

It is proved that an initial-boundary value problem for the equation ρu_tt − (F[u_x])_x = f, where F is a Prandtl-Ishlinskii operator with distribution function η, admits a unique regular solution even if ρ, η are discontinuous functions of x. The homogenization problem with spatially periodic functions ρ^ε, η^ε if is studied; the homogenized values ρ* , η* are identified and the convergence of solutions as ε → 0 is proved.

The concepts of memory and hysteresis operators acting in spaces of time-dependent functions are reviewed, and some simple examples are illustrated. Some O.D.E.s and P.D.E.s containing such operators are then briefly discussed.

In particular the initial- and boundary-value problem associated with an equation of the form is studied, assuming that is a memory operator which fulfils a suitable monotonicity condition.

We prove that for small values of the magnetic parameter b and Rayleigh numbers smaller than a critical one, the magnetic Benard equations on the infinite plate are stable.

The rigorous approximation of long wavelength motions of fluids by a pair of uncoupled Korteweg-de Vries equations is described. We then extend this result to study the motion of the Fermi-Pasta-Ulam model of coupled nonlinear oscillators.

We study the energy flow of a class of formally gradient systems on unbounded domains. We prove that the ω-limit set of each point contains an equilibrium, that all the invariant measures are supported on the set of equilibria, and that the ω-limit set of μ-almost every point in the phase space consists of equilibria, where μ is any Borel probability measure invariant for spatial translations.

We consider two aspects of pattern formation over unbounded domains. First we show the nonlinear diffusive stability of stationary spatially periodic solutions of the Swift-Hohenberg equation in two dimensions. Second we show the existence of exponentially stable modulating multi-pulse solutions for a modified Swift-Hohenberg equation with a resonant spatially periodic forcing.

Several classes of stochastic discrete dynamical systems are investigated. Dimer automata act asynchronously on grids of cells with a deterministic local function. They are suited to model reactions and diffusion on lattices. Edge processes are a generalization of this concept. They act asynchronously on directed graphs. An edge process calls an edge and changes the states of the two adjacent vertices according to a stochastic local function. A partially totalistic cellular automaton can be used to model activator-inhibitor dynamics and to generate patterns on a two-dimensional lattice.

We consider 2-d CNN and compute the lower bound of spatial entropy when the template is square-cross. We then study the structure of traveling wave solutions of 1-d CNN of the advanced type. We show the existence of monotone traveling waves, oscillating wave and eventually periodic waves by using shooting method and comparison principle.

For IPS acting on edge transitive graphs we define densities of configurations on finite subgraphs. Then existence of these densities on all finite subgraphs for a starting configuration implies the almost sure existence of them for all times. We describe the time evolution of the particle density by a differential equation and apply it to get an estimate for the parameter of the Contact Process.

It is proved the existence of an infinite number of periodic solutions of a infinite lattice of particles with external periodic forces and nearest-neighbor interaction of Toda type between particles, by using a priori bounds and topological degree together with a limiting argument.

We present a numerical approach to the inverse homogenisation problem in the context of two-dimensional linear elasticity. Our method is based on classical shape optimization techniques, in contrast with the “black and white squares” approach used in the existing literature. Our method has several advantages; it has also the disadvantage of not being able to create new holes or destroy existing ones. We overpassed the major difficulty of allowing the holes to move freely, not bounding them to the interior of the square cell.

We consider a simple variational 2D model which has been proposed to describe domain branching in microstructures arising from martensitic phase transformations, in the presence of an undeformed interface. We show the energetically optimal deformation is asymptotically self-similar, and derive local bounds on the minimizer and its energy.

We present continuum equations derived from microscopic steps-and-terraces growth models. These can be used in the modeling of epitaxial thin film growth. Assuming certain attachment and detachment mechanisms at the steps, we have obtained a Hamilton-Jacobi equation for the surface height which is coupled with a diffusion equation for the edge adatom density. These equations are supplemented with boundary conditions to describe the evolution of peaks and valleys on the surface.

Convergence theorems and asymptotic estimates (as ε → 0) are proved for eigenvalues and eigenfunctions of boundary-value problems for the Laplace operator in a plane thick periodic multi-structure with concentrated masses. An approach based on the asymptotic theory of elliptic problems in singularly perturbed domains is used.

The aim of this paper is to investigate the evolution of microstructures in crystals described by time dependent Young-measures. We consider the fully dynamic equation of elasticity with a nonconvex energy functional satisfying some regularity conditions and introduce an optional coupling with a parabolic equation describing the solid state diffusion in the crystal. We prove an existence theorem for Young-measure solutions of this system.

In our talk we observe that the relaxation theorem holds in the class of anti-plane shear or similar deformations for integrands with sufficiently fast growth at infinity. In the homogeneous case the assumption on growth can be relaxed up to the level having clear physical sense and a condition, which characterizes solvability of all boundary value minimization problems ( 1), is available.

The following sections are included:

• Introduction

• Preliminary notions

• Auxiliary results

• Main result

• References

This paper is a expanded versionof the talk delivered at EQUADIFF 99. We will review some recent advances in critical point theory and some of their applications to elliptic problems on ℝⁿ.

The abstract set up deals with the existence of critical points of C² functionals of the type

where E is a Hilbert space and f₀ possesses a finite dimensional, possibly non compact manifold Z of critical points.

The abstract results provide a unified frame for a broad variety of variational problems. Here we will focus on the problem of finding positive solutions of some elliptic equations on ℝⁿ, including the prescribed scalar curvature problem.

We investigate the existence of solutions for a class of multipoint boundary value problems for second order differential equations with nonlocal boundary conditions.

In this work we study the coupling of the Navier-Stokes equations and the Cahn-Hilliard equation which stands for a model of a multiphase flow. In a theoretical viewpoint, we study existence, uniqueness and asymptotic properties of the solutions for an alloy under shear. We also show some numerical results in more general situations.

This paper studies sequences of approximations that converge to maximal and minimal solutions of a Periodic problem for ϕ - Laplacian equation. These are build from lower and upper solutions in the reverse order. Such a result is based on an anti-maximum comparison principle.

We study the sweeping processes in a Hilbert space generated by a closed not necessarily convex moving set. Some existence results are obtained, with or without hypothesis of compactness, as well as under suitable assumptions uniqueness and regularity properties are established. One of the approaches we consider requires the variational property of closed sets generalizing convexity and having clear geometrical interpretation through the metric projection into the sets.

In this paper, we give a description of the set of solutions for the following problem:

where p > 1, 0 < β < 1 < α and λ is a real parameter

In this work we study the existence of extremal solutions for a second order integro-differential problem. In order to do this we study a linear problem, showing existence and comparison results for it. The results that we present improve recently published others ones.

In this paper λ-matrix with the properties of polynomial part of the expansion of a root of polynomial in scalar case is defined. Its existing will be proved in a terms of eigenvalues of the oldest matrix. The solution of the matrix differential equation of Riccati type will be described and the conditions for their existing will be given. For commutative matrixes the results are in the same form as in the scalar case.

We investigate critical points of the free energy E_ε(u) of the Cahn-Hilliard model over the unit square under the constraint of a mean value m. We show that for any value m in the spinodal region there are critical points of E_ε(u) having characteristic symmetries and monotonicities. As ε tends to zero these critical points have singular limits which are global minimizers of E₀(u) forming characteristic patterns.

The following equation was studied by Lazer and McKenna (see [1] and [2] for a similar problem) as a model for nonlinear oscillations in a floating beam:

where [t]⁺ = max(t, 0), [t]⁻ = −min(t, 0) ∀t ∈ ℝ and b, c are positive constants. In naval architecture a ship is frequently modelled as a floating beam, that is a vibrating beam with free-end boundary condition. In this context u represents the depth of the bottom of the ship and the restoring force due to flotation is bu when u ≥ 0 (underwater) and 0 if u ≤ 0; the equation is therefore nonlinear. The term c corresponds to the force of gravity.

We study the existence and multiplicity of positive weak solutions to the problems

with l < p < ∞, 1 < q < p < r < p*, where if 1 < p < N, p* = ∞ if p ≥ N, λ is a real parameter, a ≥ 0, h and g verify some integrability conditions.

We present general definitions of the concepts of upper and lower solutions for first order discontinuous ordinary differential equations. The arguments that show the validity of our definitions are based on classical theory of functions of one real variable. With these new definitions we study the existence of extremal solutions for a variety of nonlinear boundary value problems. Finally, we consider the situation in which the upper and the lower solution are not ordered.

We discuss the Fučík spectrum of a general Sturm-Liouville boundary value problem. In some special cases the Fučík spectrum is known explicitly and consists of a countable collection of curves, with certain geometric properties. We show that these properties hold for the spectrum of the general problem, and we also describe the asymptotic behaviour of the curves in the spectrum.

The talk is devoted to the introduction of a topological degree for densely defined operators in reflexive Banach spaces and applications of this concept to nonlinear elliptic problems with strong growth of coefficients. Results of this talk was obtained jointly with A. Kartsatos.

We prove the existence of a radial, bounded and positive solution of the nonlinear elliptic problem

under some growth and sign condition imposed on f.

We consider the existence of positive solutions of the following problem:

with Ω being a smooth bounded domain in ℝ^N, 1 < p < N, g ∈ C(ℝ) and , which changes sign.

The three-dimensional motion of an incompressible inviscid fluid is classically described by the Euler equations, but can also be seen, following Arnold, as a geodesic on a group of volume-preserving maps. Local existence and uniqueness of shortest paths have been established by Ebin and Marsden. In the large, Shnirelman exhibited a class of data for which there are no classical shortest paths. It is shown how the problem can be solved in a suitable generalized framework linked to the hydrostatic limit of the Euler equations.

Using a variational approach, the existence of smooth, time periodic solutions is shown for two types of dynamics associated to the Abelian Higgs energy functional, for second order and Schroedinger dynamics. These represent vortices in rigid rotation about one another.

In the work we investigate the boundary value problems for heat equation with integral mean-values. The problems with direct and inverse time are considered. It is established that a priori estimate is also true for the boundary value problem with inverse time, that is, the problem is correct.

In the Sobolev classes the theorems about unique solvability of boundary value problems for the loaded parabolic and hyperbolic equations with time derivatives of the first and the second order in nonlocal boundary conditions are established. The initial conditions are non-homogeneous and they aren’t always compatible on the region and it’s boundary according to the trace theorems.

By minimizing in Sobolev spaces of mappings which are equivariant with respect to certain torus actions, we construct homotopically nontrivial harmonic maps between spheres. Doing so, we can represent the nontrivial elements of π_{n + 1}(Sⁿ) (n ≥ 3) and of π_{n + 2}(Sⁿ) (n ≥ 5 odd) by harmonic maps, as well as infinitely many elements of π_n(Sⁿ) (n ∈ ℕ). The existence proof involves equivariant regularity theory.

Here we only give a summary of our construction. Details will appear in [Ga3].

We derive some new integral inequalities of the form:

where I = (α,β), −∞ ≤ α < β ≤ ∞, p > 0, H is a wide class of absolutely continuous functions h defined on I and satisfying the limits conditions h(α) = 0 or h(β) = 0, the functions r, s and u are any set of functions related by the appropriate weight functions.

I report on symmetry results for functions which yield sharp constants in Sobolev-type inequalities…

A variational problem for the bending energy of closed surfaces under the prescribed area and volume. Minimizers of this problem are interpreted as models of the shape of red blood cells. The existence and stability/instability of critical points are studied.

The following sections are included:

• Introduction

• Desingularization of constant mean curvature surfaces

• Constant mean curvature surfaces with Delaunay ends

• Minimal hypersurfaces

• Overview of the construction of Theorem 1

• References

In this paper we shall present a multiplicity result for the nonlinear elliptic equation −Δu = |u|^p−2 u + f(x) with non-homogeneous boundary conditions. In spite of the loss of symmetry, suitable perturbative arguments and variational methods allow to find infinitely many solutions of the problem.

We study the formation of singularities for hypersurfaces evolving by mean curvature flow. The hypersurfaces we consider are smooth, closed and mean convex (i.e., with positive mean curvature). We prove a priori estimates from below on the elementary symmetric polynomials of the principal curvatures. This result implies that all rescalings near a singularity are convex and yields a classification of the possible singularities.

Using topological degree methods we prove bifurcation results for some quasilinear elliptic systems defined on ℝ^N. Also, certain regularity results are obtained.

Integral inequalities of the form

are derived, where I = (α,β), −∞ < α < β < ∞, r, s, u are real functions of the variable t, H is a class of functions h absolutely continuous on I.

This paper studies the set of points of non differentiability, also called the singular set, of a concave solution to Hamilton–Jacobi equations of the form

More precisely, we prove that if (t₀, x₀) is a singular point for u, then a Lipschitz arc (t, x(t)), t ∈ [t₀, +∞[, exists such that x(t₀) = x₀ and all the points (t, x(t)) are singular.

We extend the Strong Maximum Principle to fully nonlinear degenerate elliptic operators. In particular we are interested in Hamilton-Jacobi-Bellman operators which can be expressed as the supremum or the infimum of second order linear operators. Moreover we characterize the set of propagation of the maxima of the viscosity solutions of the corresponding equations.

We briefly review some recent results concerning the Liouville property for elliptic nonlinear equations and its connection with L^∞ a priori bounds for solutions of Dirichlet problems.

We consider the homogenization problem for fully nonlinear first order scalar partial differential equations of Hamilton-Jacobi type such as

where ∊ is a small positive parameter and H is a periodic function of the second variable. Our main result is an estimate on the rate of convergence of u^∊ to the solution of the homogenized problem.

The following sections are included:

• Introduction

• Main results

• Proof of the main results

• Acknowledgment

• References

Comparison principle /CP/ for semicontinuous viscosity sub- and supersolutions is proved.

In this article we consider to extended some recent results on the regularity of the solutions to a certain class of linear elliptic equations.

The regularity and other qualities of the solution are obtained with methods employed in two relevant papers by Chiarenza, Frasca and Longo in [4] and [5].

Our motivation to use that scheme stems from its explicit formula for the derivatives of the solution.

We prove the upper semicontinuity of attractors for perturbations of dissipative evolutionary equations under the condition that the eigenvalues and eigenfunctions of the corresponding linear problems behave continuously. We apply this result to two examples of perturbations of the domain.

We discuss the fundamental concepts in the theory of dissipation, maximal compact invariant sets and global attractors with examples to illustrate the results. Counterexamples are given to show that most of the hypotheses are necessary.

We consider a Semilinear Hyperbolic Problem on all of ℝ^N. Existence of attractors and their dimensions are discussed in homogeneous Sobolev and weighted L^P-spaces.

We consider parabolic systems of reaction-diffusion type on large and unbounded domains Ω ⊂ ℝ^d. Under the assumption that there is an absorbing ball in L^∞(Ω) we discuss the existence of global attractors A^Ω and their dependence on the domain Ω. For this we derive continuity properties of the solution operator in exponentially weighted L^∞-norms.

For ε a positive small parameter, Ω a bounded domain in ℝⁿ, 1 ≤ n ≤ 4, we consider the damped wave equation , for x ∈ Ω, supplemented with homogeneous Dirichlet boundary conditions. Under some dissipation hypotheses on f, we show that this hyperbolic equation is a regular perturbation of the associated parabolic limit equation for ε = 0.

In this paper we study the existence of global compact attractors for nonlinear parabolic equations of reaction-diffusion type for which the uniqueness of solution is not guaranteed. Applications are given to differential inclusions which model combustion in porous media and the transmission of electrical impulses in nerve axons.

We prove optimal estimates for the Kolmogorov ε-entropy of the uniform attractors of non-autonomous evolution equations. We illustrate the general estimates with some examples of partial differential equations and systems.

We derive upper bounds for the Hausdorff dimension of flow invariant compact sets on Riemannian manifolds. They are formulated in terms of the eigenvalues of the symmetric part of the operator of an associated system in normal variations. For sets which possesses an equivariant tangent bundle splitting with respect to the flow we include the topological entropy and uniform Lyapunov exponents into the estimates.

Two phase-field models which account for memory effects are considered. Existence of uniform absorbing sets and attractors is discussed.

We consider some models of Cahn-Hilliard equations based on constitutive laws derived by M. Gurtin. These models take the work of internal microforces and the deformations of the material into account. We are interested here in the existence of finite dimensional attractors for these equations.

The asymptotic behaviour is studied of the solutions of nonlinear non autonomous hyperbolic equations with different types of nonlinarities.

Non-autonomous dynamical systems generate cocycles w.r.t. a flow. We give conditions for the existence of attractors for cocycles based on the so-called pull back convergence. These conditions will be applied to the non-autonomous Navier Stokes equation. In particular. we do not need compactness assumption for the time dependent coefficients of this equation.

In this paper we expose the results obtained in the joint papers of V.V.Chepyzhov and M.I.Vishik [4]-[6], [12], [ 16]-[18].

The quasilinear second order parabolic equations and systems of a reaction-diffusion type

are considered. Here Ω ⊂ ℝⁿ is an unbounded domain in ℝⁿ with a sufficiently smooth boundary, u = (u¹, …, u^k) is unknown vector-valued function, Δ_x is a Laplacian with respect to x = (x₁, …, x_n), f and g are given functions and λ₀ is fixed positive constant…

We present the general concept of an order-preserving (or monotone) random dynamical system and we apply well-developed deterministic ideas to study the long-time behaviour of the trajectories and the properties of random attractors of these systems. As an example we consider a random dynamical system generated by a coupled system of stochastic reaction–diffusion equations.

We consider a linear nonautonomous parabolic partial differential equation (PDE) of second order

where Ω ⊂ ℝⁿ is a bounded domain with boundary ∂Ω of class for each and each nonzero …

We consider the Neumann problem for time-periodic reaction-diffusion equations. We address the question whether a stable periodic solution can be subharmonic, that is, whether its minimal period can be larger than the period of the equation. We present two theorems answering this question; one dealing with spatially inhomogeneous, the other with spatially homogeneous equations. The results presented in this note have been obtained jointly with E. Yanagida and will appear with complete proofs in [11, 12].

We study the Dirichlet problem for the nonlinear degenerate and/or singular parabolic equations in a bounded, non-cylindrical and non-smooth domain Ω ⊂ ℝ^N+1, N ≥ 2. Existence and boundary regularity results are established. We introduce a notion of parabolic modulus of left-lower semicontinuity at the points of the lateral boundary manifold and show that the upper Hölder condition on it plays a crucial role for the boundary continuity of the constructed solution.

In this paper we present a new perturbation result for proving logarithmic Sobolev inequalities. By using a non-symmetric perturbation in linear Fokker-Planck equations the entropy method yields improved estimates for the logarithmic Sobolev constant in several cases.

We describe the dynamics of a class of model systems of two reaction-diffusion equations. We examine the dynamics using a new concept of ‘planar solutions’ which finally gives a Poincaré-Bendixson result.

In this paper we study a model of ferromagnetic material governed by a nonlinear Landau-Lifschitz equation coupled with Maxwell equations. After proving the existence of weak solutions, we establish that all points of the ω-limit set of any trajectory are solutions of the stationary model. Furthermore we derive rigorously the quasistatic model by an appropriate time average method. At last, we prove the existence of local strong solutions.

We discuss blow-up-connections between equilibria of a semilinear parabolic equation. By a blow-up-connection from an equilibrium ϕ⁻ to an equilibrium ϕ⁺ we mean a function u(·, t) which is a classical solution on the interval (−∞, T) for some T ∈ ℝ and blows up at T but continues to exist in an appropriate weak sense for t ∈ [T, ∞) and satisfies u(·, t) → ϕ^± as t → ±∞ in a suitable sense.

The aim of this paper is to study the blow up behaviour of radially symmetric solutions u of a semilinear parabolic equation around an unfocused blow up point x = a(a ≠ 0). We show that u behaves as if a one-dimensional problem was concerned, that is the possible blow up patterns around x = a are the ones corresponding to the case of dimension N = 1.

We study the long time behavior of layers arising in a scalar reaction-diffusion equation: u_t = ∈²u_xx + u(u − a)(1 − u), −∞ < x < ∞ (0 < a < 1/2,∈ << 1). With the aid of appropriate upper and lower solutions, we show that the collision of the layers takes place and the layers collapse in a finite time.

We use techniques based on Γ-convergence to establish sufficient conditions for existence and stability of stationary solutions to the diffusion equation , under Neumann boundary condition in two-dimensional domains. These solutions develop, inner transition layers.

Let Ω be an arbitrary sufficiently regular bounded domain in ℝ² = ℝ × ℝ. Write (x, y) for a generic point of ℝ × ℝ. Given ∈ > 0 squeeze Ω by the factor ∈ in the y-direction to obtain the squeezed domain Ω_∈. More precisely, define the map T_∈: ℝ² → ℝ² by (x, y) ↦ (x, ∈y) and set Ω_∈ := T_∈(Ω). Consider the following reaction-diffusion equation on Ω_∈:

…

A reaction-diffusion system with skew-gradient structure is a sort of activator-inhibitor system which consists of two gradient systems coupled in a skew-symmetric way. Any steady state of this system is characterized as a critical point of some functional. We study the relation between a stability property as a steady state of the reaction-diffusion system and a min maximizing property as a critical point of the functional.

As an example a weakly-coupled reaction-diffusion system on the real line is treated which is invariant under reflection. If we suppose that the critical wave number is 0, then under non-degeneracy conditions the existence of time-periodic single and multiple pulse solutions is demonstrated. Preliminary statements on the behavior of the Floquet multipliers of the linearisation on single-pulses are given.

The nonstandard finite difference method was developed empirically. This paper is an attempt towards the mathematical theory behind the success of this approach. The concept of qualitative stability is introduced. As a particular example, elementary stability of difference schemes is considered.

Pulse splitting is an important phenomena in the 1-D Gray-Scott model. We carried out an extensive analysis of slowly-modulated two pulse solutions leading up to splitting in ‘Slowly-modulated two pulse solutions and pulse splitting bifurcations” (Doelman, Eckhaus, and Kaper, submitted, April 1999, University of Utrecht Preprint 1103). Our treatment of the existence , stability, and bifurcations concerned the strong pulse interaction problem. This brief report highlights one aspect of that analysis, namely the existence of slowly-modulated two pulse solutions in a central portion of the parameter space.

In this note we study the existence and stability of asymptotically large homoclinic N-pulse solutions of the generalized Gierer-Meinhardt equation. It is shown that the stability of these patterns can be determined explicitly by the recently developed NLEP approach [2, 3, 4].

We discuss the nonlinear stability of planar travelling waves for the damped hyperbolic equation εu_tt + u_t = Δu + f(u) in ℝⁿ, where ε > 0 is a parameter. The nonlinearity f is assumed to be either of the “monostable” or of the “bistable” type. Our results hold uniformly in ε as ε → 0, and reduce to previously known statements in the parabolic limit ε = 0.

Travelling wave solutions of three-component systems with competition and diffusion are considered. Under the assumption of bistability, the existence and stability of travelling front and back solutions is shown. Next, it is shown that standing pulse solutions bifurcate from the points where travelling front and back solutions with the same zero velocity coexist. Moreover the direction of bifurcation and stability properties of bifurcated solutions are also shown.

Existence and stability of travelling waves are conveniently studied on unbounded domains, for instance, on the real line. Motivated by numerical and experimental studies, we investigate the effects of truncating the unbounded domain to a large but bounded domain. It is shown that periodic boundary conditions accurately reproduce the L²-spectrum of the unbounded domain, whereas separated boundary conditions introduce two new phenomena: Firstly, the essential spectrum is in general shifted to the so-called absolute spectrum that corresponds to optimally chosen exponential weights on the unbounded domain. Secondly, separated boundary conditions may generate additional eigenvalues.

We relate some recent stability results for spatially periodic solutions and modulating traveling waves with classical hyperbolicity results, as the Hartman-Grobman theorem and stability by linearization.

New tools (written in Mathematica) for numerical solution of linear boundary value problems (LBVPs), both regular and eigenvalue ones, are presented, which strongly enhance functionality of the built-in function NDSolve. The tools implement two basic methods - the chasing method (which was generalized by one of the authors for the multipoint problems), and the well-known shooting method. As an example of the tools application we discuss the linear stability analysis for Marangoni convection in the spherical layer.

We use the imbedding of the total differential operator D into a Heisenberg algebra to give a method to generate the transvectants and their multilinear generalizations using the coherent state method.

We determine the existence of cosymmetries for the scalar evolution equations

by using the symbolic method and emphasize the role played by the permutation group in answering divisibility questions. In some special cases, we conclude that there is only one possible cosymmetry. The method can also be used to classify the evolution equations by generating functions. As an example, we give the complete list for 7^th-order KdV-like equations with nontrivial conservation laws.

A perturbation technique is presented for computing normal forms from a general n-dimensional differential equation. The technique, without the application of center manifold theory, can be used to systematically find a simplest (or unique) normal form. Hopf bifurcation is used as an illustration for the method. The implementation of the approach, with the aid of Maple, is straightforward and the computation is efficient.

Estimates of control systems are used for the analysis of their controllability at singular points. A generic system in Rⁿ can have k-singular points with k ∈ {1,…,n}; there is small time local transitivity at any of its k-singular points for n ≥ k ≥ 3, but not at a 1-singular point. This approach is also used to obtain a sufficient conditions for local transitivity at 2-singular points.

We present an optimal control based algorithm for the computation of robust domains of attraction for perturbed systems. We give a sufficient condition for the continuity of the optimal value function and a characterization by Hamilton-Jacobi equations. A numerical scheme is presented and illustrated by an example.

We show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables.

We look at a control affine system in ℝ^d with a singular point x* ∈ ℝ^d. Motivated by the example of the perturbed Duffing-van der Pol equation we show, that under a condition on the Lyapunov exponents, there exists a control set D ⊂ ℝ^d with nonvoid interior such that x* ∈ closure(D).

In this paper we solve a general singular L-Q problem by extending both the input-trajectory map and the cost functional onto an adequate subspace of the Sobolev Space H_−r. The extended problem is equivalent to a regular L-Q problem. We describe the structure of the generalized optimal controls and the corresponding generalized trajectories. The optimal generalized solutions can be computed from the solution of a Riccati differential equation.

Controllability properties of the inverse power method on projective space are investigated. For complex eigenvalue shifts a simple characterization of the reachable sets in terms of invariant subspaces can be obtained. The real case is more complicated and is investigated in this paper. Necessary and sufficient conditions for complete controllability are obtained in terms of the solvability of a matrix equation. Partial results on the solvability of this matrix equation are given.

The present work deals with practical aspects of shape optimization for the elliptic boundary value problems. The main aim in similar problems is to find a shape of a domain D such that an object function value depended on D behaves in an appropriate way. We propose easy numerical recursive method based on the special auxiliary problem. The existence of the optimal domain and the algorithm monotonicity are examined.

On the basis of a nonlinear differential equation model of overhead traveling cranes, we derive a nonlinear feedback control law that stabilizes the system globally. The Lyapunov method is employed to derive the nonlinear stabilizing feedback.

We present three new results for general time dependent differential equations with f : D ⊂ ℝ × ℝ^N → ℝ^N “measurable in t and C^k in x”. The results are about Spectral Theory, Normal Form Theory and Smooth Foliation Theory and they are taken from Siegmund [5].

A nonautonomous differential equation is of Carathéodory type if f : D ⊂ ℝ × ℝ^N → ℝ^N is measurable in t (for fixed x) and C^k in x (for fixed t). Two examples: Random Differential Equations (RDE) driven by a metric dynamical system θ are pathwise of Carathéodory type (Arnold [1]). Bilinear Control Systems for fixed controls u : ℝ → U, U ⊂ ℝ^N compact and convex, are of Carathéodory type (Colonius, Kliemann [6]).

A short résumé of several papers is given with the following goal: For stationary and Hopf bifurcation of a general class of operator equations, including partial differential equations, we have shown that the discretization methods, based on the usual bordered systems, yield convergent bifurcation scenarios. In the following step of post-processing we transform the bifurcation scenario back to the state variables and parameter situation of the original problem. A C⁺⁺-program is given.

We study the dynamics of two-dimensional Poiseuille flow. Firstly we obtain the family of periodic solutions which bifurcates from the laminar flow, together with its stability for several values of the wave number α. The curve of periodic flows presents several Hopf bifurcations. For α = 1.02056 we follow the branches of quasi-periodic orbits that are born at one of the bifurcation points.

We outline numerical methods for the stability analysis of steady state solutions and the computation and stability analysis of periodic solutions of systems of delay differential equations with multiple fixed discrete delays.

Motivated by a wheelset application we present numerical methods for the investigation of periodic motions of mechanical multibody systems depending on parameters. The equations of motions of such rultibody systems are differential algebraic equations of index three, and require a different treatment from explicit ordinary differential equations. Using a projected collocation method we approximate periodic solutions and their Floquet multipliers, and investigate bifurcations on periodic branches.

Simulation of vector fields is ubiquitous: examples occur in every discipline of science and engineering. Periodic orbits are frequently encountered as trajectories. We use solutions of initial value problems, computed via numerical integration, as a means of finding stable periodic orbits of vector fields. We expect numerical integration algorithms to be reliable and their output to be consistent with other means of analyzing the properties of vector fields. These expectations are not always met. This lecture describes several examples of this type, followed by a brief description of a new set of boundary value solvers that appear to give significantly improved methods for computing these difficult periodic orbits.

Recently efficient set-oriented methods have been proposed for the numerical investigation of dynamical systems [1, 2, 3]. A basic question arising in implementing these methods is to compute the “set-wise image” of some set: determine all sets of some collection which intersect the image of the given set. We describe how this discretization question can be tackled rigorously and furthermore how the numerical effort can be reduced to a minimum.

We introduce a computational scheme to continue periodic orbits in Hamiltonian systems and study its bifurcations. We illustrate this method with two degrees of freedom integrable and non-integrable hamiltonian systems. In the first case the presence of continuous symmetries has to be taken into account in the algorithm.

This paper describes the adaptation of an algorithm for computing two-dimensional (un)stable manifolds of vector fields (Krauskopf & Osinga, CHAOS 9(3): 768-774, 1999) to non-orientable global manifolds of a periodic orbit. Such manifolds arise when the corresponding Floquet multiplier of the periodic orbit is negative.

The behavior of the continuous-time analogue of the classical Newton method for root-finding and optimization problems is analyzed near certain singular roots. The continuous setting provides a framework from which several results concerning iterative methods may be derived through different discretizations schemes, and points out the relation with certain singular differential-algebraic equations (DAEs).

An indirect approach to solving nonlinear smooth minimization problems is to integrate a differential algebraic equation (DAE) or an ordinary differential equation (ODE) appropriate to the underlying minimization problem. Here, local minima are computed as ω-limit sets of evolutions of these equivalent ODE’s or DAE’s. Discrete convergence theorems are then obtained by discretizing these ODE’s and DAE’s with appropriate numerical schemes.

The virtual “World of Bifurcation” (WOB) has been established as a www-domain. WOB combines a database of bifurcation problems with tutorials on nonlinear phenomena. The name of the domain is www.bifurcation.de. The emphasis is on examples that are application-oriented. The first version of WOB includes a set of 15 examples.

The splitting of separatrices of area preserving maps close to the identity is one of the most paradigmatic examples of an exponentially small or singular phenomenon. The intrinsic small parameter is the characteristic exponent h > 0 of the saddle fixed point. A standard technique to measure the splitting of separatrices is the so-called Poincaré-Melnikov method, which has several specific features in the case of analytic planar maps. The aim of this talk is to compare the predictions for the splitting of separatrices provided by the Poincare-Melnikov method, with the analytic and numerical results in a simple example, where computations in multiple-precision arithmetic are performed.

We consider general analytic families of area-preserving maps. In such families two types of bifurcations of codimension one are possible. Near a bifurcation the separatrices of the hyperbolic fixed point form a small loop. The splitting of the separatrices is exponentially small with respect to the bifurcation parameter. We provide asymptotic formulae for the splitting.

The phases of a large class of parabolic partial differential equations with rapid forcing can be separated up to exponentially small errors. This is a counterpart for partial differential equations of the theorem by Neishtadt [3] for ordinary differential equations. In our case the exponential rate depends on time t and the estimates have the form h exp(−c(t)h^−⅓).

We consider a multi-frequency slow-fast ODE system with constant unperturbed frequencies. By a near the identity change of variables we eliminate the dependence of the right-hand side of the equations on the fast variables up to exponentially small terms. Careful estimates of the exponentially small remainders are presented.

We study the stable and unstable manifolds of a partially hyperbolic invariant torus for the Generalized Arnold Model (an analytic near-integrable Hamiltonian system which displays weak hyperbolicity). Their generating functions, which are solutions of the Hamilton-Jacobi equation, have a difference that is proved to be exponentially small w.r.t. one of the perturbation parameters. Lower bounds are also discussed in a particular case.

Exponentially small phenomena occur in a wide variety of problems. Always dealing with problems which are analytical, we find these phenomena both in estimates of the remainders in normal forms of vector fields and diffeomorphisms and in averaging methods and, in general, in most of the phenomena which cannot be detected by using any fixed finite order approach of the classical perturbation theory. They also appear in phenomena like the delay of the bifurcation for fixed points and periodic solutions, in the case of slowly varying parameters, in adiabatic invariance and in the corresponding bifurcation diagrams of all these problems. They can be revealed by a suitable use of complex variable, by extending the phase space and time to suitable complex strips or by using a classical method letting the order to increase, up to some optimal value, when the small parameter tends to zero. Numerically they can be detected by a careful use of high accuracy arithmetics, leading in a natural way to the problem of optimization of all the algorithms involved.

This work develops robust contact algorithms capable of dealing with multibody nonsmooth contact geometries for which neither normals nor gap functions can be defined. Such situations arise in the early stage of fragmentation when a number of angular fragments undergo complex collision sequences before eventually scattering. Such situations precludes the application of most contact algorithms proposed to date.

A finite-difference approximation preserving the cosymmetry property for two-dimensional filtration convection problem is presented. Numerical results of the family of equilibria computation are given and family degeneration is observed when an inappropriate approximation was used.

The design of practical geometric integration algorithms is discussed. After a brief introduction to splitting algorithms, methods for varying timestep and smooth integrator switching are described. Applications for this work include the development of variable-stepsize and variable-order reversible integrators and specialized schemes for the collisional N-body problem.

“Dual composition”, a new method of constructing energy-preserving discretizations of conservative PDEs, is introduced. It extends the summation-by-parts approach to arbitrary differential operators and conserved quantities. Links to pseudospectral, Galerkin, antialiasing, and Hamiltonian methods are discussed.

I calculate the reduction of the system of three point vortices (in the plane) of strength –Γ/3 and one point vortex of strength Γ. The resulting reduced space has one degree of freedom and I reduced it by the discrete action corresponding to the exchange of the three identical-strength vortices. This system represents the internal degrees of freedom for localized 4-vortex states which have recently been obtained by collisions of 4-vortex relative equilibria.

Variational methods are a class of symplectic-momentum integrators for ODEs. Using these schemes, it is shown that the classical Newmark algorithm is structure preserving in a non-obvious way, thus explaining the observed numerical behavior. Modifications to variational methods to include forcing and dissipation are also proposed, extending the advantages of structure preserving integrators to non-conservative systems.

We present a new technique for the computation of the largest Lyapunov exponent. In contrast to the existing techniques this method is based on the spatial integration of the logarithm of the norm of certain vector fields defined over the invariant set.

We consider the problem of strong approximations of the solution of stochastic delay differential equations of Itô form with a constant delay. We indicate the nature of the equations of interest and give a convergence result for explicit single-step methods. Illustrative numerical examples using an Euler-Maruyama scheme are provided. The paper is based on joint work with C. T. H. Baker.

It is shown that one-step explicit numerical schemes with variable step size can be formulated as a discrete time cocycle system that has a numerical pullback attractor close to that of the continuous time cocycle system generated by a particular type of dissipative nonautonomous differential equation.

We discuss a conjecture concerning the existence of (nontrivial) isolated eigenvalues of the Perron-Frobenius operator by a numerical study of several one-dimensional dynamical systems.

Techniques for approximating the dynamics of deterministic systems based on a discrete Markov chain are well known and have been successfully used in the past. We now extend these techniques to random dynamical systems by defining a suitably averaged Markov model. These constructions are often numerically superior to iterative orbit based methods and provide greater theoretical control.

We consider the invariance properties of Young measures associated with weak convergence of iterates of a mapping. Numerical computation of these measures is also discussed.

Elements of the Conley decomposition of dynamical systems such as chain transitive sets, attractors and isolating blocks can be approximated by communcation classes of a Markov chain associated with Ulam’s method which is based on a finite partition of phase space. The Markov chain comes from a discrete time dynamical system (map) and its sample paths are ∈-pseudo orbits. Our results (see details in [8]) justify some of the numerical demonstrations obtained earlier by Hsu [6]. Results similar to ours on the approximation of the Conley decomposition and Morse filtrations can be found in [10]. We state two theorems giving sufficient conditions for an attracting set supporting an SBR (Sinai,Ruelle,Bowen) invariant measure to be approximated in the Hausdorff metric. The invariant measure can also be approximated in some cases. Details can be found in [7] Recently our method for approximating SBR measures as described in [7] has been used by Dellnitz and co-workers to compute the invariant measures of attracting sets e.g. the Lorenz attractor [2], [3].

The paper introduces a constructive method for localization of the Morse spectrum of a dynamical system on a vector bundle. The Morse spectrum is a limit set of Lyapunov exponents of periodic pseudotrajectories. The proposed method does not demand any preliminary information on a system. An induced dynamical system on the projective bundle is associated with a directed graph called Symbolic Image. The symbolic image can be considered as a finite discrete approximation of a dynamical system. Valuable information about the system may come from the analysis of a symbolic image. In particular, a neighborhood of the Morse spectrum can be found. By localization of the Morse spectrum, one can constructively recognize the hyperbolicity or the normal hyperbolicity of a dynamical system. The main results of this article were announced in [6], and the proofs are in [7].

We perform a numerical study of a knotted flow through a cylinder. In this example - which is due to Conley [1] - based on certain assumptions on the flow Wazewski’s Theorem guarantees the existence of an unstable invariant set inside the cylinder. We explicitly construct a flow with the desired properties and approximate the corresponding invariant set using set oriented multilevel subdivision techniques.

We study systems exhibiting a large number of coexisting attractors. An intricate fractal basin boundary separates the attractors and gives rise to an extremely high sensitivity to the final state. Applying noise to the system results in complex dynamics. We discuss noise-induced preference of attractors and the complexity of the attractor hopping dynamics.

We review an active area of recent research in nonlinear dynamics: riddling in chaotic systems. For dynamical systems with an invariant manifold in which there is a chaotic set, riddling can occur in the sense that the basin of the attracting set in the manifold can be riddled with holes that belong to the basin of another coexisting attractor off the manifold. We first consider the case where the chaotic set is attracting and give mathematical conditions and scaling laws for the bifurcation to riddling. We then discuss manifestations of riddling for situations where the chaotic set in the invariant manifold is nonattracting. These phenomena are expected to be typical and, therefore, they are observable in physical systems.

Discrete-time dynamical systems on Riemannian manifolds are investigated. Using properties of associated full variational systems and partial variational systems in transversal directions to the considered invariant submanifolds we obtain conditions for stability of such invariant sets. The contribution is also concerned with upper bounds for the fractal dimension of normally hyperbolic invariant sets.