Dynamical Systems and Applications

The following sections are included:

• Contributors

• Preface

• CONTENTS

Time-dependent invariants of a set of first-order autonomous, nonlinear ordinary differential equations (ODEs) of polynomial form are determined. The removal of self-linear terms simplifies the analysis of these ODEs. Lie point symmetries of these dynamical equations are found and used to predict the integration of the equations. Various reduction paths are analyzed and the appearance of type I hidden symmetries are indicated. Three examples are shown: the two-dimensional quadratic system, a model two-dimensional system and the three-dimensional Lotka-Volterra system. As the two dimensional systems are invariant under two Lie symmetries, they can be reduced to quadratures. The three-dimensional Lotka-Volterra equations are reduced to a two-dimensional system by elimination of one variable from a simple invariant.

We study the saddle-node bifurcation of a partially hyperbolic fixed point in a Lipschitz family of C^k diffeomorphisms on an n-dimensional manifold, in the case that the stable set and unstable set of the fixed point intersect the stable and unstable manifolds of other invariant sets in a 'critical' manner. Sufficient conditions are found which guarantee realization of only codimension one bifurcations in a family. A diffeomorphism for which the conditions are not satisfied is shown to be in the closure of the set of codimension 2 bifurcation surfaces. These results are a generalization of a one-dimensional result of Malta and Palis.

In this paper we consider optimal control problem for a class of infinite dimensional uncertain systems. Necessary conditions of optimality is presented under the assumption that the principal operator is monotone in a reflexive Banach space. A computational algorithm is also given.

This paper is concerned with linear nonautonomous differential equations in Banach spaces. We analyze spectral properties of the semigroup of operators T^h, h ≥ 0 defined by the formula (T^hν)(t) = X(t,t–h)ν(t–h), where X(t,s) is the evolution operator of the underlying differential equation and ν is an element of an appropriate function space. We present necessary and sufficient conditions for the exponential decay of all solutions or of some individual solutions of the underlying differential equation.

We establish sufficient conditions for permanence phenomena for a class of spatially heterogeneous periodic–parabolic systems arising from ecological models. The conditions are expressed in quantifiable ways in terms of the spectra of associated linear differential operators. In so doing, we connect asymptotic coexistence in such a system to the underlying biological assumptions about the model which are expressed in the parameters and coefficients of these operators.

Let and ε be a small positive real number. We consider the following singularly perturbed problem , where u(t) is subject to the impulses . We assume that the degenerate system has a orbit v₀(t) homoclinic to the hyperbolic equilibrium v = 0. We show that, under certain conditions on f, g, j, the bifurcation towards a homoclinic orbit v(t, ε) of the perturbed system depends on three functions G_i(α, ε), i = 1, 2, 3, α ∈ ℂ, that are 2πε/ω-periodic in α, for ε > 0, and satisfy , η₀ > 0, where . As a consequence we see that if the bifurcation functions are exponentially small.

For an equation in a Banach space which has a saddle-saddle equilibrium point, we prove that it appears a unique periodic orbit from a homoclinic orbit to that point when it disappears and a chaotic invariant set in the sense of Smale when there are more than one homoclinic orbit.

This paper studies the topological entropy of a particular two-parameter family of functions on the unit interval. The set of parameters for which the topological entropy is continuous is determined precisely for the family studied.

The well-known Hopfield neural network has an equilibrium set which is asymptotically stable. That network is an approximation in that the neuron has a zero threshold and, hence, charges are immediately passed on. In this expository note we present a mathematical model which suggests that for certain values of the threshold there will be periodic solutions.

A singular reaction-diffusion mixed boundary-value quenching problem is shown to have a unique critical length, a classical solution before its quenching time, finitely many quenching points which are not boundary points, and the blow-up of the time-derivative of its solution at the quenching point.

We prove that the entrance time into a small interval is asymptotically exponentially distributed. We also prove that under suitable hypothesis the sequence of visits to a small interval converges to a Poisson point process.

The following sections are included:

• Introduction

• Euler-Arnol'd and Euler's equations
  • Euler-Arnol'd equations
  • Euler's equations via reduction

• Parametrization of the heteroclinic surfaces
  • Geometric description
  • Explicit parametrization of the heteroclinic surfaces

• Integration of the Euler-Arnol'd equations

• The solid ball model of SO(3)

• The heteroclinic surfaces

• Notes and references

Starting from a topologically transitive unimodal map f on an interval I, several successive "inducing" constructions lead to maps , , where is a subset of I on which one can construct a conformal measure playing a role similar to that usually played by Lebesgue measure. The system forms a Markov fibered system with the Schweiger property and when a certain parameter takes its extreme value of zero, the measurable dynamics of is closely related to that of the system formed by Lebesgue measure on the "dynamic interval" and the original map f. In the case when the map has a "strange attractor", this setup might give a framework for studying the dynamics on the complement of the basin of the attractor.

Manev's gravitational law, given by a potential function of the form U(ρ) = ν/ρ +, μ/ρ², where ρ is the distance between particles and ν, μ are suitably chosen positive constants, is a fairly good substitute of relativity within the frame of classical mechanics. We first obtain exact formulas of the trajectory and of the nodal period in the two-body problem. Then we proceed to a qualitative study near collision and describe the geometry of the phase space. We show that a black hole effect is characteristic to this model. This implies that the set of collisions has positive measure and that collisions are not regularizable.

In this paper we consider the existence of periodic solutions for the following Hamiltonian System

The paper contains an investigation of iterations of holomorphic symplectomorphisms of ℂ². The main result is that for a dense set of maps most orbits are unbounded.

We study a free interface problem related to combustion of condensed matter and some non-equilibrium exothermal phase transitions. The problem possesses a unique classical solution globally in time. In spite of a variety of non trivial dynamical scenarios exhibited by the model the solutions are uniformly bounded and the interface velocity is a smooth function of time for arbitrary interface kinetics subject to some natural conditions. The model sustains a basic uniformly propagating wave that looses stability beyond a certain threshold value of the governing parameter resulting in Hopf bifurcation. We present numerical illustrations of complex auto-oscillatory regimes of the interface dynamics.

An Engel structure is a regular and absolutely non-integrable 2-dimensional distribution on a 4-manifold. The aim of the paper is to describe Engel structures on 4-manifolds which have simplest dynamic and algebraic properties. We consider:

- Engel distributions which admit vector field bases with simplest Lie algebraic properties;

- Engel structures with simplest (periodic) canonical foliations;

- Engel manifolds which admit a compact hypersurface transversal to the canonical foliation;

- Engel manifolds which admit co-canonical foliations.

A discrete nonlinear model of the vibrating string with a Greenspanlike approach and the corresponding continuous model are considered. The discretization error and the stability are studied and some numerical experiments are also described, illustrating the usefulness and the limits of the given stability condition.

We study non-renormalizable unimodal polynomials f(z) = h(z^ℓ), h is affine preserves the real line. We prove that there exist an infinite sequence n_i of iterates of f and an infinite sequence of pairs of neighborhoods U_i ⊂ V_i of the critical point so that the maps f^n_i : U_i → V_i are polynomial-like with the degree ℓ. The moduli of these polynomial-like maps are separated from zero depending on ℓ only. In particular, the domains U_i exist in all possible scales, i.e. U_i ⊃ U_i+1 and the intersection of all U_i is equal to the critical point of f.

For linear functional differential equations with infinite delay in a Banach space, some relationships between total stability and uniform asymptotic stability are studied. Under some mild assumptions, the equivalence of these two stabilities is proved.

A competing species problem is studied in the limiting cases of small and large diffusion μ. The question of permanence is resolved. For the case of more mathematical interest, the singular limit μ → 0, the asymptotic form of the stationary solutions is obtained. The dynamics for large t is examined using the theory of monotone dynamical systems.

A perturbation analysis, using the method of normal forms, is applied to non-autonomous oscillatory systems described by equations in which the perturbation has an explicit periodic time dependence. The method is demonstrated on two examples: An equation encountered in circular particle accelerators and the equations for the density matrix elements in NMR. The effect of the small nonlinear perturbations as well as of resonance between the natural frequency of the unperturbed system and the periodic forcing terms is analyzed.

The following sections are included:

• References

In this paper we present results concerning a chaotic behaviour of difference equations. It is proved that there exists a horseshoe for the difference equation

At the present work the method is proposed allowing to formalize the procedure of Lyapunov functionals construction. The procedure to discussed in details for stochastic difference equations. Stability conditions obtained by using the procedure are formulated immediately in terms of the coefficients of equations under consideration.

A class of Markov operators satisfies the Foguel alternative if its members are either sweeping or have stationary densities. New sufficient condition for this property is given.

We consider an established model of a thin, shallow spherical shell. Under homogeneous boundary conditions, the 'energy' remains constant in time (conservative problem). We then introduce suitable dissipative feedback controls on the boundary (forces, shears, moments). We first develop an abstract [second order, hence] first order model for the resulting (closed loop) mixed problem. Next, we show that this generates an s.c. semigroup of contractions on a natural function space. In a companion paper, we show that such semigroup is, moreover, uniformly stable.

In this work we analyze the problem of the permanence of two competing species in an inhabited region where each species possesses a refuge area. Under this assumption and using the modeling approach provided by the Lotka-Volterra model with spatialy variable coefficients we show that if the birth rates are sufficiently large, then the species are permanent independently of the competition strength in the regions where competition occurs. On the contrary, if some of the birth rates do not reach the critical value beyond which persistence occurs, then the corresponding species is shown to be driven to extinction by the other.

We weaken the hyperthesis of the Eliashberg-Floer-Gromov-McDuff's theorem by a slight modification of their arguments and as an application it implies the rigidity of symplectic blow-up.

We prove that the topological entropy as a function of a smooth (C¹) interval map with bounded modality is continuous.

A new procedure is described for the close approximation of the critical length a* in a class of nonlinear singular parabolic problems arising in applications such as polarization in ionic conductors and the kinetics of chemical catalysts. The behaviour of the solutions of the initial-boundary value problems involved is known to depend critically on the boundary length a; in any particular problem, when this length exceeds the critical length, quenching will occur; if not, ultimately a solution will approach a steady state. For the class considered, it is known that the steady state is itself a solution of an associated class of boundary value problem which has a minimal positive solution when a < a*, but none when a > a*. The procedure allows the close estimation of the steady state solution and the critical length by using a Newton sequence which converges monotonically (a < a*) and loses monotonicity (a > a*). The computational problem arising from the singularity is handled by transforming the steady state problem to a simpler form and applying a new finite difference formula to the sequence of Newton iterates used to approximate it. The discretisations used allow efficient tri-diagonal algorithms to be employed and have O(h²) accuracy for many singular problems.

Nonoscillatory second order differential equations always admit "special", principal solutions. For a certain type of oscillatory equation principal pairs of solutions were introduced by Á. Elbert, F. Neuman and J. Vosmanský, Diff. Int. Equations 5 (1992), 945–960. In this paper, the notion of principal pair is extended to a wider class of oscillatory equations. Also an interesting property of some of the principal pairs is presented that makes the notion of these "special" pairs more understandable.

Starting with sine curves one can build up more complicated behavior including sine within sine behavior, beats, as well as wave packets, by superimposing sine curves with different wave numbers (Fourier Analysis). This is not the only way we can generate such behavior starting with sine curves. We see that by altering origin point data associated with a set of nonlinear equations, where the nonlinear equations are obtained from a set of mathematically aesthetic principles, we can also obtain similar type results.

The following sections are included:

• Introduction

• Statement of the problem

• Periodic solutions and cycles

• First return function

• Simple cycles of small amplitude

• Simple cycles under large damping

• References

A complete classification of dynamics of a population of a inhibitary pulse-coupled oscillators is presented. The model is based on the work of Mirollo and Strogatz, but our model has an inhibitory coupling between oscillators which makes a sharp contrast with the dynamics of the above authors' model. The main result is that for a large class of initial conditions, the population approaches a periodic state in which all the oscillators keep finite size of phase difference (we call it "phase locking solution" here). For the remaining class of initial data except for nongeneric ones, it evolves to a periodic state with a cluster or a synchronous state depending on a size of cluster. The criterion for the classification is explicitly given and can be judged easily only by the initial condition.

In this paper we prove the stability of solitary wave solutions of the one dimensional Zakharov equations by the variational method. By virtue of a simple inequality for the energies of the Zakharov equations and the corresponding nonlinear Schrödinger equation, we reduce our problem for the Zakharov equations to the case of the single nonlinear Schrödinger equation.

A new constructive method for the study of periodic trajectories of a dynamical system is presented. The dynamical system is associated with an oriented graph called a symbolic image of the system. The symbolic image can be considered as a finite approximation of the dynamical system. Investigation of the symbolic image gives an opportunity to localize the set of periodic trajectories. Sufficient conditions for the existence of a periodic trajectory are presented. The algorithms of construction of a periodic trajectory are described.

We prove that the standard action of SL(3, ℤ) on 𝕋³ is infinitesimally rigid.

Existence of solutions, usually in L^α for some α > 1, is established for integral equations of Volterra and Hammerstein type. The analysis relies on using either Schauder's fixed point theorem or a nonlinear alternative of Leray–Schauder type.

The paper considers a scenario for arising chaos in difference equations with continuous time. The chaos at issue is caused by the fact that the equation has solutions tending (in some special metric) to random functions. We call such a phenomena as self-stochastisity.

In recent years more and more mathematicians, physicists and engineers have realized that computer algebra is a useful tool to tackle mathematical problems. Here we give a survey of how computer algebra can be used in the investigation of differential equations. Mostly nonlinear ordinary differential equations are considered.

We present the uniqueness and existence theorems for (global) minimax solutions of first-order nonlinear partial differential equations satisfying Carathéodory's conditions. The results in this paper generalize that of Adiatullina and Subbotin in [1] and are new even in the case where the Hamiltonians are continuous functions of their arguments.

A first-order finite-difference method is developed, analysed and implemented for the numerical solution of a second-order initial-value problem. The differential equation in this problem exhibits cubic damping, a cubic restoring force and a decaying forcing-term which is periodic with constant frequency. The method is compared with the Euler method and a second-order method is also considered.

A harmonic-balance analysis shows that the initial-value problem discussed in detail is asymptotically equivalent to a more complicated problem.

We describe a normal form theory for parametrized families of symplectic linear operators, and show how on can obtain versal unfoldings of a given symplectic linear operator. As an example we apply the results to the case of a Krein collision.