Maslov index for homoclinic orbits of Hamiltonian systems

A useful tool for studying nonlinear differential equations is index theory. For symplectic paths on bounded intervals, the index theory has been completely established, which revealed tremendous applications in the study of periodic orbits of Hamiltonian systems. Nevertheless, analogous questions concerning homoclinic orbits are still left open. In this paper we use a geometric approach to set up Maslov index for homoclinic orbits of Hamiltonian systems. On the other hand, a relative Morse index for homoclinic orbits will be derived through Fredholm index theory. It will be shown that these two indices coincide.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Periodic solutions to impulsive differential inclusions with constraints

The existence of a periodic solution to an impulsive differential inclusion being invariant with respect to a non-convex set of state constraints is established by the use a Lefschetz type fixed-point theorem for set-valued maps.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

Melnikov theory for nonlinear implicit ODEs

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some implicit differential equations. In particular we show persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds.     

Levitan/Bohr almost periodic and almost automorphic solutions of second order monotone differential equations

The aim of this paper is to prove the existence of Levitan/Bohr almost periodic, almost automorphic, recurrent and Poisson stable solutions of the second order differential equation

(1)     

Asymmetric type II periodic motions for nonlinear impact oscillators

In this paper a general class of nonlinear impact oscillators is considered for Type II periodic motions. This system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation. The unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law. By approximating the Poincaré map to O(ε)$O\left(\epsilon \right)$ directly, a general method of Melnikov type for detecting the existence of asymmetric Type II subharmonic orbits outside the homoclinic cycles is presented.     

Bifurcation of periodic orbits in a class of planar Filippov systems

In this paper we discuss the perturbations of a general planar Filippov system with exactly one switching line. When the system has a limit cycle, we give a condition for its persistence; when the system has an annulus of periodic orbits, we give a condition under which limit cycles are bifurcated from the annulus. We also further discuss the stability and bifurcations of a nonhyperbolic limit cycle. When the system has an annulus of periodic orbits, we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching.     

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.     

Positive Periodic and Homoclinic Solutions for Nonlinear Differential Equations with Nonsmooth Potential

We study the existence of positive solutions and of positive homoclinic (to zero) solutions for a class of periodic problems driven by the scalar ordinary p-Laplacian and having a nonsmooth potential. Our approach is variational based on the nonsmooth critical point theory and our results extend the recent works of Korman–Lazer (Electronic JDE (1994)) and of Grossinho–Minhos–Tersian (J. Math. Anal. Appl. 240 (1999)).     

The differential equations on time scales through impulsive differential equations

In this paper we investigate differential equations on certain time scales with transition conditions (DETC) on the basis of reduction to the impulsive differential equations (IDE). DETC are in some sense more general than dynamic equations on time scales [M. Bohner, A. Peterson, Dynamic equations on time scales, in: An Introduction With Applications, Birkhäuser Boston, Inc., Boston, MA, 2001, p. x+358; V. Laksmikantham, S. Sivasundaram, B. Kaymakcalan, Dynamical Systems on Measure Chains, in: Math. and its Appl., vol. 370, Kluwer Academic, Dordrecht, 1996]. The basic properties of linear systems, the existence and stability of periodic solutions, and almost periodic solutions are considered. Appropriate examples are given to illustrate the theory.     

On the asymptotic stability of discontinuous systems analysed via the averaging method

The averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems. The idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions. A large number of practical problems lead to differential equations with discontinuous right-hand sides. In a rigorous theory of such systems, developed by Filippov, solutions of a differential equation with discontinuous right-hand side are regarded as being solutions to a special differential inclusion with upper semi-continuous right-hand side. The averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion. In this paper we prove natural extensions of Bogolyubov’s first theorem and the Samoilenko-Stanzhitskii theorem to differential inclusions with an upper semi-continuous right-hand side. We prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one. The extension of Bogolyubov’s theorem concerns finite time intervals, while the extension of the Samoilenko-Stanzhitskii theorem deals with solutions defined on the infinite interval. The averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required.     

Stability of nonlinear waves in a ring of neurons with delays

In this paper, we consider a ring of identical neurons with self-feedback and delays. Based on the normal form approach and the center manifold theory, we derive some formula to determine the direction of Hopf bifurcation and stability of the Hopf bifurcated synchronous periodic orbits, phase-locked oscillatory waves, standing waves, mirror-reflecting waves, and so on. In addition, under general conditions, such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation. Despite the fact that the slowly oscillatory synchronous periodic solution of the scalar equation is stable, we show that the corresponding synchronized periodic solution is unstable if the number of the neurons is large or arbitrary even.     

Nonsmooth generalized guiding functions for periodic differential inclusions

We consider the periodic problem for differential inclusions in $$ \user2{\mathbb{R}}^{\rm N} $$ with a nonconvex-valued orientor field F(t, ζ), which is lower semicontinuous in $$ \zeta \in \user2{\mathbb{R}}^{\rm N} $$ Using the notion of a nonsmooth, locally Lipschitz generalized guiding function, we prove that the inclusion has periodic solutions. We have two such existence theorems. We also study the “convex” periodic problem and prove an existence result under upper semicontinuity hypothesis on F(t, ·) and using a nonsmooth guiding function. Our work was motivated by the recent paper of Mawhin-Ward [23] and extends the single-valued results of Mawhin [19] and the multivalued results of De Blasi-Górniewicz-Pianigiani [4], where either the guiding function is C¹ or the conditions on F are more restrictive and more difficult to verify.     

Nonautonomous bifurcation patterns for one-dimensional differential equations

Although, bifurcation theory of ordinary differential equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this article, two different approaches are discussed which are based on special notions of attractivity and repulsivity. Generalizations of the well-known one-dimensional transcritical and pitchfork bifurcation are obtained.     

Low-order resonances in weakly dissipative spin-orbit models

Second-order differential equations with small nonlinearity and weak dissipation, such as the spin-orbit model of celestial mechanics, are considered. Explicit conditions for the coexistence of periodic orbits and estimates on the measure of the basins of attraction of stable periodic orbits are discussed.     

Long periodic shadowing

A general theorem for establishing the existence of a true periodic orbit near a numerically computed pseudoperiodic orbit of an autonomous system of ordinary differential equations is presented. For practical applications, a Newton method is devised to compute appropriate pseudoperiodic orbits. Then numerical considerations for checking the hypotheses of the theorem in terms of quantities which can be computed directly from the pseudoperiodic orbit and the vector field are addressed. Finally, a numerical method for estimating the Lyapunov exponents of the true periodic orbit is given. The theory and computations are designed to be applicable for unstable periodic orbits with long periods. The existence of several such periodic orbits of the Lorenz equations is exhibited. This revised version was published online in June 2006 with corrections to the Cover Date.     

Limit cycles and Lie symmetries

Given a planar vector field U which generates the Lie symmetry of some other vector field X, we prove a new criterion to control the stability of the periodic orbits of U. The problem is linked to a classical problem proposed by A.T. Winfree in the seventies about the existence of isochrons of limit cycles (the question suggested by the study of biological clocks), already answered by Guckenheimer using a different terminology. We apply our criterion to give upper bounds of the number of limit cycles for some families of vector fields as well as to provide a class of vector fields with a prescribed number of hyperbolic limit cycles. Finally we show how this procedure solves the problem of the hyperbolicity of periodic orbits in problems where other criteria, like the classical one of the divergence, fail.     

Borg’s criterion for almost periodic differential equations

Borg’s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction. In this article, this method is generalized to almost periodic differential equations. Both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution. The condition uses a Riemannian metric, and an example for the explicit construction of such a metric is presented.     

Almost periodic solutions for a class of non-instantaneous impulsive differential equations

Abstract

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations.