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 Orbits for Some Systems of Delay Differential Equations

We provide sufficient conditions for the existence of periodic orbits of some systems of delay differential equations with a unique delay. We extend Kaplan-Yorke＇s method for finding periodic orbits from a delay differential equation with several delays to a system of delay differential equations with a unique delay.     

Periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces

This paper discusses the existence and multiplicity of periodic orbits of Hamiltonian systems on symmetric positive-type hypersurfaces. We prove that each such energy hypersurface carries at least one symmetric periodic orbit. Under some suitable pinching conditions, we also get an existence result of multiple symmetric periodic orbits.     

Periodic orbits and expansiveness

We prove that if a local diffeomorphism has expanding periodic points robustly then it is an expanding map. Using this, we reobtain a result due to Sakai: generic positively expansive maps are expanding. Our methods also show a global version of a result by Gan and Yang: generic expansive diffeomorphisms are Axiom A without cycles.     

On the existence and uniqueness of connecting orbits for cooperative systems

This paper considers the problem of the existence and uniqueness of the connecting orbit for a cooperative vector fieldF in the retangular boxB. Suppose that the origin O and the pointP with positive components are vertices ofB, only containing the equilibria O andP, and that the Jacobian matrices ofF at O andP are irreducible. Then there is a unique orbit ofF contained inB which joins O andP.     

Periodic orbits for a class ofC 1 three-dimensional systems

We studyC ¹ perturbations of a reversible polynomial differential system of degree 4 in\(\mathbb{R}^3 \). We introduce the concept of strongly reversible vector field. If the perturbation is strongly reversible, the dynamics of the perturbed system does not change. For non-strongly reversible perturbations we prove the existence of an arbitrary number of symmetric periodic orbits. Additionally, we provide a polynomial vector field of degree 4 in\(\mathbb{R}^3 \) with infinitely many limit cycles in a bounded domain if a generic assumption is satisfied.     

Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems

A Lagrangian system describing a motion of a charged particle on a Riemannian manifold is studied. For this flow an analog of a Ricci curvature is introduced, and for Ricci positively curved flows the existence of periodic orbits is proved.

     

Qualitative analysis of a class of competitive and cooperative systems

Hirsch[1,2] studied the limiting behavior of solutions of competitive or cooperative systems, and showed that ifL is an ω-limit set of a three-dimensional cooperative system, which contains no equilibrium, thenL is a nonattracting closed orbit. Smith^[3] considered a three-dimensional irreducible competitive system and showed that an ω-limit set containing no equilibrium must be a closed orbit which has a simple Floquet multiplier λ<1, and may be attracting. In this paper we carry out the qualitative analysis of a class of competitive and cooperative systems, and a generalization of the result of Levine^[4] is given. The stability problem of closed orbits raised in [5] and [6] is resolved.     

Periodic orbits and chaos in fast-slow systems with Bogdanov-Takens type fold points

The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast-slow type having Bogdanov-Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov-Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronquée solution of the first Painlevé equation in the blow-up space.     

Periodic orbits from nonperiodic orbits on an interval

This paper demonstrates that any continuous real-valued function which has an orbit with infinitely many limit points must necessarily have periodic cycles of arbitrarily large prime period. We present an example of a function with an orbit whose limit points are exactly Z⁺.     

Periodic orbits of radially symmetric Keplerian-like systems: A topological degree approach

We are concerned with non-autonomous radially symmetric systems with a singularity, which are T-periodic in time. By the use of topological degree theory, we prove the existence of large-amplitude periodic solutions whose minimal period is an integer multiple of T. Precise estimates are then given in the case of Keplerian-like systems, showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system.     

Periodic oscillatory solution in delayed competitive–cooperative neural networks: A decomposition approach

In this paper, the problems of exponential convergence and the exponential stability of the periodic solution for a general class of non-autonomous competitive–cooperative neural networks are analyzed via the decomposition approach. The idea is to divide the connection weights into inhibitory or excitatory types and thereby to embed a competitive–cooperative delayed neural network into an augmented cooperative delay system through a symmetric transformation. Some simple necessary and sufficient conditions are derived to ensure the componentwise exponential convergence and the exponential stability of the periodic solution of the considered neural networks. These results generalize and improve the previous works, and they are easy to check and apply in practice.     

Periodic orbits for additive cellular automata

We formulate and study a necessary and sufficient condition for a configuration of any type of infinite additive cellular automata to have periodic behavior in time. The number of orbits with periodn is counted. Relations between spatial and temporal periods are discussed.Supported in part by G.M.C.I., DEEE-LNETI (Portugal).     

Periodic orbits in complex Abel equations

This paper is devoted to prove two unexpected properties of the Abel equation dz/dt=z³+B(t)z²+C(t)z, where B and C are smooth, 2π-periodic complex valuated functions, t∈R and z∈C. The first one is that there is no upper bound for its number of isolated 2π-periodic solutions. In contrast, recall that if the functions B and C are real valuated then the number of complex 2π-periodic solutions is at most three. The second property is that there are examples of the above equation with B and C being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of B and C. This result is also in contrast with the characterization of the center variety for the examples of Abel equations dz/dt=A(t)z³+B(t)z² studied in the literature, where the center variety is located in a finite number of connected components.     

Planar competitive and cooperative difference equations

This paper focuses on the asymptotic behavior of planner order-preserving difference equations with particular attention to those arising from models of two-species competition.     

Periodic orbits of the restricted three-body problem

We prove, using a variational formulation, the existence of an infinity of periodic solutions of the restricted three-body problem. When the problem has some additional symmetry (in particular, in the autonomous case), we prove the existence of at least two periodic solutions of minimal period , for every . We also study the bifurcation problem in a neighborhood of each closed orbit of the autonomous restricted three-body problem.

     

Periodic orbits and chain-transitive sets of C1-diffeomorphisms

We prove that the chain-transitive sets of C¹-generic diffeomorphisms are approximated in the Hausdorff topology by periodic orbits. This implies that the homoclinic classes are dense among the chain-recurrence classes. This result is a consequence of a global connecting lemma, which allows to build by a C¹-perturbation an orbit connecting several prescribed points. One deduces a weak shadowing property satisfied by C¹-generic diffeomorphisms: any pseudo-orbit is approximated in the Hausdorff topology by a finite segment of a genuine orbit. As a consequence, we obtain a criterion for proving the tolerance stability conjecture in Diff¹(M).