Systematic Computer Assisted Proofs of periodic orbits of Hamiltonian systems

The numerical study of Dynamical Systems leads to obtain invariant objects of the systems such as periodic orbits, invariant tori, attractors and so on, that helps to the global understanding of the problem. In this paper we focus on the rigorous computation of periodic orbits and their distribution on the phase space, which configures the so called skeleton of the system. We use Computer Assisted Proof techniques to make a rigorous proof of the existence and the stability of families of periodic orbits in two-degrees of freedom Hamiltonian systems, which provide rigorous skeletons of periodic orbits. To that goal we show how to prove the existence and stability of a huge set of discrete initial conditions of periodic orbits, and later, how to prove the existence and stability of continuous families of periodic orbits. We illustrate the approach with two paradigmatic problems: the Hénon–Heiles Hamiltonian and the Diamagnetic Kepler problem.     

Existence of periodic orbits for high-dimensional autonomous systems

We give a result on existence of periodic orbits for autonomous differential systems with arbitrary finite dimension. It is based on a Poincaré-Bendixson property enjoyed by a new class of monotone systems introduced in [L.A. Sanchez, Cones of rank 2 and the Poincaré-Bendixson property for a new class of monotone systems, J. Differential Equations 216 (2009) 1170-1190]. A concrete application is done to a scalar differential equation of order 4.     

Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

This paper analyses the existence of invariant manifolds ofperiodic orbits for a specific piecewise linear three-dimensionalsystem with two zones, whose linear parts share a pair of imaginaryeigenvalues. This degenerate situation is obtained from thelack of controllability. The analysis proceeds by its reductionto a periodic one-dimensional equation for which some resultsof the Ambrosetti–Prodi type are given.     

A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards

We prove a recent theorem by C. Culter every polygonal outer billiard in the affine plane has a periodic trajectory.      

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.     

Globally exponential stability of periodic solutions for nonlinear impulsive delay systems

In this paper, the existence and globally exponential stability of periodic solutions for nonlinear impulsive delay systems are studied. Our results improve and generalize some of the previous results found in the literature. Two examples are discussed to illustrate our results.     

PERSISTENCE AND PERIODIC ORBITS FOR TWO-SPECIES NON-AUTONOMOUS DIFFUSION LOTKA-VOLTERRA MODELS

A non-autonomous competing system is investigated in this paper,where the species x can diffuse between two patches of a heterogeneous environment with barriers between patches,but for species y,the diffusion does not involve a barrier between patches,further it is assumed that all the parameters are time dependent. It is shown that the system can be made persistent under some appropriate conditions. Moreover,sufficient conditions that guarantee the existence of a unique positive periodic orbit which is globally asymptotic stable are derived.     

On the determination of the basin of attraction of periodic orbits in three- and higher-dimensional systems

The determination of the basin of attraction of a periodic orbit can be achieved using a Lyapunov function. A Lyapunov function can be constructed by approximation of a first-order linear PDE for the orbital derivative via meshless collocation. However, if the periodic orbit is only accessible numerically, a different method has to be used near the periodic orbit. Borg's criterion provides a method to obtain information about the basin of attraction by measuring whether adjacent solutions approach each other with respect to a Riemannian metric. Using a numerical approximation of the periodic orbit and its first variation equation, a suitable Riemannian metric is constructed.     

Infinitely many homoclinic orbits for the second-order Hamiltonian systems

In this paper, the existence of homoclinic orbits for the second-order Hamiltonian systems without periodicity is studied and infinitely many homoclinic orbits for both superlinear and asymptotically linear cases are obtained.     

Monotone periodic orbits for torus homeomorphisms

Let be a homeomorphism of the torus isotopic to the identity and suppose that there exists a periodic orbit with a non-zero rotation vector . Then has a topologically monotone periodic orbit with the same rotation vector.

     

Periodic orbits of a one-dimensional non-autonomous Hamiltonian system

In this paper we study the properties of the periodic orbits of with x∈S¹ and a T₀ periodic potential. Called the frequency of windings of an orbit in S¹ we show that exists an infinite number of periodic solutions with a given ρ. We give a lower bound on the number of periodic orbits with a given period and ρ by means of the Morse theory.     

Computation of periodic solutions of Hamiltonian systems

This paper attempts to give a practical method to compute global periodic solutions of autonomous Hamiltonian systems of arbitrary finite order. The proposed numerical method is based on continuation of solutions branching from equlibrium points and requires no iterations. Moreover, during computation of one-parameter families of periodic orbits, their possible bifurcations are determined as well.     

The existence, uniqueness and stability of positive periodic solution for periodic reaction-diffusion system

1. IntroductionLet fi e RN be a bounded open domain with smooth boundary afl, and for eachi E {1, 2,', m}, let Li be a second order differelltial operator defined byand Bi be a boundary operator given bywhere % denotes the outward normal derivative of m on an.We consider the following boundary value problem of the reaction-diffusion system withtime delaywhere i = 1, 2,'. I m) mr = "i(x, t -- r), r 2 0 is a constant. If r ~ 0, it means that system(I) does not include the terms of time lag…     

Formal first integrals for periodic systems

In the present paper, we give an elementary proof for the result of Li et al. (2003) [6] about nonexistence of formal first integrals for periodic systems in a neighborhood of a constant solution. Moreover, we present a criterion about partial existence of formal first integrals for the periodic system, by using the Floquet's theory.     

An extension of Sharkovsky's theorem to periodic difference equations

We present an extension of Sharkovsky's theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties.     

Hamiltonian weakly damped autonomous systems exhibiting periodic attractors

The dynamic local stability of autonomous Hamiltonian, weakly damped, lumped-mass (discrete) systems is reconsidered. For such potential(conservative) systems conditions for the existence of limit cycles are discussed by studying the effect of the damping matrix on the Jacobian eigenvalues. New findings that contradict existing results are presented. Thus, undamped stable symmetric systems with the inclusion of slight damping may experience: (a) a double zero eigenvalue bifurcation, a degenerate Hopf bifurcation and a generic (usual) Hopf bifurcation, and (b) a limit cycle (dynamic) mode of instability prior to the static (divergence) mode of instability (failure of Zieglers kinetic criterion). A variety of numerical examples verified by a nonlinear analysis confirm the validity of the theoretical findings presented herein. Received: January 3, 2003; revised: July 14, 2003 and February 17, 2004     

Existence of periodic orbits and chaos in a class of three-dimensional piecewise linear systems with two virtual stable node-foci

In this paper, we consider a new class of piecewise linear (PWL) systems with two virtual stable node-foci (the meaning of “virtual” is from Bernardo et al. (2008)) which exhibits periodic orbits and chaos. This fact that PWL systems have no unstable equilibria but has chaos will unavoidably make the exploration of this chaos more complicated. Particular values for bifurcation diagram are provided. Based on mathematical analysis and Poincaré map, periodic orbits of this kind of system without unstable equilibrium points are derived, the corresponding existence theorems are given, and the obtained results are applied to specific examples.