Hyperbolic Periodic Orbits Research Articles

Splitting of separatrices in the resonances of nearly integrable Hamiltonian systems of one and a half degrees of freedom

In this paper we consider general nearly integrable analytic Hamiltonian systems of one and a half degrees of freedom which are a trigonometric polynomial in the angular state variable. In the resonances of these systems generically appear hyperbolic periodic orbits. We study the possible transversal intersections of their invariant manifolds, which is exponentially small, and we give an asymptotic formula for the measure of the splitting. We see that its asymptotic first order is of the form $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$ and we identify the constants $K,\beta,a$ in terms of the system features. We compare our results with the classical Melnikov Theory and we show that, typically, in the resonances of nearly integrable systems Melnikov Theory fails to predict correctly the constants $K$ and $\beta$ involved in the formula.

Quenching in a class of singularly perturbed mechanical systems

In this paper we get some results about the dynamics of a class of mechanical systems under strong dissipation. We have obtained the stability of the semi-trivial solutions in this class, its asymptotic expansion, and the existence of hyperbolic periodic orbits and their stability was analysed. From these results we have showed the existence of quenching in a particular case.

The rate of convergence of new Lax–Oleinik type operators for time-periodic positive definite Lagrangian systems

Assume that the Aubry set of the time-periodic positive definite Lagrangian L consists of one hyperbolic 1-periodic orbit. We provide an upper bound estimate of the rate of convergence of the family of new Lax–Oleinik type operators associated with L introduced by the authors in Wang and Yan (2012 Commun. Math. Phys. 309 663–91). In addition, we construct an example where the Aubry set of a time-independent positive definite Lagrangian system consists of one hyperbolic periodic orbit and the rate of convergence of the Lax–Oleinik semigroup cannot be better than O(1/t) as t → +∞.

A NONLINEAR CONTROL SCHEME FOR DISCRETE TIME CHAOTIC SYSTEMS

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems. We consider both one dimensional and higher dimensional cases. We propose a nonlinear feedback law and present some stability results. These results show that for period 1 all hyperbolic periodic orbits can be stabilized with the proposed method. By restricting the gain matrix to a special form we obtain some novel stability results. The stability proofs also give the possible feedback gains which achieve stabilization. We also present some simulation results.

Invariant relations for the Bowen-Series transform

Consider the Bowen-Series transform T T associated with an even corners fundamental domain of finite volume for some Fuchsian group Γ \Gamma . We prove a generic invariance result that abstracts Series’ orbit-equivalence theorem to families of relations on the unit circle. Two applications of this result are developed. We first prove that T T satisfies a strong-orbit equivalence property, which allows to identify its hyperbolic periodic orbits with primitive hyperbolic conjugacy classes of Γ \Gamma . Then, we show thanks to the invariance theorem that the eigendistributions for the eigenvalue 1 1 of the transfer operator of T T with spectral parameter s ∈ C s \in \mathbb {C} are in bijection with smooth bounded eigenfunctions for the eigenvalue s ( 1 − s ) s(1-s) of the hyperbolic Laplacian on the quotient D / Γ \mathbb {D} / \Gamma .

Phase space structure of the hydrogen atom in a circularly polarized microwave field

We consider the problem of the hydrogen atom interacting with a circularly polarized microwave field, modeled as a perturbed Kepler problem. A remarkable feature of this system is that the electron can follow what we term erratic orbits before ionizing. In an erratic orbit the electron makes multiple large distance excursions from the nucleus with each excursion being followed by a close approach to the nucleus, where the interaction is large. Here we are interested in the mechanisms that explain this observation. We find that the manifolds associated with certain hyperbolic periodic orbits may play an important role, despite the fact that, in some respects, the dynamics is almost Keplerian. A study of some relevant invariant objects is carried out for different system parameters. The consequences of our findings for ionization of an electron by the external field are also discussed.

Lyapunov Exponents, Periodic Orbits and Horseshoes for Mappings of Hilbert Spaces

We consider smooth (not necessarily invertible) maps of Hilbert spaces preserving ergodic Borel probability measures, and prove the existence of hyperbolic periodic orbits and horseshoes in the absence of zero Lyapunov exponents. These results extend Katok’s work on diffeomorphisms of compact manifolds to infinite dimensions, with potential applications to some classes of periodically forced PDEs.

Generic Morse–Smale property for the parabolic equation on the circle

In this paper, we show that, for scalar reaction–diffusion equations ut=uxx+f(x,u,ux) on the circle S1, the Morse–Smale property is generic with respect to the non-linearity f. In Czaja and Rocha (2008) [13], Czaja and Rocha have proved that any connecting orbit, which connects two hyperbolic periodic orbits, is transverse and that there does not exist any homoclinic orbit, connecting a hyperbolic periodic orbit to itself. In Joly and Raugel (2010) [31], we have shown that, generically with respect to the non-linearity f, all the equilibria and periodic orbits are hyperbolic. Here we complete these results by showing that any connecting orbit between two hyperbolic equilibria with distinct Morse indices or between a hyperbolic equilibrium and a hyperbolic periodic orbit is automatically transverse. We also show that, generically with respect to f, there does not exist any connection between equilibria with the same Morse index. The above properties, together with the existence of a compact global attractor and the Poincaré–Bendixson property, allow us to deduce that, generically with respect to f, the non-wandering set consists in a finite number of hyperbolic equilibria and periodic orbits. The main tools in the proofs include the lap number property, exponential dichotomies and the Sard–Smale theorem. The proofs also require a careful analysis of the asymptotic behavior of solutions of the linearized equations along the connecting orbits.

A NEW PERIODIC CONTROLLER FOR DISCRETE TIME CHAOTIC SYSTEMS

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems. For simplicity we consider only one dimensional case. We propose a novel periodic feedback controller law and present some stability results. This scheme may be considered as a novel generalization of the classical delayed feedback scheme, which is also known as Pyragas scheme. The stability results show that all hyperbolic periodic orbits can be stabilized with the proposed method. The stability proofs also give the possible feedback gains which achieve stabilization. We will also present some simulation results.

Sufficient conditions for a partially hyperbolic attractor to be a homoclinic class

We prove that a partially hyperbolic attractor with two-dimensional central direction Λ is a homoclinic class if it exhibits a hyperbolic periodic orbit O and a Lorenz-like singularity σ with W u ( σ ) ∩ W s ( O ) ≠ ∅ such that W s ( σ ) is dense in Λ.

The Lyapunov exponents of C 1 hyperbolic systems

Let f be a C 1 diffeomorphisim of smooth Riemannian manifold and preserve a hyperbolic ergodic measure µ. We prove that if the Osledec splitting is dominated, then the Lyapunov exponents of µ can be approximated by the exponents of atomic measures on hyperbolic periodic orbits.

Geometric shadowing in slow–fast Hamiltonian systems

We study a class of slow–fast Hamiltonian systems with any finite number of degrees of freedom, but with at least one slow one and two fast ones. At ε = 0 the slow dynamics is frozen. We assume that the frozen system (i.e. the unperturbed fast dynamics) has families of hyperbolic periodic orbits with transversal heteroclinics.For each periodic orbit we define an action J. This action may be viewed as an action Hamiltonian (in the slow variables). It has been shown in Brännström and Gelfreich (2008 Physica D 237 2913–21) that there are orbits of the full dynamics which shadow any finite combination of forward orbits of J for a time t = O(ε−1).We introduce an assumption on the actions of periodic orbits which enables us to shadow any continuous curve (of arbitrary length) in the slow phase space for any time. The slow dynamics shadows the curve as a purely geometrical object, thus the time on the slow dynamics has to be reparametrized.

Lyapunov exponents of hyperbolic measures and hyperbolic periodic orbits

Lyapunov exponents of a hyperbolic ergodic measure are approximated by Lyapunov exponents of hyperbolic atomic measures on periodic orbits.

Integration of Dissipative Partial Differential Equations: A Case Study

We develop a computer-assisted technique for constructing and analyzing orbits of dissipative evolution equations. As a case study, the methods are applied to the Kuramoto–Sivashinski equation, for which we prove the existence of a hyperbolic periodic orbit.

Lin's method for heteroclinic chains involving periodic orbits

We present an extension of the theory known as Lin's method to heteroclinic chains that connect hyperbolic equilibria and hyperbolic periodic orbits. Based on the construction of a so-called Lin orbit, that is a sequence of continuous partial orbits that only have jumps in a certain prescribed linear subspace, estimates for these jumps are derived. We use the jump estimates to discuss bifurcation equations for homoclinic orbits near heteroclinic cycles between an equilibrium and a periodic orbit (EtoP cycles).

Asymptotic orbits in the (N+1)-body ring problem

In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m 0=β m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov periodic orbits, which emanate from the collinear equilibrium points L 1 and L 2. We construct numerically, from the intersection points of the appropriate Poincare cuts, homoclinic symmetric asymptotic orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points. All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented.

The Aubry set for periodic Lagrangians on the circle

For periodic convex Lagrangians on \( \mathbb{S}^1 \), we show that, generically, in the sense of Mane, there exists a dense open set of cohomology classes such that the Aubry set of these Lagrangians is a hyperbolic periodic orbit. This allows us to prove Mane’s conjecture on \( \mathbb{S}^1 \).

Resonant zones, inner and outer splittings in generic and low order resonances of area preserving maps

We consider a one-parameter family of area preserving maps in a neighbourhood of an elliptic fixed point. As the parameter evolves hyperbolic and elliptic periodic orbits of different periods are created. The exceptional resonances of order less than 5 have to be considered separately. The invariant manifolds of the hyperbolic periodic points bound islands containing the elliptic periodic points. Generically, these manifolds split. It turns out that the inner and outer splittings are different under suitable conditions. We provide accurate formulae describing the splittings of these manifolds as a function of the parameter and the relative values of these magnitudes as a function of geometric properties. The numerical agreement is illustrated using mainly the Hénon map as an example.

Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a one-dimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs when there are two codimension-one heteroclinic connections, with the connection from the periodic orbit to the equilibrium corresponding to a tangency between the two relevant manifolds. The results are restricted to $\mathbb{R}^3$, the lowest possible dimension in which such a heteroclinic cycle can occur, but are expected to be applicable to systems of higher dimensions as well. A geometric analysis is used to partially unfold the dynamics near such a heteroclinic cycle by constructing a leading-order expression for the Poincare map in a full neighborhood of the cycle in both phase and parameter space. Curves of orbits homoclin...

A NEW DELAYED FEEDBACK CONTROL SCHEME FOR DISCRETE TIME CHAOTIC SYSTEMS

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems. We consider both one dimensional and higher dimensional cases. We propose a novel generalization of the classical delayed feedback law and present some stability results. These results show that for period 1 all hyperbolic periodic orbits can be stabilized with the proposed method. Although for higher order periods the proposed scheme may possess some limitations, some improvement over the classical delayed feedback scheme still can be achieved with the proposed scheme. The stability proofs also give the possible feedback gains which achieve stabilization. We will also present some simulation results.