Nonhyperbolic Periodic Orbits Research Articles

Dynamical implications of convexity beyond dynamical convexity

We establish sharp dynamical implications of convexity on symmetric spheres that do not follow from dynamical convexity. It allows us to show the existence of elliptic and non-hyperbolic periodic orbits and to furnish new examples of dynamically convex contact forms, in any dimension, that are not equivalent to convex ones via contactomorphisms that preserve the symmetry. Moreover, these examples are $$C^1$$ -stable in the sense that they are actually not equivalent to convex ones via contactomorphisms that are $$C^1$$ -close to those preserving the symmetry. We also show the multiplicity of symmetric non-hyperbolic and symmetric (not necessarily non-hyperbolic) closed Reeb orbits under suitable pinching conditions.

On a One-Parameter Discrete-Time Z4-Equivariant Cubic Dynamical System

The lemniscate sine and cosine are solutions of a [Formula: see text]-equivariant planar Hamiltonian system for all of which nontrivial solutions are nonhyperbolic periodic orbits. The forward Euler scheme is applied to this system and the one-parameter discrete-time [Formula: see text]-equivariant cubic dynamical system is obtained. The discrete-time system depending upon a parameter exhibits rich dynamics: numerical simulation shows that the system has attracting closed invariant curves, multiple periodic orbits and attracting sets exhibiting chaotic behavior. The approximating system of ordinary differential equations is constructed. We discuss the existence of closed invariant curves for the discrete-time system.

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.

Multiplicity of periodic orbits for dynamically convex contact forms

We give a sharp lower bound for the number of geometrically distinct contractible periodic orbits of dynamically convex Reeb flows on prequantizations of symplectic manifolds that are not aspherical. Several consequences of this result are obtained, like a new proof that every bumpy Finsler metric on $$S^n$$ carries at least two prime closed geodesics, multiplicity of elliptic and non-hyperbolic periodic orbits for dynamically convex contact forms with finitely many geometrically distinct contractible closed orbits and precise estimates of the number of even periodic orbits of perfect contact forms. We also slightly relax the hypothesis of dynamical convexity. A fundamental ingredient in our proofs is the common index jump theorem due to Y. Long and C. Zhu.

Local contact homology and applications

We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations. Several applications are obtained, including a generalization of Gromoll–Meyer's theorem on the existence of infinitely many simple periodic orbits, resonance relations and conditions for the existence of non-hyperbolic periodic orbits. Most of the results of this paper remain conjectural until the foundational issues of Symplectic Field Theory are resolved.

Nonhyperbolic Periodic Orbits of Vector Fields in the Plane Revisited

The main goal of this paper is to present a theory of approximation of periodic orbits of vector fields in the plane. From the theory developed here, it is possible to obtain an approximation to the curve of nonhyperbolic periodic orbits in the bifurcation diagram of a family of differential equations that has a transversal Hopf point of codimension two. Applications of the developed theory are made in Liénard-type equations and in Bazykin’s predator-prey system.

Dominated splitting of differentiable dynamics with $\mathrm{C}^1$-topological weak-star property

We study weak hyperbolicity of a differentiable dynamical system which is robustly free of non-hyperbolic periodic orbits of Markus type. Let S be a $\mathrm{C}^1$-class vector field on a closed manifold $M^n$, which is free of any singularities. It is of $\mathrm{C}^1$-weak-star in case there exists a $\mathrm{C}^1$-neighborhood $\mathscr{U}$ of S such that for any X$\in\mathscr{U}$, if $P$ is a common periodic orbit of X and S with S$_{\upharpoonright P}=$X$_{\upharpoonright P}$, then $P$ is hyperbolic with respect to X. We show, in the framework of Liao theory, that S possesses the $\mathrm{C}^1$-weak-star property if and only if it has a natural and nonuniformly hyperbolic dominated splitting on the set of periodic points $\mathrm{Per}$(S), for the case $n=3$.

Box dimension and bifurcations of one-dimensional discrete dynamical systems

This paper is devoted to study the box dimension of the orbits of one-dimensional discrete dynamical systems and their bifurcations in nonhyperbolic fixed points. It is already known that there is a connection between some bifurcations in a nonhyperbolic fixed point of one-dimensional maps, and the box dimension of the orbits near that point. The main purpose of this paper is to generalize that result to the one-dimensional maps of class $C^{k}$ and apply it to one and two-parameter bifurcations of maps with the generalized nondegeneracy conditions. These results show that the value of the box dimension changes at the bifurcation point, and depends only on the order of the nondegeneracy condition. Furthermore, we obtain the reverse result, that is, the order of the nondegeneracy of a map in a nonhyperbolic fixed point can be obtained from the box dimension of a orbit near that point. This reverse result can be applied to the continuous planar dynamical systems by using the Poincare map, in order to get the multiplicity of a weak focus or nonhyperbolic limit cycle. We also apply the main result to the bifurcations of nonhyperbolic periodic orbits in the plane.

On the fold-Hopf bifurcation for continuous piecewise linear differential systems with symmetry

In this paper a partial unfolding for an analog to the fold-Hopf bifurcation in three-dimensional symmetric piecewise linear differential systems is obtained. A particular biparametric family of such systems is studied starting from a very degenerate configuration of nonhyperbolic periodic orbits and looking for the possible bifurcation of limit cycles. It is proved that four limit cycles can coexist after perturbation of the original configuration, and other two limit cycles are conjectured. It is shown that the described bifurcation scenario appears for appropriate values of parameters in the celebrated Chua's oscillator.

ON THE STABILIZATION OF PERIODIC ORBITS FOR DISCRETE TIME CHAOTIC SYSTEMS BY USING SCALAR FEEDBACK

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems by using a scalar input. We use a simple periodic delayed feedback law and present some stability results. These results show that all hyperbolic periodic orbits as well as some nonhyperbolic periodic orbits can be stabilized with the proposed method by using a scalar input, provided that some controllability or observability conditions are satisfied. The stability proofs also lead to the possible feedback gains which achieve stabilization. We will present some simulation results as well.

On the stabilization of periodic orbits for discrete time chaotic systems

In this Letter we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems. We propose a novel and simple periodic delayed feedback law and present some stability results. These results show that all hyperbolic periodic orbits as well as some non-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.

Asymptotic phase revisited

We discuss the classical problem on asymptotic phase of periodic orbits of planar systems. The existence of asymptotic phase for non-hyperbolic periodic orbits is completely determined with hypotheses on the derivatives of a Poincaré map and a return-time map. Smoothness of the vector field turns out to be crucial for existence of asymptotic phase. For hyperbolic periodic orbits, a new proof for the existence of invariant foliations is provided.

Asymptotic phase revisited

We discuss the classical problem on asymptotic phase of periodic orbits of planar systems. The existence of asymptotic phase for non-hyperbolic periodic orbits is completely determined with hypotheses on the derivatives of a Poincaré map and a return-time map. Smoothness of the vector field turns out to be crucial for existence of asymptotic phase. For hyperbolic periodic orbits, a new proof for the existence of invariant foliations is provided.

Quasi-periodic Saddle-Node Bifurcations Near a Differentiable Singularity for Forced Oscillations

In the two-parameter unfolding of a Bogdanov-Takens singularity for autonomous differential equations in the plane with reflection symmetry, it is known in one case that there is a curve Γ in parameter space that corresponds to nonhyperbolic periodic orbits, and all one-parameter paths that cross Γ transversally give saddle-node bifurcations of periodic orbits. In the analogous situation for periodically forced systems, the curve Γ is replaced by a Cantor set Γ̃ of parameter values that corresponds to nonhyperbolic quasi-periodic tori, and there is a restricted set of one-parameter paths that give quasi-periodic saddle-node bifurcations of tori. We require only finite differentiability of the system ( C 2 dependence on parameters, C k dependence on state variables, k ≥ 29). The proof of this result uses a version of the Nash-Moser implicit function theorem that obtains C 2 dependence of the implicitly defined function on parameters.