Reversible Vector Fields Research Articles

Invariant manifolds in a reversible Hamiltonian system: The tentacle-like geometry

We study a one-parameter family of time-reversible Hamiltonian vector fields in R4, which has received great attention in the literature. On the one hand, it is due to the role it plays in the context of certain applications in the field of Physics or Engineering and, on the other hand, we especially highlight its relevance within the framework of generic unfoldings of the four-dimensional nilpotent singularity of codimension four.The system exhibits a bifocal equilibrium point for a range of parameter values. The associated two-dimensional invariant manifolds, stable and unstable, fold into the phase space in such a way that they produce intricate patterns. This entangled geometry has previously been called tentacular geometry.We consider a three-dimensional level set containing the bifocal equilibrium point to gain insight into the folding behavior of these invariant manifolds. Our method consists of describing the traces left by invariant manifolds when crossing an invariant cross section by the reversibility map. With this new approach, we provide a better understanding of how tentacular geometry evolves with respect to the parameter.Our techniques enables us to link the tentacular geometry on the cross section with the study of cocooning cascades of homoclinic tangencies. Indeed, we present a general theory to extend to R4 the phenomena related to cocoon bifurcation that classically develop within families of three-dimensional reversible vector fields. On the basis of these results, we conjecture the existence of heteroclinic cycles consisting of two orbits connecting the bifocus with a saddle node periodic orbit.

Stability conditions for reversible and partially integrable systems

The purpose of this paper is to provide results like Peixoto’s Theorem inside a class \mathfrak{X} of 3-dimensional reversible systems. We deal with reversible systems associated to an involution \varphi_1 such that \operatorname{Dim}\operatorname{Fix}(\varphi_1) is equal to one. We further assume the existence of a first integral for any X\in\mathfrak{X} that leaves invariant any sphere centered at the origin. The main result establishes generic necessary conditions for the structural stability in \mathfrak{X} . Stability conditions on generic one-parameter families of reversible vector fields on the two-dimensional sphere are also stated.

Phase portraits of (2;0) reversible vector fields with symmetrical singularities

In this paper we study the phase portraits in the Poincaré disk of the reversible vector fields of type (2;0) having generic bifurcations around a symmetric singular point p. We also prove the nonexistence of any periodic orbit surrounding p. We point out that some numerical computations were necessary in order to control the number of limit cycles.

On the Hamiltonian structure of normal forms at elliptic equilibria of reversible vector fields in [formula omitted

This paper addresses the question whether normal forms of smooth reversible vector fields in R4 at an elliptic equilibrium possess a formal Hamiltonian structure. In the non-resonant case we establish a formal conjugacy between reversible and Hamiltonian normal forms. In the case of non-semi-simple 1:1 resonance and p:q resonance with p+q>2 we establish a weaker form of equivalence, namely that of a formal orbital equivalence to a Hamiltonian normal form that involves an additional time-reparametrization of orbits. Moreover, in case p+q>3 we show that no formal conjugacy to a Hamiltonian normal form exists.

Chaos in periodically forced reversible vector fields

We discuss the appearance of chaos in time-periodic perturbations of reversible vector fields in the plane. We use the normal forms of codimension~$1$ reversible vector fields and discuss the ways a time-dependent periodic forcing term of pulse form may be added to them to yield topological chaotic behaviour. Chaos here means that the resulting dynamics is semi-conjugate to a shift in a finite alphabet. The results rely on the classification of reversible vector fields and on the theory of topological horseshoes. This work is part of a project of studying periodic forcing of symmetric vector fields.

Hopf Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

This paper is concerned with small quadratic perturbations to one parameter family of generic reversible quadratic vector fields with a simple center. The first objective is to show that this system exhibits two small amplitude limit cycles emerging from a Hopf bifurcation. The second one we prove that the system has no limit cycle around the weak focus of order two. The results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev (1998).

Cyclicity of a Family of Generic Reversible Quadratic Systems with One Center

This paper is concerned with the quadratic perturbations from one parameter family of generic reversible quadratic vector fields having a simple center and an invariant straight line. It is shown that the system can generate at least two limit cycles. As the parameter is rational, we propose a procedure for finding the upper bound to cyclicity of period annulus based on the Chebyshev criterion for Abelian integrals together with one rationalizing transformation. To illustrate our approach, we determine the cyclicity of three representative reversible systems. Our results may be viewed as a contribution to proving the conjecture on cyclicity proposed by Iliev [1998].

Normal form theory for reversible equivariant vector fields

We give a method to obtain formal normal forms of reversible equivariant vector fields. The procedurewe present is based on the classical method of normal forms combined with tools from invariant theory. Normal forms of two classes of resonant cases are presented, both with linearization having a 2-dimensional nilpotent part and a semisimple part with purely imaginary eigenvalues.

Formal equivalence between normal forms of reversible and hamiltonian dynamical systems

We show the existence of formal equivalences between $2n$-dimensional reversible and Hamiltonian vector fields. The main tool we employ is the normal form theory.

Dynamical stability of quasi-periodic response solutions in planar conservative systems

We study non-autonomous planar Hamiltonian or reversible vector fields that vanish at the origin. The time-dependence is quasi-periodic with strongly non-resonant frequencies. First, we give a simple criterion in terms of the averaged system for the trivial solution to be dynamically stable. Then we obtain destabilizations for classes of examples where the conditions of the criterion are not satisfied. We end with possible ways to stabilize an unstable trivial solution by means of vector fields with zero average.

On the similarity of Hamiltonian and reversible vector fields in 4D

We study the existence of formal conjugacies between reversible vector fields and Hamiltonian vector fields in 4D around a generic singularity. We construct conjugacies for a generic class of reversible vector fields. We also show that reversible vector fields are formally orbitally equivalent to polynomial decoupled Hamiltonian vector fields. The main tool we employ is the normal formtheory.

On the birth of minimal sets for perturbed reversible vector fields

The results in this paper fit into a program to study the existenceof periodic orbits, invariant cylinders and tori filled with periodicorbits in perturbed reversible systems. Here we focus on bifurcationsof one-parameter families of periodic orbits for reversible vectorfields in $\mathbb{R}^4$. The main used tools are normal forms theory,Lyapunov-Schmidt method and averaging theory.

Reversibility and quasi-homogeneous normal forms of vector fields

This paper uses tools in quasi-homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible.

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p0 of a Z2(ℝ)-reversible vector field in ℝ2n with reversing symmetry R satisfying R2 = I and dimFix(R) = n. We deal with one-parameter families of such systems Xλ such that X0 presents at p0 a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X0 (at p0) has as eigenvalues: λ1 = 0 and λj = ±iαj, j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.

Homoclinic and heteroclinic orbits for the 0² or 0²𝑖𝜔 singularity in the presence of two reversibility symmetries

This paper is devoted to the study of the dynamics of reversible vector fields close to an equilibrium when a 0 2 0^2 or a 0 2 i ω 0^2i\omega resonance occurs in the presence of two symmetries of reversibility. In the presence of a unique reversibility symmetry the existence of a homoclinic connection to 0 is known for the 0 2 0^2 resonance whereas for the 0 2 i ω 0^2i\omega resonance there is generically no homoclinic connection to 0 but there is always a homoclinic connection to an exponentially small periodic orbit. In the presence of a second symmetry of reversibility, the situation is more degenerate. Indeed, because of the second symmetry the quadratic part of the normal forms vanishes, and so the dynamics of the normal forms is governed by the cubic part. For the 0 2 0^2 resonance we prove the existence of homoclinic connections to 0 and of heteroclinic orbits. For the 0 2 i ω 0^2i\omega resonance we prove that in most of the cases the second symmetry induces the existence of homoclinic connections to 0 and of heteroclinic orbits whereas with a unique symmetry there is generically no homoclinic connection to 0. Such a reversible vector field with two reversibility symmetries occurs for instance after center manifold reduction when studying 2-dimensional waves in NLS type systems with one-dimensional potential or when studying localized waves in nonlinear chains of coupled oscillators. It also occurs when studying localized buckling in rods with noncircular cross section.

Hopf Bifurcation in Presence of 1 : 3 Resonance

Abstract Assume that the linear part at a singular point p0 of a C4 differential system Y0 in ℝ4 has eigenvalues ±αi and ±βi such that β/α = 1/3. In the main result of the paper we exhibit a one-parameter family of systems Yε for ε ∈ (-δ0;+δ0) where is shown that the original vector field around p0 can bifurcate in 0, 1, 2, 3 or 4 one-parameter families of periodic orbits. The tool for proving such a result is the averaging theory for non-C1 differentiable system. Moreover, assuming now that Yε is a one-parameter family of ℤ2- reversible polynomial vector fields of degree 5, we show that it can bifurcate in 0 or 2 one-parameter families of periodic orbits.

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.

Cocoon bifurcation in three-dimensional reversible vector fields

The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed by Lau (1992 Int. J. Bifurc. Chaos 2 543–58) in the Michelson system, a three-dimensional ODE system describing travelling waves of the Kuramoto–Sivashinsky equation. In this paper, we present an organizing centre of the principal part of the cocoon bifurcation in more general terms in the setting of reversible vector fields on . We prove that in a generic unfolding of an organizing centre called the cusp-transverse heteroclinic chain, there is a cascade of heteroclinic bifurcations with an increasing length close to the organizing centre, which resembles the principal part of the cocoon bifurcation.We also study a heteroclinic cycle called the reversible Bykov cycle. Such a cycle is believed to occur in the Michelson system, as well as in a model equation of a Josephson Junction (van den Berg et al 2003 Nonlinearity 16 707–17). We conjecture that a reversible Bykov cycle is, in its unfolding, an accumulation point of a sequence of cusp-transverse heteroclinic chains. As a first result in this direction, we show that a reversible Bykov cycle is an accumulation point of reversible generic saddle-node bifurcations of periodic orbits, the main ingredient of the cusp-transverse heteroclinic chain.

Singular perturbation problems for time-reversible systems

In this paper singularly perturbed reversible vector fields defined in R n R^n without normal hyperbolicity conditions are discussed. The main results give conditions for the existence of infinitely many periodic orbits and heteroclinic cycles converging to singular orbits with respect to the Hausdorff distance.

Heteroclinic bifurcations near Hopf-zero bifurcation in reversible vector fields in [formula omitted

We study the dynamics near a symmetric Hopf-zero (also known as saddle-node Hopf or fold-Hopf) bifurcation in a reversible vector field in R 3 , with involutory an reversing symmetry whose fixed point subspace is one-dimensional. We focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle-foci. We study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets (horseshoes), independent of the eigenvalues of the saddle-foci, as well as cascades of bifurcations of periodic, heteroclinic and homoclinic orbits. Finally, we discuss the application of our results to the Michelson system, describing stationary states and travelling waves of the Kuramoto–Sivashinsky PDE.