Reversible Vector Fields Research Articles

Vector fields on manifolds with boundary and reversibility — An expository account —

In this paper an expository account on singularities of reversible vector fields on manifolds and boundary singularities is presented. Also we present the bifurcation diagram of a boundary cusp of codimension three, i.e, a Bogdanov-Takens singular point in the boundary of the semi plane {(x, y)∈R2:x>-0} whose topological unfolding is given by the quadratic three parameter family\(y\frac{\partial }{{\partial x}} + (x^2 + ax + c + \alpha x(x + b))\frac{\partial }{{\partial y}}\),\(\alpha = \pm 1\). This study can be applied to the analysis of the behavior of singularity of the germ of vector fieldX0(x, y)=(y, 2x(x4+x2y)) in the class of reversible vector fields.

Bifurcation of Homoclinic Orbits to a Saddle-Center in Reversible Systems

We consider two-parameter families of reversible vector fields having (at the critical parameter value) a homoclinic orbit to a nonhyperbolic fixed point. The nonhyperbolicity is due to a pair of purely imaginary eigenvalues. We give a complete description of the bifurcating one-homoclinic orbits to the center manifold. For that purpose we adapt Lin's method.

Résonance 1 : 2 pour les champs de vecteurs réversibles

We study the 1:2 resonance for reversible vector fields in R 4 , depending on a parameter μ and we show the bifurcation of two families of periodic solutions and of two reversible solutions which are homoclinic to some of the periodic solutions belonging to one or the other of the previous families, depending on the sign of a coefficient. To cite this article: M. Barrandon, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 7–10

Codimension-two singularities of reversible vector fields in 3D

This paper is concerned with the dynamics near an equilibrium point of reversible systems. For a large class of reversible vector fields on the three dimensional space we present all the topological types and their respective normal forms of the codimension-two symmetric singularities. Such classification comes from useful new results, also proved here, on dynamical systems defined on manifolds with boundary.

Hopf-zero bifurcations of reversible vector fields

We study the dynamics of a class of reversible vectorfields having eigenvalues (0,αi,-αi) around their symmetricequilibria. We give a complete list of all normal forms for such vectorfields, their versal unfoldings, and the corresponding bifurcation diagramsof the codimensional-one case. We also obtain some important conclusions onthe existence of homoclinic and heteroclinic orbits, invariant tori andsymmetric periodic orbits.

Multi-bump patterns by a normal form approach

In this paper the behaviour of small solutions in a reaction-diffusionmodel problem is studied near a co-dimension 2 point. The normal form theory for reversible vector fields is applied on the stationary part of the reaction-diffusion system. This normal form is reduced to a 3-dimensional ODE that iscompletely integrable. An explicit expression for the solutions to the ODE andtherefore for the reaction-diffusion system is given under certain conditions.These solutions have the same multi-bump pattern as the asymptotically stablestationary multi-bump solutions that were found in the numerical simulationsof the full reaction-diffusion system.

Symmetric and reversible families of vector fields near a partially hyperbolic singularity

We study smooth local models of families of symmetric and reversible vector fields near a partially hyperbolic singularity. Special attention is given to the question of whether the involved changes of variables commute with the symmetry.

Non-persistence of Homoclinic Connections for Perturbed Integrable Reversible Systems

The dynamics of an analytic reversible vector field \(V\)(X,μ) is studied in \(\mathbb{R}^4 \) with one real parameter μ close to 0; X=0 is a fixed point. The differential Dx\(V\)(0,0) generates an “oscillatory” dynamics with a frequency of order 1—due to two simple, opposite eigenvalues lying on the imaginary axis—and it also generates a “slow” dynamics which changes from a hyperbolic type—eigenvalues are \( \pm \sqrt { - \mu } \)—to an elliptic type—eigenvalues are \( \pm {\text{ }}i{\text{ }}\sqrt \mu \)—as μ passes trough 0. The existence of reversible homoclinic connections to periodic orbits is known for such vector fields. In this paper we study a particular subclass of such vector fields, obtained by small reversible perturbations of the normal form. We give an explicit condition on the perturbation, generically satisfied, which prevents the existence of a homoclinic connections to 0 for the perturbed system. The normal form system of any order admits a reversible homoclinic connection to 0, which then does not survive under perturbation of higher order. It will be seen that normal form essentially decouples the hyperbolic and elliptic part of the linearization to any chosen algebraic order. However, this decoupling does not persist arbitrary reversible perturbation, which finally causes the appearance of small amplitude oscillations.

Symmetric singularities of reversible vector fields in dimension three

For a large class of reversible vector fields on the three-dimensional space we present all the topological types and their respective normal forms of the symmetric singularities of codimension 0, 1.

Existence d'orbites homoclines à un équilibre elliptique, pour un système réversible

We consider a reversible vector field in ℝ 4, where the origin is a critical point, and where the differential at the origin has a pair of double non semisimple pure imaginary eigenvalues ±iw. We assume that the coefficient ε of a cubic term of the normal form is positive and close to 0, and that a certain coefficient of order 5 is negative. Then we show that there exist two reversible orbits homoclinic to the origin, of size √ε and such that they oscillate with a damping in 1/t when t tends towards ±∞. For obtaining such a result, we give explicitly the inverse of the linearized operator around the reversible homoclinics of the normal form, and solve the problem by a fixed point argument.

Generic bifurcation of reversible vector fields on a 2-dimensional manifold

In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed.

Generic bifurcation of reversible vector fields on a 2-dimensional manifold

Generic bifurcation of reversible vector fields on a 2-dimensional manifold

Singularities of reversible vector fields

For a large class of reversible vector fields on the plane we present all the topological types and their respective normal forms of the symmetric singularities of codimensions 0, 1 and 2.

Homoclinic orbits to small periodic orbits for a class of reversible systems

In this paper a one-parameter class of four-dimensional, reversible vector fields is investigated near an equilibrium. We call the parameter μ and place the equilibrium at 0. The differential at 0 is supposed to have ±iq, q > 0, as simple eigenvalues and 0 as a double, nonsemisimple eigenvalue. Our ultimate goal is to construct homoclinic connections of periodic orbits of arbitrary small size, in fact we shall show that the oscillations of the homoclinic orbits at infinity are bounded by a flat function of μ. This result receives its significance from the still unsolved question as to whether solutions exist which are homoclinic to the equilibrium or whether the amplitudes of the oscillations at infinity have a positive infimum. First we construct the periodic solutions. In contrast to previous work, we find these in a full rectangle [0, K0] × ]0,μ0], where K measures the amplitude of the periodic orbits. Then we show that for each n ∈ ℕ there is a μn and a family of periodic solutions X(μ), μ ∈]0,μn[, of Size μn. To each of these solutions, we can find two homoclinic orbits, which are distinguished by their phase shift at infinity. One example of such a vector field occurs when describing the flow of an inviscid irrotational fluid layer under the influence of gravity and small surface tension (Bond number b < ⅓), for a Froude number F close to 1. In this context a homoclinic solution to a periodic orbit is called a generalised solitary wave. Our work shows that there exist solitary waves with oscillations at infinity of order less than |μ|n for every n.

Bifurcation from a Periodic Orbit for a Strongly Resonant Reversible Autonomous Vector Field

The author considers a reversible system admitting a symmetric periodic orbit such that the Jordan block belonging to the Floquet exponent zero is four-dimensional, nonsemisimple. Using the normal form theory around closed orbits, it is shown that, generically, such a solution is part of a one-parameter family of symmetric periodic orbits. The existence in such a system of two one-parameter families of symmetric solutions homoclinic to some periodic orbits is also proven. Finally, the author shows how this problem is related to the perturbed reversible 1-1 resonance vector fields, and allows its study to be completed.

Perturbed Homoclinic Solutions in Reversible 1:1 Resonance Vector Fields

We consider a smooth reversible vector field in R^4, such that the origin is a fixed point. The differential at the origin has two double pure imaginary eigenvalues ±iq for the critical value 0 of the parameter µ. We show, by a normal form analysis, that the vector field can be approximated by an integrable field in R^4, for which we know all solutions. Specially interesting ones are the homoclinics to 0, and homoclinics to periodic solutions, depending on the sign of a leading nonlinear coefficient. We prove in particular the persistence of these homoclinics for the full vector field.

Bifurcations of families of invariant curves in four-dimensional reversible mappings

The bifurcation of families of closed invariant curves near a fixed point of a four-dimensional reversible mapping as two pairs of its multipliers collide and move off the unit circle, is studied with reference to an analogous bifurcation in reversible vector fields. A description of the principal features of the bifurcation is presented by way of a conjecture, and for a particular class of maps an analytic expression is derived for distinguishing normal from inverted bifurcations. Evidence in support of the conjecture is indicated in terms of an order-by-order perturbation scheme as well as of numerical computations.

Homoclinic and heteroclinic orbits of reversible vectorfields under perturbation

SynopsisAveraging techniques on Hamiltonian dynamical systems can often be used to establish the existence of hyperbolic periodic orbits. In equilibrium situations, it is then often difficult to show that there are homoclinic/heteroclinic connections between these hyperbolic orbits in the original unaveraged system. This existence problem is solved in this paper for a class of Hamiltonian systems admitting a sufficient number of symmetries (including reversing symmetries). Under isoenergetic reduction, the problem is reduced to one involving reversible vector fields under time-dependent perturbations admitting the same reversing symmetries. Applications are made to the one-parameter Hénon-Heiles family. The paper concludes with remarks on the problem of showing transversality of these homoclinic/heteroclinic orbits.