Detection of symmetric homoclinic orbits to saddle-centres in reversible systems

Abstract

Detection of symmetric homoclinic orbits to saddle-centres in reversible systems. AbstractWe present a perturbation technique for the detection of symmetric homoclinic orbits to saddle-centre equilibria in reversible systems of ordinary differential equations. We assume that the unperturbed system has primary, symmetric homoclinic orbits, which may be either isolated or appear in a family, and use an idea similar to that of Melnikov's method to detect homoclinic orbits in their neighbourhood. This technique also allows us to identify bifurcations of unperturbed or perturbed, symmetric homoclinic orbits. Our technique is of importance in applications such as nonlinear optics and water waves since homoclinic orbits to saddle-centre equilibria describe embedded solitons (ESs) in systems of partial differential equations representing physical models, and except for special cases their existence has been previously studied only numerically using shooting methods and continuation techniques. We apply the general theory to two examples, a four-dimensional system describing ESs in nonlinear optical media and a six-dimensional system which can possess a one-parameter family of symmetric homoclinic orbits in the unperturbed case. For these examples, the analysis is compared with numerical computations and an excellent agreement between both results is found.

Detecting stable–unstable nonlinear invariant manifold and homoclinic orbits in mechanical systems

“…It is immediate to recognize that, as expected, (26) reduces to (14) for ε = 0. For our purpose, it is important to study the possible existence of the homoclinic loop on Σ ε , i.e., to determine whether (26) …”

“…Since in the polynomial in right-hand side of (24) 2 the unknown x 2 appears only with even powers (see (26)), we found it convenient to assume x 2 = √ 2g(x 1 ) instead of x 2 = h(x 1 ). By differentiating this expression with respect to time, we geṫ…”

“…The homoclinc behavior of saddle-centers has been the subject of several studies in the recent years, especially in the mathematical literature [3,7,8,10,14,15,20,25,26], because of its very complicated behavior which follows from having hyperbolicity in some directions and ellipticity in others.…”

Homoclinic and Heteroclinic Bifurcations in Vector Fields

“…It is worthwhile to remark that it is the assumption (5.22) that discriminates between the two cases: indeed, (5.17) shows that the level set H −1 (0) is tangent to W cs (0) in Hamiltonian system; see [72,Lemma 2]. We refer to [72] for a comprehensive discussion and unfolding results in the situation where the Hamiltonian structure is broken while reversibility is retained; see also [219,427] for results on homoclinic orbits to saddle-centers in reversible systems and to [361] for infinite-dimensional conservative systems.…”

Resonant chaotic motions of a buckled rectangular thin plate with parametrically and externally excitations

“…A series of works have been achieved to demonstrate that the non-hyperbolic modes do not affect the critical conditions for the occurrence of nonresonant chaotic dynamics [1][2][3][4][5]. The theory has been applied to study the non-resonant chaotic motions of a buckled elastic beam [1,2,5].…”