Singular Perturbation Theory for Homoclinic Orbits in a Class of Near- Integrable Dissipative Systems

Abstract

This paper presents a new unified theory of orbits homoclinic to resonance bands in a class of near-integrable dissipative systems. It describes three sets of conditions, each of which implies the existence of homoclinic or heteroclinic orbits that connect equilibria or periodic orbits in a resonance band. These homoclinic and heteroclinic orbits are born under a given small dissipative perturbation out of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria in the phase space of the nearby integrable system. The result is a constructive method that may be used to ascertain the existence of orbits homoclinic to objects in a resonance band, as well as to determine their precise shape, asymptotic behavior, and bifurcations in a given example. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.

Foliations of isonergy surfaces and singularities of curves

“…Notably, the appearance of a circle of hyperbolic fixed points with nonorientable separatrices (type A * ) had emerged as a "generic" scenario which had not been studied yet; under small perturbations we expect to obtain various multipulse orbits to some resonance zone which will replace the circle of fixed points. While the tools developed by Haller [34,32] and Kovačič [36] to study the orientable case should apply, the nature of the chaotic set may be quite different. Second, the complete listing of the structure of the isoenergy surfaces near all non-degenerate global bifurcations may lead to a systematic study of the emerging chaotic sets of the various cases under small perturbations.…”

“…The normal forms near the different singularities of the bifurcation set can be now derived and used to classify the various instabilities that may develop under small perturbations. The analysis of each of these near-integrable scenarios is far from being complete; beyond KAM theory, there is a large body of literature dealing with persistence results for lower dimensional tori (namely circles in the 2 degrees of freedom case) -see, for example, [17,24,30,43,53], and another large body of results describing the instabilities arising near singular circles or fixed points that are not elliptic (see [36,32,55] and references therein). Our classification reveals several new generic cases that were not studied in this near-integrable context.…”

Homoclinic orbits and chaos in a second-harmonic generating optical cavity

“…, n − 1. Here, the phase difference ∆(I) is given by equation (6). These heteroclinic orbits are called pulses.…”

“…First, its second piece, or rather the corresponding trajectory of the system (7) that connects the point A 2 (∞,Ī,φ 0 − (n − 1)∆(Ī)) to the equilibrium (8), must avoid the circle 2I = |A 2 | 2 = κ 2 1 and its interior. This is because, otherwise, the mathematical arguments [6][7][8] used to establish the existence of such homoclinic orbits cease to be valid. Second, we must choose the parameters α, γ 2 , κ 1 and κ 2 so that the n-pulse Melnikov function M n (Ī,φ 0 ), to be described next, vanishes along the string of heteroclinic orbits (…”

“…We use the results of [6][7][8] to compute two large families of homoclinic orbits that the model (1) supports. All these orbits are ofŠilnikov type [9][10][11][12][13], and bring along with them chaotic dynamics arising from a Smale horseshoe construction [14].…”