Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation

Kovačič, Gregor; Wiggins, Stephen

doi:10.1016/0167-2789(92)90092-2

Cited by 183 publications

(135 citation statements)

References 18 publications

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“…Similar to that of Theorem 4.3, (4.14) has a unique solution 15) in the small neighborhood of s 12 = s 2 = s 21 = 0. Thus, we get the uniqueness and non-coexistence.…”

Section: 12supporting

confidence: 48%

See 1 more Smart Citation

Bifurcations of twisted double homoclinic loops with resonant condition

Jin¹,

Zhu²,

Li³

et al. 2016

J. Nonlinear Sci. Appl.

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In this paper, the bifurcation problems of twisted double homoclinic loops with resonant condition are studied for (m + n)-dimensional nonlinear dynamic systems. In the small tubular neighborhoods of the homoclinic orbits, the foundational solutions of the linear variational systems are selected as the local coordinate systems. The Poincaré maps are constructed by using the composition of two maps, one is in the small tubular neighborhood of the homoclinic orbit, and another is in the small neighborhood of the equilibrium point of system. By the analysis of bifurcation equations, the existence, uniqueness and existence regions of the large homoclinic loops, large periodic orbits are obtained, respectively. Moreover, the corresponding bifurcation diagrams are given. c 2016 all rights reserved.

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“…Similar to that of Theorem 4.3, (4.14) has a unique solution 15) in the small neighborhood of s 12 = s 2 = s 21 = 0. Thus, we get the uniqueness and non-coexistence.…”

Section: 12supporting

confidence: 48%

“…Since the 1980s, Wiggins, Kovacic, Luo, Han, et al studied some low dimensional systems and some systems with special forms (e.g. Hamilton) by using the well-known Melnikov methods [2,3,15,18,20,21]. In 1990, Chow et al studied the high dimensional non-degenerate homoclinic orbits bifurcations in [1].…”

Section: Introduction and Hypothesesmentioning

confidence: 99%

Bifurcations of twisted double homoclinic loops with resonant condition

Jin¹,

Zhu²,

Li³

et al. 2016

J. Nonlinear Sci. Appl.

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“…In particular, theorem 9.1 in Fenichel [1979] Theorems similar to proposition 6.2 are also stated in Sakamoto [1990], Jones and Kopell [1994], and Kovačič and Wiggins [1992]. Proposition 6.2 implies that one can think of the stable and unstable fibers as "traveling" stable and unstable manifolds of their base points.…”

Section: Geometric Singular Perturbation Theorymentioning

confidence: 70%

“…(A special case of the theory of orbits homoclinic to resonance bands in a dissipative system was first discussed in Kovačič and Wiggins [1992], a general theory is presented in Kovačič [1992c]. Examples, special cases, and parallel developments of the theory are given in Feng and Wiggins [1992], Kovačič [1992b], andMcLaughlin et.…”

Section: Introductionmentioning

confidence: 99%

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

Kovačič

1993

J Dyn Diff Equat

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This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.

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“…The relation between the type of theory described here and earlicr work (Kovacic and Wiggins [1992]; Wiggins [1993, 1995a,b]) on the origin of complex dynamics in the forced weakly damped nonlinear Sehrodinger equation is also of interest. This work focused on the ncar-Hamiltonian limit and exploited generalizations of the Mel'nikov theory to PDEs to establish the presence of a variety of multipulse orbits homoclinic or heteroclinic to a slow manifold.…”

Section: Discussionmentioning

confidence: 99%

Nearly Inviscid Faraday Waves

Knobloch¹,

Vega²

Geometry, Mechanics, and Dynamics

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Many powerful techniques from Hamiltonian mechanics are available for the study of ideal hydrodynamics. This articte explores some of the consequences of including srnall viscosity in a study of surface gravitycapillary waves excited hy the vertical vibration of a container. It is shown that in this system, as in others, the addition of smatl viscosity provides a singular perturbation of the ideal fluid problem, and that as a result its effects are nontrivial. The relevance of cxisting studies of ideal fluid problems is discussed from this point of vicw.

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Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation

Cited by 183 publications

References 18 publications

Bifurcations of twisted double homoclinic loops with resonant condition

Bifurcations of twisted double homoclinic loops with resonant condition

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

Nearly Inviscid Faraday Waves

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