Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center

Mielke, Alexander; Holmes, Philip; O’Reilly, Oliver M.

doi:10.1007/bf01048157

Cited by 115 publications

(117 citation statements)

References 19 publications

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“…From the analytical results in Grotta Ragazzo (1997); Mielke et al (1992) a cascade phenomenon of multi-round homoclinic orbits in the parameter value µ follows, in the sense that there are not only 2-round, but also 3-round,. .…”

Section: Multi-round Homoclinic Orbitsmentioning

confidence: 97%

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

Barrabés

Mondelo

Ollé

2009

Celest Mech Dyn Astr

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We consider the planar Restricted Three-Body problem and the collinear equilibrium point L 3 , as an example of a center×saddle equilibrium point in a Hamiltonian with two degrees of freedom. We explore the existence of symmetric and non-symmetric homoclinic orbits to L 3 , when varying the mass parameter µ. Concerning the symmetric homoclinic orbits (SHO), we study the multi-round, m-round, SHO for m ≥ 2. More precisely, given a transversal value of µ for which there is a 1-round SHO, say µ 1 , we show that for any m ≥ 2, there are countable sets of values of µ, tending to µ 1 , corresponding to m-round SHO. Some comments on related analytical results are also made.

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Section: Multi-round Homoclinic Orbitsmentioning

confidence: 97%

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

Barrabés

Mondelo

Ollé

2009

Celest Mech Dyn Astr

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“…Note that all branches can be continued into the limit s → 0. Also for each 4π solution one can appeal to the theory of Mielke et al [24] that gives the existence of a sequence of parameter values that converge exponentially to that of the kink in question at which 6π-kinks exist. More generally there are similar sequences of 2(n + 1)π-kinks converging on each 2nπ-kink for all n ≥ 2.…”

Section: Embedded Soliton Theorymentioning

confidence: 99%

A new barrier to the existence of moving kinks in Frenkel–Kontorova lattices

Aigner

Champneys

Rothos

2003

Physica D: Nonlinear Phenomena

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An explanation is offered for an observed lower bound on the wave speed of travelling kinks in Frenkel-Kontorova lattices. Kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves (phonons). However, they fail to exist even in this codimension-one sense whenever there is more than one phonon branch in the dispersion relation; inside such bands only quasi-kinks with nondecaying oscillatory tails are possible. The results are presented for a discrete sine-Gordon lattice with an onsite potential that has a tunable amount of anharmonicity. Novel numerical methods are used to trace kinks with topological charge Q = 1 and 2 in three parameters representing the propagation speed, lattice discreteness and anharmonicity. Although none of the analysis is presented as rigorous mathematics, numerical results suggest that the bound on allowable wavespeeds is sharp. The results also explain why the vanishing, at discrete values of anharmonicity, of the Peierls-Nabarro barrier between stationary kinks as discovered by Savin et al., does not lead to bifurcation of kinks with small wave speed.

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“…Here, general results are available which explain the accumulation of such solutions on parameter values where the primary orbit exists, see [7] for the reversible case and [27,23] for the case of systems that are additionally Hamiltonian. An interesting point is that these studies show differences in the behaviour of systems that are purely reversible and of those are also Hamiltonian.…”

Section: Discussionmentioning

confidence: 99%

“…General theories explain how the multi-pulses accumulate on the parameter values at which the 1-pulses exist [27,23]. This paper concerns only the single humped variety, since to date all non-fundamental embedded solitons have been found to be (exponentially) unstable as solutions of the underlying PDEs, see e.g.…”

Section: Overviewmentioning

confidence: 99%

When gap solitons become embedded solitons: a generic unfolding

Wagenknecht

Champneys

2003

Physica D: Nonlinear Phenomena

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A two-parameter unfolding is considered of single-pulsed homoclinic orbits to an equilibrium with two real and two zero eigenvalues in fourth-order reversible dynamical systems. One parameter controls the linearisation, with a transition occurring between a saddle-centre and a hyperbolic equilibrium. In the saddle-centre region, the homoclinic orbit is of codimension-one, which is controlled by the second generic parameter, whereas when the equilibrium is hyperbolic the homoclinic orbit is structurally stable. A geometric approach reveals the homoclinic orbits to the saddle to be generically destroyed either by developing an algebraically decaying tail or through a fold, depending on the sign of the perturbation of the second parameter. Special cases of different actions of Z 2 -symmetry are considered, as is the case of the system being Hamiltonian. Application of these results is considered to the transition between embedded solitons (corresponding to the codimension-one homoclinic orbits) and gap solitons (the structurally stable ones) in nonlinear wave systems. The theory is shown to match numerical experiments on two models arising in nonlinear optics and on a form of 5th-order Korteweg de Vries equation.

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Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center

Cited by 115 publications

References 19 publications

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

Dynamical aspects of multi-round horseshoe-shaped homoclinic orbits in the RTBP

A new barrier to the existence of moving kinks in Frenkel–Kontorova lattices

When gap solitons become embedded solitons: a generic unfolding

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