Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics

Champneys, Alan R

doi:10.1016/s0167-2789(97)00209-1

Cited by 256 publications

(355 citation statements)

References 111 publications

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“…spatiotemporally chaotic. Spatially localised solutions are of importance in many different areas, such as the study of localised buckling of long struts [1,2], nonlinear optics [3], vibrating granular media [4], convection problems [5] and neuroscience [6,7]. In neural field models, stationary spatially localised regions of high activity ("bumps") have been studied in the context of working memory, as single-bump steady state solutions are believed to be the analogue of shortterm memory [6,[8][9][10].…”

Section: Introductionmentioning

confidence: 99%

“…The time-independent system can often be written as a dynamical system in space, where spatially localised solutions correspond to homoclinic orbits to the fixed point at the origin. Homoclinic snaking is also a feature in many systems [2,[11][12][13][14][15], with some of the best studied examples being fourth-order partial differential equations [1,2,15].…”

Section: Introductionmentioning

confidence: 99%

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Exploiting the Hamiltonian structure of a neural field model

Elvin

Laing

McLachlan

et al. 2010

Physica D: Nonlinear Phenomena

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We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model's behaviour. The lack of robustness of the model to small perturbations in parameters is surprising and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the timeindependent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves.By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exploiting the Hamiltonian structure of a neural field model

Elvin

Laing

McLachlan

et al. 2010

Physica D: Nonlinear Phenomena

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“…Crossing this line gives a situation which -for homoclinic orbits in non-reversible, non-Hamiltonian systems -is sometimes referred to as "broom-bifurcation" (see [7] and references therein). This bifurcation leads to a dramatic change in the dynamics near the cycle from tame to chaotic.…”

Section: Discussionmentioning

confidence: 99%

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

Wagenknecht¹

2002

Dynamical Systems

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We study bifurcations from a fixed point with fourfold eigenvalue zero occurring in a two degrees of freedom Hamiltonian system of second order ODEs which is additionally reversible with respect to two different linear involutions. Using techniques from Catastrophe Theory we are led to a codimension 2 problem and obtain two different unfoldings of the singularity related to the hyperbolic and elliptic umbilic, respectively.The analysis of the unfolded systems is essentially concerned with the existence and properties of homoclinic and heteroclinic orbits. Our studies are motivated by a problem from nonlinear optics concerning the existence of solitons in a χ 2 -medium.

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“…If x(t) = Rx(−t), that is, if for the corresponding orbit X(t) := {x(t) : t ∈ R} we have RX = X then we call the solution or the orbit symmetric. As general references concerning the theory of reversible systems we refer to [5,13,31].…”

Section: The Mathematical Settingmentioning

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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Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics

Cited by 256 publications

References 111 publications

Exploiting the Hamiltonian structure of a neural field model

Exploiting the Hamiltonian structure of a neural field model

Bifurcation of a reversible Hamiltonian system from a fixed point with fourfold eigenvalue zero

When gap solitons become embedded solitons: a generic unfolding

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