Homoclinic Branch Switching: A Numerical Implementation of Lin's Method

Oldeman, Bart E.; Champneys, Alan R; Krauskopf, Bernd

doi:10.1142/s0218127403008326

Cited by 35 publications

(32 citation statements)

References 29 publications

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“…Thus the homoclinic banana is now split. Similar behavior was found in [33] and [30]. Using HomCont we were able to find the critical value of ε where the behavior changes and the two Belyakov points appear.…”

Section: Numerical Continuation Resultssupporting

confidence: 71%

When Shil'nikov Meets Hopf in Excitable Systems

Champneys¹,

Kirk²,

Knobloch³

et al. 2007

SIAM J. Appl. Dyn. Syst.

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111

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Abstract. This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh-Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a Ushaped locus of Hopf bifurcations that corresponds to the onset of small-amplitude linear waves. The central question addressed is how the Hopf and solitary wave (homoclinic orbit in a moving frame) bifurcation curves interact in these "CU systems." A variety of possible codimension-two mechanisms is reviewed through which such Hopf and homoclinic bifurcation curves can interact. These include Shil'nikov-Hopf bifurcations and the local birth of homoclinic chaos from a saddle-node/Hopf (Gavrilov-Guckenheimer) point. Alternatively, there may be barriers in phase space that prevent the homoclinic curve from reaching the Hopf bifurcation. For example, the homoclinic orbit may bump into another equilibrium at a so-called T-point, or it may terminate by forming a heteroclinic cycle with a periodic orbit. This paper presents the results of detailed numerical continuation results on different CU systems, thereby illustrating various mechanisms by which Hopf and homoclinic curves interact in CU systems. Owing to a separation of time scales in these systems, considerable care has to be taken with the numerics in order to reveal the true nature of the bifurcation curves observed.

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Section: Numerical Continuation Resultssupporting

confidence: 71%

When Shil'nikov Meets Hopf in Excitable Systems

Champneys¹,

Kirk²,

Knobloch³

et al. 2007

SIAM J. Appl. Dyn. Syst.

Self Cite

111

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“…auto also allows users to switch to N -homoclinic orbits at bifurcation points using a numerical implementation of Lin's method developed in [297]. dde-biftool is a matlab package that implements a similar functionality for delay differential equations [125].…”

Section: Numerical Techniquesmentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horseshoes and attractors are stated. Bifurcations of homoclinic orbits from equilibria in local bifurcations are also considered. The main analytic and geometric techniques such as Lin's method, Shil'nikov variables and homoclinic center manifolds for analyzing these bifurcations are discussed. Finally, a few related topics, such as topological moduli, numerical algorithms, variational methods, and extensions to singularly perturbed and infinitedimensional systems, are reviewed briefly.

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“…However, that is not the case when the kinks are well-separated and w is small enough. Indeed, the matrix 16) in the left-hand side of (2.15) coincides with the Jacobi matrix (2.6) and similarly the matrix is close to the identity if the kinks are well-separated. On the other hand, the matrix G(d, w) may become very small or even singular if some distances between kinks become small which indicates that the projection method is no longer applicable.…”

Section: Projection System (Ps)mentioning

confidence: 87%

“…Several advances in the numerical computation of stationary one-dimensional multi-pulses have been made in ODEs. One approach to address this problem has been to use Lin's method to set-up a boundary value problem where one looks for zeros of an algebraic function using path-following routines and has been implemented in HOMCONT; see [16].…”

Section: Introductionmentioning

confidence: 99%

Computing Interacting Multi-fronts in One Dimensional Real Ginzburg Landau Equations

2014

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We develop an efficient and robust numerical scheme to compute multi-fronts in one-dimensional Real Ginzburg-Landau equations that range from well-separated to strongly interacting and colliding. The scheme is based on the global centre-manifold reduction where one considers an initial sum of fronts plus a remainder function (not necessarily small) and applying a suitable projection based on the neutral Eigenmodes of each front. Such a scheme efficiently captures the weakly interacting tails of the fronts. Furthermore, as the fronts become strongly interacting, we show how they may be added to the remainder function to accurately compute through collisions. We then present results of our numerical scheme applied to various real Ginzburg Landau equations where we observe colliding fronts, travelling fronts and fronts converging to bound states. Finally, we discuss how this numerical scheme can be extended to general PDE systems and other multi-localised structures.

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Homoclinic Branch Switching: A Numerical Implementation of Lin's Method

Cited by 35 publications

References 29 publications

When Shil'nikov Meets Hopf in Excitable Systems

When Shil'nikov Meets Hopf in Excitable Systems

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Computing Interacting Multi-fronts in One Dimensional Real Ginzburg Landau Equations

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