Death of period-doublings: locating the homoclinic-doubling cascade

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

doi:10.1016/s0167-2789(00)00133-0

Cited by 31 publications

(23 citation statements)

References 23 publications

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“…An extensive numerical investigation of homoclinic-doubling cascades can be found in [298], and further numerical evidence for the existence of homoclinic-doubling cascades has been provided in the Shimizu-Morioka model…”

Section: Homoclinic-doubling Cascadesmentioning

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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Section: Homoclinic-doubling Cascadesmentioning

confidence: 99%

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homburg

Sandstede

2010

Handbook of Dynamical Systems

165

229

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“…The cases A (no additional bifurcations), B (homoclinic doubling) and C (fan of many N-homoclinic orbits, n-periodic orbits for any n and chaos) depend on the eigenvalues at the equilibrium (the origin), as given in Figure 4. For an overview of the fan depicted in Figure 5 and the other possibilities for type C see (Homburg and Krauskopf 2000, Oldeman et. al.…”

Section: Test Examplesmentioning

confidence: 99%

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

Oldeman

Champneys

Krauskopf

2003

Int. J. Bifurcation Chaos

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N -homoclinic orbit, for any N > 1 (if it exists nearby). This scheme is based on Lin's method and it is robust and reliable.The method is implemented in AUTO/HomCont. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multi-hump travelling wave solutions in the FitzHughNagumo nerve-axon equations and in a 4th-order Hamiltonian system arising as a model for water waves.

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“…It is caused by a degeneracy in the global twistedness of the stable and unstable manifolds W s,u (u 0 ) of a saddle u 0 around its homoclinic orbit q(x); see e.g. [36] and references therein. At each point q(x) along the homoclinic orbit, the normal bundle…”

Section: Saddle-homoclinic Orbits and Their Orientationmentioning

confidence: 99%

Numerical Computation of Coherent Structures

Champneys

Sandstede

2007

Understanding Complex Systems

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In many applications one is interested in finding solutions to nonlinear evolution equations with a particular spatial and temporal structure. For instance, solitons in optical fibers and wave guides or buckling modes of long structures can be interpreted as localised travelling or standing waves of an appropriate underlying partial differential equation (PDE) posed on an unbounded domain. Spiral waves or other defects in oscillatory media are time-periodic waves with an asymptotic spatially periodic structure. All of these examples are referred to as coherent structure. They represent relative equilibria, that is, their temporal evolution is determined by a symmetry of the underlying PDE: namely, translational symmetry for travelling waves, rotational symmetry for spiral waves, and phase symmetry for oscillatory structures.Given the complexity of typical PDE models, these nonlinear waves are in general accessible only through numerical computations. One possible approach is via direct simulation which is, however, expensive and fails to capture solutions that are either unstable or may have a small basin of stability. Simulation also fails to provide valuable information on how branches of solutions are organised in parameter space.In this chapter, we give an overview of boundary-value problem formulations for coherent structures which provide a robust and less expensive alternative to simulation. Moreover, setting up well-posed boundary-value problems allows us to continue solutions in parameters space, investigate their spectral stability directly, and continue branches of solutions efficiently as parameters vary.In the next section we outline how PDEs can be supplemented by phase conditions that allow us to compute nonlinear waves as regular zeros of the resulting nonlinear system. In the remaining sections, we treat different kinds of coherent structures, namely travelling waves, time-periodic structures, and planar localised patterns. In each case we explain how to set up a well-posed boundary-value problem and illustrate the theory with the results of an example computation.

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Death of period-doublings: locating the homoclinic-doubling cascade

Cited by 31 publications

References 23 publications

Homoclinic and Heteroclinic Bifurcations in Vector Fields

Homoclinic and Heteroclinic Bifurcations in Vector Fields

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

Numerical Computation of Coherent Structures

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