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MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS
International Journal of Bifurcation and Chaos, 1996The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main
Kokubu, Hiroshi +2 more
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N-Homoclinic bifurcations for homoclinic orbits changing their twisting
Journal of Dynamics and Differential Equations, 1996The author considers two-parameter families of vector fields possessing a homoclinic orbit along a path in the parameter plane. These homoclinic orbits are homoclinic to a hyperbolic singularity that has a one-dimensional unstable manifold. The weakest stable and unstable eigenvalues of the linearized vector field at the singularity are supposed to be ...
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Bifurcation of degenerate homoclinics
Results in Mathematics, 1992The continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit is analyzed. It is shown that the existence of such degenerate homoclinic orbits is a codimension three phenomenon and that generically the set of parameter values at which a homoclinic solution exists forms a codimension one surface which shows a singularity
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Poincaré sequences, homoclinic bifurcation, and chaos
1999Abstract Important features can be totally obscured in such a diagram, but Poincare maps can be used to detect underlying structure, such as periodic solutions having the forcing or a subharmonic frequency. In this context the investigation of periodic solutions, nearly periodic solutions, and similar phenomena is to a considerable ...
D W Jordan, P Smith
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Homoclinic bifurcation with nonhyperbolic equilibria
Nonlinear Analysis: Theory, Methods & Applications, 2007The authors study the bifurcation of a homoclinic or heteroclinic orbit with a nonhyperbolic equilibrium, which is a pitchfork bifurcation point. The unperturbed system is assumed to possess a homoclinic orbit \(\Gamma\). Combining two discrete maps in the vicinity of \(\Gamma\), one of which describes the flow close to the equilibrium, while the other
Liu, Xingbo, Fu, Xianlong, Zhu, Deming
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Cascades of Homoclinic Doubling Bifurcations
2001We present an overview of the theory of homoclinic doubling cascades, describing bifurcation theory and discussing universal scaling properties obtained from a renormalization theory.
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Homoclinic Bifurcation of a Grid-Forming Voltage Source Converter
IEEE transactions on power electronics, 2021Jingxi Yang +3 more
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Low-Dimensional Homoclinic Bifurcations of Repellers
International Journal of Bifurcation and Chaos, 2014We study the relationship between homoclinic orbits associated with repellers, usually called snap-back repellers, and expanding sets of smooth endomorphisms. We also consider critical homoclinic orbits and prove that the bifurcations they create constitute a codimension one phenomenon.
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Homoclinic flip bifurcations accompanied by transcritical bifurcation
Chinese Annals of Mathematics, Series B, 2011The author investigates the bifurcation of a homoclinic loop with a nonhyperbolic equilibrium by constructing a suitable Poincaré map. Using the fundamental solutions to linear variational equations as an active coordinate system, he constructs a global map which is composed of a regular map in the tubular neighborhood of the homoclinic orbit and a ...
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