The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit
- 1 December 1994
- journal article
- research article
- Published by Cambridge University Press (CUP) in Ergodic Theory and Dynamical Systems
- Vol. 14 (4) , 667-693
- https://doi.org/10.1017/s0143385700008117
Abstract
Deng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2λu > min{−λs, λuu} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.Keywords
This publication has 9 references indexed in Scilit:
- Homoclinic twist bifurcations with ℤ2 symmetryJournal of Nonlinear Science, 1994
- Homoclinic twisting bifurcations and cusp horseshoe mapsJournal of Dynamics and Differential Equations, 1993
- Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fieldsJournal of Dynamics and Differential Equations, 1993
- Coupled arrays of Josephson junctions and bifurcation of maps with SNsymmetryNonlinearity, 1991
- Lorenz attractors through Šil'nikov-type bifurcation. Part IErgodic Theory and Dynamical Systems, 1990
- Homoclinic bifurcation at resonant eigenvaluesJournal of Dynamics and Differential Equations, 1990
- Homoclinic bifurcation to a transitive attractor of Lorenz typeNonlinearity, 1989
- Branching of double pulse solutions from single pulse solutions in nerve axon equationsJournal of Differential Equations, 1987
- ON THE GENERATION OF A PERIODIC MOTION FROM TRAJECTORIES DOUBLY ASYMPTOTIC TO AN EQUILIBRIUM STATE OF SADDLE TYPEMathematics of the USSR-Sbornik, 1968