A non - transverse homoclinic orbit to a saddle - node equilibrium .
نویسنده
چکیده
Abst ract A homoclinic orbit is considered for which the center-stable and center-unstable manifolds of a saddle-node equilibrium have a quadratic tangency. This bifurcation is of codimension two and leads generically to the creation of a bifurcation curve deening two independent transverse homoclinic orbits to a saddle-node. This latter case was shown by L.P. Shilnikov to imply shift dynamics. It is proved here that in a large open parameter region of the codimension-two singularity, the dynamics are completely described by a perturbation of the H enon-map giving strange attractors, Newhouse sinks and the creation of the shift dynamics. In addition, an example system admitting this bifurcation is constructed and numerical computations are performed on it.
منابع مشابه
Homoclinic Bifurcations that Give Rise to Heterodimensional Cycles near A Saddle-Focus Equilibrium
We show that heterodimensional cycles can be born at the bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium for flows in dimension 4 and higher. In addition to the classical heterodimensional connection between two periodic orbits, we found, in intermediate steps, two new types of heterodimensional connections: one is a heteroclinic between a homoclinic loop and a periodic...
متن کاملNumerical Detection and Continuation of Saddle-node Homoclinic Bifurcations of Codimension One and Two
An extension of an existing truncated boundary-value method for the numerical continuation of connecting orbits is proposed to deal with homoclinic orbits to a saddle-node equilibrium. In contrast to previous numerical work by Schecter and Friedman & Doedel, the method is based on (linear) projection boundary conditions. These boundary conditions , with extra deening conditions for a saddle-nod...
متن کاملMulti-round Homoclinic Orbits to a Hamiltonian with Saddle-center
We consider a real analytic, two degrees of freedom Hamiltonian system possessing a homoclinic orbit to a saddle-center equilibrium p (two nonzero real and two nonzero imaginary eigenvalues). We take a two-parameter unfolding for such the system and show that in nonresonance case there are countable sets of multiround homoclinic orbits to p. We also find families of periodic orbits, accumulatin...
متن کاملBifurcations of global reinjection orbits near a saddle-node Hopf bifurcation
The saddle-node Hopf bifurcation (SNH) is a generic codimensiontwo bifurcation of equilibria of vector fields in dimension at least three. It has been identified as an organizing centre in numerous vector field models arising in applications. We consider here the case that there is a global reinjection mechanism, because the centre manifold of the zero eigenvalue returns to a neighbourhood of t...
متن کاملGlobal Invariant Manifolds Near Homoclinic Orbits to a Real Saddle: (Non)Orientability and Flip Bifurcation
Homoclinic bifurcations are important phenomena that cause global re-arrangements of the dynamics in phase space, including changes to basins of attractions and the generation of chaotic dynamics. We consider here a homoclinic (or connecting) orbit that converges in both forward and backward time to a saddle equilibrium of a three-dimensional vector field. We assume that the saddle is such that...
متن کامل