Inclination-flips in the Unfolding of a Singular Heteroclinic Cycle
نویسنده
چکیده
We study bifurcations from a singular heteroclinic cycle in R 3. This heteroclinic cycle contains a hyperbolic singularity and a saddle-node. At the saddle-node, the linearized vector eld has two complex conjugate eigenvalues in addition to an eigenvalue 0. This implies the existence of a cascade of inclination-ip homoclinic bifurcations to the heteroclinic bifurcation.
منابع مشابه
An Equivariant, Inclination-flip, Heteroclinic Bifurcation
We examine a heteroclinic bifurcation occurring in families of equivariant vector elds. Within these families, the ows contain structurally stable heteroclinic cycles. The ow can twist around the cycle to produce what is the equivalent of an \inclination-ip" homoclinic orbit. An unfolding of this bifurcation in a generic one-parameter equivariant family shows that Smale horseshoes are embedded ...
متن کاملBifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria
The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and nonco...
متن کاملBreakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity
In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in R3. This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit betw...
متن کاملMelnikov Functions for Period Annulus, Nondegenerate Centers, Heteroclinic and Homoclinic Cycles
We give sufficient conditions in terms of the Melnikov functions in order that an analytic or a polynomial differential system in the real plane has a period annulus. We study the first nonzero Melnikov function of the analytic differential systems in the real plane obtained by perturbing a Hamiltonian system having either a nondegenerate center, a heteroclinic cycle, a homoclinic cycle, or thr...
متن کاملResonant Homoclinic Bifurcations with Orbit Flips and Inclination Flips
Homoclinic bifurcation with one orbit flip, two inclination flips and resonance in the tangent directions of homoclinic orbit is considered. By studying the associated successor functions constructed from a local active coordinate system, we prove the existence of double 1-periodic orbit, 1-homoclinic orbit, and also some coexistence conditions of 1-periodic orbit and 1-homoclinic orbit.
متن کامل