Construction of Codimension One Homoclinic Cycles
نویسندگان
چکیده
We give an explicit construction of families of Dm-equivariant polynomial vector fields in R possessing a codimension-one homoclinic cycle. The homoclinic cycle consist of m homoclinic trajectories all connected to the equilibrium at the origin. The constructed vector fields can provide a setting for a (numerical) bifurcation study of these homoclinic cycles, in particular for m a multiple of four, where the bifurcations form an open problem.
منابع مشابه
Bifurcation from Codimension One Relative Homoclinic Cycles
We study bifurcations of relative homoclinic cycles in flows that are equivariant under the action of a finite group. The relative homoclinic cycles we consider are not robust, but have codimension one. We assume real leading eigenvalues and connecting trajectories that approach the equilibria along leading directions. We show how suspensions of subshifts of finite type generically appear in th...
متن کاملCoexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate
Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135–163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov–Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, on...
متن کاملBifurcation of Codimension 3 in a Predator-Prey System of Leslie Type with Simplified Holling Type IV Functional Response
It was shown in [Li & Xiao, 2007] that in a predator–prey model of Leslie type with simplified Holling type IV functional response some complex bifurcations can occur simultaneously for some values of parameters, such as codimension 1 subcritical Hopf bifurcation and codimension 2 Bogdanov–Takens bifurcation. In this paper, we show that for the same model there exists a unique degenerate positi...
متن کاملUnfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle
Abstract. The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a onedimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs when t...
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 کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013