In this work, we study the bifurcation problems of double homoclinic loops with resonant condition for higher dimensional systems. The Poincaré maps are constructed by using the foundational solutions of the linear variational systems as the local coordinate systems in the small tubular neighborhoods of the homoclinic orbits. We obtain the existence, number and existence regions of the small homoclinic loops, small periodic orbits, and the large homoclinic loops, large periodic orbits, respectively. Moreover, the complete bifurcation diagrams are given.
In this paper, perturbed polynomial Moon-Rand systems are considered. The Padé approximant and analytic solution in the neighborhood of the initial value are introduced into the process of constructing the Shilnikov type homoclinic orbits for three dimensional nonlinear dynamical systems. In order to get real bifurcation parameters, four undetermined coefficients are introduced including three initial values about position and the value of bifurcation parameter. By the eigenvectors of its all eigenvalues, the value of the bifurcation parameter and three initial values about position are obtained directly. And, the analytical expressions of the Shilnikov type homoclinic orbits are achieved and the deletion errors relative to the practical system are given. In the end, we roughly predict when the horseshoe chaos occurs.
The bifurcation problems of twisted heteroclinic loop with two hyperbolic critical points are studied for the case [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] are the pair of principal eigenvalues of unperturbed system at the critical point [Formula: see text], [Formula: see text]. Under the transversal conditions, the authors obtained some results of the existence and the number of 1-homoclinic loop, 1-periodic orbit, double 1-periodic orbit, 2-homoclinic loop and 2-periodic orbit. Moreover, the relative bifurcation surfaces and the existence regions are given, and the corresponding bifurcation graphs are drawn.
In this paper, we use the way of local coordinates instead of the Floquet method to study the problems of homoclinic and periodic orbits bifurcated from heteroclinic loop for high-dimensional system. Under some transversal conditions and the non-twisted or twisted conditions, we discuss the existence, uniqueness, coexistence, and non-coexistence of 1-periodic orbit, 1-homoclinic orbit, and 1-heteroclinic orbit near the heteroclinic loop. We get some general conclusions only under the basic hypotheses, and the other conclusions under the two hyperbolic ratios of the heteroclinic loop are greater than 1. Meanwhile, the bifurcation surfaces and existence regions are given.
In this paper, the bifurcation problems of twisted double homoclinic loops with resonant condition are studied for (m + n)-dimensional nonlinear dynamic systems. In the small tubular neighborhoods of the homoclinic orbits, the foundational solutions of the linear variational systems are selected as the local coordinate systems. The Poincaré maps are constructed by using the composition of two maps, one is in the small tubular neighborhood of the homoclinic orbit, and another is in the small neighborhood of the equilibrium point of system. By the analysis of bifurcation equations, the existence, uniqueness and existence regions of the large homoclinic loops, large periodic orbits are obtained, respectively. Moreover, the corresponding bifurcation diagrams are given. c 2016 all rights reserved.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.