Simultaneous equilibrium and heteroclinic bifurcation of planar vector fields via the Melnikov integral
نویسنده
چکیده
We study the unfoldings of planar vector fields in which a semihyperbolic equilibrium p o is connected to a hyperbolic saddle qo by a heteroclinic orbit that lies in the strong unstable manifold of po. We show how to produce normal forms for this situation using singularity theory and a version of the Melnikov integral. The normal forms consist ot two polynomials, one to describe bifurcation of the semihyperbolic equilibrium and one to describe bifurcation of the heteroclinic orbit. We show how to explicitly compute, to first order, the relation between parameters in a given problem and parameters in the normal form. We also discuss problems in which there is a known equilibrium near p o for all values of the parameters. AMS classification scheme number: 58F14
منابع مشابه
Bifurcation of limit cycles in 3rd-order Z2 Hamiltonian planar vector fields with 3rd-order perturbations
In this paper, we show that a Z2-equivariant 3rd-order Hamiltonian planar vector fields with 3rd-order symmetric perturbations can have at least 10 limit cycles. The method combines the general perturbation to the vector field and the perturbation to the Hamiltonian function. The Melnikov function is evaluated near the center of vector field, as well as near homoclinic and heteroclinic orbits. ...
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 کاملBifurcation of Limit Cycles in a Class of Liénard Systems with a Cusp and Nilpotent Saddle
In this paper the asymptotic expansion of first-order Melnikov function of a heteroclinic loop connecting a cusp and a nilpotent saddle both of order one for a planar near-Hamiltonian system are given. Next, we consider the bifurcation of limit cycles of a class of hyper-elliptic Liénard system with this kind of heteroclinic loop. It is shown that this system can undergo Poincarè bifurcation fr...
متن کاملBertrand-nash Equilibrium in the Retail Duopoly Model under Asymmetric Costs
In this paper, the Bertrand's price competition in the retail duopoly with asymmetric costs is analyzed. Retailers sell substitute products in the framework of the classical economic order quantity (EOQ) model with linear demand function. The market potential and competitor price are considered to be the bifurcation parameters of retailers. Levels of the barriers to market penetration depending...
متن کاملHomoclinic and Heteroclinic Bifurcations in Vector Fields
An overview of homoclinic and heteroclinic bifurcation theory for autonomous vector fields is given. Specifically, homoclinic and heteroclinic bifurcations of codimension one and two in generic, equivariant, reversible, and conservative systems are reviewed, and results pertaining to the existence of multi-round homoclinic and periodic orbits and of complicated dynamics such as suspended horses...
متن کامل