Homoclinic Branch Switching: a Numerical Implementation of Lin's Method
نویسندگان
چکیده
We present a numerical method for branch switching between homoclinic orbits to equilibria of ODEs computed via numerical continuation. Starting from a 1-homoclinic orbit our method allows us to find and follow an N -homoclinic orbit, for any N > 1 (if it exists nearby). This scheme is based on Lin’s method and it is robust and reliable. The method is implemented in AUTO/HomCont. A system of ordinary differential equations introduced by Sandstede featuring inclination and orbit flip bifurcations and homoclinic-doubling cascades, is used as a test bed for the algorithm. It is also successfully applied to reliably find multi-hump travelling wave solutions in the FitzHughNagumo nerve-axon equations and in a 4th-order Hamiltonian system arising as a model for water waves.
منابع مشابه
Initialization of Homoclinic Solutions near Bogdanov-Takens Points: Lindstedt-Poincaré Compared with Regular Perturbation Method
To continue a branch of homoclinic solutions starting from a Bogdanov–Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt–Poinca...
متن کاملA Numerical Bifurcation Function for Homoclinic Orbits
We present a numerical method to locate periodic orbits near homoclinic orbits. Using a method of X.-B. Lin and solutions of the adjoint variational equation, we get a bifurcation function for periodic orbits whose period is asymptotic to innnity on approaching a homoclinic orbit. As a bonus, a linear predictor for continuation of the homoclinic orbit is easily available. Numerical approximatio...
متن کاملImproved Homoclinic Predictor for Bogdanov-Takens Bifurcation
An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical “blow-up” technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit firstand second-order corrections of the unperturbed homoclinic orbit and parameter value. To ob...
متن کاملComputation and Continuation of Homoclinic and Heteroclinic Orbits with Arclength Parameterization
In this paper, we study a numerical method for the computation and continuation of homoclinic and heteroclinic orbits based upon the arclength parameterization of the orbits. Unlike most other methods, this method utilizes the geometric structure of the homoclinic and heteroclinic orbits and does not require solving a boundary value problem on an innnite interval. However, the boundary value pr...
متن کاملThe Numerical Computation of Homoclinic Orbits for Maps
Transversal homoclinic orbits of maps are known to generate shift dynamics on a set with Cantor like structure. In this paper a numerical method is developed for computation of the corresponding homoclinic orbits. They are approximated by nite orbit segments subject to asymptotic boundary conditions. We provide a detailed error analysis including a shadowing type result by which one can infer t...
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:
- I. J. Bifurcation and Chaos
دوره 13 شماره
صفحات -
تاریخ انتشار 2003