A Numerical Bifurcation Function for Homoclinic Orbits
Ashwin, Peter and Mei, Zhen (1998) A Numerical Bifurcation Function for Homoclinic Orbits SIAM Journal on Numerical Analysis, 35. pp. 2055-2069.
We present a numerical method to locate periodic orbits near homoclinic orbits. Using a method of [X.-B. Lin, Proc. Roy. Soc. Edinburgh, 116A (1990), pp. 295--325] and solutions of the adjoint variational equation, we get a bifurcation function for periodic orbits, whose periods are asymptotic to infinity on approaching a homoclinic orbit. As a bonus, a linear predictor for continuation of the homoclinic orbit is easily available.
Numerical approximation of the homoclinic orbit and the solution of the adjoint variational equation are discussed. We consider a class of methods for approximating the latter equation such that a scalar quantity is preserved. We also consider a context where the effects of continuous symmetries of equations can be incorporated.
Applying the method to an ordinary differential equation on R3 studied by [E. Freire, A. Rodriguez-Luis, and E. Ponce, Phys. D, 62 (1993), pp. 230--253] we show the bifurcation function gives good agreement with path-followed solutions even down to low period. As an example application to a parabolic partial differential equation (PDE), we examine the bifurcation function for a homoclinic orbit in the Kuramoto--Sivashinsky equation.
|Additional Information:||Published in SIAM Journal on Numerical Analysis, 35, 2055-2069. © 1998 Society for Industrial and Applied Mathematics.|
|Uncontrolled Keywords:||homoclinic orbit, numerical bifurcation function, periodic solutions|
|Divisions:||Faculty of Engineering and Physical Sciences > Mathematics|
|Depositing User:||Mr Adam Field|
|Date Deposited:||27 May 2010 14:40|
|Last Modified:||23 Sep 2013 18:32|
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