Numerical Computation Of Connecting Orbits On A Manifold

Globally structural stability properties and numerical methods of dynamical sys-tems on Rmspace have been widely studied, many classical results have been proposed,see [4,5,6,33,65,62]. With the rapid development of science and technology, theapplication of dynamical systems on manifolds has made a great progress in many ar-eas, for example, networks analysis, chemical engineering systems, ecosystems, optimumcontrol, constrained mechanical system, etc. Compared with Rmspace, the study of dy-namical systems on a manifold and its numerical computation methods can be said tobe relatively late, as well as the limitations of a manifold bring certain difculties tostudy the dynamical systems on a manifold, and many results on Rmspace can not bedirectly extent to the similar case on a manifold.In resent years, a large amount of results about the stability analysis and numericalmethods of dynamical systems on a manifold have been obtained, see [20,27,58,59,34,25]. However it is still lack of profound insight, and a lot of work need to be done.In this paper, we mainly study the numerical computation methods for approxi-mating homoclinic orbits and heteroclinic orbits on a smooth manifold.We first consider the numerical computation methods and the bifurcation proper-ties of homoclinic orbits and heteroclinic orbits for continuous dynamical systems ona smooth manifold. We construct a matrix-valued function, which is smooth and in-verse. By using of this matrix-valued function, we can construct equivalent diferential-algebraic equations. We define nondegeneracy conditions of a connecting orbit with re-spect to parameters for the dynamical systems on a manifold, which ensure the regularityof connecting orbit and its bifurcation parameters of a suitable constructed extendingequation. In the case of numerical computations, we construct a modified projectionboundary conditions which is used for truncating the connecting orbits on a manifold on a finite interval. We prove the regularity of the truncated connecting orbit andparameters, and we also estimate the truncation errors.We also study the retentivity of connecting orbits in numerical discrete scheme ofthe d-order for diferential equations on a manifold. We define the γ-tangential andnondegenerate conditions of discrete connecting orbit on a manifold. We prove thatthe sampling of the exact homoclinic orbits are1-tangential connecting orbits and theyare nondegenerate with respect to bifurcation parameter λ. Besides, we proved thatthe connecting orbits and their bifurcation parameters are regulation solutions of someoperator equation. Furthermore, we proved the preservation properties of connectingorbits for numerical method of d-order. Especially, we prove the discrete connectingorbit of numerical scheme is either transversal or1-tangential and nondegenerate. Wealso prove that this discrete connecting orbit tends to the continuous one by the rate ofO(εd) as the step-size ε goes to0. And the corresponding parameters vary periodicallywith respect to a phase parameter with period ε while the orbit shifts its index afterone revolution. We also show that at least two1-tangential connecting orbits exist onthis loop. At last, we estimate the splitting distance of bifurcation parameters whichmeasures the size of the parameter region in which the numerical discrete scheme hasthe connecting orbits.