A Method for Homoclinic and Heteroclinic Continuation in Two and Three Dimensions

A numerical method for the detection and continuation of homoclinic and heteroclinic orbits is developed for the case of biparametric dynamical systems in two and three dimensions. We formulate a continuation problem for which the regularity conditions are studied. The numerical method is applied to several systems, some of them well-known.

[1]  Joseph Gruendler,et al.  The Existence of Homoclinic Orbits and the Method of Melnikov for Systems in $R^n$ , 1985 .

[2]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[4]  J. Craggs.  Applied Mathematical Sciences , 1973 .

[5]  Yu. A. Kuznetsov.  Computation of Invariant Manifold Bifurcations , 1990 .

[6]  Raymond Kapral,et al.  Bifurcation phenomena near homoclinic systems: A two-parameter analysis , 1984 .

[7]  C. Sparrow.  The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors , 1982 .

[8]  S. Levin.  Lectu re Notes in Biomathematics , 1983 .

[9]  Werner C. Rheinboldt,et al.  A locally parameterized continuation process , 1983, TOMS.

[10]  Kenneth J. Palmer,et al.  Exponential dichotomies and transversal homoclinic points , 1984 .

[11]  A. Spence,et al.  Continuation and Bifurcations: Numerical Techniques and Applications , 1990 .

[12]  Colin Sparrow,et al.  Local and global behavior near homoclinic orbits , 1984 .

[13]  Leopoldo García Franquelo,et al.  An Algorithm for Symbolic Computation of Center Manifolds , 1988, ISSAC.

[14]  Stephen Wiggins,et al.  Global Bifurcations and Chaos , 1988 .

[15]  Colin Sparrow,et al.  The Lorenz equations , 1982 .

[16]  W. A. Coppel.  Dichotomies in Stability Theory , 1978 .

[17]  Paul C. Fife,et al.  Mathematical Aspects of Reacting and Diffusing Systems , 1979 .

[18]  Stephen Wiggins.  Global Bifurcations and Chaos: Analytical Methods , 1988 .