About Convergence of Numerical Approximations to Homoclinic Twist Bifurcation Points in Z2-Symmetric Systems in ℝ4

Abstract

An algorithm to detect homoclinic twist bifurcation points in Z2-symmetric autonomous systems of ordinary differential equations in ℝ4 along curves of symmetric homoclinic orbits to hyperbolic equilibria has been developed. We show convergence of numerical approximations to homoclinic twist bifurcation points in such systems. A test function is defined on the homoclinic solutions, which has a regular zero at the codimension-two bifurcation points. This codimension-two singularity can be continued appending the test function to a three-parameter vector field. We demonstrate the use of the test function on an example of two-coupled Josephson junctions.