An Adaptive Homotopy Method for Computing Bifurcations of Nonlinear Parametric Systems

In this paper, we present an adaptive step-size homotopy tracking method for computing bifurcation points of nonlinear systems. There are four components in this new method: (1) an adaptive tracking technique is developed near bifurcation points; (2) an inflation technique is backed up when the adaptive tracking fails; (3) Puiseux series interpolation is used to compute bifurcation points; and (4) the tangent cone structure of the bifurcation point is approximated numerically to compute solutions on different branches. Various numerical examples of nonlinear systems are given to illustrate the efficiency of this new approach. This new adaptive homotopy tracking method is also applied to a system of nonlinear PDEs and shows robustness and efficiency for large-scale nonlinear discretized systems.