An adaptive homotopy tracking algorithm for solving nonlinear parametric systems with applications in nonlinear ODEs

Homotopy continuation has played an important role in solving nonlinear parametric systems. The homotopy tracking algorithm is the key part of this method. In this paper, we develop an adaptive stepsize control strategy for the homotopy tracking algorithm. We also extend this algorithm to the predictor/corrector methods for solving systems of nonlinear ODEs. The adaptive strategy for adjusting the stepsize is to guarantee the convergence of Newton's method at each iteration along the solution path. Several numerical examples are used to show the efficiency of this new adaptive strategy by comparing it with other existing methods.