Finding zeros of nonlinear functions using the hybrid parallel cell mapping method

• Develop a new method for finding zeros of nonlinear vector functions
• Extend the method to find stability boundary conditions
• Compute zeros of challenging nonlinear functions
• Compute stability boundaries in potential fields

Analysis of nonlinear dynamical systems including finding equilibrium states and stability boundaries often leads to a problem of finding zeros of vector functions. However, finding all the zeros of a set of vector functions in the domain of interest is quite a challenging task. This paper proposes a zero finding algorithm that combines the cell mapping methods and the subdivision techniques. Both the simple cell mapping (SCM) and generalized cell mapping (GCM) methods are used to identify a covering set of zeros. The subdivision technique is applied to enhance the solution resolution. The parallel implementation of the proposed method is discussed extensively. Several examples are presented to demonstrate the application and effectiveness of the proposed method. We then extend the study of finding zeros to the problem of finding stability boundaries of potential fields. Examples of two and three dimensional potential fields are studied. In addition to the effectiveness in finding the stability boundaries, the proposed method can handle several millions of cells in just a few seconds with the help of parallel computing in graphics processing units (GPUs).