Analysis of Nonlinear Systems via Bernstein Expansions

This paper presents an analytical approach for the stability and bifurcation analyses of nonlinear dynamic systems that have polynomial and trigonometric nonlinearities. The method ensures that all equilibrium points in a bounded region of the state space are found and yields arbitrarily tight bounds to the parameter points where the system exhibits local bifurcations. The solution method requires calculating and sequentially refining polynomial bounds of the trigonometric nonlinearities using Taylor expansions and Bernstein polynomials. The bounds supporting the resulting analyses, which are exempt from approximation and numerical error, can be made arbitrarily tight with additional computational effort. Due to the exponential complexity of the Bernstein expansion, the methodology is applicable to systems with a moderate number of states and bifurcating parameters.