On the Number of Unstable Equilibrium Points on Spatially-Periodic Stability Boundary

Abstract

Unstable equilibrium points are fundamental in the study of dynamical systems, and can have various implications for nonlinear physical and engineering systems. In the present technical note, we derive lower bound as well as upper bound on the number of unstable equilibrium points on the stability boundary. For a class of nonlinear dynamical systems, by taking advantage of the spatial-periodicity, it is shown that there are at least $(k+1)C_{n}^{k}$ type- $k$ equilibrium points on a stability boundary, where $C_{n}^{k}=n!/k!(n-k)!$ . Meanwhile, an upper bound is obtained by applying the Bezout's Theorem, when the system can be converted to polynomials by variable substitution.