Detecting oscillatory behavior using Lyapunov functions

Conditions are derived which guarantee the existence of oscillatory behavior for general nonlinear, high-dimensional systems. The first result is motivated by the energy transfer which occurs for example in an oscillatory LC circuit. A simple generalization of this energy transfer mechanism leads to conditions which guarantee the existence of oscillations and which can be expressed in terms of Lyapunov-like functions. The second result in this paper are new computationally tractable conditions which allow to use numerical methods, like sum of squares techniques, to verify oscillatory behavior for polynomial systems. Moreover, a simple condition for the nonexistence of oscillatory behavior is pointed out. The applicability of the results is demonstrated by several examples.