Proving the Convergence to Limit Cycles using Periodically Decreasing Jacobian Matrix Measures

Methods based on “(Jacobian) matrix measure” to show the convergence of a dynamical system to a limit cycle (LC), generally assume that the measure is negative everywhere on the LC. We relax this assumption by assuming that the matrix measure is negative “on average” over one period of LC. Using an approximate Euler trajectory, we thus present a method that guarantees the LC existence, and allows us to construct a basin of attraction. This is illustrated on the example of the Van der Pol system.