Predicting chaos with second method of Lyapunov

We overview several analytic methods of predicting the emergence of chaotic motion in nonlinear oscillatory systems. A special attention is given to the second method of Lyapunov, a technique that has been widely used in the analysis of stability of motion in the theory of dynamical systems but received little attention in the context of chaotic systems analysis. We show that the method allows formulating a necessary condition for the appearance of chaos in nonlinear systems. In other terms, it provides an analytic estimate of an area in the space of control parameters where the largest Lyapunov exponent is strictly negative. A complementary area thus comprises the values of controls, where the exponent can take positive values, and hence the motion can become chaotic. Contrary to other commonly used methods based on perturbation analysis, such as e.g., Melnikov criterion, harmonic balance, or averaging, our approach demonstrates superior performance at large values of the parameters of dissipation and nonlinearity. Several classical examples including mathematical pendulum, Duffing oscillator, and a system of two coupled oscillators, are analyzed in detail demonstrating advantages of the proposed method compared to other existing techniques.