THE SECOND-ORDER APPROXIMATION OF THRESHOLD VALUE OF CHAOS

For the Melnikov theory of chaos, the second-order approximation is given. Applying the result to the dynamics system with quadratic nonlinear term, it is shown that the threshold of chaos depends on initial condition and can be greater than that of the Melnikov method. The first order variational equations of some nonlinear dynamical systems are all the second-order ordinary differential equations with hyperbolic cosine function, its solution is given.