Analysis of spurious synchronization with positive conditional Lyapunov exponents in computer simulations

Synchronization of chaotic systems has been a field of great interest and potential applications. The necessary condition for synchronization is the negativity of the largest conditional Lyapunov exponent. Some researches have shown that negativity of the largest conditional Lyapunov exponent is not a sufficient condition for high-quality synchronization in the presence of small perturbations. However, it was reported that synchronization can be achieved with positive conditional Lyapunov exponents based on numerical simulations. In this paper, we first analyze the behavior of synchronization with positive conditional Lyapunov exponents in computer simulations of the synchronization of chaotic systems with various couplings, and demonstrate that synchronization is an outcome of finite precision in numerical simulations. It is shown that such a numerical artifact is quite common and easily occurs in numerical simulations, thus can be confusing and misleading. Some behavior and properties of the synchronized system with slightly positive conditional Lyapunov exponents can be understood based on the theory of on-off intermittency. We also study the effects of finite precision on numerical simulation of on-off intermittency. Special care should be taken in numerical simulations of chaotic systems in order not to mistake numerical artifacts as physical phenomena.