A chaos detectable and time step-size adaptive numerical scheme for nonlinear dynamical systems

The first step in investigation the dynamics of a continuous time system described by ordinary differential equations is to integrate them to obtain trajectories. In this paper, we convert the group-preserving scheme (GPS) developed by Liu [International Journal of Non-Linear Mechanics 36 (2001) 1047–1068] to a time step-size adaptive scheme, x ℓ + 1 = x ℓ + h f ( x ℓ , t ℓ ) , where x ∈ R n is the system variables we are concerned with, and f ( x , t ) ∈ R n is a time-varying vector field. The scheme has the form similar to the Euler scheme, x ℓ + 1 = x ℓ + Δ t f ( x ℓ , t ℓ ) , but our step-size h is adaptive automatically. Very interestingly, the ratio h / Δ t , which we call the adaptive factor, can forecast the appearance of chaos if the considered dynamical system becomes chaotical. The numerical examples of the Duffing equation, the Lorenz equation and the Rossler equation, which may exhibit chaotic behaviors under certain parameters values, are used to demonstrate these phenomena. Two other non-chaotic examples are included to compare the performance of the GPS and the adaptive one.