THREE CONDITIONS LYAPUNOV EXPONENTS SHOULD SATISFY

ABSTRACT In contrast to the unilateral claim in some papers that a positive Lyapunov exponent means chaos, it was claimed in this paper that this is just one of the three conditions that Lyapunov exponent should satisfy in a dissipative dynamical system when the chaotic motion appears. The other two conditions, any continuous dynamical system without a fixed point has at least one zero exponent, and any dissipative dynamical system has at least one negative exponent and the sum of all of the 1-dimensional Lyapunov exponents id negative, are also discussed. In order to verify the conclusion, a MATLAB scheme was developed for the computation of the 1-dimensional and 3-dimensional Lyapunov exponents of the Duffing system with square and cubic nonlinearity. KEYWORDS: Lyapunov exponent, chaos, three conditions, Duffing system INTRODUCTION Recently, chaotic motions that arise from the nonlinearity of dissipative dynamical systems have received a great concern in both physical and non-physical fields. The most striking feature of chaos is the unpredictability of the future despite a deterministic time evolution. This unpredictability is a consequence of the inherent instability of the solutions, reflected by what is called sensitive dependence on initial conditions. The tiny deviations between the initial conditions of all the trajectories are blown up after a short time. A more careful investigation of this instability leads to two different, although related, concepts. One is the loss of information related to unpredictability, quantified by the Kolmogorov-Sinai entropy. The other is a simple geometric one, namely, that nearby trajectories separate very fast, or more precisely, separate exponentially over time. The properly averaged exponent of this increase is the characteristics for th