1605Based on the Lyapunov characteristic exponents, the ergodic property of dissipative dynamical systems with a few degrees of freedom is studied numerically by employing, as an example, the Lorenz system. The Lorenz system shows the spectra of ( +, 0, -) type concerning the 1-dimensional Lyapunov exponents, and the exponents take the same values for orbits starting from almost of all initial points on the attractor.This result suggests that the ergodic property for general dynamical systems not necessarily belonging to the category of the axiom-A may also be characterized in the framework of the spectra of the Lyapunov characteristic exponents. § I. Introduction Recently, chaotic motions that arise clue to non-linearities of dissipative dynamical systems have received a great concern in physical and non-physical fields. 1 ) However, in general dynamical systems which do not satisfy the axiom-A, little progress has been made to analyse those chaotic motions by theoretically wellesta blishecl methods. 2 )~4 )One of the purposes of this paper is to present numerical methods, by which wide-spread chaotic motions in dissipative dynamical systems would be characterized in a systematic manner. Our basic idea for this aim is to utilize the complete set of 1-dimensional Lyapunov exponents, which characterize the asymptotic orbital instability of dynamical systems. 5 )~g) The second purpose is to show that the concept of Lyapunov exponents presents a practical tool to discuss problems of bifurcation of those chaotic solutions.For these purposes, it becomes an important problem to estimate the Lyapunov exponents by some numerical methods, because it may not be expected, in general, that the equations for orbits exhibiting chaotic motions have globally single-valued analytic solutions. In case of measure preservig diffeomorphisms, Benettin et al. !1), *l have recently pointed out almost the same method as developed in this paper.
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Since the epoch-making work of Lorenz, the turbulent phenomena of dissipative dynamical systems have received a great concern from the standpoint of the theory of bifurcation. It is known that the erratic solutions also exist in various kinds of dynamical systems described by nonlinear dissipative equations. It may be understood that these erratic solutions are due to the wandering motion on the high-dimensional attractor which can appear in dissipative dynamical systems with more than 3-degrees of freedom.The purpose of this short note is to clarify some of the dynamical properties of these so-called erratic solutions. It will be demonstrated that a quantity k described in the following is very powerful not only to characterize the global orbital instability on the high-dimensional attractor but also to give the degree of stochasticity of such at
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