Analytic Properties of Characteristic Exponents for Chaotic Dynamical Systems

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Abstract
A study on an analogy in mathematical formalism between a characteristic exponent λq=(1/q) lim j →∞(1/j) ln ≪exp (qΣj-1s=0Δs)>, where { Δj } is a steady one-dimensional time sequence, and other functions such as the Helmholtz free energy and a set of dimensions of a strange attractor, is carried out, and a new concept “filtering parameter” is proposed. The inverse temperature acts as the filtering parameter in the Helmholtz free energy. A complex partition function Z(z)=exp (zλz) is introduced in order to classfy chaos transitions. Distribution of the solutions of Z(z)=0 is calculated for several time sequences which are generated by chaotic dynamical systems. It is found that there are two types of transitions; (a) the solutions accumulate at the north pole of Riemann sphere and (b) the solutions approach the real axis of the complex z-plane.

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