Analytic Properties of Characteristic Exponents for Chaotic Dynamical Systems

• Published in: Progress of Theoretical Physics
• Main Authors: Inoue, Masayoshi, Fujisaka, Hirokazu
• Format: Text
• Language: English
• Published: Oxford University Press 1987
• Subjects: Articles; North Pole
• Online Access: http://ptp.oxfordjournals.org/cgi/content/short/77/5/1077; https://doi.org/10.1143/PTP.77.1077

A study on an analogy in mathematical formalism between a characteristic exponent λ q =(1/ q ) lim j →∞ (1/ j ) ln ≪exp ( q Σ j -1 s =0 Δ s )>, where { Δ j } 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 ( z λ z ) 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.