Analytic Properties of Characteristic Exponents for Chaotic Dynamical Systems
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 attra...
| Published in: | Progress of Theoretical Physics |
|---|---|
| Main Authors: | , |
| Format: | Text |
| Language: | English |
| Published: |
Oxford University Press
1987
|
| Subjects: | |
| Online Access: | http://ptp.oxfordjournals.org/cgi/content/short/77/5/1077 https://doi.org/10.1143/PTP.77.1077 |
| Summary: | 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. |
|---|