Logarithmic periodicities in the bifurcations of type-I intermittent chaos

• Published in: Physical review letters 2004-06, Vol.92 (25), p.254102.1-254102.4, Article 254102
• Main Authors: CAVALCANTE, Hugo L. D. De S, RIOS LEITE, J. R
• Format: Article
• Language: English
• Subjects: Condensed matter: structure, mechanical and thermal properties; Equations of state, phase equilibria, and phase transitions; Equilibrium properties near critical points, critical exponents; Exact sciences and technology; General studies of phase transitions; Nonlinear dynamics and nonlinear dynamical systems; Physics; Statistical physics, thermodynamics, and nonlinear dynamical systems
• ISSN: 0031-9007; 1079-7114
• DOI: 10.1103/PhysRevLett.92.254102

The critical relations for statistical properties on saddle-node bifurcations are shown to display undulating fine structure, in addition to their known smooth dependence on the control parameter. A piecewise linear map with the type-I intermittency is studied and a log-periodic dependence is numerically obtained for the average time between laminar events, the Lyapunov exponent, and attractor moments. The origin of the oscillations is built in the natural probabilistic measure of the map and can be traced back to the existence of logarithmically distributed discrete values of the control parameter giving Markov partition. Reinjection and noise effect dependences are discussed and indications are given on how the oscillations are potentially applicable to complement predictions made with the usual critical exponents, taken from data in critical phenomena.