Abstract
Transitions to chaos in archetypal low-dimensional nonlinear maps offer real and precise model systems in which to assess proposed generalizations of statistical mechanics. The known association of chaotic dynamics with the structure of Boltzmann–Gibbs (BG) statistical mechanics has suggested the potential verification of these generalizations at the onset of chaos, when the only Lyapunov exponent vanishes and ergodic and mixing properties cease to hold. There are three well-known routes to chaos in these deterministic dissipative systems, period-doubling, quasi-periodicity and intermittency, which provide the setting in which to explore the limit of validity of the standard BG structure. It has been shown that there is a rich and intricate behavior for both the dynamics within and towards the attractors at the onset of chaos and that these two kinds of properties are linked via generalized statistical-mechanical expressions. Amongst the topics presented are: (i) permanently growing sensitivity fluctuations and their infinite family of generalized Pesin identities; (ii) the emergence of statistical-mechanical structures in the dynamics along the routes to chaos; (iii) dynamical hierarchies with modular organization; and (iv) limit distributions of sums of deterministic variables. The occurrence of generalized entropy properties in condensed-matter physical systems is illustrated by considering critical fluctuations, localization transition and glass formation. We complete our presentation with the description of the manifestations of the dynamics at the transitions to chaos in various kinds of complex systems, such as, frequency and size rank distributions and complex network images of time series. We discuss the results.
Highlights
Chaotic dynamical systems, such as chaotic attractors in one-dimensional nonlinear iterated maps, accept a statistical-mechanical description with an entropy expression of the Boltzmann–Gibbs (BG)type [1]
The chaotic attractors generated by these maps have ergodic and mixing properties, and not surprisingly, they can be described by a thermodynamic formalism compatible with BG statistics [1]
As it is generally suggested, the standard exponential divergence of trajectories in chaotic attractors provides a mechanism to justify the assumption of irreversibility in the BG statistical mechanics [1]
Summary
Chaotic dynamical systems, such as chaotic attractors in one-dimensional nonlinear iterated maps, accept a statistical-mechanical description with an entropy expression of the Boltzmann–Gibbs (BG). The attractors at the transitions to chaos in one-dimensional nonlinear maps (that we shall refer to as critical attractors) have a vanishing ordinary Lyapunov coefficient, λ1 , and the sensitivity to initial conditions, ξt , for large iteration time t ceases to obey exponential behavior, exhibiting, instead, power-law or faster than exponential growth behavior [2,3,4,5,6,7] As it is generally suggested, the standard exponential divergence of trajectories in chaotic attractors provides a mechanism to justify the assumption of irreversibility in the BG statistical mechanics [1].
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