Chaos, the Critical Phenomenon in Phase Space: Feigenbaum Constants and Critical Exponents

TL;DR

Chaos is treated as a phase-space critical phenomenon for both dissipative and conservative systems using renormalization-group ideas. Using the logistic map as a canonical dissipative example, the work identifies the universal constants $\delta$ and $\alpha$ with $\delta=4.6692...$ and $\alpha=2.5029...$, and derives the independent exponents $\nu_1=\log_{\delta}2$ and $\nu_2=\log_{\alpha}2$, establishing a two-exponent classification of chaos. In conservative chaos, the analysis shows that chaotic bands form fat fractals with nonzero Lebesgue measure and that Liouville ergodicity induces inter-parameter correlations, leading to permanent information loss and effective irreversibility, akin to decoherence. The findings provide a unified equilibrium-statistics perspective on chaos, linking Feigenbaum universality to phase-transition-like critical points, and offer a resolution to thermodynamic reversibility paradoxes in deterministic dynamics.

Abstract

Chaos in both dissipative systems and conservative systems is investigated on the approach of renormalization group. It is found that the chaos is regarded as the critical phenomenon of equilibrium statistics in phase space. The two Feigenbaum constants in the period-doubling bifurcation systems correspond to two independent critical exponents, which are universal and can be adopted to distinguish the classes of chaos. For the conservative systems, due to the critical nature of the chaos, the isolated systems with different parameters are correlated in the phase space, and therefore the isolated system is no longer isolated in the phase space. The information of conservative systems is irreversibly lost over time, which leads to the increase entropy in an isolated system, and the contradiction between the second law of thermodynamics and the reversibility of isolated systems can be resolved.

Figures (1)

• Figure 1: (a). The sketch of the period--doubling process in the continuous dynamical system, period--$2^{n}$ cycle $\rightarrow$ period--$2^{n+1}$ cycle. (b). The bifurcation diagram given by the logistic map $x_{i+1}=rx_{i}(1-x_{i})$.