A mechanism for growth of topological entropy and global changes of the shape of chaotic attractors

TL;DR

This work develops a computer-assisted framework to prove global changes in attractor geometry and unbounded growth of topological entropy in a continuous-time dynamical system, exemplified by the Rössler model. By combining a simple sine-map toy model to illustrate the mechanism with a rigorous Poincaré-map analysis and validated numerics, the authors prove the existence of a chaotic attractor for a parameter range and establish a sequence of saddle-node bifurcations that induce semiconjugacy to symbolic dynamics with increasing numbers of symbols. The methodology leverages validated integration, interval Newton methods, and covering-relations to obtain guaranteed lower bounds on entropy via subshifts of finite type, with entropy rising from 2 up to 13 symbols as parameters vary. The results demonstrate a concrete pathway for global attractor remodeling and entropy amplification in continuous-time systems, supported by substantial computational proofs and supplementary data.

Abstract

The theoretical and numerical understanding of the key concept of topological entropy is an important problem in dynamical systems. Most studies have been carried out on maps (discrete-time systems). We analyse a scenario of global changes of the structure of an attractor in continuous-time systems leading to an unbounded growth of the topological entropy of the underlying dynamical system. As an example, we consider the classical Roessler system. We show that for an explicit range of parameters a chaotic attractor exists. We also prove the existence of a sequence of bifurcations leading to the growth of the topological entropy. The proofs are computer-aided.

Figures (11)

• Figure 1: (a): Lower and upper bounds for the values of the topological entropy on the range $a \in [0, 7.5]$ and the value of $\log_2(a)$. (b) Lyapunov exponent of the map (\ref{['eq:SinModel']}). (c): bifurcation diagram. (d): 3D plot of the function map (\ref{['eq:SinModel']}) depending on the parameter $a$.
• Figure 2: Plot of $f_a(x)$ for different parameter values. A saddle-node bifurcation creates a pair of fixed points (left column). The stable fixed point losses stability via period doubling bifurcation (middle column). Further growth of parameter leads to creation of new symbol for conjugacy to symbolic dynamics (right column).
• Figure 3: Graph of symbolic dynamics of the sine model (\ref{['eq:SinModel']}) for $a=3$.
• Figure 4: Biparametric plot of the largest Lyapunov exponents of (\ref{['eq:rossler']}) with $b=0.2$. Color (from green to red) represents chaotic regimes shown by the maximum Lyapunov exponent. In black and white, regular dynamics, first exponent is null and the second is represented. Blue curves mark saddle node bifurcations, while red curves indicate period doubling bifurcations. The white and grey dashed segment marks the line (with $c=15$) that we will study in more detail in the rest of the paper.
• Figure 5: Top: Poincaré section and a typical trajectory of (\ref{['eq:rossler']}) for $a=a_{\mathrm{min}}=0.12$ (left) and $a=a_{\mathrm{max}}=0.3659$ (right). Bottom: The corresponding FRM.

Theorems & Definitions (15)

• Theorem 1
• Definition 1
• Theorem 2
• Theorem 3
• proof
• Theorem 4
• Theorem 5: Katok_Hasselblatt_1995
• Theorem 6
• Remark 7
• Remark 8