Defining chaos: expansion entropy

[Auto-Didact]

Hunt & Ott 2015, Defining Chaos

In this paper we propose, discuss and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered, including attractors, repellers and non-periodically forced systems. This definition is based on an entropy-like quantity, which we call "expansion entropy", and we define chaos as occurring when this quantity is positive. We relate and compare expansion entropy to the well-known concept of topological entropy, to which it is equivalent under appropriate conditions. We also present example illustrations, discuss computational implementations, and point out issues arising from attempts at giving definitions of chaos that are not entropy-based.

NB: For a more introductory version, phys.org ran a piece on this article two summers ago

This paper was published as a review of the concept of chaos in the journal Chaos for the 25th anniversary of that journal. The abstract is extended with a clearer motivation within the paper:

Toward the end of the 19th century, Poincaré demonstrated the occurrence of extremely complicated orbits in the Newtonian dynamics of three gravitationally attracting bodies. This complexity is now called chaos and has received a vast amount of attention since Poincaré's early discovery. In spite of this abundant past and current work, there is still no broadly applicable, convenient, generally accepted definition of the term chaos. In this paper, we advocate a particular entropy-based definition that appears to be very simple, while, at the same time, is readily accessible to numerical computation, and can be very generally applied to a variety of often-encountered situations, including attractors, repellers, and non-periodically forced systems. We also review and compare various previous definitions of chaos.

Any chaotic/nonlinear dynamics experts here which find fault with this particular expansion entropy definition of chaos?

 

[eljose79]

As a scientist with expertise in chaotic/nonlinear dynamics, I find this expansion entropy definition of chaos to be a valuable contribution to the field. The concept of entropy has been widely used in studying complex systems, and the application of this concept to define chaos is a logical and practical approach. The authors have also provided a clear motivation for their definition, which is based on the historical discovery of chaos in three-body dynamics by Poincaré.

I do not find any major faults with this definition, as it appears to be well-supported and applicable to a broad range of systems. However, as with any definition, there may be some limitations and potential criticisms that could arise from further research and application. For example, some may argue that the concept of expansion entropy may not capture all aspects of chaos, and that other measures such as Lyapunov exponents or fractal dimension may also be important in defining chaos. Additionally, there may be challenges in applying this definition to more complex systems with multiple attractors or chaotic regions.

Overall, I believe this paper provides a valuable contribution to the understanding and study of chaos, and I look forward to seeing further research and discussion on this topic.