Brian R. Hunt 1 and Edward Ott 2
1 Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742, USA
2 Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA
Chaos 25, 097618 (2015);
Abstract
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.
Key Topics
EntropyChaosAttractorsManifoldsSingular values
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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.
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