^{1}and Edward Ott

^{2}

### 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.

The work of E. Ott was supported by the U.S. Army Research Office under Grant No. W911NF-12-1-0101. We thank S. Newhouse for pointing out earlier work related to expansion entropy, and J. Yorke and the reviewers for helpful comments.

I. INITIAL DISCUSSION II. EXPANSION ENTROPY A. Definition B. Expansion entropy of the inverse system C. Discussion of numerical evaluation of expansion entropy D. Generalization to

*q*-order expansion entropy III. ILLUSTRATIVE EXAMPLES A. Attracting and repelling fixed points B. Example: A one-dimensional map with a chaotic repeller and an attracting fixed point C. Example: A random one-dimensional map D. Example: Shear map on the 2-torus E. Example: Horseshoe and Hénon map IV. TOPOLOGICAL ENTROPY V. DEFINITIONS OF CHAOS THAT DO NOT INVOLVE ENTROPY: SENSITIVE DEPENDENCE, LYAPUNOV EXPONENTS, AND CHAOTIC ATTRACTORS VI. CONCLUSION

^{∞}maps,” Erg. Theory Dyn. Syst. 18, 405–424 (1998).

^{2}Riemannian manifolds.

^{14,15}

^{∞}map on a compact manifold, and S is the entire manifold, then the limit has been proved to exist.

^{27}More generally, H

_{0}could be defined as the lim sup, as in other definitions of entropy (see Sec. IV).

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### 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.

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