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5. Y.-C. Lai and T. Tél, Transient Chaos: Complex Dynamics on Finite Time Scales, Applied Mathematical Sciences Vol. 173 ( Springer, New York, 2011).
11.For example, R. E. Gillian and G. S. Ezra, “ Transport and turnstiles in multidimensional Hamiltonian mappings for unimolecular fragmentation: Application to van der Waals predissociation,” J. Chem. Phys. 94, 2648–2668 (1991).
12.For example, M. L. Du and J. Delos, “ Effect of closed classical orbits on quantum spectra: Ionization of atoms in a magnetic field. I. Physical picture and calculations,” Phys. Rev. A 38, 1896–1912 (1988).
13. U. Feudel, S. Kuznetsov, and A. Pikovsky, Strange Nonchaotic Attractors ( World Scientific, Singapore, 2006).
17. I. I. Rypina, F. J. Beron-Vera, M. G. Brown, H. Kocak, M. J. Olascoaga, and I. A. Udovydchenkov, “ On the Lagrangian dynamics of atmospheric zonal jets and the permeability of the stratospheric polar vortex,” J. Atmos. Sci. 64, 3595–3610 (2007).
20. F. Varosi, T. M. Antonsen, and E. Ott, “ The spectrum of fractal dimensions of passively convected scalar gradients in chaotic fluid flows,” Phys. Fluids A 3, 1017–1028 (1991).
28. If M is a Riemannian manifold, then it has a canonical volume that is equivalent to Lebesgue measure in appropriate local coordinates. Furthermore, when we treat the derivative of a map as a matrix, or use (small) distances in M, we assume the use of “normal coordinates,” which exist at least for C2 Riemannian manifolds.
29. Though we define expansion entropy only for a particular realization of a stochastic system, we expect that under appropriate hypotheses it has the same value for almost every realization. At a minimum, this should be the case when S is invariant for all realizations of a system forced by an IID process, because in this case the expansion entropy depends only on the tail of the IID process. A suitable setting for the rigorous study of expansion entropy in random systems would be that of Ledrappier and Young.14,15
30. If f is a C∞ map on a compact manifold, and S is the entire manifold, then the limit has been proved to exist.27 More generally, H0 could be defined as the lim sup, as in other definitions of entropy (see Sec. IV).
31. J. Jacobs, E. Ott, and B. R. Hunt, “ Calculating topological entropy for transient chaos with an application to communicating with chaos,” Phys. Rev. E 57, 6577–6588 (1998).
32. J. Balatoni and A. Rényi, “ Remarks on entropy,” Pub. Math. Inst. Hung. Acad. Sci. 1, 9–37 (1956). Translated in Selected Papers of A. Rényi (Akadémiai Kiadó, Budapest, 1976), Vol. 1, p. 558.
37. Ya. B. Pesin, “ Lyapunov characteristic exponents and ergodic properties of smooth dynamical systems with an invariant measure,” Dokl. Akad. Nauk SSSR 226, 774–777 (1976)
37. Ya. B. Pesin, [Sov. Math. Dokl. 17, 196–199 (1976)].
42. E. I. Dinaburg, “ The relation between topological entropy and metric entropy,” Dokl. Akad. Nauk SSSR 190, 19–22 (1970)
42. E. I. Dinaburg, [Sov. Math. Dokl. 11, 13–16 (1970)].
47. M. Misiurewicz
and W. Szlenk
, “ Entropy of piecewise monotone mappings
,” Stud. Math. 67
), see https://eudml.org/doc/218304
54. R. L. Devaney, An Introduction to Chaotic Dynamical Systems ( Addison-Wesley, New York and Reading, 1989).
55. J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacy, “ On Devaney's definition of chaos,” Am. Math. Mon. 99, 332–334 (1992).
56. S. Wiggins, Chaotic Transport in Dynamical Systems, Interdisciplinary Applied Mathematics Series Vol. 2 ( Springer-Verlag, Berlin, 1992).
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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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