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T. Tél, in Directions in Chaos, edited by Bai-lin Hao (World Scientific, Singapore, 1990), Vol. 3
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4
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0004210476
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Bai-lin Hao, World Scientific, Singapore
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in STATPHYS 19, edited by Bai-lin Hao (World Scientific, Singapore, 1996).
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(1996)
STATPHYS 19
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6
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85036409711
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Chaos 3 (1993), focus issue on chaotic scattering
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Chaos 3 (1993), focus issue on chaotic scattering.
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7
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S. W. McDonald, C. Grebogi, E. Ott, and J. A. Yorke, Physica D 17, 125 (1985).
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McDonald, S.W.1
Grebogi, C.2
Ott, E.3
Yorke, J.A.4
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9
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0001587809
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Á. Péntek, Z. Toroczkai, T. Tél, C. Grebogi, and J. Yorke, Phys. Rev. E 51, 4076 (1995)
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(1995)
Phys. Rev. E
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Péntek, Á.1
Toroczkai, Z.2
Tél, T.3
Grebogi, C.4
Yorke, J.5
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11
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0000853548
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and references therein
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Z. Toroczkai, G. Károlyi, Á. Péntek, T. Tél, and C. Grebogi, Phys. Rev. Lett. 80, 500 (1998), and references therein.
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Phys. Rev. Lett.
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Toroczkai, Z.1
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Péntek, Á.3
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13
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D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. H. Gunaratne, and I. Procaccia, Phys. Rev. Lett. 58, 2387 (1988)
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Procaccia, I.5
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14
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85036308802
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focus issue on periodic orbit theory;
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D. Auerbach, P. Cvitanović, J.-P. Eckmann, G. H. Gunaratne, and I. Procaccia, Chaos 2, (1993), focus issue on periodic orbit
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(1993)
Chaos
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Auerbach, D.1
Cvitanović, P.2
Eckmann, J.-P.3
Gunaratne, G.H.4
Procaccia, I.5
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16
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85036211786
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At present, it is not clear how our work can be extended to higher dimensions due to the difficulties in the systematic computation of unstable periodic orbits
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At present, it is not clear how our work can be extended to higher dimensions due to the difficulties in the systematic computation of unstable periodic orbits.
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17
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85036291219
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The dynamics is hyperbolic on a chaotic set if at each point of the trajectory the phase space can be split into expanding and contracting subspaces and the angle between them is bounded away from zero. Furthermore, the expanding subspace evolves into expansion along the trajectory and the same is true for the contracting subspace. Otherwise the set is nonhyperbolic
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The dynamics is hyperbolic on a chaotic set if at each point of the trajectory the phase space can be split into expanding and contracting subspaces and the angle between them is bounded away from zero. Furthermore, the expanding subspace evolves into expansion along the trajectory and the same is true for the contracting subspace. Otherwise the set is nonhyperbolic.
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18
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0020814903
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Chaotic saddles in Hamiltonian systems, on the other hand, are generally nonhyperbolic due to the presence of hierarchies of Kol’mogorov-Arnol’d-Moser (KAM) tori. Due to the stickiness effect of KAM tori, the decay of the number of chaotic trajectories in a phase-space region containing both the chaotic saddle and some KAM tori is algebraic in time [C. F. F. Karney, Physica D 8, 360 (1983)
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Karney, C.F.F.1
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29
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85036407278
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dissipative systems, the sole source of nonhyperbolicity is the tangencies between the stable and the unstable manifolds. A trajectory on a nonhyperbolic chaotic saddle typically spends most of the time in regions where the dynamics is hyperbolic 13141516171820. Thus, we expect the decay of transient chaotic trajectories to be exponential [Eq. (1)] and the natural measure decays exponentially with the period as well [Eq. (6)]
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In dissipative systems, the sole source of nonhyperbolicity is the tangencies between the stable and the unstable manifolds. A trajectory on a nonhyperbolic chaotic saddle typically spends most of the time in regions where the dynamics is hyperbolic 13141516171820. Thus, we expect the decay of transient chaotic trajectories to be exponential [Eq. (1)] and the natural measure decays exponentially with the period as well [Eq. (6)].
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30
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0000906011
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Y.-C. Lai, C. Grebogi, J. A. Yorke, and I. Kan, Nonlinearity 6, 779 (1993).
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Kan, I.4
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32
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45149143640
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It is believed that the PIM-triple algorithm typically generates the natural measure of the chaotic saddle [H. E. Nusse and J. A. Yorke, Physica D 36, 137 (1989)
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Nusse, H.E.1
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So, P.1
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Grebogi, C.6
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