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E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993); K. T. Alligood, T. Saure, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York, 1996)
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E. Ott, Chaos in Dynamical Systems (Cambridge University Press, Cambridge, 1993); K. T. Alligood, T. Saure, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, New York, 1996).
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3
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A crisis is triggered by the collision of a periodic orbit, usually of low period, embedded in the chaotic attractor, either with the basin boundary (boundary crisis) or with a nonattracting chaotic saddle (interior crisis). See C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 48, 1507 (1982); PRLTAO
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5
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Basin boundaries in nonlinear systems with multiple coexisting attractors can undergo metamorphoses when an unstable periodic orbit on the boundary collides with a nonattracting chaotic saddle. See C. Grebogi, E. Ott, and J. A. Yorke, Phys. Rev. Lett. 56, 1011 (1986); PRLTAO
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9
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85037232742
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“Riddling” means that for every point in the basin of the chaotic attractor in the invariant subspace there are points arbitrarily nearby that belong to the basins of other attractors. See, for example, J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, Int. J. Bifurcation Chaos 2, 795 (1992)
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“Riddling” means that for every point in the basin of the chaotic attractor in the invariant subspace there are points arbitrarily nearby that belong to the basins of other attractors. See, for example, J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, Int. J. Bifurcation Chaos 2, 795 (1992);
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Y.-C. Lai, 53, R4267 (1996);
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54, 321 (1996);
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S. C. Venkataramani, T. M. Antonsen, Jr., E. Ott, and J. C. Sommerer, Physica D 96, 66 (1996).PDNPDT
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At the bifurcation, the natural measure of the chaotic attractor in S becomes unstable
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At the bifurcation, the natural measure of the chaotic attractor in S becomes unstable.
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The dynamics is hyperbolic on a chaotic attractor if at each point of the trajectory the phase space can be split into an expanding and a contracting subspace and the angle between them is bounded away from zero. Furthermore, the expanding subspace evolves into the expanding one along the trajectory and the same is true for the contracting subspace. Otherwise the set is nonhyperbolic. In general, nonhyperbolicity is a complicating feature because it can cause fundamental difficulties in the study of the chaotic systems, a known one being the shadowability of numerical trajectories by true trajectories [C. Grebogi, S. M. Hammel, and J. A. Yorke, J. Complexity 3, 136 (1987)
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The dynamics is hyperbolic on a chaotic attractor if at each point of the trajectory the phase space can be split into an expanding and a contracting subspace and the angle between them is bounded away from zero. Furthermore, the expanding subspace evolves into the expanding one along the trajectory and the same is true for the contracting subspace. Otherwise the set is nonhyperbolic. In general, nonhyperbolicity is a complicating feature because it can cause fundamental difficulties in the study of the chaotic systems, a known one being the shadowability of numerical trajectories by true trajectories [C. Grebogi, S. M. Hammel, and J. A. Yorke, J. Complexity 3, 136 (1987);
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J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Lecture Notes in Mathematics Vol. 730 (Springer, Berlin, 1979)
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J. L. Kaplan and J. A. Yorke, in Functional Differential Equations and Approximations of Fixed Points, edited by H.-O. Peitgen and H.-O. Walter, Lecture Notes in Mathematics Vol. 730 (Springer, Berlin, 1979).
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85037214107
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The mechanism leading to this scaling behavior is similar to that in the doubling transformation because the chaotic attractor in Fig. 55 is hyperbolic and its unstable direction is the (Formula presented) axis along which the dynamics is the doubling transformation
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The mechanism leading to this scaling behavior is similar to that in the doubling transformation because the chaotic attractor in Fig. 55 is hyperbolic and its unstable direction is the (Formula presented) axis along which the dynamics is the doubling transformation.
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Alligood, K.T.1
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85037210906
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(private communication)
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H. S. Greenside (private communication).
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Greenside, H.S.1
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