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Volumn 56, Issue 4, 1997, Pages 4031-4041

Periodic-orbit theory of the blowout bifurcation

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EID: 4243299076     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.56.4031     Document Type: Article
Times cited : (61)

References (67)
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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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    • 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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    • “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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    • 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)
    • 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 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
    • 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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    • (private communication)
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* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.