Period-doubling cascades galore
Evelyn Sander,James A. Yorke +1 more
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In this paper, a general theory of cascades for generic parametrized maps is presented, and it is shown that there is a close connection between the transition through innitely many cascades and the creation of a horseshoe.Abstract:
The appearance of numerous period-doubling cascades is among the most prominent features of parametrized maps, that is, smooth one-parameter families of maps F : R M ! M, where M is a smooth locally compact manifold without boundary, typically R N . Each cascade has innitely many period-doubling bifurcations, and it is typical to observe { such as in all the examples we investigate here { that whenever there are any cascades, there are innitely many cascades. We develop a general theory of cascades for generic F . We illustrate this theory with several examples. We show that there is a close connection between the transition through innitely many cascades and the creation of a horseshoe.read more
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Replication of chaos
Marat Akhmet,Mehmet Onur Fen +1 more
TL;DR: A rigorous method for replication of chaos from a prior one to systems with large dimensions is proposed and new definitions such as chaotic sets of functions, the generator and replicator of chaos, and precise description of ingredients for Devaney and Li-Yorke chaos in continuous dynamics are provided.
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Global dynamics in a stage-structured discrete-time population model with harvesting
Eduardo Liz,Paweł Pilarczyk +1 more
TL;DR: The range of parameters for which the population abundance gets larger in spite of an increase in the harvest rate is determined, and for which subsequent increases in harvesting effort can magnify fluctuations in population abundance.
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The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting
Eduardo Liz,Alfonso Ruiz-Herrera +1 more
TL;DR: It is shown that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates and the phenomenon of bubbling and the hydra effect, which means that the stock size gets larger as harvesting rate increases.
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Connecting period-doubling cascades to chaos
Evelyn Sander,James A. Yorke +1 more
TL;DR: In this paper, it was shown that often virtually all (i.e., all but finitely many) "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path of regular periodic orbits; and virtually all cascades are either paired or solitary.
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Connecting period-doubling cascades to chaos
Evelyn Sander,James A. Yorke +1 more
TL;DR: The investigation of infinitely many cascades is essentially reduced to studying the regular periodic orbits of F(μ2, ⋅), which shows that often virtually all "regular" periodic orbits at μ2 are each connected to exactly one cascade by a path ofregular periodic orbits.
References
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