Map with more than 100 coexisting low-period periodic attractors

Feudel, Ulrike; Grebogi, Celso; Hunt, Brian R.; Yorke, James A.

doi:10.1103/physreve.54.71

Cited by 158 publications

(133 citation statements)

References 25 publications

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“…We also make a connection between the Standard Mapping and the waveguide models in order to localize the position of the invariant spanning curves in the phase space. The Standard Mapping is useful to describe the dynamics of a single kicked rotor [13] and it is given by S :…”

Section: The Standard Modelmentioning

confidence: 99%

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Finding invariant tori in the problem of a periodically corrugated waveguide

Rabelo

Leonel

2008

Braz. J. Phys.

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Some dynamic properties for a light ray suffering specular reflections inside a periodically corrugated waveguide are studied. The dynamics of the model is described in terms of a two dimensional nonlinear area preserving map. We show that the phase space is mixed in the sense that there are KAM islands surrounded by a large chaotic sea that is confined by two invariant spanning curves. We have used a connection with the Standard Mapping near a transition from local to global chaos and found the position of these two invariant spanning curves limiting the size of the chaotic sea as function of the control parameter.

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Section: The Standard Modelmentioning

confidence: 99%

“…(5)) and the mapping (13) it is easy to see that there is an effective control parameter K e f f which is given by…”

Section: The Standard Modelmentioning

confidence: 99%

Finding invariant tori in the problem of a periodically corrugated waveguide

Rabelo

Leonel

2008

Braz. J. Phys.

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“…In particular, many observed attractors are characterized by low periods. We must emphasize that studies of nonlinear dynamical systems considering a single mechanical rotor model [19] had shown that, in the Hamiltonian case of the purely rotor map, it exhibits a huge number of stable periodic orbits and each of them turn into an attracting sink when a small amount of dissipation is applied. Moreover, it is expected that the complexity of such model should be extended for higher-dimensional systems, as for example a double rotor [20], where it was found many more than 1000 coexisting low-period periodic attractors.…”

Section: Introductionmentioning

confidence: 99%

A simplified Fermi Accelerator Model under quadratic frictional force

Tavares

Leonel

2008

Braz. J. Phys.

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Some dynamical properties for a simplified version of a one-dimensional Fermi Accelerator Model under the action of a small dissipation is studied. The dissipation is introduced via a damping force which is assumed to be proportional to the square particle's velocity. The dynamics of the model is described by using a twodimensional, nonlinear area contracting mapping for the variables velocity of the particle and time. Our results confirm that the structure of the phase space of the conservative version is replaced by a large number of attracting periodic orbits. For a fixed set of control parameters, we obtain many periodic attractors and show that most of them posses low period. The stable orbits produce a complex structure of basin of attraction whose limit cover almost all phase space, thus suggesting a fractality.

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“…We modeled the system to be immersed in a thermal bath by adding a small damping and noise [see the third line in Eq. (4)], the effects of which have been studied in [24,25]. The point relevant for the following analysis is that when the driving force amplitude (henceforth called "kicking strength") is large, the rotor dynamics are fully chaotic, but if the kicking strength drops below a critical value (K 5) periodic orbits appear in the configuration space, and are made globally attractive in the presence of damping, thus quickly making the dynamics integrable (we refer to this phenomenon below as "dynamical regularization").…”

Section: Toy Modelmentioning

confidence: 99%

Least-rattling feedback from strong time-scale separation

Chvykov

England

2018

Phys. Rev. E

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In most interacting many-body systems associated with some "emergent phenomena," we can identify subgroups of degrees of freedom that relax on dramatically different time scales. Time-scale separation of this kind is particularly helpful in nonequilibrium systems where only the fast variables are subjected to external driving; in such a case, it may be shown through elimination of fast variables that the slow coordinates effectively experience a thermal bath of spatially varying temperature. In this paper, we investigate how such a temperature landscape arises according to how the slow variables affect the character of the driven quasisteady state reached by the fast variables. Brownian motion in the presence of spatial temperature gradients is known to lead to the accumulation of probability density in low-temperature regions. Here, we focus on the implications of attraction to low effective temperature for the long-term evolution of slow variables. After quantitatively deriving the temperature landscape for a general class of overdamped systems using a path-integral technique, we then illustrate in a simple dynamical system how the attraction to low effective temperature has a fine-tuning effect on the slow variable, selecting configurations that bring about exceptionally low force fluctuation in the fast-variable steady state. We furthermore demonstrate that a particularly strong effect of this kind can take place when the slow variable is tuned to bring about orderly, integrable motion in the fast dynamics that avoids thermalizing energy absorbed from the drive. We thus point to a potentially general feedback mechanism in multi-time-scale active systems, that leads to the exploration of slow variable space, as if in search of fine tuning for a "least-rattling" response in the fast coordinates.

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Map with more than 100 coexisting low-period periodic attractors

Cited by 158 publications

References 25 publications

Finding invariant tori in the problem of a periodically corrugated waveguide

Finding invariant tori in the problem of a periodically corrugated waveguide

A simplified Fermi Accelerator Model under quadratic frictional force

Least-rattling feedback from strong time-scale separation

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