A physical system with qualitatively uncertain dynamics

Sommerer, John C.; Ott, Edward

doi:10.1038/365138a0

Cited by 146 publications

(88 citation statements)

References 3 publications

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“…A riddled basin has a complicated geometric structure in that each open set that intersects the attractor also intersects its complement in a set of positive measure. Such basins have been studied, for example, in [2][3][4][5][6][7][8][9]. While most previous work has focussed on the global structure of riddled basins and bifurcations that create riddled basins, in this paper we turn our focus to the local geometry of a riddled basin.…”

Section: Motivationmentioning

confidence: 99%

Local and global stability indices for a riddled basin attractor of a piecewise linear map

Roslan

Ashwin

2016

Dynamical Systems

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We consider a piecewise expanding linear map with a Milnor attractor whose basin is riddled with the basin of a second attractor. To characterize the local geometry of this riddled basin, we calculate a stability index for points within the attractor as well as introducing a global stability index for the attractor as a set. Our results show that for Lebesgue almost all points in attractor the index is positive and we characterise a parameter region where some points have negative index. We show there exists a dense set of points for which the index is not converge. Comparing to recent results of Keller, we show that the stability index for points in the attractor can be expressed in terms of a global stability index for the attractor and Lyapunov exponents for this point.

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Section: Motivationmentioning

confidence: 99%

Local and global stability indices for a riddled basin attractor of a piecewise linear map

Roslan

Ashwin

2016

Dynamical Systems

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“…4a Fig. 4b, demonstrates that riddled basins may occur not only on chaotic and strange attractors [17] but also on stable limit cycles.…”

Section: Riddlingmentioning

confidence: 99%

“…We have numerically found two, three, and more coexisting stable limit cycles with the same and different periods, whose basins of attraction are riddled by each other. The basin is said to be riddled [17], if any point in the basin has in its arbitrary small vicinity points of other attractor basins.…”

Section: Riddlingmentioning

confidence: 99%

Synchronization of internal and external degrees of freedom of atoms in a standing laser wave

Argonov

Prants

2005

Phys. Rev. A

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We consider dissipative dynamics of atoms in a strong standing laser wave and find a nonlinear dynamical effect of synchronization between center-of-mass motion and internal Rabi oscillations. The synchronization manifests itself in the phase space as limit cycles which may have different periods and riddled basins of attraction. The effect can be detected in the fluorescence spectra of atoms as equidistant sideband frequencies with the space between adjacent peaks to be inversely proportional to the value of the period of the respective limit cycle. With increasing the intensity of the laser field, we observe numerically cascades of bifurcations that eventually end up in settling a strange chaotic attractor. A broadband noise is shown to destroy a fine structure of the bifurcation scenario, but prominent features of period-1 and period-3 limit cycles survive under a weak noise. The character of the atomic motion is analyzed with the help of the friction force whose zeroes are attractor or repellor points in the velocity space. We find ranges of the laser parameters where the atomic motion resembles a random but deterministic walking of atoms erratically jumping between different wells of the optical potential. Such a random walking is shown to be fractal in the sense that the measured characteristic of the motion, time of exit of atoms from a given space of the standing wave, is a complicated function that has a self-similar structure with singularities on a Cantor set of values of one of the control parameters.

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“…The spatial evolution and transient phenomena before a trajectory may settle onto the subset is largely unknown, especially since complex phenomena such as intermittency Heagy et al 1994a, Ott & Sommerer 1994 and riddled basins Alexander et al 1992, Sommerer & Ott 1993] may occur. In this paper we report some detailed simulations which focus on the convergence of the dynamics onto the invariant subset, looking at possible relationships between exponents of scaling laws and the relative change of parameters and/or initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Transient Time in Unidirectional Synchronization

Santoboni

Bishop

Varone

1999

Int. J. Bifurcation Chaos

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We consider here the behaviour of a dynamical system consisting of unidirectionally coupled Du ng oscillators. Under certain conditions the subsystems may become synchronised corresponding to a stable invariant subset of the full dimensional phase space. The distribution for the transient time that trajectories take to converge onto the synchronised state is investigated via numerical simulations. Some initial conditions undergo a very long transient motion distinct from the drive, before synchronisation occurrs. The dependence of this time on the governing system parameters and on the initial conditions of the driving system is discussed.Short title: Transient Time to synchronisation.

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A physical system with qualitatively uncertain dynamics

Cited by 146 publications

References 3 publications

Local and global stability indices for a riddled basin attractor of a piecewise linear map

Local and global stability indices for a riddled basin attractor of a piecewise linear map

Synchronization of internal and external degrees of freedom of atoms in a standing laser wave

Transient Time in Unidirectional Synchronization

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