Real-World Existence and Origins of the Spiral Organization of Shrimp-Shaped Domains

Stoop, Ruedi; Benner, Philipp; Uwate, Yoko

doi:10.1103/physrevlett.105.074102

Cited by 84 publications

(51 citation statements)

References 18 publications

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“…Interesting to observe that for F = 0 the boundaries W 1 (a, b) and W 1→2 (a, b) become coupled in Eq. (2). In other words, the two independent conditions W 1 (a, b) = 0 for saddle-node bifurcation and W 1→2 (a, b) = 0 for PDB, are transformed in one saddlenode bifurcation condition W p F =2 (a, b, F ) = 0.…”

Section: Analytical Results For Pf =mentioning

confidence: 99%

“…ISSs were found in many systems, and we would like to mention some of them. In theoretical [1] and experimental [2] electronic circuits, continuous systems [3][4][5][6][7][8][9], maps [3,[10][11][12][13][14][15] lasers models [16], cancer models [17], classical [18][19][20] and quantum ratchet systems [21][22][23]. For the description of nature processes it is essential to discover generic properties for parameter combinations in nonlinear dynamical systems which can be applied to any realistic situation, independent of the specific physical system.…”

Section: Introductionmentioning

confidence: 99%

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Proliferation of stability in phase and parameter spaces of nonlinear systems

Manchein

Silva

Beims

2017

Chaos

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In this work we show how the composition of maps allows us to multiply, enlarge and move stable domains in phase and parameter spaces of discrete nonlinear systems. Using Hénon maps with distinct parameters we generate many identical copies of isoperiodic stable structures (ISSs) in the parameter space and attractors in phase space. The equivalence of the identical ISSs is checked by the largest Lyapunov exponent analysis and the multiplied basins of attraction become riddled. Our proliferation procedure should be applicable to any two-dimensional nonlinear system. In high-dimensional dynamical systems the possibility of controlling the dynamics through parametric changes is of great interest. Adding a time dependent parameter it is possible to control the dynamics of the paradigmatic Hénon map generating multiply Isoperiodic Stable Structures (ISSs) on the parameter space and consequently increasing the number of attractors on the phase space. Numerical simulations and analytical results explain the origin of new stable domains due to saddle-node bifurcations for specific parametric combinations. The distance between the multiplied ISSs can be controlled by the intensity of the time dependent parameter and general rules for the occurrence of proliferation are treated in details. We believe that the present study represents a significantly new insight on the use of alternating forces to control the dynamics of complex nonlinear systems modeled by twodimensional maps and can be extended to various applications ranging from physics, biology to engineering.

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Section: Analytical Results For Pf =mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Proliferation of stability in phase and parameter spaces of nonlinear systems

Manchein

Silva

Beims

2017

Chaos

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“…In the past two decades, periodic windows have been numerically obtained for a large number of applied dynamical systems like lasers [8,9], electronic circuits [9][10][11], mechanical oscillators [12], and also in population dynamics [13]. Periodic windows have also been identified in experiments with electronic circuits [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Replicate periodic windows in the parameter space of driven oscillators

Medeiros

Souza

Medrano-T

et al. 2011

Chaos, Solitons & Fractals

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In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have periodicity and pattern similar to stable and unstable periodic orbits already existent for the unperturbed oscillator. These features indicate that the reported replicate periodic windows are associated with chaos control of the considered oscillators.

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“…These structures are called "shrimps" due to their characteristic shape consisting of a main central body and four elongated "antennae" or "legs" 22,23 , and are globally organized along straight lines, ellipses, and spirals. A theoretical explanation of the spiral organization of shrimps was presented by Stoop et al 24 . They performed numerical simulations and laboratory experiments of the Nishio-Inaba electronic circuits to demonstrate that the spiral emergence of shrimps is due to the presence of a homoclinic saddle-focus point in the parameter space.…”

Section: Introductionmentioning

confidence: 99%

“…They performed numerical simulations and laboratory experiments of the Nishio-Inaba electronic circuits to demonstrate that the spiral emergence of shrimps is due to the presence of a homoclinic saddle-focus point in the parameter space. Shrimps have been found in several models including discrete-time maps 22,25 , continuous-time models represented by ordinary differential equations 23,26 and in a number of experiments using nonlinear circuits 24,27 . Our analysis is then focused on a periodic window, in which we perform a numerical detection of chaotic saddles, which are nonattracting chaotic sets responsible for chaotic transients in nonlinear systems [28][29][30][31] .…”

Section: Introductionmentioning

confidence: 99%

Chaotic saddles in nonlinear modulational interactions in a plasma

2012

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A nonlinear model of modulational processes in the subsonic regime involving a linearly unstable wave and two linearly damped waves with different damping rates in a plasma is studied numerically. We compute the maximum Lyapunov exponent as a function of the damping rates in a two-parameter space, and identify shrimp-shaped self-similar structures in the parameter space. By varying the damping rate of the low-frequency wave, we construct bifurcation diagrams and focus on a saddle-node bifurcation and an interior crisis associated with a periodic window. We detect chaotic saddles and their stable and unstable manifolds, and demonstrate how the connection between two chaotic saddles via coupling unstable periodic orbits can result in a crisis-induced intermittency. The relevance of this work for the understanding of modulational processes observed in plasmas and fluids is discussed.

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Real-World Existence and Origins of the Spiral Organization of Shrimp-Shaped Domains

Cited by 84 publications

References 18 publications

Proliferation of stability in phase and parameter spaces of nonlinear systems

Proliferation of stability in phase and parameter spaces of nonlinear systems

Replicate periodic windows in the parameter space of driven oscillators

Chaotic saddles in nonlinear modulational interactions in a plasma

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