Stable structures in parameter space and optimal ratchet transport

Celestino, Alan; Manchein, Cesar; Albuquerque, Holokx A.; Beims, Marcus W.

doi:10.1016/j.cnsns.2013.06.020

Cited by 38 publications

(41 citation statements)

References 62 publications

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“…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%

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: Introductionmentioning

confidence: 99%

Proliferation of stability in phase and parameter spaces of nonlinear systems

Manchein

Silva

Beims

2017

Chaos

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“…The quantum-classical transition was studied by means of a comparison between the C S and C C . We show how and to which extent the classical generic isoperiodic stable structures [13,14,20] are destroyed by quantum fluctuations. Even though these structures become blurred and start to disappear in the quantum world, their sharp borders tend to persist when the correlation between quantum and classical currents is investigated.…”

Section: Discussionmentioning

confidence: 99%

“…Two main regions with distinct behaviors can be observed: first, the "cloudy" background, identified as A in Fig. 1(d) showing a mixture of zero, small negative and positive currents, where the dynamics is chaotic [14]. Second, the four ISSs, B 1 , B 2 , C −1 and F .…”

Section: A Ratchet Currentmentioning

confidence: 99%

“…Second, the four ISSs, B 1 , B 2 , C −1 and F . Besides F , which has zero current, the ISSs are responsible for the optimal C C and can be recognized by their sharp borders (for a detailed explanation see [13,14]). We note that at the crossing regions of B 1 and F , positive currents are observed as a consequence of multistable attractors.…”

Section: A Ratchet Currentmentioning

confidence: 99%

“…It is known [10] that in classical conservative systems the parameters must be chosen so that the underlying dynamics presents a mixture of regular and chaotic motion. In dissipative inertial systems the Classical Ratchet Current (CRC) is more efficient when parameters are chosen inside the Isoperiodic Stable Structures (ISSs), which appear in the parameter space of ratchet models [13,14]. Such ISSs are generic Lyapunov stable islands with dynamics globally structurally stable and come along in many dynamical systems [15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

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Quantum-classical transition and quantum activation of ratchet currents in the parameter space

et al. 2015

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The quantum ratchet current is studied in the parameter space of the dissipative kicked rotor model coupled to a zero temperature quantum environment. We show that vacuum fluctuations blur the generic isoperiodic stable structures found in the classical case. Such structures tend to survive when a measure of statistical dependence between the quantum and classical currents are displayed in the parameter space. In addition, we show that quantum fluctuations can be used to overcome transport barriers in the phase space. Related quantum ratchet current activation regions are spotted in the parameter space. Results are discussed based on quantum, semiclassical and classical calculations. While the semiclassical dynamics involves vacuum fluctuations, the classical map is driven by thermal noise.

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Non-feedback technique to directly control multistability in nonlinear oscillators by dual-frequency driving

et al. 2018

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A novel method to control multistability of nonlinear oscillators by applying dual-frequency driving is presented. The test model is the Keller-Miksis equation describing the oscillation of a bubble in a liquid. It is solved by an in-house initial-value problem solver capable to exploit the high computational resources of professional graphics cards. Dur-Electronic supplementary material The online version of this article (

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Stable structures in parameter space and optimal ratchet transport

Cited by 38 publications

References 62 publications

Proliferation of stability in phase and parameter spaces of nonlinear systems

Proliferation of stability in phase and parameter spaces of nonlinear systems

Quantum-classical transition and quantum activation of ratchet currents in the parameter space

Non-feedback technique to directly control multistability in nonlinear oscillators by dual-frequency driving

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