The effect of temperature on generic stable periodic structures in the parameter space of dissipative relativistic standard map

Horstmann, Ana C.C.; Albuquerque, Holokx A.; Manchein, Cesar

doi:10.1140/epjb/e2017-70529-6

Cited by 24 publications

(11 citation statements)

References 39 publications

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“…Consequently occurs a considerable enlargement of the stable domains in phase and parameter spaces. This is crucial for the survival of the desired dynamics under noise and temperature effects, which usually destroy the ISSs starting from their borders [20] (also observed in the parameter space of the relativistic standard map [32]). Future contributions intend to verify the enlargement of stable domains for practical applications submitted to thermal effects.…”

Section: Discussionmentioning

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: Discussionmentioning

confidence: 99%

Proliferation of stability in phase and parameter spaces of nonlinear systems

Manchein

Silva

Beims

2017

Chaos

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“…As shown and discussed elsewhere [Viana et al, 2010[Viana et al, , 2012Boccaletti et al, 2000], noise disturbs the basin of attractions of periodic attractors of nonlinear systems , usually inducing periodic behaviors to chaotic ones if the noise strength is high-enough, and specifically in the parameterspaces, deforming and destroying the periodic structures [Horstmann et al, 2017]. That effect is shown in Fig.…”

Section: Resultsmentioning

confidence: 73%

Extensive Numerical Study and Circuitry Implementation of the Watt Governor Model

Marcôndes

Comassetto

Pedro

et al. 2017

Int. J. Bifurcation Chaos

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In this work we carry out extensive numerical study of a Watt-centrifugal-governor system model, and we also implement an electronic circuit by analog computation to experimentally solve the model. Our numerical results show the existence of self-organized stable periodic structures (SPSs) on parameter-space of the largest Lyapunov exponent and isospikes of time series of the Watt governor system model. A peculiar hierarchical organization and period-adding bifurcation cascade of the SPSs are observed, and this self-organized cascade accumulates on a periodic boundary. It is also shown that the periods of these structures organize themselves obeying the solutions of Diophantine equations. In addition, an experimental setup is implemented by a circuitry analogy of mechanical systems using analog computing technique to characterize the robustness of our numerical results. After applying an active control of chaos in the experiment, the effect of intrinsic experimental noise was minimized such that, the experimental results are in astonishing well agreement with our numerical findings. We can also mention as another remarkable result, the application of analog computing technique to perform an experimental circuitry analysis in real mechanical problems.

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“…To show that noise distorts the Lyapunov diagram of the real experiment (Fig. 3), corroding the periodic structures from their borders, as recently reported in references [37,40], we added a Gaussian noise in the model ( 1)-( 3), and simulate the Lyapunov diagrams for different noise intensities A (see Fig. 8).…”

Section: Discussionmentioning

confidence: 97%

Exploring an experimental analog Chua’s circuit

et al. 2019

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In this work we carry out experimental studies of the paradigmatic Chua's circuit using an approach of the analog computation instead of performing experiments in the canonical circuit. This means that we have built an electronic circuit that integrates (analog computation), in continuous time, the equations of motion of the canonical Chua's circuit. The equations of motion of the analogical circuit are equivalent to the canonical circuit, so that the dynamical behaviour is the same. With this approach, we successfully obtain an experimental parameter plane using the largest Lyapunov exponent (here named Lyapunov diagram), directly calculated from the experimental time series, with a good precision, so that different types of dynamical behaviours were characterized in this diagram. Results are in very good agreement with numerical simulation with an additional Gaussian noise. The approach by analog computation used here can be extended to a wide range of dynamical systems, once that the analog circuit simulates, by circuitry implementation, the dynamics of these systems from an experimental point of view.

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The effect of temperature on generic stable periodic structures in the parameter space of dissipative relativistic standard map

Cited by 24 publications

References 39 publications

Proliferation of stability in phase and parameter spaces of nonlinear systems

Proliferation of stability in phase and parameter spaces of nonlinear systems

Extensive Numerical Study and Circuitry Implementation of the Watt Governor Model

Exploring an experimental analog Chua’s circuit

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