The role of extreme orbits in the global organization of periodic regions in parameter space for one dimensional maps

Costa, Diogo Ricardo da; Hansen, Matheus; Guarise, Gustavo; Medrano-T, Rene O.; Leonel, Edson D.

doi:10.1016/j.physleta.2016.02.049

Cited by 23 publications

(11 citation statements)

References 31 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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“…4 below), which are standard ingredients for bifurcation diagrams; furthermore, the maximum bubble radius and the maximum absolute value of the bubble wall velocity, which are important for applications; finally, the period, the Lyapunov exponent and the winding number of the attractors found, quantities that are essential for a detailed analysis of bifurcation structures. A strategy to represent the results of parametric studies involving high-dimensional parameter spaces consists in creating high-resolution bi-parametric plots, a rapidly spreading technique in the investigation of nonlinear systems with a high-dimensional parameter space [39][40][41][42][43][44][45][46][47][48][49]. The system studied here, a bubble in water with dual-frequency acoustic excitation, has a four-dimensional driving parameter space (P A1 , P A2 , ω R1 , ω R2 ).…”

Section: Numerical Implementation and Parameter Choicementioning

confidence: 99%

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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“…That is, they require relatively low computational resources compared to an up to date personal computer. However, in order to explore the complex bifurcation structure in parameter space with high resolution [28][29][30][31][32], the necessary computational power can increase by orders of magnitude. For instance, even in a two dimensional parameter plane-employing an initial value problem solver (IVP) with a resolution of 1000 Â 1000-the computational requirements are increased by three orders of magnitude compared to conventional 1D bifurcation plots with the same resolution of 1000.…”

Section: Introductionmentioning

confidence: 99%

High-performance GPU computations in nonlinear dynamics: an efficient tool for new discoveries

et al. 2020

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The main aim of this paper is to demonstrate the benefit of the application of high-performance computing techniques in the field of non-linear science through two kinds of dynamical systems as test models. It is shown that high-resolution, multi-dimensional parameter scans (in the order of millions of parameter combinations) via an initial value problem solver are an efficient tool to discover new features of dynamical systems that are hard to find by other means. The employed initial value problem solver is an in-house code written in C++ and CUDA C software environments, which can exploit the high processing power of professional graphics cards (GPUs). The first test model is the Keller–Miksis equation, a non-linear oscillator describing the dynamics of a driven single spherical gas bubble placed in an infinite domain of liquid. This equation is important in the field of cavitation and sonochemistry. Here, the high-resolution parameter scans gave us the opportunity to lay down the basis of a non-feedback technique to control multi-stability in which direct selection of the desired attractor is possible. The second test model is related to a pressure relief valve that can exhibit a special kind of impact dynamics called grazing impact. A fine scan of the initial conditions revealed a second focal point of the grazing lines in the initial-condition space that was hidden in previous studies.

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The role of extreme orbits in the global organization of periodic regions in parameter space for one dimensional maps

Cited by 23 publications

References 31 publications

Proliferation of stability in phase and parameter spaces of nonlinear systems

Proliferation of stability in phase and parameter spaces of nonlinear systems

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

High-performance GPU computations in nonlinear dynamics: an efficient tool for new discoveries

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