Odd periodic window and bifurcations on 2D parameter space of low frequency oscillations

Silva, S.L. da; Rubinger, R. M.; Oliveira, A. G. de; Viana, E. R.

doi:10.1016/j.physd.2009.07.002

Cited by 3 publications

(3 citation statements)

References 16 publications

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“…In the bifurcation diagram, we show the well-known period-doubling bifurcation route-to-chaos: (period-1) -(period-2) -(period-4) -... -(chaos). Besides that, is also possible to observe stable odd-periodic windows, as (period-3) and (period-5), embedded in the chaotic region, which was also observed experimentally for the SI-GaAs system [02,29] and for other systems [07,18].…”

Section: Attractors and Bifurcation Diagramssupporting

confidence: 69%

“…This structure established a direct communication between the windows in order within chaotic regions, producing new routes of bifurcation. These intricate self-organized patterns were experimentally observed in the literature [da Silva et al, 2004;Rubinger et al, 2003;Albuquerque et al, 2003;da Silva et al, 2006da Silva et al, , 2009, but still needed a full theoretical description of how they emerge, what their size is in the parameter-space, if they have a fractal structure, based on the new model presented.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

High Resolution Parameter-Space from a Two-Level Model on Semi-Insulating GaAs

Silva

Viana

Oliveira

et al. 2015

Int. J. Bifurcation Chaos

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Semi-insulating Gallium Arsenide (SI-GaAs) samples experimentally show, under high electric fields and even at room temperature, negative differential conductivity in N-shaped form (NNDC). Since the most consolidated model for n-GaAs, namely, "the model", proposed by E. Schöll was not capable to generate the NNDC curve for SI-GaAs, in this work we have proposed an alternative model. The model proposed, "the two-valley model" is based on the minimal set of generation-recombination equations for two valleys inside of the conduction band, and an equation for the drift velocity as a function of the applied electric field, that covers the physical properties of the nonlinear electrical conduction of the SI-GaAs system. The "two-valley model" was capable to generate theoretically the NNDC region for the first time, and with that, we were able to build a high resolution parameter-space of the periodicity (PSP) using a Periodicity-Detection (PD) routine. In the parameter-space were observed self-organized periodic structures immersed in chaotic regions. The complex regions are presented in a "shrimp" shape rotated around a focal point, which forms in large-scale a "snail shell" shape, with intricate connections between different "shrimps". The knowledge of detailed information on parameter spaces is crucial to localize wide regions of smooth and continuous chaos.

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Section: Attractors and Bifurcation Diagramssupporting

confidence: 69%

Section: Discussionmentioning

confidence: 99%

High Resolution Parameter-Space from a Two-Level Model on Semi-Insulating GaAs

Silva

Viana

Oliveira

et al. 2015

Int. J. Bifurcation Chaos

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“…Period-adding [Kaneko, 1982] is a phenomenon observed in the parameter space, where, in a cascade of periodic orbits, the revolution difference between two consecutive periodic attractors is given by a constant ρ. The cascade can appear in continuous or discrete sets of periodicity and have been recently the subject of several works in different areas [Bonatto & Gallas, 2007;da Silva et al, 2009;de Souza et al, 2012;Medeiros et al, 2013;So et al, 2014;Rech, 2013;Stegemann & Rech, 2014].…”

Section: Period-adding In Sets Of Periodicitymentioning

confidence: 99%

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

Medrano-T

Rocha

2014

Int. J. Bifurcation Chaos

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Although Chua's circuit is one of the most studied nonlinear dynamical systems, its version with negative parameters remains practically untouched. This work reports an interesting and rich dynamic scenery that was hidden in this almost unexplored region. The study is focused on 2D parameter space and presents an analysis of stability based on describing functions. Numerical investigations present a gallery of period-adding cascades and a strong presence of basin boundary metamorphoses. The key to this new scenario is that for negative parameters, Chua's system does not satisfy the Shilnikov condition and it is shown that the homoclinic orbit organizes the parameter space completely different from as known. The obtained experimental results corroborate with the numerical and theoretical investigations.

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Exploring the dynamics of a four-dimensional Chua-like system: Numerical simulations and hardware investigations

Antunes,

Manchein,

Albuquerque

et al. 2024

Int. J. Mod. Phys. C

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In this work, we employ simulations to investigate deterministic and stochastic dynamics of a system governed by four coupled nonlinear ordinary differential equations (ODEs) derived from the canonical three-dimensional Chua model, replacing the piecewise nonlinearity with a cubic function. We employed two simulation approaches: (i) using software — where we performed numerical simulations (digital computation) in the ODEs model, and (ii) using hardware — where we performed experimental studies (analog computation) in the ODEs model. In the initial software-based approach for simulations, we performed stability and bifurcation analyses of the trivial equilibrium point, which were analytically conducted, along with the numerical continuation method. We solved the model using digital computation and explored its intricate dynamical behavior through nonlinear dynamics techniques applied to the parameter planes. In the second approach, employing hardware for simulations through analog computation, we designed an electronic circuit using analog electronics to solve the model in a continuous-time integration. The experimental setup mirrored the numerical simulations, yielding comparable results in terms of dynamical behaviors. This study underscores significant agreement between numerical and experimental findings, highlighting the robustness of the model across digital and analog contexts and offering practical insights into the various nonlinear dynamics presented in this system.

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Odd periodic window and bifurcations on 2D parameter space of low frequency oscillations

Cited by 3 publications

References 16 publications

High Resolution Parameter-Space from a Two-Level Model on Semi-Insulating GaAs

High Resolution Parameter-Space from a Two-Level Model on Semi-Insulating GaAs

The Negative Side of Chua's Circuit Parameter Space: Stability Analysis, Period-Adding, Basin of Attraction Metamorphoses, and Experimental Investigation

Exploring the dynamics of a four-dimensional Chua-like system: Numerical simulations and hardware investigations

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