Distribution of chaos and periodic spikes in a three-cell population model of cancer

Gallas, Michelle; Gallas, Márcia Russman; Gallas, Jason A. C.

doi:10.1140/epjst/e2014-02254-3

Cited by 43 publications

(29 citation statements)

References 43 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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“…Considering the relevance of the CO 2 laser with optoelectronic feedback in nonlinear dynamics, a detailed investigation taking advantage from the novel isospike technique [39][40][41][42][43][44][45][46] has been proposed. In isospike diagrams, new dynamical features related to the organization of stable spiking and bursting behavior emerge for accessible parameter values.…”

Section: Discussionmentioning

confidence: 99%

“…Figure 8(d) illustrates an infinite sequence of spirals of chaos and spirals of regularity that arise around certain exceptional points in control space, called periodicity hubs, well-known to organize the dynamics over extended parameter regions. 44,[53][54][55][56][57] The exceptional point responsible for the large anti-clockwise spiraling in Fig. 8(d) is located at the periodicity hub F, numerically estimated to be near F ¼ ða; bÞ ¼ ð297:85; 0:3431Þ.…”

Section: Stability Diagramsmentioning

confidence: 92%

“…1 contain triplets of dots labeled A, B, C; D, E, F; and G, H, I. Such points are the first ones of an apparently infinite sequence of analogous points lying inside certain complex structures (shrimps 13,44,47,[50][51][52] ). These sequences of points accumulate towards a large region on the right-hand-side containing the number 3 and representing periodic laser oscillations with three-spikes per period.…”

Section: Stability Diagramsmentioning

confidence: 99%

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Self-organization of pulsing and bursting in a CO2 laser with opto-electronic feedback

Freire

Meucci

Arecchi

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We report a detailed investigation of the stability of a CO2 laser with feedback as described by a six-dimensional rate-equations model which provides satisfactory agreement between numerical and experimental results. We focus on experimentally accessible parameters, like bias voltage, feedback gain, and the bandwidth of the feedback loop. The impact of decay rates and parameters controlling cavity losses are also investigated as well as control planes which imply changes of the laser physical medium. For several parameter combinations, we report stability diagrams detailing how laser spiking and bursting is organized over extended intervals. Laser pulsations are shown to emerge organized in several hitherto unseen regular and irregular phases and to exhibit a much richer and complex range of behaviors than described thus far. A significant observation is that qualitatively similar organization of laser spiking and bursting can be obtained by tuning rather distinct control parameters, suggesting the existence of unexpected symmetries in the laser control space.

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“…Figure 1 illustrates the typical situation studied in this paper. For a given window in the frequency versus the parameter A control space, it shows two distinct stability diagrams: a typical Lyapunov stability diagram is shown in Figure 1a, while an isospike diagram [17,[22][23][24] is shown in Figure 1b. This latter diagram represents in colors parameter regions characterized by periodic oscillations having the same number of spikes per period, as recorded in the colorbar under the diagram.…”

Section: The Driven Brusselatormentioning

confidence: 99%

Stability mosaics in a forced Brusselator

Freire

Gallas

2017

Eur. Phys. J. Spec. Top.

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Abstract. We report a startling mosaic-like organization of stability phases found in the low-frequency limit of a driven Brusselator. Such phases correspond to periodic oscillations having a constant number of spikes per period. The mosaic is free from chaotic oscillations and is formed by an apparently infinite cascade of oscillations whose number of spikes grow without bound. Wide windows free from chaos but supporting unbounded quantities of complex oscillations are potentially of interest to operate driven oscillators such as lasers, electronic circuits, and biochemical pacemakers.

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Distribution of chaos and periodic spikes in a three-cell population model of cancer

Cited by 43 publications

References 43 publications

Proliferation of stability in phase and parameter spaces of nonlinear systems

Proliferation of stability in phase and parameter spaces of nonlinear systems

Self-organization of pulsing and bursting in a CO2 laser with opto-electronic feedback

Stability mosaics in a forced Brusselator

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