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Анищенко, В. С.; Astakhov, V. V.; Вадивасова, Т. Е.; Neiman, Alexander; Schimansky‐Geier, Lutz

doi:10.1007/978-3-540-38168-6_1

Cited by 50 publications

(88 citation statements)

References 81 publications

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“…The bifurcation line on the left denotes the passage from a single limit cycle to three limit cycles, while the right line denotes the reverse passage from three limit cycles to a single solution. At the conjunction, a codimension-two bifurcation, or cusp [43], appears . The first bifurcation encountered increasing α corresponds to the saddle-node bifurcation of the outer, or larger amplitude cycle, while the second bifurcation occurs in correspondence of a saddle-node bifurcation of the inner, or smaller amplitude, cycle.…”

Section: Dynamical Attractors and Birhythmicity Propertiesmentioning

confidence: 99%

Global stability analysis of birhythmicity in a self-sustained oscillator

Yamapi

Филатрелла

Aziz-Alaoui

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We analyze global stability properties of birhythmicity in a self-sustained system with random excitations. The model is a multi-limit cycles variation of the van der Pol oscillator introduced to analyze enzymatic substrate reactions in brain waves. We show that the two frequencies are strongly influenced by the nonlinear coefficients α and β. With a random excitation, such as a Gaussian white noise, the attractor's global stability is measured by the mean escape time τ from one limit-cycle. An effective activation energy barrier is obtained by the slope of the linear part of the variation of the escape time τ versus the inverse noise-intensity 1/D. We find that the trapping barriers of the two frequencies can be very different, thus leaving the system on the same attractor for an overwhelming time. However, we also find that the system is nearly symmetric in a narrow range of the parameters. Some models employed to describe natural systems, such as for instance glycolysis reactions and circadian proteins rhythmics, exhibit spontaneous oscillations at two distinct frequencies. The phenomenon is known as birhythmicity, and the underlying dynamical structure is characterized by the coexistence of two stable attractors, each displaying a different frequency. Being the attractors locally stable, the system would however stay at a single frequency, the one selected by the choice of the initial conditions, unless an external source disturbs the evolution and causes a switch to the other attractor. To investigate such process, we have focused on a particular system of biological interest, a modified van der Pol oscillator (that displays birhythmicity), to determine the global stability properties of the attractors under the influence of noise. More specifically, we have characterized the stability of the attractors with the escape times, or the average time that the system requires to switch from an attractor to the other under the influence of random fluctuations. Such analysis reveals that the two attractors can possess very different properties, with very different relative residence times. Even excluding the most asymmetric cases, the system can spend something like 10 years on one attractor for each second spent on the other. We conclude that although a system can be structurally biorhythmic for the contemporary presence of two locally stable attractors at two different frequencies, actual switch from one frequency to the other could be very difficult to observe. A global stability analysis can therefore help to determine the region of the parameter space in which birhythmic behavior will be genuinely observed.

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Section: Dynamical Attractors and Birhythmicity Propertiesmentioning

confidence: 99%

Global stability analysis of birhythmicity in a self-sustained oscillator

Yamapi

Филатрелла

Aziz-Alaoui

2010

Chaos: An Interdisciplinary Journal of Nonlinear Science

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“…Nowadays, the collection of models with strange attractors is very rich, including mathematical examples, as well as models of physical, chemical, and biological systems [1][2][3][4][5][6][7].…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

2011

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“…(7) depends on the assumption that the expression systems are always exposed to identical external variables in Eq. (5). Is this always true or can there be exceptions?…”

Section: Intrinsic and Extrinsic Noise: An Overviewmentioning

confidence: 99%

“…Biology has been an immense source of problems demanding quantitative understanding. Efforts to take up this challenge have fueled the emergence of novel areas in mathematics and physics, including theories of random walks, 1,2 synchronization 3 deterministic chaos, 4,5 biological networks, 6,7 and stochastic resonance. 8 Without exception, each of these fields has been confronted with the issue of noise, defined as random and unpredictable fluctuations in the variables or parameters of the system being investigated.…”

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confidence: 99%

A common repressor pool results in indeterminacy of extrinsic noise

Stamatakis

Adams

Balázsi

2011

Chaos: An Interdisciplinary Journal of Nonlinear Science

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For just over a decade, stochastic gene expression has been the focus of many experimental and theoretical studies. It is now widely accepted that noise in gene expression can be decomposed into extrinsic and intrinsic components, which have orthogonal contributions to the total noise. Intrinsic noise stems from the random occurrence of biochemical reactions and is inherent to gene expression. Extrinsic noise originates from fluctuations in the concentrations of regulatory components or random transitions in the cell’s state and is imposed to the gene of interest by the intra- and extra-cellular environment. The basic assumption has been that extrinsic noise acts as a pure input on the gene of interest, which exerts no feedback on the extrinsic noise source. Thus, multiple copies of a gene would be uniformly influenced by an extrinsic noise source. Here, we report that this assumption falls short when multiple genes share a common pool of a regulatory molecule. Due to the competitive utilization of the molecules existing in this pool, genes are no longer uniformly influenced by the extrinsic noise source. Rather, they exert negative regulation on each other and thus extrinsic noise cannot be determined by the currently established method.

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Tutorial

Cited by 50 publications

References 81 publications

Global stability analysis of birhythmicity in a self-sustained oscillator

Global stability analysis of birhythmicity in a self-sustained oscillator

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

A common repressor pool results in indeterminacy of extrinsic noise

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