The influence of noise on the logistic model

Mayer‐Kress, Gottfried; Haken, Hermann

doi:10.1007/bf01106791

Cited by 146 publications

(55 citation statements)

References 36 publications

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“…Известен широкий круг явлений, связанных с воздействием случайных помех: стохастический ре-зонанс [5,6], индуцированные шумами переходы [2], порождаемый шумами порядок [7,8] и хаос [9]. Хорошо известно, что фазовый портрет нелинейной системы может быть суще-ственно изменен под воздействием шума [10,11,12]. Эффекты, связанные с присутствием шума, значительно усиливаются вблизи точек бифуркации.…”

Section: § 1 введениеunclassified

“…При увеличении интенсивности шума происходит слияние пиков функции плотности распределения [10]. В результате такого слияния форма стохастических аттрак-торов упрощается: 2-цикл трансформируется в 1-цикл, 4-цикл -в 2-цикл, и т. д. Такое явление уменьшения кратности стохастических циклов (обратная стохастическая бифур-кация) для модели Ресслера изучалось в [28].…”

Section: § 1 введениеunclassified

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Backward stochastic bifurcations of the discrete system cycles

Bashkirtseva¹,

Ryashko²,

Fedotov³

et al. 2010

Nelin. Dinam.

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В работе рассматриваются стохастически возмущенные предельные циклы дискретных динамических систем в зоне удвоения периода. Исследуется явление обратных стохастиче-ских бифуркаций (ОСБ) -уменьшения кратности цикла при увеличении интенсивности шума. Предлагается метод анализа ОСБ на основе техники функции стохастической чув-ствительности. Конструктивные возможности данного метода демонстрируются на примере анализа ОСБ стохастических циклов систем Ферхюльста и Риккера.Ключевые слова: бифуркации, дискретные системы, модель Ферхюльста, стохастиче-ская чувствительность I. A. Bashkirtseva, L. B. Ryashko, S. P. Fedotov, I. N. Tsvetkov Backward stochastic bifurcations of the discrete system cycles We study stochastically forced limit cycles of discrete dynamical systems in a perioddoubling bifurcation zone. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation (BSB). In this paper, for the BSB analysis we suggest a stochastic sensitivity function technique. As a result, a method for the estimation of critical values of noise intensity corresponding to BSB is proposed. The constructive possibilities of this general method for the detailed BSB analysis of the multiple stochastic cycles of the forced Verhulst and Ricker systems are demonstrated.

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Section: § 1 введениеunclassified

Backward stochastic bifurcations of the discrete system cycles

Bashkirtseva¹,

Ryashko²,

Fedotov³

et al. 2010

Nelin. Dinam.

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“…The effect of noise on period-doubling transition to chaos was studied in [Crutchfield et al, 1981;Crutchfield et al, 1982] where for scaling analysis of Lyapunov exponents, a renormalization group approach was used. The influence of noise on the periodic attractors of the logistic model was investigated in [Mayer-Kress & Haken, 1981] on the base of the analysis of stationary distributions of corresponding stochastic attractors. Main approaches suggested in these pioneer works were widely developed by many researchers [Landa & McClintock, 2000;Anishchenko et al, 2007].…”

Section: Introductionmentioning

confidence: 99%

Noise-Induced Chaos and Backward Stochastic Bifurcations in the Lorenz Model

Bashkirtseva

Ryashko

Stikhin

2013

Int. J. Bifurcation Chaos

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We study the phenomena of stochastic D-and P-bifurcations of randomly forced limit cycles for the Lorenz model. As noise intensity increases, regular multiple limit cycles of this model in a period-doubling bifurcations zone are deformed to be stochastic attractors that look chaotic (D-bifurcation) and their multiplicity is reduced (P-bifurcation). In this paper for the comparative investigation of these bifurcations, the analysis of Lyapunov exponents and stochastic sensitivity function technique are used. A probabilistic mechanism of backward stochastic bifurcations for cycles of high multiplicity is analyzed in detail. We show that for a limit cycle with multiplicity two and higher, a threshold value of the noise intensity which marks the onset of chaos agrees with the first backward stochastic bifurcation.

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“…Indeed, for nonlinear system with single attractor in a sufficiently complicated spatial form, the stochastic hopping between its different but close portions can be induced. [29][30][31] As a consequence of this hopping, the dynamics of the perturbed system geometrically become chaos-like. The change of the phase space arrangement of the correspondent stochastic attractor is associated with the change of its dynamical properties.…”

Section: Introductionmentioning

confidence: 99%

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

Bashkirtseva¹,

Chen

Ryashko³

2012

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noiseinduced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noiseinduced hopping which generates chaos are demonstrated on the stochastic Chen system. Noise-induced chaos in stochastically perturbed nonlinear deterministic systems with regular attractors has been studied extensively. This phenomenon has received a great deal of interest among researchers from various fields of physics, life sciences, and engineering. Nonlinearity, even in a non-chaotic regime, can imply a non-uniformity of phase portraits, diverse forms of coexicting regular attractors with a complicated geometry of basins of attraction. Under random disturbances, a phase trajectory can cross separatices and induce stochastic hopping between both coexisting regular attractors and their different but close portions. As a consequence of this hopping, the random trajectories with a high probability fall within zones of divergency (local instability), and the dynamics of the perturbed system as a whole becomes chaotic. The positivity of the largest Lyapunov exponent can be used as a measure that this new noise-induced regime is chaotic.In this paper, for the constructive prediction of the noise-induced transitions from regular to chaotic oscillations in a dynamical system, a new stochastic sensitivity function technique is developed and applied. This technique enables to find dispersion ellipses of the random states for the stochastically perturbed dynamical system around deterministic attractors. The sizes of these ellipses are enlarged as noise intensity increases. The constructive method of the dispersion ellipses allows to estimate the threshold of the noise intensity corresponding to the deformation of a regular stochastic attractor to a chaotic one. The effectiveness of this approach is demonstrated by the stochastically perturbed Chen system.

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The influence of noise on the logistic model

Cited by 146 publications

References 36 publications

Backward stochastic bifurcations of the discrete system cycles

Backward stochastic bifurcations of the discrete system cycles

Noise-Induced Chaos and Backward Stochastic Bifurcations in the Lorenz Model

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

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