Most probable dynamics of some nonlinear systems under noisy fluctuations

Cheng, Zhuan; Duan, Jinqiao; Wang, Liang

doi:10.1016/j.cnsns.2015.06.016

Cited by 33 publications

(30 citation statements)

References 11 publications

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“…Now, the question arises, which trajectory does the system most probably to evolve in phase plane? Our geometric tool, the most probable trajectories [3,55], will tackle this problem. As is known, the solution of a stochastic system (1.2) is a stochastic process, and the probability density function of the stochastic process is governed by the corresponding Fokker-Planck equation [42,57].…”

Section: Methodsmentioning

confidence: 99%

Most probable dynamics of a genetic regulatory network under stable Lévy noise

Chen

Duan

et al. 2019

Applied Mathematics and Computation

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Numerous studies have demonstrated the important role of noise in the dynamical behaviour of a complex system. The most probable trajectories of nonlinear systems under the influence of Gaussian noise have recently been studied already. However, there has been only a few works that examine how most probable trajectories in the two-dimensional system (MeKS network) are influenced under non-Gaussian stable Lévy noise. Therefore, we discuss the most probable trajectories of a two-dimensional model depicting the competence behaviour in B. subtilis under the influence of stable Lévy noise. On the basis of the Fokker-Planck equation, we describe the noise-induced most probable trajectories of the MeKS network from the low ComK protein concentration (vegetative state) to the high ComK protein concentration (competence state) under stable Lévy noise.We demonstrate choices of the non-Gaussianity index α and the noise intensity ǫ which generate the ComK protein escape from the low concentration to the high concentration. We also reveal the optimal combination of both parameters α and ǫ making the tipping time shortest. Moreover, we find that different initial concentrations around the low ComK protein concentration evolve to a metastable state, and provide the optimal α and ǫ such that the distance between the deterministic competence state and the metastable state is smallest.

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Section: Methodsmentioning

confidence: 99%

Most probable dynamics of a genetic regulatory network under stable Lévy noise

Chen

Duan

et al. 2019

Applied Mathematics and Computation

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“…Unlike the phase portraits for deterministic dynamical systems [56,57], these sample paths ('orbit' or 'trajectories') mingled together and can not provide much information. Most probable phase portraits [48,49,52] for the stochastic gene regulation system (3) is a powerful tool to understand the stochastic system.…”

Section: Resultsmentioning

confidence: 99%

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

Cheng

Wang

et al. 2019

Physica A: Statistical Mechanics and its Applications

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We study the most probable transition pathways and maximal likely trajectories in a genetic regulation model of the transcription factor activator's concentration evolution, with Gaussian noise and non-Gaussian stable Lévy noise in the synthesis reaction rate taking into account, respectively. We compute the most probable transition pathways by the Onsager-Machlup least action principle, and calculate the maximal likely trajectories by spatially maximizing the probability density of the system path, i.e., the solution of the associated nonlocal Fokker-Planck equation. We have observed the rare most probable transition pathways in the case of Gaussian noise, for certain noise intensity, evolution time scale and system parameters. We have especially studied the maximal likely trajectories starting at the low concentration metastable state, and examined whether they evolve to or near the high concentration metastable state (i.e., the likely transcription regime) for certain parameters, in order to gain insights into the transcription processes and the tipping time for the transcription likely to occur. This enables us: (i) to visualize the progress of concentration evolution (i.e., observe whether the system enters the transcription regime within a given time period); (ii) to predict or avoid certain transcriptions via selecting specific noise parameters in particular regions in the parameter space. Moreover, we have found some peculiar or counter-intuitive phenomena in this gene model system, including: (a) A smaller noise intensity may trigger the transcription process, while a larger noise intensity can not, under the same asymmetric Lévy noise. This phenomenon does not occur in the case of symmetric Lévy noise; (b) The symmetric Lévy motion always induces transition to high concentration, but certain asymmetric Lévy motions do not trigger the switch to transcription.These findings provide insights for further experimental research, in order to achieve or to avoid specific gene transcriptions, with possible relevance for medical advances.

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“…One promising option for phase portraits of stochastic dynamical systems is the so-called most probable phase portraits [Duan , 2015;Cheng et al , 2016]. We thus conduct stochastic bifurcation study with the help of these phase portraits.…”

Section: Resultsmentioning

confidence: 99%

A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

Wang

Chen

Duan

2018

Int. J. Bifurcation Chaos

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We study stochastic bifurcation for a system under multiplicative stable Lévy noise (an important class of non-Gaussian noise), by examining the qualitative changes of equilibrium states in its most probable phase portraits. We have found some peculiar bifurcation phenomena in contrast to the deterministic counterpart: (i) When the non-Gaussianity parameter in Lévy noise varies, there is either one, two or none backward pitchfork type bifurcations; (ii) When a parameter in the vector field varies, there are two or three forward pitchfork bifurcations; (iii) The non-Gaussian Lévy noise clearly leads to fundamentally more complex bifurcation scenarios, since in the special case of Gaussian noise, there is only one pitchfork bifurcation which is reminiscent of the deterministic situation.

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Most probable dynamics of some nonlinear systems under noisy fluctuations

Cited by 33 publications

References 11 publications

Most probable dynamics of a genetic regulatory network under stable Lévy noise

Most probable dynamics of a genetic regulatory network under stable Lévy noise

Most probable transition pathways and maximal likely trajectories in a genetic regulatory system

A Stochastic Pitchfork Bifurcation in Most Probable Phase Portraits

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