Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

Yue, Xiaole; Xu, Yong; Xu, Wei; Sun, Jian‐Qiao

doi:10.1016/j.physa.2018.10.034

Cited by 16 publications

(2 citation statements)

References 50 publications

(64 reference statements)

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“…Han and coworkers explored this strategy extensively, considering nonautonomous cases [18] under colored noise [19], stochastic bifurcations in a turbulent swirling flow [20], and a combination with digraph algorithms [21]. The simple and generalized cell-mapping was recently reformulated by Yue et al [22], the socalled compatible cell-mapping method. This method employs adaptative refinement of the phase-space, increasing the resolution of global attractors of random dynamical systems.…”

Section: Introductionmentioning

confidence: 99%

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Benedetti

Gonçalves

Lenci

et al. 2022

Preprint

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An adaptative phase-space discretization strategy for the global analysis of stochastic nonlinear dynamical systems with competing attractors considering parameter uncertainty or noise is proposed. The strategy is based on the classical Ulam method. The appropriate transfer operators for a given dynamical system are derived and applied to obtain and refine the basins of attraction boundaries and attractors distributions and densities. A review of the main concepts of parameter uncertainty and stochasticity from a global dynamics perspective is given, and the necessary modifications to the Ulam method are addressed. The stochastic basin of attraction definition here used replaces the usual basin concept. It quantifies the probability of the response associated with a given set of initial conditions converging to a particular attractor. The phase-space dimension is augmented to include the extra dimensions associated with the parameter space for the case of parameter uncertainty, being a function of the number of uncertain parameters. The expanded space is discretized, resulting in a collection of transfer operators that enable obtaining the required statistics. Finally, a Monte Carlo procedure is conducted for the stochastic case to construct the proper transfer operator. The present formulation is employed in Part II to investigate the global dynamics of two archetypal nonlinear oscillators.

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Section: Introductionmentioning

confidence: 99%

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Benedetti

Gonçalves

Lenci

et al. 2022

Preprint

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“…The GCM method is confirmed as a valid tool to analyze the global properties and bifurcation of deterministic and stochastic dynamical systems [39]. In order to broaden the range of application of this method, many improvements have been carried out [40][41][42][43]. In 1990, a short-time Gaussian approximation (STGA) scheme was proposed to improve the efficiency of the GCM method [44].…”

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confidence: 99%

Transient responses of nonlinear dynamical systems under colored noise

Yue

Wang

Han

et al. 2019

EPL

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The influence of colored noise on the transient response of nonlinear dynamical systems is investigated by the generalized cell mapping (GCM) based on the short-time Gaussian approximation (STGA) scheme. The block matrix procedure is introduced into the GCM/STGA method to solve the storage problem caused by the dimensionality of the system. In addition, a parallel calculation strategy can be implemented due to the independence of the storage and the computation of the block matrixes. Taking the well-known Mathieu-Duffing oscillator as an example, the deterministic global properties are first computed with the digraph cell mapping method, in which the coexistence of multiple attractors is observed. Then, the evolutionary processes of transient probability density functions (PDFs) of the response under colored noise from different initial distributions are revealed. Due to the attractive characteristics of the attractor and disturbance generated by colored noise, it can be seen that it takes longer for transient responses from the attractor to achieve the steady-state than the case of the initial distribution which locates around the saddle. The evolutionary processes of transient responses are quite disparate from different initial distributions with the influence of the colored noise, although they both concentrate around the attractor after enough time, which is consistent with the global structure without noise. Furthermore, the effects of the intensity and the correlation time of the colored noise on the mulistability are discussed. The stochastic P-bifurcation occurs with the increase of these two parameters, respectively. The evolutionary directions of the colored-noise–induced bifurcations are opposite in the two cases. Monte Carlo (MC) simulations are in good agreement with the results obtained by the GCM/STGA method based on the block matrix procedure.

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Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method

Ning

Wang

2019

Computational and Experimental Simulations in Engineering

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Probabilistic response of dynamical systems based on the global attractor with the compatible cell mapping method

Cited by 16 publications

References 50 publications

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Global Analysis of Stochastic Nonlinear Dynamical Systems. Part 1: Adaptative Phase-Space Discretization Strategy

Transient responses of nonlinear dynamical systems under colored noise

Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method

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