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

2020 ◽  
Vol 122 (1) ◽  
pp. 49-59
Author(s):  
Shichao Ma ◽  
Xin Ning ◽  
Liang Wang
Keyword(s):  
Dynamic Analysis ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping

Studying the Global Bifurcation Involving Wada Boundary Metamorphosis by a Method of Generalized Cell Mapping with Sampling-Adaptive Interpolation

2018 ◽  
Vol 28 (02) ◽  
pp. 1830003 ◽  
Author(s):  
Xiao-Ming Liu ◽  
Jun Jiang ◽  
Ling Hong ◽  
Dafeng Tang
Keyword(s):  
Global Bifurcation ◽  
Mapping Method ◽  
Computational Time ◽  
Cell Mapping ◽  
Third Order ◽  
Pendulum System ◽  
Generalized Cell Mapping ◽  
Adaptive Interpolation ◽  
One Step ◽  
Probability Transition Matrix

In this paper, a new method of Generalized Cell Mapping with Sampling-Adaptive Interpolation (GCMSAI) is presented in order to enhance the efficiency of the computation of one-step probability transition matrix of the Generalized Cell Mapping method (GCM). Integrations with one mapping step are replaced by sampling-adaptive interpolations of third order. An explicit formula of interpolation error is derived for a sampling-adaptive control to switch on integrations for the accuracy of computations with GCMSAI. By applying the proposed method to a two-dimensional forced damped pendulum system, global bifurcations are investigated with observations of boundary metamorphoses including full to partial and partial to partial as well as the birth of fully Wada boundary. Moreover GCMSAI requires a computational time of one thirtieth up to one fiftieth compared to that of the previous GCM.

A Cell Mapping Method for Nonlinear Deterministic and Stochastic Systems—Part II: Examples of Application

1986 ◽  
Vol 53 (3) ◽  
pp. 702-710 ◽  
Author(s):  
H. M. Chiu ◽  
C. S. Hsu
Keyword(s):  
Present Method ◽  
Stochastic Systems ◽  
Random Excitation ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Deterministic Systems ◽  
Strongly Nonlinear ◽  
Stochastic Problems ◽  
A Cell

In this second part of the two-part paper we demonstrate the viability of the compatible simple and generalized cell mapping method by applying it to various deterministic and stochastic problems. First we consider deterministic problems with non-chaotic responses. For this class of problems we show how system responses and domains of attraction can be obtained by a refining procedure of the present method. Then, we consider stochastic problems with stochasticity lying in system parameters or excitation. Next, deterministic systems with chaotic responses are considered. By the present method, finding the statistical responses of such systems under random excitation also presents no difficulties. Some of the systems studied here are well-known. New results are, however, also obtained. These are results on Duffing systems with a stochastic coefficient, the global results of a Duffing system shown in Section 4, the results on strongly nonlinear Duffing systems under random excitations reported in Section 7.2, and the strange attractor results for systems subjected to random excitations.

GLOBAL ANALYSIS OF STOCHASTIC BIFURCATION IN DUFFING SYSTEM

2003 ◽  
Vol 13 (10) ◽  
pp. 3115-3123 ◽  
Author(s):  
WEI XU ◽  
QUN HE ◽  
TONG FANG ◽  
HAIWU RONG
Keyword(s):  
Global Analysis ◽  
Sudden Change ◽  
Harmonic Excitation ◽  
Mapping Method ◽  
Invariant Sets ◽  
Stochastic Bifurcation ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Duffing System ◽  
Stable Invariant

Stochastic bifurcation of a Duffing system subject to a combination of a deterministic harmonic excitation and a white noise excitation is studied in detail by the generalized cell mapping method using digraph. It is found that under certain conditions there exist two stable invariant sets in the phase space, associated with the randomly perturbed steady-state motions, which may be called stochastic attractors. Each attractor owns its attractive basin, and the attractive basins are separated by boundaries. Along with attractors there also exists an unstable invariant set, which might be called a stochastic saddle as well, and stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. This definition applies equally well either to randomly perturbed motions, or to purely deterministic motions. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence and evolution of stochastic bifurcation can be explored clearly and vividly.

Global Invariant Manifolds of Dynamical Systems with the Compatible Cell Mapping Method

2019 ◽  
Vol 29 (08) ◽  
pp. 1950105 ◽  
Author(s):  
Xiao-Le Yue ◽  
Yong Xu ◽  
Wei Xu ◽  
Jian-Qiao Sun
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Invariant Manifolds ◽  
High Efficiency ◽  
Search Algorithm ◽  
Mapping Method ◽  
Cell Mapping ◽  
Stable And Unstable Manifolds ◽  
Generalized Cell Mapping ◽  
Global Invariant

An iterative compatible cell mapping (CCM) method with the digraph theory is presented in this paper to compute the global invariant manifolds of dynamical systems with high precision and high efficiency. The accurate attractors and saddles can be simultaneously obtained. The simple cell mapping (SCM) method is first used to obtain the periodic solutions. The results obtained by the generalized cell mapping (GCM) method are treated as a database. The SCM and GCM are compatible in the sense that the SCM is a subset of the GCM. The depth-first search algorithm is utilized to find the coarse coverings of global stable and unstable manifolds based on this database. The digraph GCM method is used if the saddle-like periodic solutions cannot be obtained with the SCM method. By taking this coarse covering as a new cell state space, an efficient iterative procedure of the CCM method is proposed by combining sort, search and digraph algorithms. To demonstrate the effectiveness of the proposed method, the classical Hénon map with periodic or chaotic saddles is studied in far more depth than reported in the literature. Not only the global invariant manifolds, but also the attractors and saddles are computed. The computational efficiency can be improved by up to 200 times compared to the traditional GCM method.

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