The Application of Cell Mapping Method to High-Dimensional Nonlinear System

2005 ◽  
Author(s):  
Xiang Yu ◽  
Shi-Jian Zhu ◽  
Shu-Yong Liu
Keyword(s):  
Global Analysis ◽  
Mapping Method ◽  
High Dimensional ◽  
Dimensional System ◽  
Cell Mapping ◽  
Chaotic Motions ◽  
Point Mapping ◽  
Modified Method ◽  
Conventional Cell ◽  
Selection Of

After analyzing the inefficiency of the conventional Cell Mapping Methods in global analysis for high-dimensional nonlinear systems, several principles should be followed for these methods’ implementations in high-dimensional systems are proposed in this paper. Those are: appropriate selection of investigating plane, reduction of data size, and projection of attractors to the investigating plane. According to these, the idea of dynamic array is introduced to the method of Point Mapping Under Cell Reference (PMUCR) to improve computing efficiency. The comparison of the CPU time between the applications of this modified method to a 2-dimensional system and to a 4-dimensional one is carried out, and the results confirm this modified method can be utilized to analyze high-dimensional systems effectively. Finally, as examples, the periodic and chaotic motions of a coupled Duffing system are investigated through this method and some diagrams of global characteristics are presented.

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.

A Theory of Cell-to-Cell Mapping Dynamical Systems

1980 ◽  
Vol 47 (4) ◽  
pp. 931-939 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical Systems ◽  
Numerical Evaluation ◽  
Global Analysis ◽  
Cell Mapping ◽  
Point Mapping ◽  
State Variable ◽  
Physical Measurements ◽  
Point To Point ◽  
Discretization Process ◽  
Mapping Systems

The method of point-to-point mappings has been receiving increasing attention in recent years. In this paper we discuss instead dynamical systems governed by cell-to-cell mappings. The justifications of considering such mappings come from the unavoidable accuracy limitations of both physical measurements and numerical evaluation. Because of these limitations one is not really able to treat a state variable as a continuum of points but rather only as a collection of very small intervals. The introduction of the idea of cell-to-cell mappings has led to an algorithm which is found to be potentially a very powerful tool for global analysis of dynamical systems. In this paper an introductory theory of cell-to-cell mappings is offered. The theory provides a basis for the algorithm presented in [14]. In the first half of the paper we discuss the analysis of cell-to-cell mappings in their own right. In the second half the cell-to-cell mappings which are obtained from point-to-point mappings by discretization are examined in order to see what properties of the point mapping systems are preserved in the discretization process.

Global Analysis of Stochastic Systems by the Digraph Cell Mapping Method Based on Short-Time Gaussian Approximation

2020 ◽  
Vol 30 (05) ◽  
pp. 2050071
Author(s):  
Qun Han ◽  
Wei Xu ◽  
Huibing Hao ◽  
Xiaole Yue
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Stochastic Systems ◽  
Gaussian Approximation ◽  
Global Analysis ◽  
Mapping Method ◽  
Periodic Excitation ◽  
Cell Mapping ◽  
One Step ◽  
Short Time

The digraph cell mapping method is popular in the global analysis of stochastic systems. Traditionally, the Monte Carlo simulation is used in finding the image cells of one-step mapping, and it is notably costly in the computation time. In this paper, a novel short-time Gaussian approximation (STGA) scheme is incorporated into the digraph cell mapping method to study the global analysis of nonlinear dynamical systems under Gaussian white noise excitations. In order to find out all the active image cells in one-step cell mapping quickly, the STGA scheme together with a probability truncation method is introduced for systems without periodic excitation, and then in the case with periodic excitation. The global structures, such as the stochastic attractors, stochastic basins of attraction and stochastic saddles, are calculated by the digraph analysis algorithm. The proposed methodology has been applied to three typical stochastic dynamical systems. For each system, the effectiveness and superiority of the proposed STGA scheme are verified by checking the image cells of one-step mapping and comparing with the results of Monte Carlo simulation. It is found in the global analysis that the change of the amplitude of periodic excitation induces stochastic bifurcations in the stochastic Duffing system. Moreover, a stochastic bifurcation occurs in the stochastic Lorenz system with the increase of noise intensities.

Poincare Linear Interpolated Cell Mapping: Method for Global Analysis of Oscillating Systems

1995 ◽  
Vol 62 (2) ◽  
pp. 489-495 ◽  
Author(s):  
J. Levitas ◽  
T. Weller
Keyword(s):  
Global Analysis ◽  
Mapping Method ◽  
Cell Mapping ◽  
Significant Saving ◽  
Nonlinear Dynamical ◽  
Oscillating Systems ◽  
Simple Cell Mapping ◽  
Interpolation Procedure ◽  
Van Der Pol Oscillators ◽  
Direct Numerical Integration

A method for global analysis of nonlinear dynamical oscillating systems was developed. The method is based on the idea of introducing a Poincare section into a multidimensional state space of the dynamical system and combine it with an interpolation procedure within the cells which constitute the discretized problem domain of interest. The proposed method was applied to study the global behavior of two nonlinear coupled van der Pol oscillators. Significant saving in calculation time, in comparison with both direct numerical integration and Poincare-like simple cell mapping, is demonstrated.

On the Data-Driven Generalized Cell Mapping Method

2019 ◽  
Vol 29 (14) ◽  
pp. 1950204 ◽  
Author(s):  
Zigang Li ◽  
Jun Jiang ◽  
Ling Hong ◽  
Jian-Qiao Sun
Keyword(s):  
Global Analysis ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Mapping Method ◽  
Data Driven ◽  
Cell Mapping ◽  
Time Step ◽  
Generalized Cell Mapping ◽  
Step Transition ◽  
The One

Global analysis is often necessary for exploiting various applications or understanding the mechanisms of many dynamical phenomena in engineering practice where the underlying system model is too complex to analyze or even unavailable. Without a mathematical model, however, it is very difficult to apply cell mapping for global analysis. This paper for the first time proposes a data-driven generalized cell mapping to investigate the global properties of nonlinear systems from a sequence of measurement data, without prior knowledge of the underlying system. The proposed method includes the estimation of the state dimension of the system and time step for creating a mapping from the data. With the knowledge of the estimated state dimension and proper mapping time step, the one-step transition probability matrix can be computed from a statistical approach. The global properties of the underlying system can be uncovered with the one-step transition probability matrix. Three examples from applications are presented to illustrate a quality global analysis with the proposed data-driven generalized cell mapping method.

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