A Cell Mapping Method for Nonlinear Deterministic and Stochastic Systems—Part I: The Method of Analysis

1986 ◽  
Vol 53 (3) ◽  
pp. 695-701 ◽  
Author(s):  
C. S. Hsu ◽  
H. M. Chiu
Keyword(s):  
Stochastic Systems ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Strongly Nonlinear ◽  
Cell Mapping Methods ◽  
The Past ◽  
Simple Cell Mapping ◽  
A Cell ◽  
Vibration Problems

In the past few years as an attempt to devise more efficient and more practical ways of determining the global behavior of strongly nonlinear systems, two cell-to-cell mapping methods have been proposed, namely, the simple cell mapping and the generalized cell mapping. In this first part of the two-part paper we present a different and more efficient cell mapping method for treating nonlinear vibration problems. The vibratory systems may be deterministic or stochastic. The method utilizes compatible simple and generalized cell mapping and it combines the advantages of both. Applications to various systems will be presented in the second part of the paper.

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.

A Statistical Study of Generalized Cell Mapping

1988 ◽  
Vol 55 (3) ◽  
pp. 694-701 ◽  
Author(s):  
Jian-Qiao Sun ◽  
C. S. Hsu
Keyword(s):  
Stochastic Systems ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
One Step ◽  
The Mean ◽  
Step Transition ◽  
The One

In this paper a statistical error analysis of the generalized cell mapping method for both deterministic and stochastic dynamical systems is examined, based upon the statistical analogy of the generalized cell mapping method to the density estimation. The convergence of the mean square error of the one step transition probability matrix of generalized cell mapping for deterministic and stochastic systems is studied. For stochastic systems, a well-known trade-off feature of the density estimation exists in the mean square error of the one step transition probability matrix, which leads to an optimal design of generalized cell mapping for stochastic systems. The conclusions of the study are illustrated with some examples.

GLOBAL ANALYSIS BY CELL MAPPING

1992 ◽  
Vol 02 (04) ◽  
pp. 727-771 ◽  
Author(s):  
C.S. HSU
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Mapping Method ◽  
Simple Cell ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
New Developments ◽  
Nonlinear Dynamical ◽  
Simple Cell Mapping ◽  
Computation Algorithms

This paper deals with cell mapping methodology for global analysis of nonlinear dynamical systems. It serves a mixed set of purposes. It is basically a tutorial paper on cell mapping. But, it also reports on certain new developments in cell mapping and includes a summary of recent publications on the topic. Presented in Sec. 1–3 and 5 are the basic concepts and theory of cell mapping. Two types of cell-to-cell mapping are discussed, namely: simple cell mapping and generalized cell mapping. Once a dynamical system has been cast in the form of a cell mapping, one needs to extract the system behavior from the mapping. In Secs. 4 and 6, computation algorithms for simple cell mapping and generalized cell mapping are discussed in detail. Moreover, a workshop-type example is included to guide the reader if he wishes to gain a working knowledge of the methodology. The new developments presented in the paper include an algorithm for processing simple cell mappings and a theory of subdomain-to-subdomain global transient analysis of generalized cell mapping. These are reported in Secs. 4–6. Listed in the last section are publications on cell mapping, including applications in many rather diversified areas of dynamics. Hopefully, the breadth of the examples of application will indicate the potential of the cell mapping method.

scholarly journals An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Yu Wang ◽  
Heng Cao ◽  
JinLin Jiang
Keyword(s):  
Domain Of Attraction ◽  
Mapping Method ◽  
Simple Cell ◽  
Iteration Algorithm ◽  
Poincaré Mapping ◽  
Cell Mapping ◽  
Improved Method ◽  
Stable Domain ◽  
Simple Cell Mapping ◽  
Poincare Mapping

An indicator of a passive biped walker’s global stability is its domain of attraction, which is usually estimated by the simple cell mapping method. It needs to calculate a large number of cells’ Poincare mapping result in the estimating process. However, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of the passive biped walker. How to estimate the domain of attraction efficiently and reliably is a problem to be solved. Based on the simple cell mapping method, an improved method is proposed to solve it. The proposed method uses the multiple iteration algorithm to calculate a stable domain of attraction and effectively decreases the total number of Poincare mappings. Through the simulation of the simplest passive biped walker, the improved method can obtain the same domain of attraction as that calculated using the simple cell mapping method and reduce calculation time significantly. Furthermore, this improved method not only proposes a way of rapid estimating the domain of attraction, but also provides a feasible tool for selecting the domain of interest and its discretization level.

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.

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.

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