Parallel Cell Mapping Method for Global Analysis of High-Dimensional Nonlinear Dynamical Systems1

2015 ◽  
Vol 82 (11) ◽  
Author(s):  
Fu-Rui Xiong ◽  
Zhi-Chang Qin ◽  
Qian Ding ◽  
Carlos Hernández ◽  
Jesús Fernandez ◽  
...  
Keyword(s):  
Steady State ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Mapping Method ◽  
Cell Mapping ◽  
Nonlinear Dynamical ◽  
Cell Mapping Methods ◽  
Steady State Responses ◽  
State Responses

The cell mapping methods were originated by Hsu in 1980s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states, periodic motions, and chaotic attractors. The cell mapping methods have been applied to deterministic, stochastic, and fuzzy dynamical systems. Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space: the interpolated cell mapping (ICM) and the set-oriented method with subdivision technique. For a long time, the cell mapping methods have been applied to dynamical systems with low dimension until now. With the advent of cheap dynamic memory and massively parallel computing technologies, such as the graphical processing units (GPUs), global analysis of moderate- to high-dimensional nonlinear dynamical systems becomes feasible. This paper presents a parallel cell mapping method for global analysis of nonlinear dynamical systems. The simple cell mapping (SCM) and generalized cell mapping (GCM) are implemented in a hybrid manner. The solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses, followed by the subdivision technique to enhance the accuracy of the steady-state responses. When the cells are small enough, no further subdivision is necessary. We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database, which provides a basis for the ICM to generate the pointwise approximation of the solutions without additional numerical integrations of differential equations. A modified global analysis of nonlinear systems with transient states is developed by taking advantage of parallel computing without subdivision. To validate the parallelized cell mapping techniques and to demonstrate the effectiveness of the proposed method, a low-dimensional dynamical system governed by implicit mappings is first presented, followed by the global analysis of a three-dimensional plasma model and a six-dimensional Lorenz system. For the six-dimensional example, an error analysis of the ICM is conducted with the Hausdorff distance as a metric.

GLOBAL ANALYSIS OF DYNAMICAL SYSTEMS USING POSETS AND DIGRAPHS

1995 ◽  
Vol 05 (04) ◽  
pp. 1085-1118 ◽  
Author(s):  
C. S. HSU
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Ordered Sets ◽  
Cell Mapping ◽  
Basic Notion ◽  
Nonlinear Dynamical ◽  
Global Evolution ◽  
Computation Algorithms ◽  
Primitive Notion

In this paper the resources of the theory of partially ordered sets (posets) and the theory of digraphs are used to aid the task of global analysis of nonlinear dynamical systems. The basic idea underpinning this approach is the primitive notion that a dynamical systems is simply an ordering machine which assigns fore-and-after relations for pairs of states. In order to make the linkage between the theory of posets and digraphs and dynamical systems, cell mapping is used to put dynamical systems in their discretized form and an essential concept of self-cycling sets is used. After a discussion of the basic notion of ordering, appropriate results from the theory of posets and digraphs are adapted for the purpose of determining the global evolution properties of dynamical systems. In terms of posets, evolution processes and strange attractors can be studied in a new light. It is believed that this approach offers us a new way to examine the multifaceted complex behavior of nonlinear systems. Computation algorithms are also discussed and an example of application is included.

Estimating Ljapunov-Exponents Using the Generalized Cell Mapping Method

1995 ◽  
Author(s):  
Swen Schaub ◽  
Werner Schiehlen
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Mapping Method ◽  
Cell Mapping ◽  
Coexisting Attractors ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Global Stability Analysis ◽  
Probabilistic Description ◽  
Stability Of Dynamical Systems

Abstract Ljapunov-Exponents are widely used to characterize the local stability of dynamical systems. On the other hand, Cell Mapping methods provide an effective numerical tool for global study by a probabilistic description of the time evolution. Using this description together with powerful interpolation techniques, an iterative method for global stability analysis with estimated Ljapunov-Exponents for all coexisting attractors of nonlinear dynamical systems is presented.

Global Analysis of Nonlinear Time-Delayed Dynamical Systems

2011 ◽  
Author(s):  
Jian-Qiao Sun ◽  
Bo Song ◽  
Jie Yang
Keyword(s):  
Steady State ◽  
Dynamical Systems ◽  
State Space ◽  
Global Analysis ◽  
Stable Steady State ◽  
Domains Of Attraction ◽  
Infinite Dimensional ◽  
Dimensional State Space ◽  
Steady State Responses ◽  
State Responses

Time-delayed dynamical systems are defined in an infinite dimensional state-space. When the system has multiple stable steady-state responses, the global analysis of the system such as finding the domains of attraction and boundary basin has to be done in the infinite dimensional state-space. This paper examines this issue and shows that there are rich opportunities for global analysis research of nonlinear time-delayed dynamical systems.

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.

Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds

1997 ◽  
Vol 07 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
Rabbijah Guder ◽  
Edwin Kreuzer
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Nonlinear Dynamical Systems ◽  
Powerful Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Long Term Behavior ◽  
Mapping Theory

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

A Probabilistic Theory of Nonlinear Dynamical Systems Based on the Cell State Space Concept

1982 ◽  
Vol 49 (4) ◽  
pp. 895-902 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Dynamical Systems ◽  
State Space ◽  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Cell Structure ◽  
The State ◽  
Probabilistic Theory ◽  
Nonlinear Dynamical ◽  
Linear Ordinary Differential Equations ◽  
Highly Nonlinear

Developed in the paper is a probabilistic theory for nonlinear dynamical systems. The theory is based on discretizing the state space into a cell structure and using the cell probability functions to describe the state of a system. Although the dynamical system may be highly nonlinear the probabilistic formulation always leads to a set of linear ordinary differential equations. The evolution of the probability distribution among the cells can then be studied by applying the theory of Markov processes to this set of equations. It is believed that this development possibly offers a new approach to the global analysis of nonlinear systems.

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