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 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.

Generalized Cell Mapping Method with Deep Learning for Global Analysis and Response Prediction of Dynamical Systems

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Xiao-Le Yue ◽  
Su-Ping Cui ◽  
Hao Zhang ◽  
Jian-Qiao Sun ◽  
Yong Xu
Keyword(s):  
Neural Network ◽  
Deep Learning ◽  
Dynamical Systems ◽  
Global Analysis ◽  
Transition Probability ◽  
Response Prediction ◽  
Training Data ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Global Properties

A novel method that combines generalized cell mapping and deep learning is developed to analyze the global properties and predict the responses of dynamical systems. The proposed method only requires some prior knowledge of the system governing equations and obtains dynamical properties of the system from observed data. By combining the theoretical demonstration and empirical inference results, appropriate network structure and training hyperparameters are computed. Then a robust and efficient neural network approximation with the estimated mapping parameters is obtained. By using the approximate dynamical system model, we construct the one-step transition probability matrix and introduce the digraph analysis method to analyze the global properties. System responses at any time can be obtained with the trained model on the basis of the property of Markov chain. Several examples with periodic or chaotic attractors are presented to validate the proposed method. The influence of the number of hidden layers and the size of training data on calculated results is discussed, and an admissible architecture of the neural network is found. Numerical results indicate that the proposed method is quite effective for both global analysis and response prediction.

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.

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.

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