Generalized Cell Mapping for Randomly Perturbed Dynamical Systems

2001 ◽  
Vol 81 (11) ◽  
pp. 769 ◽  
Author(s):  
J. Fischer ◽  
E. Kreuzer
Keyword(s):  
Dynamical Systems ◽  
Cell Mapping ◽  
Generalized Cell Mapping

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.

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.

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.

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 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.

Fuzzy Noise-Induced Codimension-Two Bifurcations Captured by Fuzzy Generalized Cell Mapping with Adaptive Interpolation

2019 ◽  
Vol 29 (11) ◽  
pp. 1950151
Author(s):  
Xiao-Ming Liu ◽  
Jun Jiang ◽  
Ling Hong ◽  
Zigang Li ◽  
Dafeng Tang
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Three Dimensional ◽  
Maximum Efficiency ◽  
Global Changes ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Codimension Two ◽  
Adaptive Interpolation ◽  
Membership Matrix ◽  
Fuzzy Noise

In this paper, the Fuzzy Generalized Cell Mapping (FGCM) method is developed with the help of the Adaptive Interpolation (AI) in the space of fuzzy parameters. The adaptive interpolation on the set-valued fuzzy parameter is introduced in computing the one-step transition membership matrix to enhance the efficiency of the FGCM. For each of initial points in the state space, a coarse database is constructed at first, and then interpolation nodes are inserted into the database iteratively each time errors are examined with the explicit formula of interpolation error until the maximal errors are just under the error bound. With such an adaptively expanded database on hand, interpolating calculations assure the required accuracy with maximum efficiency gains. The new method is termed as Fuzzy Generalized Cell Mapping with Adaptive Interpolation (FGCM with AI), and is used to investigate codimension-two bifurcations in two-dimensional and three-dimensional nonlinear dynamical systems with fuzzy noise. It is found that global changes in fuzzy dynamics are dominated by the underlying deterministic counterparts, and the fuzzy attractor expands along the unstable manifold leading to a collision with a saddle when a bifurcation occurs. The examples show that the FGCM with AI has a thirtyfold to fiftyfold efficiency over the traditional FGCM to achieve the same analyzing accuracy.

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