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.

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.

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.

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.

Fuzzy Codimension Two Bifurcations

2011 ◽  
Author(s):  
Ling Hong ◽  
Jian-Qiao Sun
Keyword(s):  
Symmetry Breaking ◽  
Mapping Method ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Generalized Cell Mapping ◽  
Codimension Two ◽  
Deterministic Systems ◽  
Breaking Parameter ◽  
Fuzzy Noise ◽  
Codimension Two Bifurcation

By means of fuzzy generalized cell mapping method, a Duffing-Van der Pol oscillator in the presence of fuzzy noise is studied in a regime where two symmetrically related fuzzy period-one attractors grow and merge as the intensity of fuzzy noise is increased. By introducing a small symmetry-breaking parameter to break the symmetry, the merging explosion bifurcation unfolds to a pattern of two catastrophic and explosive bifurcations. Considering both the intensity of fuzzy noise and the symmetry-breaking parameter together as controls, a codimension two bifurcation of fuzzy attractors is defined, and two examples of additive and multiplicative fuzzy noise are given. Such a codimension two bifurcation is fuzzy noise-induced effects which cannot be seen in the deterministic systems.

A Generalized Theory of Cell-to-Cell Mapping for Nonlinear Dynamical Systems

1981 ◽  
Vol 48 (3) ◽  
pp. 634-642 ◽  
Author(s):  
C. S. Hsu
Keyword(s):  
Nonlinear Dynamical Systems ◽  
Global Analysis ◽  
Simple Theory ◽  
Behavior Patterns ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Multiple Image ◽  
A Cell ◽  
Mapping Theory

The simple theory of cell mapping and the associated algorithm presented in [1, 2] have been found to form a very effective tool for the global analysis of nonlinear systems. In this paper we generalize the theory by allowing the mapping of a cell to have multiple image cells with appropriate individual mapping probabilities. This generalized theory will be able to deal with very fine and complicated global behavior patterns, if they exist, in a more attractive way without having to utilize extremely small cell sizes. It is found that such a generalized cell mapping can be identified with a Markov chain and the well-developed mathematical theory of Markov chains can be immediately applied. Similar to the simple theory of [1], the generalized cell mapping theory is also eminently suited as a theoretic base for computer alogorithms which will be needed when dealing with systems involving a very large number of cells.

BIFURCATIONS OF A FORCED DUFFING OSCILLATOR IN THE PRESENCE OF FUZZY NOISE BY THE GENERALIZED CELL MAPPING METHOD

2006 ◽  
Vol 16 (10) ◽  
pp. 3043-3051 ◽  
Author(s):  
LING HONG ◽  
JIAN-QIAO SUN
Keyword(s):  
Steady State ◽  
Duffing Oscillator ◽  
Mapping Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Deterministic Systems ◽  
Fuzzy Noise ◽  
Topological Changes

Bifurcations of a forced Duffing oscillator in the presence of fuzzy noise are studied by means of the fuzzy generalized cell mapping (FGCM) method. The FGCM method is first introduced. Two categories of bifurcations are investigated, namely catastrophic and explosive bifurcations. Fuzzy bifurcations are characterized by topological changes of the attractors of the system, represented by the persistent groups of cells in the context of the FGCM method, and by changes of the steady state membership distribution. The fuzzy noise-induced bifurcations studied herein are not commonly seen in the deterministic systems, and can be well described by the FGCM method. Furthermore, two conjectures are proposed regarding the condition under which the steady state membership distribution of a fuzzy attractor is invariant or not.

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.

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