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.

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.

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.

Stochastic dynamics of dielectric elastomer balloon with viscoelasticity under pressure disturbance

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hao Dong ◽  
Lin Du ◽  
Rongchun Hu ◽  
Shuo Zhang ◽  
Zichen Deng
Keyword(s):  
High Efficiency ◽  
Stochastic Dynamics ◽  
Stretch Ratio ◽  
Dielectric Elastomer ◽  
Mapping Method ◽  
The State ◽  
Stochastic Dynamic ◽  
Steady State Probability ◽  
Generalized Cell Mapping ◽  
Biological Compatibility

Abstract Dielectric elastomers are widely used in many fields due to their advantages of high deformability, light weight, biological compatibility, and high efficiency. In this study, the stochastic dynamic response and bifurcation of a dielectric elastomer balloon (DEB) with viscoelasticity are investigated. Firstly, the rheological model is adopted to describe the viscoelasticity of the DEB, and the dynamic model is deduced by using the free energy method. The effect of viscoelasticity on the state of equilibrium with static pressure and voltage is analysed. Then, the stochastic differential equation about the perturbation around the state of equilibrium is derived when the DEB is under random pressure and static voltage. The steady-state probability densities of the perturbation stretch ratio are determined by the generalized cell mapping method. The effects of parameter conditions on the mean value of the perturbation stretch ratio are calculated. Finally, sinusoidal voltage and random pressure are applied to the viscoelastic DEB, and the phenomenon of P-bifurcation is observed. Our results are compared with those obtained from Monte Carlo simulation to verify their accuracy. This work provides a potential theoretical reference for the design and application of DEs.

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.

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.

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.

scholarly journals COMPUTING TWO-DIMENSIONAL GLOBAL INVARIANT MANIFOLDS IN SLOW–FAST SYSTEMS

2007 ◽  
Vol 17 (03) ◽  
pp. 805-822 ◽  
Author(s):  
J. P. ENGLAND ◽  
B. KRAUSKOPF ◽  
H. M. OSINGA
Keyword(s):  
Invariant Manifolds ◽  
Multiple Time Scales ◽  
Test Case ◽  
Multiple Time ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Boundary Problems ◽  
Case Examples ◽  
Unstable Manifolds ◽  
Global Invariant

We present the GLOBALIZEBVP algorithm for the computation of two-dimensional stable and unstable manifolds of a vector field. Specifically, we use the collocation routines of AUTO to solve boundary problems that are used during the computation to find the next approximate geodesic level set on the manifold. The resulting implementation is numerically very stable and well suited for systems with multiple time scales. This is illustrated with the test-case examples of the Lorenz and Chua systems, and with a slow–fast model of a somatotroph cell.

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