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.

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.

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.

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.

BIFURCATION STRUCTURE OF A DRIVEN MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR (II): SYMMETRY-BREAKING CRISIS AND INTERMITTENCY

1991 ◽  
Vol 01 (03) ◽  
pp. 711-715 ◽  
Author(s):  
C. EICHWALD ◽  
F. KAISER
Keyword(s):  
Symmetry Breaking ◽  
Limit Cycles ◽  
Critical Exponents ◽  
The Self ◽  
Van Der Pol Oscillator ◽  
Chaotic Attractors ◽  
Bifurcation Structure ◽  
Van Der Pol ◽  
Sustained Oscillations ◽  
Symmetry Properties

Bifurcations in the superharmonic region of a generalized version of the van der Pol oscillator which exhibits three limit cycles are investigated. An external force causes the subsequent breakdown of the self-sustained oscillations. Beyond these series of bifurcations chaotic solutions also exist. They display a symmetry-breaking crisis followed by a type III intermittency transition. The bifurcations are discussed with respect to the symmetry properties of chaotic attractors. The critical exponents connected with the bifurcations offer a scaling which partially contradicts that known from literature. An explanation for this behavior is given.

Codimension two bifurcations of discrete Bonhoeffer–van der Pol oscillator model

2021 ◽  
Vol 25 (7) ◽  
pp. 5261-5276
Author(s):  
J. Alidousti ◽  
Z. Eskandari ◽  
M. Fardi ◽  
M. Asadipour
Keyword(s):  
Van Der Pol Oscillator ◽  
Oscillator Model ◽  
Van Der Pol ◽  
Codimension Two

Long-Term Dynamics of Autonomous Fractional Differential Equations

2016 ◽  
Vol 26 (04) ◽  
pp. 1650055 ◽  
Author(s):  
Tao Liu ◽  
Wei Xu ◽  
Yong Xu ◽  
Qun Han
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Mapping Method ◽  
Van Der Pol Oscillator ◽  
Cell Mapping ◽  
Time Duration ◽  
Generalized Cell Mapping ◽  
Dynamic Behaviors ◽  
Fractional Differential

This paper aims to investigate long-term dynamic behaviors of autonomous fractional differential equations with effective numerical method. The long-term dynamic behaviors predict where systems are heading after long-term evolution. We make some modification and transplant cell mapping methods to autonomous fractional differential equations. The mapping time duration of cell mapping is enlarged to deal with the long memory effect. Three illustrative examples, i.e. fractional Lotka–Volterra equation, fractional van der Pol oscillator and fractional Duffing equation, are studied with our revised generalized cell mapping method. We obtain long-term dynamics, such as attractors, basins of attraction, and saddles. Compared with some existing stability and numerical results, the validity of our method is verified. Furthermore, we find that the fractional order has its effect on the long-term dynamics of autonomous fractional differential equations.

Stochastic Response of a Vibro-Impact System by Path Integration Based on Generalized Cell Mapping Method

2014 ◽  
Vol 24 (10) ◽  
pp. 1450129 ◽  
Author(s):  
Chao Li ◽  
Wei Xu ◽  
Xiaole Yue
Keyword(s):  
White Noise ◽  
Mapping Method ◽  
Cell Mapping ◽  
Van Der Pol ◽  
Generalized Cell Mapping ◽  
White Noise Excitation ◽  
Impact System ◽  
Steady State Responses ◽  
State Responses ◽  
Transient And Steady State

The generalized cell mapping method is extended to study the response of a vibro-impact system with white noise excitation. The transient and steady-state responses of a Duffing–van der Pol vibro-impact system under white noise excitation are obtained by using this method. The accuracy of the method is verified by comparison with Monte Carlo simulation results. In addition, stochastic P-bifurcation for different parameters is considered, and several special forms are observed in this paper.

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