Generalized Cell Mapping Digraph Research Articles

Analysis of supercritical pitchfork bifurcation in active magnetic bearing-rotor system with current saturation

The bifurcation characteristics of the active magnetic bearing-rotor system subjected to the external excitation were investigated analytically when it was operating at a speed far away from its natural frequencies. During operation of the system, some nonlinear factors may be prominent, for example, the nonlinearity of bearing force and current saturation. Nonlinear factors can lead to some complicated behaviors, which have negative effects on the operating performance and stability. To analyze the bifurcations happening at the speed far away from harmonic resonances, an approximate analytical method that can be applicable to the bifurcation analysis of the forced vibration system was proposed. By applying it to the active magnetic bearing-rotor system, multiple static equilibriums and periodic solutions were obtained, and then, the stability analysis was conducted based on Floquet theory. The validity and accuracy of the approximate analytical method were verified by the numerical integration method and generalized cell mapping digraph method. It was found that there was supercritical pitchfork bifurcation of static equilibrium in the active magnetic bearing-rotor system. The influences of external excitation and controller parameters on dynamical characteristics were discussed. Based on analysis results, controller parameters were also improved to prevent nonlinear behaviors and improve system performance.

Noise-Induced Chaos in a Piecewise Linear System

In the present paper, the phenomenon of noise-induced chaos in a piecewise linear system that is excited by Gaussian white noise is investigated. Firstly, the global dynamical behaviors of the deterministic piecewise linear system are investigated numerically in advance by using the generalized cell-mapping digraph (GCMD) method. Then, based on these global properties, the system that is excited by Gaussian white noise is introduced. Then, it is simplified by the stochastic averaging method, through which, a four-dimensional averaged Itô system is finally obtained. In order to reveal the phenomenon of noise-induced chaos quantitatively, MFPT (the mean first-passage time) is selected as the measure. The expression for MFPT is formulated by using the singular perturbation method and then a rather simple representation is obtained via the Laplace approximation, and within which, the concept of quasi-potential is introduced. Furthermore, with the rays method, the MFPT under a certain set of parameters is estimated. However, within the process of analysis, the authors had to face a difficult problem concerning the ill-conditioned matrix, which is the obstacle for the estimation of MFPT, which was then solved by applying one more approximation. Finally, the result is compared with the numerical one that is obtained by the Monte Carlo simulation.

Global analysis of stochastic bifurcation in permanent magnet synchronous generator for wind turbine system

The permanent magnet synchronous generator (PMSG) for wind turbine system operating under inevitable stochastic disturbance from wind power is a nonlinear stochastic dynamical system. With the random interaction and nonlinearity, the intense nonlinear stochastic oscillation is likely to happen in such a system, causing the system to be unstable or even collapse. However, the PMSG is usually considered as a deterministic system when analyzing its nonlinear dynamic behaviors in the past researches. Such a simplification can lead to wrong predictions for the system stability and reliability. This paper aims to discuss the effect of the stochastic disturbance on the nonlinear dynamic behavior of the PMSG. Based on the derived PMSG model considering the stochastic disturbance from the input mechanical torque, the evolution of the system global structure with the stochastic intensity is investigated using the generalized cell mapping digraph method. Meanwhile, the occurrence process and development process of the stochastic bifurcation are illustrated. Based on this global analysis, the intrinsic mechanism for the effect of the stochastic disturbance on the operating performances of the PMSG is revealed. Finally, the numerical simulations based on the Euler-Maruyama algorithm are carried out to validate the results of the theoretical analysis. The results present that as the intensity of the stochastic disturbance increases, two kinds of stochastic bifurcations can be observed in the PMSG system according to the definition of a sudden change in characteristic of the stochastic attractor. One is the stochastic interior crisis that occurs when a stochastic attractor collides with a stochastic saddle in its attraction basin interior, leading to the abrupt increase of the attractor and the disappearance of the saddle. This kind of bifurcation results in the intense stochastic oscillation and instability of the PMSG system. Another stochastic bifurcation is the stochastic boundary crisis which occurs when a stochastic attractor collides with the boundary of its attraction basin and results in the disappearance of the attractor. This sudden change of the number of stochastic attractors induces the stable solution set to vanish and thus the PMSG system to collapse. In a word, even the stochastic disturbance with small intensity may lead to the complete destruction of the stable structure of the PMSG, inducing the system to suffer a strong disordered oscillation or the operation to collapse. The results of this paper can provide significant theoretic reference for both practically operating and designing the PMSG for wind turbine systems.

On the Global Analysis of a Piecewise Linear System that is excited by a Gaussian White Noise

The global analysis is very important for a nonlinear dynamical system which possesses a chaotic saddle and a nonchaotic attractor, especially for the one that is driven by a noise. For a random dynamical system, within which, chaotic saddles exist, it is found that if the noise intensity exceeds a critical value, the so called “noise-induced chaos” is observed. Meanwhile, the exit behavior is also found to be influenced significantly by the existence of chaotic saddles. In the present paper, based on the generalized cell-mapping digraph (GCMD) method, the global dynamical behaviors of a piecewise linear system, wherein a chaotic saddle exists and consists of subharmonic solutions in a wide frequency range, are investigated numerically. Further, in order to simplify the system that is driven by a Gaussian white noise excitation, the stochastic averaging method is applied and through which, a five-dimensional Itô system is obtained. Some of the global dynamical behaviors of the original system are retained in the averaged one and then are analyzed. The researches in this paper show that GCMD method is a good numerical tool to investigate the global behaviors of a nonlinear random dynamical system, and the stochastic averaging method is effective for solving the global problems.

Chaotic transients and generalized crises of a Duffing-van der Pol oscillator with two external periodic excitations

Global property of a Duffing-van der Pol oscillator with two external periodic excitations is investigated by generalized cell mapping digraph method. As the bifurcation parameter varies, a chaotic transient appears in a regular boundary crisis. Two kinds of transient boundary crises are discovered to reveal some reasons for the discontinuous changes for domains of attraction and boundaries. A chaotic saddle collides with the stable manifold of a periodic saddle at the fractal boundary of domains when the crisis occurs, if the chaotic saddle lies in the basin of attraction, the basin of attraction decreases suddenly while the boundary increases after the crisis; if the chaotic saddle is at a boundary, the two boundaries merge into one because of the crisis. In addition, two chaotic saddles can be merged into a new one, when they touch each other in a transient merging crisis. Finally the chaotic transient disappears in an interior crisis. The characteristics of these generalized crises are quite important for the study of chaotic transients.

A HYPERCHAOTIC CRISIS

A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary. Two examples of such a hyperchaotic crisis are given in Kawakami map.

A chaotic crisis between chaotic saddle and attractor in forced Duffing oscillators

We investigate crises in forced Duffing oscillators by the generalized cell mapping digraph method to efficiently complete the global analysis of non-linear systems, which includes global transient analysis through digraphic algorithms based on a strictly theoretical correspondence between generalized cell mappings and digraphs. A process of generalized cell mapping method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space. A chaotic boundary crisis and a chaotic interior crisis are discovered. A chaotic boundary crisis owing to a collision between chaotic attractor and saddle occurs in its basin boundary possessing a fractal structure. In such a case the chaotic attractor together with its basin of attraction is suddenly destroyed as the parameter passes through the critical value, and the chaotic saddle also undergoes an abrupt enlargement in its size. Namely, the chaotic attractor is converted into an incremental portion of the chaotic saddle after the collision. For a chaotic interior crisis, there is a sudden increase in the size of a chaotic attractor as the parameter passes through the critical value. For the chaotic interior crisis, it is demonstrated that the chaotic attractor collides with a chaotic saddle in its basin interior when the crisis occurs. This chaotic saddle is an invariant and non-attracting set. The origin and evolution of the chaotic saddle are investigated as well.

Chaotic Saddles in Wada Basin Boundaries and Their Bifurcations by the Generalized Cell-Mapping Digraph (GCMD) Method

By means of the generalized cell-mapping digraph (GCMD) method, we studybifurcations governing the escape of periodically forced oscillatorsfrom a potential well, in which a chaotic saddle plays an extremelyimportant role. In this paper, we find the chaotic saddle anddemonstrate that it is embedded in a strange fractalbasin boundary which has the Wada property that any point that is on theboundary of that basin is also simultaneously on the boundary of atleast two other basins. The chaotic saddle in the Wada basin boundary,by colliding with a chaotic attractor, leads to a chaotic boundarycrisis with indeterminate outcome. A local saddle-node fold bifurcation,if the saddle of the saddle-node fold is located in tangency with thechaotic saddle in the Wada basin boundary, also results in a strangeglobal phenomenon, namely that the local saddle-node fold bifurcation hasglobally indeterminate outcome. We also investigate the origin andevolution of the chaotic saddle in the Wada basin boundary, particularlyconcentrating on its discontinuous bifurcations (metamorphoses). Wedemonstrate that the chaotic saddle in the Wada basin boundary iscreated by a collision between two chaotic saddles in differentfractal basin boundaries. After a final escape bifurcation, there onlyexists the attractor at infinity and a chaotic saddle with a beautifulpattern is left behind in the phase space.

Study of the Wada fractal boundary and indeterminate crisis

By using the generalized cell mapping digraph (GCMD) method, we study bifurcations governing the escape of periodically forced oscillators in a potential well, in which a chaotic saddle plays an extremely important role. In this paper, we find the chaotic saddle and we demonstrate that the chaotic saddle is embedded in a strange fractal boundary which has the Wada property, that any point on the boundary of that basin is also simultaneously on the boundary of at least two other basins. The chaotic saddle in the Wada fractal boundary, by colliding with a chaotic attractor, leads to a chaotic boundary crisis with a global indeterminate outcome which presents an extreme form of indeterminacy in a dynamical system. We also investigate the origin and evolution of the chaotic saddle in the Wada fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the chaotic saddle in the Wada fractal boundary is created by the collision between two chaotic saddles in different fractal boundaries. After a final escape bifurcation, there only exists the attractor at infinity; a chaotic saddle with a beautiful pattern is left behind in phase space.

Double crises in two-parameter forced oscillators by generalized cell mapping digraph method

By means of generalized cell mapping digraph method, a crisis of sinusoidally forced oscillators is observed as a system parameter is varied. The crisis happens when a chaotic attractor collides with a regular saddle or a chaotic saddle, and is called a regular crisis or a chaotic crisis, respectively. By increasing a small constant bias in the forcing and considering both the system parameter and bias together as controls, a double crisis vertex in a two-parameter plane is determined, at which four curves of crises meet and four distinct crises coincide. The two patterns of such a double crisis vertex are investigated, namely, two regular boundary crises compounded with two chaotic interior crises and two chaotic boundary crises compounded with two chaotic interior crises.

DISCONTINUOUS BIFURCATIONS OF CHAOTIC ATTRACTORS IN FORCED OSCILLATORS BY GENERALIZED CELL MAPPING DIGRAPH (GCMD) METHOD

Discontinuous bifurcations of chaotic attractors are investigated by means of Generalized Cell Mapping Digraph (GCMD) method. The two categories of discontinuous bifurcations of chaotic attractors are found in typical systems described by dissipative ordinary differential equations, namely catastrophic and explosive bifurcations. For the case of a catastrophe, the two different types, namely a regular saddle and a chaotic saddle catastrophe, are studied, in particular, the subtle distinction between them can be explicitly observed in phase portrait. We show that a regular saddle catastrophe results from a collision between a chaotic attractor and a regular saddle on its smooth basin boundary. In such a case the chaotic attractor is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the original chaotic attractor in phase space. We demonstrate that a chaotic saddle catastrophe results from a collision between a chaotic attractor and a chaotic saddle in its fractal basin boundary. In such a case the chaotic attractor is suddenly destroyed as the parameter passes through a critical value, simultaneously the chaotic saddle also undergoes an abrupt enlargement in its size, namely, the chaotic attractor is converted into a new incremental portion of the chaotic saddle after the collision. For the case of an explosion, a chaotic saddle explosion is studied, in which there is a sudden increase in the size of a chaotic attractor as the parameter passes through a critical value. We demonstrate that at the chaotic saddle explosion the chaotic attractor collides with a chaotic saddle already existent when the explosion occurs. This chaotic saddle is an invariant and nonattracting set. We also investigate the origin and evolution of the chaotic saddle. A locally refining process of Generalized Cell Mapping (GCM) method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space.

A NEW TYPE OF BOUNDARY CRISES: CHAOTIC BOUNDARY CRISES

Crises of chaotic attractors are typical phenomena in nonlinear dynamical systems. By means of generalized cell mapping digraph (GCMD) method, we show that a chaotic boundary crisis results from a collision between a chaotic attractor and a chaotic saddle in the fractal basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, simultaneously the chaotic saddle also undergoes an abrupt enlargement in its size.

FURTHER STUDY OF INTERIOR CRISES IN SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Crises in systems of ordinary differential equations are investigated by means o f Generalized Cell Mapping Digraph (GCMD) method. We show that a boundary crisis results from a collision between a chaotic attractor and a periodic saddle on i ts basin boundary. In such a case the chaotic attractor, together with its basin of attraction, is suddenly destroyed as the parameter passes through a critical value, leaving behind a nonattracting chaotic saddle in the place of the origin al chaotic attractor in phase space. We focus here on a sudden change in the siz e of a chaotic attractor, namely an interior crisis. We demonstrate that at an i nterior crisis the chaotic attractor collides with a chaotic saddle within its b asin of attraction. This chaotic saddle is an invariant and nonattracting set an d resembles the new portion of the larger chaotic attractor just after the inter ior crisis. We also investigate the origin and evolution of the chaotic saddle. The local refining procedures of persistent and transient self-cycling sets are given.

Crises and chaotic transients studied by the generalized cell mapping digraph method

In this Letter, a generalized cell mapping digraph method is presented on the basis of a correspondence between the generalized cell mapping of dynamical systems and digraphs, and this correspondence is theoretically proved on the basis of set theory in the cell state space. State cells are classified afresh, and self-cycling sets, persistent self-cycling sets and transient self-cycling sets are defined. The algorithms of digraphs are adopted for the purpose of determining the global evolution properties of the systems. After all the self-cycling sets are condensed by using the digraphic condensation method, a topological sorting of the global transient state cells can be efficiently achieved. Based on the different treatments, the global properties can be divided into qualitative and quantitative properties. In the analysis of the qualitative properties, only Boolean operations are used. As a result, the complicated behavior of nonlinear dynamical systems can be efficiently studied in a new way. A boundary crisis is studied by means of the generalized cell mapping digraph method. Attractors, basins, basin boundaries and unstable solutions are obtained once through a global analysis at low computational cost. Moreover, the approach of a chaotic attractor to an unstable periodic orbit at its basin boundary before a boundary crisis, the collision of the chaotic attractor with the unstable periodic orbit when the crisis occurs, and a chaotic transient after the crisis, are explicitly shown. The limiting probability distribution of the chaotic attractor is calculated.