Jun Jiang

Global bifurcations in fractional-order chaotic systems with an extended generalized cell mapping method

Global bifurcations include sudden changes in chaotic sets due to crises. There are three types of crises defined by Grebogi et al. [Physica D 7, 181 (1983)]: boundary crisis, interior crisis, and metamorphosis. In this paper, by means of the extended generalized cell mapping (EGCM), boundary and interior crises of a fractional-order Duffing system are studied as one of the system parameters or the fractional derivative order is varied. It is found that a crisis can be generally defined as a collision between a chaotic basic set and a basic set, either periodic or chaotic, to cause a sudden discontinuous change in chaotic sets. Here chaotic sets involve three different kinds: a chaotic attractor, a chaotic saddle on a fractal basin boundary, and a chaotic saddle in the interior of a basin and disjoint from the attractor. A boundary crisis results from the collision of a periodic (or chaotic) attractor with a chaotic (or regular) saddle in the fractal (or smooth) boundary. In such a case, the attractor, together with its basin of attraction, is suddenly destroyed as the control parameter passes through a critical value, leaving behind a chaotic saddle in the place of the original attractor and saddle after the crisis. An interior crisis happens when an unstable chaotic set in the basin of attraction collides with a periodic attractor, which causes the appearance of a new chaotic attractor, while the original attractor and the unstable chaotic set are converted to the part of the chaotic attractor after the crisis. These results further demonstrate that the EGCM is a powerful tool to reveal the mechanism of crises in fractional-order systems.

Optimization of actuator placement in a truss-cored sandwich plate with independent modal space control

The determination of optimal locations of actuators is an important issue in the field of vibration control in structures. In this paper, a practical method for the optimal actuator placement in a truss-cored sandwich plate based on independent modal space control (IMSC) is proposed. It is shown that the modal strain energy (MSE) alone is not enough to determine the actuator locations since the system may become uncontrollable, especially when the structure is symmetrical. Moreover, it is found that the controllability can be measured through the singular values of the input matrix. Therefore the singular values of the input matrix together with the modal strain energy may serve as an effective optimization index for actuator placement. Numerical examples are presented to show that the maximal actuator force can be significantly reduced after the optimization in order to achieve an equal modal control force. (Some figures in this article are in colour only in the electronic version)

An Iterative Method of Point Mapping under Cell Reference for the Global Analysis: Theory and a Multiscale Reference Technique

In iterative method of Point Mapping under Cell Reference, a cell co-ordinate system, called cell reference, is built to identify the subregions (cells) in the state space. When the cell reference is equipped with the so-called characteristic functions, it can work as an ‘inspector’ or a ‘recorder’ to derive the local dynamics of the subregions from the information provided by the trajectories passing through them. This method can retain the accuracy of the Point Mapping Method but greatly reduce the computational work. In this paper, the theoretic basis for this method is first discussed and a multiscale reference technique is then devised which can select an optimal cell reference and make the method more practicable. Finally, an example for application is presented. It is shown that the present method cannot only accurately and efficiently depict the basins of attraction of a dynamical system but also potentially detect other characteristics of the system.

Fuzzy Responses and Bifurcations of a Forced Duffing Oscillator with a Triple-Well Potential

Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

Estimation of Critical Conditions for Noise-Induced Bifurcation in Nonautonomous Nonlinear Systems by Stochastic Sensitivity Function

This paper proposes an efficient but simple method to determine the approximate stationary probability distribution around periodic attractors of nonautonomous nonlinear systems under multiple time-dependent parametric noises and estimate the critical noise intensity for noise-induced explosive bifurcations under a given confidence probability. After adopting a stroboscopic map constructed by a method with higher accuracy and efficiency, nonautonomous dynamical systems around periodic attractors are transformed into mapping ones. Then the mean-square analysis method of discrete systems is used to derive the stochastic sensitivity function. Based on the confidence ellipses of stochastic attractors and the global structure of deterministic nonlinear systems, the critical noise intensity of noise-induced explosive bifurcations under a given confidence probability is estimated. A Mathieu–Duffing oscillator under both multiplicative and additive noises is studied to show the validity of the proposed method.

Study of Evolution of Global Structure Into Chaotic Itinerancy by Point Mapping Under Cell Reference Method

Chaotic itinerancy is a complex phenomenon in high-dimensional dynamic systems, by which an orbit successively itinerates over low-dimensional semi-stable invariant sets in a chaotic manner. In this paper the evolution of the global structure of the coupled neural oscillators into the chaotic itinerancy is investigated by using an extended Point Mapping under Cell Reference Method. The method of Point Mapping under Cell Reference, denoted as PMUCR in short, was a numerical method developed for the global analysis of nonlinear dynamical systems with the aim to retain the accuracy of Point Mapping Method but enhance its computational efficiency. The method is extended to be able compute both stable and unstable invariant sets in highdimensional dynamical systems by taking the virtue of the cell structure and incorporate with PIM-triple method. By applying extended PMUCR method to the coupled Morris-Lecar neuron model, some important global structure changes in invariant sets during the evolution into the chaotic itinerancy are demonstrated.Copyright

The Influence of the Cross-Coupling Effects on the Dynamics of Rotor/Stator Rubbing

In this chapter, the influence of cross-coupling effects on the rubbing-related dynamics of rotor/stator systems is investigated. The model considered in this chapter is a four-dof rotor/stator system, which takes into account the dynamics of the stator and the deformation on the contact surface as well as the cross-coupling effects. The stability of the synchronous full annular rub solution of the model is first analyzed. Then, the cross-coupling effects on the stability of the system at different system parameter planes are studied. It is found that the cross-coupling damping of the stator benefits the synchronous full annular rubs and that of the rotor has a little influence on the response. While the cross-coupling stiffness of the stator always reduces the stability domain of the response, the cross-coupling stiffness of the rotor may either increase or decrease the stability domain depending upon its value.

Study on Critical Conditions and Transient Behavior in Noise-Induced Bifurcations

In this work, the stochastic sensitivity function method , which can describe the probabilistic distribution range of a stochastic attractor, is extended to the non-autonomous dynamical systems by constructing a 1/N-period stroboscopic map to discretize a continuous cycle into a discrete one. With confidence ranges of a stochastic attractor and the global structure of the deterministic nonlinear system, like chaotic saddle in basin of attraction and/or saddle on basin boundary as well as its stable and unstable manifolds, the critical noise intensity for the occurrence of transition behavior due to noise-induced bifurcations may be estimated. Furthermore, to efficiently capture the stochastic transient behaviors after the critical conditions, an idea of evolving probabilistic vector (EPV) is introduced into the Generalized Cell Mapping method (GCM) in order to enhance the computation efficiency of the numerical method. A Mathieu-Duffing oscillator under external and parametric excitation as well as additive noise is studied as an example of application to show the validity of the proposed methods and the interesting phenomena in noise-induced explosive and dangerous bifurcations of the oscillator that are characterized respectively by an abrupt enlargement and a sudden fast jump of the response probability distribution are demonstrated. The insight into the roles of deterministic global structure and noise as well as their interplay is gained.

Response Analysis of a Forced Duffing Oscillator with Fuzzy Uncertainty

The transient and steady-state membership distribution functions (MDFs) of fuzzy response of a forced Duffing oscillator with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. The FGCM method is first introduced. A rigorous mathematical foundation of the FGCM is established with a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min-max operator of fuzzy logic. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The evolutionary process of transient and steady-state MDFs is determined by the quantitative analysis of the FGCM with the min-max calculations. It is found that the evolutionary orientation of MDFs is in accordance with invariant manifolds leading to invariant sets. In the evolutionary process of a steady-state fuzzy response with an increase of the intensity of fuzzy noise, a merging bifurcation is observed in a sudden change of the MDFs from two sharp peaks of most possibility to one peak band around unstable manifolds.

Dynamical Characteristics of the Fractional-Order FitzHugh-Nagumo Model Neuron

Through the research on the fractional-order FitzHugh-Nagumo model, it is found that the Hopf bifurcation point in such a model, where the state of the model neuron changes from the quiescence into periodic spiking, is different from that of the corresponding integer-order model when the externally applied current is considered to be the bifurcation parameter. Moreover, we demonstrate that the range of periodic spiking of the fractional-order model neuron is clearly smaller than that of the corresponding integer-order model neuron, that is, the range of periodic spiking of the former is just embedded in that of the latter. In addition, we show that the firing frequency of the fractional-order model neuron is evidently larger than that of the integer-order counterpart. The Adomian decomposition method is used to calculate fractional-order differential equations numerically due to its rapid convergence and high accuracy.