Controlling chaos for the dynamical system of coupled dynamos

This work is a tutorial on the different methods to control chaotic behaviour of the coupled dynamos system. Feedback and nonfeedback control techniques are proposed to suppress chaos to unstable equilibrium or unstable periodic solution. The stabilization of unstable fixed point of the chaotic behaviours is achieved also by bounded feedback method. Stability of the controlled systems are studied by Routh–Hurwitz criterion. Nonfeedback method and a derived method based on the delay feedback control are used to control chaos to periodic orbits. Numerical simulation results are included to show the control process of the different methods.     

Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays

Time delays are often sources of complex behavior in dynamic systems. Yet its complexity needs to be further explored, particularly when multiple time delays are present. As a purpose to gain insight into such complexity under multiple time delays, we investigate the mechanism for the action of multiple time delays on a particular non-autonomous system in this paper. The original mathematical model under consideration is a Duffing oscillator with harmonic excitation. A delayed system is obtained by adding delayed feedbacks to the original system. Two time delays are involved in such system, one of which in the displacement feedback and the other in the velocity feedback. The time delays are taken as adjustable parameters to study their effects on the dynamics of the system. Firstly, the stability of the trivial equilibrium of the linearized system is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or double Hopf bifurcation may occur. Then, the chaotic behavior of such system is investigated in detail. Particular emphasis is laid on the effect of delay difference between two time delays on the chaotic properties. A Melnikov’s analysis is employed to obtain the necessary condition for onset of chaos resulting from homoclinic bifurcation. And numerical analyses via the bifurcation diagram and the top Lyapunov exponent are carried out to show the actual time delay effect. Both the results obtained by the two analyses show that the delay difference between two time delays plays a very important role in inducing or suppressing chaos, so that it can be taken as a simple but efficient “switch” to control the motion of a system: either from order to chaos or from chaos to order.     

Control, anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay

Chaos, control, anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor and spring for which time-delay effect is considered are studied in the paper. By applying numerical results, phase diagram and power spectrum are presented to observe periodic and chaotic motions. Linear feedback control and adaptive control algorithm are used to control chaos effectively. Linear and nonlinear feedback synchronization and phase synchronization for the coupled systems are presented. Finally, anticontrol of chaos for this system is also studied.     

Resonance,stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback

This paper examines dynamical behavior of a nonlinear oscillator with a symmetric potential that models a quarter-car forced by the road profile. The primary, superharmonic and subharmonic resonances of a harmonically excited nonlinear quarter-car model with linear time delayed active control are investigated. The method of multiple scales is utilized to obtain first order approximation of response. We focus on the influence of delay in the system. This naturally gives rise to a delay deferential equation (DDE) model of the system. The effect of time delay and feedback gains of the steady state responses of primary, superharmonic and subharmonic resonances are investigated. By means of Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. We describe a method to identify the critical forcing function and time delay above which the system becomes unstable. It is found that proper selection of time-delay shows optimum dynamical behavior. The accuracy of the method is obtained from the fractal basin boundaries.     

Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback

The chaotic behavior of a double-well Duffing oscillator with both delayed displacement and velocity feedbacks under a harmonic excitation is investigated. By means of the Melnikov technique, necessary condition for onset of chaos resulting from homoclinic bifurcation is derived analytically. The analytical results reveal that for negative feedback the presence of time delay lowers the threshold and enlarges the possible chaotic domain in parameter space; while for positive feedback the presence of time delay enhances the threshold and reduces the possible chaotic domain in parameter space, which are further verified numerically through Poincare maps of the original system. Furthermore, the effect of the control gain parameters on the chaotic motion of the original system is studied in detail.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

Steady state bifurcation of a periodically excited system under delayed feedback controls

This paper investigates the steady state bifurcation of a periodically excited system subject to time-delayed feedback controls by the combined method of residue harmonic balance and polynomial homotopy continuation. Three kinds of delayed feedback controls are considered to examine the effects of different delayed feedback controls and delay time on the steady state response. By means of polynomial homotopy continuation, all the possible steady state solutions corresponding the third-order superharmonic and second-subharmonic responses are derived analytically, i.e. without numerical integration. It is found that the delayed feedback changes the bifurcating curves qualitatively and possibly eliminates the saddle-node bifurcation during resonant. The delayed position-velocity coupling and the delayed velocity feedback controls can destabilize the steady state responses. Coexisting periodic solutions, period-doubling bifurcation and even chaos are found in these control systems. The neighborhood of the periodic solutions is verified numerically in the phase portraits. The various effects of time delay on the steady state response are investigated. Many new phenomena are observed.     

Intelligent control of chaos using linear feedback controller and neural network identifier

A method for controlling chaos when the mathematical model of the system is unknown is presented in this paper. The controller is designed by the pole placement algorithm which provides a linear feedback control method. For calculating the feedback gain, a neural network is used for identification of the system from which the Jacobian of the system in its fixed point can be approximated. The weights of the neural network are adjusted online by the gradient descent algorithm in which the difference between the system output and the network output is considered as the error to be decreased. The method is applied on both discrete-time and continuous-time systems. For continuous-time systems, equivalent discrete-time systems are constructed by using the Poincare map concept. Two discrete-time systems and one continuous-time system are tested as examples for simulation and the results show good functionality of the proposed method. It can be concluded that the chaos in systems with unknown dynamics may be eliminated by the presented intelligent control system based on pole placement and neural network.     

Control of chaos in atomic force microscopes using delayed feedback based on entropy minimization

Active chaos control of a tapping mode atomic force microscope (AFM) model via delayed feedback method is presented. The feedback gain is obtained and adapted according to a minimum entropy (ME) algorithm. In this method, stabilizing an unstable fixed point of the system Poincare map is achieved by minimizing the entropy of points distribution on the Poincare section. Simulation results show the feasibility of the proposed method in applying the delayed feedback technique for chaos control of an AFM system.     

Complex Dynamics in a Hysteretic Relay Feedback System with Delay

Relay feedback systems are used to control engineering devices. In practice the switching between different functional forms of the system is never instantaneous, but takes place after a small delay. In this paper we analyse the dynamics and bifurcations of a representative example of such systems. In the absence of delay, negative feedback results only in unimodal symmetric limit cycles, but positive feedback can lead to aperiodic trajectories and chaos. In the presence of delay, the system can behave as an equivalent system without delay, provided that the delay is small in a sense which we define precisely. For larger delays, we identify a new bifurcation phenomenon, an event collision, where the delayed switching manifold intersects the relay hysteretic lines. In this case the dynamics become much more complicated.     

A simple adaptive feedback control method for chaos and hyper-chaos control

This paper investigates the problem of chaos and hyper-chaos control, and proposes a simple adaptive feedback control method for chaos control under a reasonable assumption. In comparison with previous methods, the present control technique is simple both in the form of the controller and its application. Several illustrative examples with numerical simulations are studied by using the results obtained in this paper. Study of examples shows that our control method works very well in chaos control.     

Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.     

Stabilization problem of stochastic time-varying coupled systems with time delay and feedback control

The stabilization of stochastic coupled systems with time delay and time-varying coupling structure (SCSTT) via feedback control is investigated. We generalize systems with constant coupling structure to the time-varying coupling structure. Combining the graph theory with the Lyapunov method, a systematic method is provided to construct a Lyapunov function for SCSTT, and a Lyapunov-type theorem and a coefficient-type criterion are obtained to guarantee the stabilization in the sense of pth moment exponential stability. Furthermore, theoretical results are applied to analyze the stabilization of stochastic-coupled oscillators with time delay and time-varying coupling structure in order to illustrate the practicability of the results. Finally, two numerical examples are given to illustrate the effectiveness and feasibility of theoretical results.     

Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control

A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.     

Heteroclinic orbits in Chen circuit with time delay

Existence of Shil’nikov type of heteroclinic orbit in Chen circuit with direct time delay feedback is proved using the undetermined coefficient method. As a result, Shil’nikov criterion guarantees that the circuit has Smale horseshoes and the circuit demonstrates chaos in a rigorous analytical sense. The geometric structure of the generated chaos is determined by the heteroclinic orbits. Both the simulation and the experimental results show that chaos is indeed generated in the non-chaotic Chen circuit with the direct time delay feedback.     

Synthesis of optimal feedback and the stabilization of systems with a delay in the control

The linear problem of the optimal control of systems in which the input signals contain a time delay is considered. The method of realizing optimal feedback control that is proposed is based on a special procedure for correcting the current optimal programme controls, realized by an optimal controller using a dual linear programming method. The results are used to construct two types of stabilizer of systems with a delay in the control.     

Delayed feedback control of chaotic spinning disk via minimum entropy approach

Chaos control of a spinning disk model via delayed feedback method is presented. The feedback gain is obtained and adapted according to a minimum entropy (ME) algorithm. In this method, stabilizing an unstable fixed point of the system Poincare map is achieved by minimizing the entropy of point distribution on the Poincare section. Simulation results show the feasibility of the proposed method in applying the delayed feedback technique for chaos control of spinning disks.     

Robust Stability and Stabilizability of Markov Jump Linear Uncertain Systems with Mode-Dependent Time Delays

This paper studies the class of uncertain linear systems with time delay and Markov jump disturbance, in which the time delay is assumed to be dependent on the system mode. An LMI-based condition for this class of systems to be robustly stable is established. Sufficient conditions for the robust stabilizability under a state feedback controller are developed, and an LMI-based method to design the state feedback is proposed. Numerical examples are worked out to show the usefulness of the theoretical results.