Dynamic modelling and chaos control for a thin plate oscillator using Bubnov–Galerkin integral method

The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin plate system driven by acoustic wave signals. By combining the mechanical analysis of thin plate microelements with the Bubnov–Galerkin integral method, the governing equation for the forced vibration of a square thin plate is derived. Notably,the reaction force of the thin plate vibration system is defined as f=α|w|, resembling Hooke’s law. The energy function and energy level curve of the system are also analyzed. Subsequently, the amplitude–frequency response function of the thin plate oscillator is solved using the harmonic balance method. Through numerical simulations, the amplitude–frequency curves are analyzed for different vibration modes under the influence of various parameters. Furthermore, the paper demonstrates the occurrence of conservative chaotic motions in the thin plate oscillator using theoretical and numerical methods. Dynamics maps illustrating the system’s states are presented to reveal the evolution laws of the system. By exploring the effects of force fields and system energy, the underlying mechanism of chaos is interpreted. Additionally, the phenomenon of chaos in the oscillator can be controlled through the method of velocity and displacement states feedback, which holds significance for engineering applications.

MODEL FOR CHAOS CONTROL OF REGIONAL Ec-R-Ev SYSTEM

In order to realize the coordination oriented management of the economy resource environment(Ec R Ev) composite system,its optimal aggregate coordinating measurement model is established.A self learning method is presented,which combines the optimizing technology and stability analysis together to achieve a tradeoff between the optimizing objective and system stability.So a model for chaos control of the composite system based on coordination is proposed.

Fuzzy Chaos Control for Vehicle Lateral Dynamics Based on Active Suspension System

The existing research of the active suspension system（ASS） mainly focuses on the different evaluation indexes and control strategies. Among the different components, the nonlinear characteristics of practical systems and control are usually not considered for vehicle lateral dynamics. But the vehicle model has some shortages on tyre model with side-slip angle, road adhesion coefficient, vertical load and velocity. In this paper, the nonlinear dynamic model of lateral system is considered and also the adaptive neural network of tire is introduced. By nonlinear analysis methods, such as the bifurcation diagram and Lyapunov exponent, it has shown that the lateral dynamics exhibits complicated motions with the forward speed. Then, a fuzzy control method is applied to the lateral system aiming to convert chaos into periodic motion using the linear-state feedback of an available lateral force with changing tire load. Finally, the rapid control prototyping is built to conduct the real vehicle test. By comparison of time response diagram, phase portraits and Lyapunov exponents at different work conditions, the results on step input and S-shaped road indicate that the slip angle and yaw velocity of lateral dynamics enter into stable domain and the results of test are consistent to the simulation and verified the correctness of simulation. And the Lyapunov exponents of the closed-loop system are becoming from positive to negative. This research proposes a fuzzy control method which has sufficient suppress chaotic motions as an effective active suspension system.

Chaos Control in a New Three-Dimensional Chaotic T System

In this paper, we study chaos control of the new 3D chaotic system. We use three feedback methods （the linear, speed, doubly-periodic function controller） to suppress the chaos to unstable equilibrium. As a result, some controllers are obtained. Moreover, numerical simulations are used to verify the effectiveness of the obtained controllers.

Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems

In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.

Chaos control and reduced-order generalized synchronization for the Chen-Liao system

This paper deals with the problem of chaos control and synchronization of the Chen-Liao system. From rigorous mathematic justification, the chaotic trajectories of the Chen-Liao system are led to a type of points whose fourdimensional coordinates have a particular functional relation among them. Meanwhile, a new synchronization manner, reduced-order generalized synchronization （RGS）, is proposed which has the characteristic of having a functional relation between the slave and the partial master systems. It is shown that this new synchronization phenomenon can be realized by a novel technique. Numerical simulations have verified the effectiveness of the proposed scheme.

Parrondo's paradox for chaos control and anticontrol of fractional-order systems

We present the generalized forms of Parrondo＇s paradox existing in fractional-order nonlinear systems. The gener- alization is implemented by applying a parameter switching （PS） algorithm to the corresponding initial value problems associated with the fractional-order nonlinear systems. The PS algorithm switches a system parameter within a specific set of N 〉 2 values when solving the system with some numerical integration method. It is proven that any attractor of the concerned system can be approximated numerically. By replacing the words ＂winning＂ and ＂loosing＂ in the classical Parrondo＇s paradox with ＂order＂ and ＂chaos＂, respectively, the PS algorithm leads to the generalized Parrondo＇s paradox： chaos1 ＋ chaos2 ＋..- ＋ chaosN = order and order1 ＋ order2 ＋.-. ＋ orderN = chaos. Finally, the concept is well demon- strated with the results based on the fractional-order Chen system.

Coordinated chaos control of urban expressway based on synchronization of complex networks

We investigate the problem of coordinated chaos control on an urban expressway based on pinning synchronization of complex networks. A node coupling model of an urban expressway based on complex networks has been established using the cell transmission model（CTM）. The pinning controller corresponding to multi-ramp coordinated controller was designed by using the delayed feedback control（DFC） method, whose objective is to realize periodical orbits from chaotic states. The concrete pinning control nodes corresponding to the subsystems of regulating the inflows from the on-ramps to the mainline were obtained and the parameters of the controller were optimized by using the stability theory of complex networks to ensure the network synchronization. The validity of the proposed coordinated chaos control method was proven via the simulation experiment. The results of the examples indicated that the order motion on urban expressway can be realized, the wide-moving traffic jam can be suppressed, and the operating efficiency is superior to that of the traditional control methods.

Chaos control in the nonlinear Schrdinger equation with Kerr law nonlinearity

The nonlinear Schr6dinger equation with Kerr law nonlinearity in the two-frequency interference is studied by the numerical method. Chaos occurs easily due to the absence of damping. This phenomenon will cause the distortion in the process of information transmission. We find that fiber-optic transmit signals still present chaotic phenomena if the control intensity is smaller. With the increase of intensity, the fiber-optic signal can stay in a stable state in some regions. When the strength is suppressed to a certain value, an unstable phenomenon of the fiber-optic signal occurs. Moreover we discuss the sensitivities of the parameters to be controlled. The results show that the linear term coefficient and the environment of two quite different frequences have less effects on the fiber-optic transmission. Meanwhile the phenomena of vibration, attenuation and escape occur in some regions.

Chaos control of gear system with elastomeric web based on multi-parameter multi-step method

This paper employs a multi-parameter multi-step chaos control method, which is built up on the OGY method, to stabilize desirable UPOs of a gear system with elastomeric web as a high-dimensional and non-hyperbolic chaotic system, and the analyses are carried out. Three types of relations between components of a certain control parameter combination are defined in a certain control process. Special emphasis is put on the comparison of control efficiencies of the multi-parameter multi-step method and single-parameter multi-step method. The numerical experiments show the ability to switch between different orbits and the method can be a good chaos control alternative since it provides a more effective UPOs stabilization of high-dimensional and non-hyperbolic chaotic systems than the single-parameter chaos control, and according to the relation between components of each parameter combination, the best combination for chaos control in a certain UPO stabilization process are obtained.

Fixed Point Iteration Chaos Controlled ZCDPLL

The stable operation of first and second order Zero Crossing Digital Phase Locked Loop (ZCDPLL) is extended by using a Fixed Point Iteration (FPI) method with relaxation. The non-linear components of ZCDPLL such as sampler phase detector and Digital Controlled Oscillator (DCO) lead to unstable and chaotic operation when the filter gains are high. FPI will be used to stabilize the chaotic operation and consequently extend the lock range of the loop. The proposed stabilized loop can work in higher filter gains which are needed for faster signal acquisition.

Chaos Control and Modified Projective Synchronization of Chaotic Dissipative Gyroscope Systems

This paper proposes the chaos control and the modified projective synchronization methods for chaotic dissipative gyroscope systems. Because of the nonlinear terms of the gyroscope system, the system exhibits chaotic motions. Occasionally, the extreme sensitivity to initial states in a system operating in chaotic mode can be very destructive to the system because of unpredictable behavior. In order to improve the performance of a dynamic system or avoid the chaotic phenomena, it is necessary to control a chaotic system with a periodic motion beneficial for working with a particular condition. As chaotic signals are usually broadband and noise like, synchronized chaotic systems can be used as cipher generators for secure communication. This paper presents chaos synchronization of two identical chaotic motions of symmetric gyroscopes. Using the variable structure control technique, control laws are established which guarantees the chaos control and the modified projective synchronization. By Lyapunov stability theory, control lows are proposed to ensure the stability of the controlled and synchronized system. Numerical simulations are presented to verify the proposed control and the synchronization approach. This paper demonstrates that synchronization and anti-synchronization can coexist in dissipative gyroscope systems via variable structure control.

Bifurcation analysis for Hindmarsh-Rose neuronal model with time-delayed feedback control and application to chaos control

This paper is concerned with bifurcations and chaos control of the Hindmarsh-Rose(HR)neuronal model with the time-delayed feedback control.By stability and bifurcation analysis,we find that the excitable neuron can emit spikes via the subcritical Hopf bifurcation,and exhibits periodic or chaotic spiking/bursting behaviors with the increase of external current.For the purpose of control of chaos,we adopt the time-delayed feedback control,and convert chaos control to the Hopf bifurcation of the delayed feedback system.Then the analytical conditions under which the Hopf bifurcation occurs are given with an explicit formula.Based on this,we show the Hopf bifurcation curves in the two-parameter plane.Finally,some numerical simulations are carried out to support the theoretical results.It is shown that by appropriate choice of feedback gain and time delay,the chaotic orbit can be controlled to be stable.The adopted method in this paper is general and can be applied to other neuronal models.It may help us better understand the bifurcation mechanisms of neural behaviors.

Controlling chaos in permanent magnet synchronous motor based on finite-time stability theory

This paper reports that the performance of permanent magnet synchronous motor （PMSM） degrades due to chaos when its systemic parameters fall into a certain area. To control the undesirable chaos in PMSM, a nonlinear controller, which is simple and easy to be constructed, is presented to achieve finite-time chaos control based on the finite-time stability theory. Computer simulation results show that the proposed controller is very effective. The obtained results may help to maintain the industrial servo driven system＇s security operation.

Controlling chaos in RCL-shunted Josephson junction by delayed linear feedback

The resistively-capacitively-inductively-shunted （RCL-shunted） Josephson junction （RCLSJJ） shows chaotic behaviour under some parameter conditions. Here a scheme for controlling chaos in the RCLSJJ is presented based on the linear feedback theory. Numerical simulations show that this scheme can be effectively used to control chaotic states in this junction into stable periodic states. Moreover, the different stable period states with different period numbers can be obtained by appropriately adjusting the feedback intensity and delay time without any pre-knowledge of this system required.

Controlling chaos in power system based on finite-time stability theory

Recent investigations show that a power system is a highly nonlinear system and can exhibit chaotic behaviour leading to a voltage collapse, which severely threatens the secure and stable operation of the power system. Based on the finite-time stability theory, two control strategies are presented to achieve finite-time chaos control. In addition, the problem of how to stabilize an unstable nonzero equilibrium point in a finite time is solved by coordinate transformation for the first time. Numerical simulations are presented to demonstrate the effectiveness and the robustness of the proposed scheme. The research in this paper may help to maintain the secure operation of power systems.

Passive adaptive control of chaos in synchronous reluctance motor

The performance of synchronous reluctance motor （SynRM） degrades due to chaos when its systemic parameters fall into a certain area. To control the undesirable chaos in SynRM, a passive control law is presented in this paper, which transforms the chaotic SynRM into an equivalent passive system. It is proved that the equivalent system can be asymptotically stabilized at the set equilibrium point, namely, chaos in SynRM can be controlled. Moreover, in order to eliminate the influence of undeterministic parameters, an adaptive law is introduced into the designed controller. Computer simulation results show that the proposed controller is very effective and robust against the uncertainties in systemic parameters. The present study may help to maintain the secure operation of industrial servo drive system.

A survey on delayed feedback control of chaos

This paper introduces the basic idea and provides the mathematical formulation of the delayed feedback control （DFC） methodology, which has been widely used in chaos control. Stability analysis including the well-known odd number linfitation of the DFC is reviewed. Some new developments in characterizing the limitation of the DFC are presented. Various modified DFC methods, which are developed in order to overcome the odd number limitation, are also described. Finally, some open problems in this research field are discussed.

Pole placement method of controlling chaos in DC-DC buck converters

Based on the mechanism for the generation of chaos in a buck converter, a pole placement method is proposed and applied to controlling the chaos in a circuit. The control circuit is designed and tested. Numerical calculation and circuit implementation demonstrate the validity of this chaos control method.

Hybrid control of bifurcation and chaos in stroboscopic model of Internet congestion control system

Interaction between transmission control protocol （TCP） and random early detection （RED） gateway in the Internet congestion control system has been modelled as a discrete-time dynamic system which exhibits complex bifurcating and chaotic behaviours. In this paper, a hybrid control strategy using both state feedback and parameter perturbation is employed to control the bifurcation and stabilize the chaotic orbits embedded in this discrete-time dynamic system of TCP/RED. Theoretical analysis and numerical simulations show that the bifurcation is delayed and the chaotic orbits are stabilized to a fixed point, which reliably achieves a stable average queue size in an extended range of parameters and even completely eliminates the chaotic behaviour in a particular range of parameters. Therefore it is possible to decrease the sensitivity of RED to parameters. By using the hybrid strategy, we may improve the stability and performance of TCP/RED congestion control system significantly.