Steady-state,self-oscillating and chaotic behavior of a PID controlled nonlinear servomechanism by using Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations

This paper analyzes a controlled servomechanism with feedback and a cubic nonlinearity by means of the Bogdanov–Takens and Andronov–Poincaré–Hopf bifurcations, from which steady-state, self-oscillating and chaotic behaviors will be investigated using the center manifold theorem. The system controller is formed by a Proportional plus Integral plus Derivative action (PID) that allows to stabilize and drive to a prescribed set point a body connected to the shaft of a DC motor. The Bogdanov–Takens bifurcation is analyzed through the second Lyapunov stability method and the harmonic-balance method, whereas the first Lyapunov value is used for the Andronov–Poincaré–Hopf bifurcation. On the basis of the results deduced from the bifurcation analysis, we show a procedure to select the parameters of the PID controller so that an arbitrary steady-state position of the servomechanism can be reached even in presence of noise. We also show how chaotic behavior can be obtained by applying a harmonical external torque to the device in self-oscillating regime. The advantage of achieving chaotic behavior is that it can be used so that the system reaches a set point inside a strange attractor with a small control effort. The analytical calculations have been verified through detailed numerical simulations.     

Quick estimation of escape rate with the help of fractal dimension

The non-linearity caused by the application of digital control may lead to transient chaotic behaviour. In the present paper, we analyse a simple model of a digitally controlled mechanical system with dry friction, which may perform transient chaotic vibrations. As a consequence of the digital effects, the behaviour of this system can be described by a 1D piecewise linear map. The fractal dimension of the so-called chaotic repeller set is calculated and the results are used for the quick estimation of the mean lifetime of chaotic transients.     

Simple adaptive variable structure control for unknown chaotic systems

This paper proposes two novel adaptive variable structure tracking controllers for a large class of chaotic systems with unknown dynamics in presence of both external disturbances and input nonlinearities. The pros and cons of each proposed methodology is also represented. In order to eliminate the chattering effect in the former controlled system, two corresponding fuzzy adaptive controllers are presented. Besides, synchronization of two non-identical uncertain chaotic systems is investigated using our proposed methods in both full and reduced-order forms. It can be seen that not only our proposed control schemes can be applied to a wide class of uncertain chaotic systems but also it is simple to implement in practical application. Finally, the proposed methods are applied to some famous chaotic systems to verify the effectiveness of the proposed methods.     

Controlling chaos in space-clamped FitzHugh–Nagumo neuron by adaptive passive method

The space-clamped FitzHugh–Nagumo (SCFHN) neuron exhibits complex chaotic firing when the amplitude of the external current falls into a certain area. To control the undesirable chaos in SCFHN neuron, a passive control law is presented in this paper, which transforms the chaotic SCFHN neuron into an equivalent passive system. It is proved that the equivalent system can be asymptotically stabilized at any desired fixed state, namely, chaos in SCFHN neuron can be controlled. Moreover, 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 uncertainty in systemic parameters.     

Controlling the chaotic response to a prospective external signal using back-propagation neural networks

For a chaotic system, a control scheme is presented, based on the back-propagation neural network (BPNN). The scheme can control the chaotic response to a prospective external signal, which can be periodic, nonlinear or even a non-analytical discontinuous function. For a chaotic system with high dimensions, each variable can be controlled for the different signals. For Lorenz, Rossler and Duffing systems, simulations are carried out and the proposed scheme is proved to be effective within a short control time.     

Chaotic Behavior in Slowly Varying Systems of Nonlinear Differential Equations

A technique is developed to find parameter regions of chaotic behavior in certain systems of nonlinear differential equations with slowly varying periodic coefficients. The technique combines previous results on how to find branches of periodic solutions which terminate with a homoclinic orbit and results on how to find chaotic trajectories in the neighborhood of homoclinic trajectories of the autonomous system. The technique is applied to the continuous stirred tank reaction A → B, for which it is shown that a slowly varying periodic flow rate can yield aperiodic temperature fluctuations.     

Feedback control and adaptive control of the energy resource chaotic system

In this paper, the problem of control for the energy resource chaotic system is considered. Two different method of control, feedback control (include linear feedback control, non-autonomous feedback control) and adaptive control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. The Routh–Hurwitz criteria and Lyapunov direct method are used to study the conditions of the asymptotic stability of the steady states of the controlled system. The designed adaptive controller is robust with respect to certain class of disturbances in the energy resource chaotic system. Numerical simulations are presented to show these results.     

Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.     

Global dynamics for a model of a class of continuous‐time dynamical systems

In this paper, the global asympotic behavior of solutions of a class of continuous‐time dynamical system is studied. Not only do we obtain the ultimate boundedness of solutions of the system but we also obtain the rate of the trajectories of the system going from the exterior of the trapping set to the interior of the trapping set, which can be applied to study chaotic control and chaotic synchronization of the system. Copyright © 2015 John Wiley & Sons, Ltd.     

Study on chaos control of second-order non-autonomous phase-locked loop based on state observer

With system parameters falling into a certain area, the second-order non-autonomous phase locked loop (PLL) is experiencing chaotic behavior which is undesirable in system, where it is necessary to estimate the phase of a received signal. In order to control chaos in PLL and drive it to the locked state, dynamical equation for phase error model of PLL is firstly derived. Then, the state values of phase and transient frequency errors were estimated by a state observer. Moreover, by exploiting these state estimations, a non-linear feedback controller is designed. Since the presented controller does not need to change the controlled system structure and not to use any information of system except the system state variables, the designed controller is simple and desirable. Simulation results show that the presented control law is very effective.     

States on systems of sets that are closed under symmetric difference

We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set‐representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non‐Boolean situations. On the other hand, a difference‐closed system can be constructed such that even two‐valued states do not allow for extensions. Finally, we consider these questions in a σ‐complete setup and find a large class of such systems with rather interesting state properties.     

Adaptive control of discrete-time chaotic systems: a fuzzy control approach

This paper discusses adaptive control of a class of discrete-time chaotic systems from a fuzzy control approach. Using the T–S model of discrete-time chaotic systems, an adaptive control algorithm is developed based on some conventional adaptive control techniques. The resulting adaptively controlled chaotic system is shown to be globally stable, and its robustness is discussed. A simulation example of the chaotic Henon map control is finally presented, to illustrate an application and the performance of the proposed control algorithm.     

Hopf bifurcation in a control system for the Washout filter-based delayed neural equation

Washout filter is a simple filter that can be designed easily. In this paper, a system for controlling a neural equation with discrete time delay based on Washout filter is presented. The transcendental equation of the corresponding linearized system is analyzed. In this control system, it is found that Hopf bifurcation occurs when the control parameters are chosen properly and that a chaotic orbit can be controlled to a stable periodic solution. The stability condition for bifurcating periodic solutions and the direction of Hopf bifurcation are studied by applying the normal form theory and the center manifold theorem. Some numerical results are also presented to illustrate the correctness of our results.     

A novel GCM chaotic neural network for information processing

A new chaotic neural network named “globally coupled map using sine map(SI-GCM)”, which is a modified Kaneko’s globally coupled map model, is proposed. With the introduction of sine map and chaotic neurons’ different way of coupling, it exhibits rich dynamic behaviors. By adopting a variable threshold parameter control method, it can be controlled to specified-period orbit. Furthermore, the controlled SI-GCM has excellent associative memory performance. It can not only output unique fixed pattern, but also output periodic patterns which contain the stored pattern closest to the initial pattern. Simulation results suggest that SI-GCM is fit for information processing.     

Complex dynamics of an eco-epidemiological model with different competition coefficients and weak Allee in the predator

The paper explores an eco-epidemiological model with weak Allee in predator, and the disease in the prey population. We consider a predator-prey model with type II functional response. The curiosity of this paper is to consider different competition coefficients within the prey population, which leads to the emergent carrying capacity. We perform the local and global stability analysis of the equilibrium points and the Hopf bifurcation analysis around the endemic equilibrium point. Further we pay attention to the chaotic dynamics which is produced by disease. Our numerical simulations reveal that the three species eco-epidemiological system without weak-Allee induced chaos from stable focus for increasing the force of infection, whereas in the presence of the weak-Allee effect, it exhibits stable solution. We conclude that chaotic dynamics can be controlled by the Allee parameter as well as the competition coefficients. We apply basic tools of non-linear dynamics such as Poincare section and maximum Lyapunov exponent to identify chaotic behavior of the system.     

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

A non-linear controlled dynamical system that describes the dynamics of a broad class of non-linear mechanical and electromechanical systems (in particular, electromechanical robot manipulators) is considered. It is proposed that the real parameter vector of a non-linear controlled dynamical system belongs to an assigned (admissible) constrained closed set and is assumed to be unknown. The programmed motion of the non-linear controlled dynamical system and the programmed control that produces it are assigned (constructed) by using an estimate, that is, the nominal value of the parameter vector of the non-linear controlled dynamical system, which differs from its actual value. A procedure for synthesizing stabilizing control laws with linear feedback with respect to the state that ensure stabilization of the programmed motions of the non-linear controlled dynamical system under parametric perturbations is proposed. A non-singular linear transformation of the coordinates of the state space that transforms the original non-linear controlled dynamical system in deviations (from the programmed motion and programmed control) into a certain non-linear controlled dynamical system of special form, which is convenient for analysing and synthesizing laws for controlling the motion of the system, is constructed. A certain non-linear controlled dynamical system of canonical form is derived in the original non-linear controlled dynamical system in deviations. The transformation of the coordinates of the state space constructed and the Lyapunov function methodology are used to synthesize stabilizing control laws with linear feedback with respect to the state, which ensure asymptotic stability as a whole of the equilibrium position of the non-linear controlled dynamical system of canonical form and dissipativity “in the large” of the non-linear controlled dynamical system of special form and of the original non-linear controlled dynamical system in deviations. In the control laws synthesized, the formulae for the elements of their matrices of the feedback loop gains do not depend on the real parameter vector of the non-linear controlled dynamical system, and they depend solely on the constants from certain estimates that hold for all of its possible values from an assigned set. Estimates of the region of dissipativity “in the large” of the non-linear controlled dynamical system of special form and the original non-linear controlled dynamical system in deviations closed by the stabilizing control laws synthesized are given, and estimates for their limit sets and regions of attraction are presented.     

Chaos in temporarily destabilized regular systems with the slow passage effect

We provide evidences for chaotic behaviour in temporarily destabilized regular systems. In particular, we focus on time-continuous systems with the slow passage effect. The extreme sensitivity of the slow passage phase enables the existence of long chaotic transients induced by random pulsatile perturbations, thereby evoking chaotic behaviour in an initially regular system. We confirm the chaotic behaviour of the temporarily destabilized system by calculating the largest Lyapunov exponent. Moreover, we show that the newly obtained unstable periodic orbits can be easily controlled with conventional chaos control techniques, thereby guaranteeing a rich diversity of accessible dynamical states that is usually expected only in intrinsically chaotic systems. Additionally, we discuss the biological importance of presented results.     

Chaos control and generalized projective synchronization of heavy symmetric chaotic gyroscope systems via Gaussian radial basis adaptive variable structure control

This paper proposes the chaos control and the generalized projective synchronization methods for heavy symmetric gyroscope systems via Gaussian radial basis adaptive variable structure control. 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. In this paper, the switching surfaces are adopted to ensure the stability of the error dynamics in variable structure control. Using the neural variable structure control technique, control laws are established which guarantees the chaos control and the generalized projective synchronization of unknown gyroscope systems. In the neural variable structure control, Gaussian radial basis functions are utilized to on-line estimate the system dynamic functions. Also, the adaptation laws of the on-line estimator are derived in the sense of Lyapunov function. Thus, the unknown gyro systems can be guaranteed to be asymptotically stable. Also, the proposed method can achieve the control objectives. Numerical simulations are presented to verify the proposed control and synchronization methods. Finally, the effectiveness of the proposed methods is discussed.     

Chaos suppression on a class of uncertain nonlinear chaotic systems using an optimal H∞ adaptive PID controller

This paper introduces an optimal H_∞ adaptive PID (OHAPID) control scheme for a class of nonlinear chaotic system in the presence system uncertainties and external disturbances. Based on Lyapunov stability theory, it is shown that the proposed control scheme can guarantee the stability robustness of closed-loop system with H_∞ tracking performance. In the core of proposed controller, to achieve an optimal performance of OHAPID, the Particle Swarm Optimization (PSO) algorithm is utilized. To show the feasibility of proposed OHAPID controller, it is applied on the chaotic gyro system. Simulation results demonstrate that it has highly effective in providing an optimal performance.     

Dynamical behavior,chaos control and synchronization of a memristor-based ADVP circuit

This paper is devoted to study the dynamical behavior of a modified Autonomous Van der Pol-Duffing (ADVP) circuit when its nonlinear element is replaced by a flux controlled memristor. The bifurcation diagrams, Lyapunov exponents, and phase portraits of the state variables are presented. Then, the chaos which appears at certain values of the system’s parameters is controlled using linear feedback control. Finally, the synchronization between two chaotic modified ADVP circuits is achieved in the case of fully unknown parameters of the system using adaptive synchronization.