Chaos generalized synchronization of new Mathieu–Van der Pol systems with new Duffing–Van der Pol systems as functional system by GYC partial region stability theory

In this paper, a new strategy by using GYC partial region stability theory is proposed to achieve generalized chaos synchronization. via using the GYC partial region stability theory, the new Lyapunov function used is a simple linear homogeneous function of states and the lower order controllers are much more simple and introduce less simulation error. Numerical simulations are given for new Mathieu–Van der Pol system and new Duffing–Van der Pol system to show the effectiveness of this strategy.     

Chaos and synchronization of the fractional-order Chua’s system

Chaotic synchronization of fractional-order Chua’s system is further studied. An algorithm for numerical solution of fractional-order differential equations is presented; the chaos in a fractional-order Chua system with some parameters is discussed. The scheme of synchronization system consist of fractional-order Chua’s system is constructed. The synchronization conditions are investigated theoretically. And the synchronization thresholds are discussed by utilizing bifurcation graphs.     

Exact Solutions and Integrability of the Duffing – Van der Pol Equation

The force-free Duffing–Van der Pol oscillator is considered. The truncated expansions for finding the solutions are used to look for exact solutions of this nonlinear ordinary differential equation. Conditions on parameter values of the equation are found to have the linearization of the Duffing–Van der Pol equation. The Painlevé test for this equation is used to study the integrability of the model. Exact solutions of this differential equation are found. In the special case the approach is simplified to demonstrate that some well-known methods can be used for finding exact solutions of nonlinear differential equations. The first integral of the Duffing–Van der Pol equation is found and the general solution of the equation is given in the special case for parameters of the equation. We also demonstrate the efficiency of the method for finding the first integral and the general solution for one of nonlinear second-order ordinary differential equations.     

Chaos control of Bonhoeffer–van der Pol oscillator using neural networks

This paper addresses the control of chaos using a neural network for a continuous time dynamical system. The neural network is trained on both the Ott–Grebogi–Yorke (OGY) control algorithm and the Pyragas's delayed feedback control algorithm. The system considered for this study is a Bonhoeffer–van der Pol (BVP) oscillator. A feed-forward backpropagating neural network is used for the control application. It is found that the control effected by the neural network trained on the OGY control algorithm results in smaller control transients than when the control is effected directly by the OGY algorithm itself. The control transients are of the same order in the case of the Pyragas method.     

Suppression of hysteresis in a forced van der Pol–Duffing oscillator

This paper examines the suppression of hysteresis in a forced nonlinear self-sustained oscillator near the fundamental resonance. The suppression is studied by applying a rapid forcing on the oscillator. Analytical treatment based on perturbation analysis is performed to capture the entrainment zone, the quasiperiodic modulation domain and the hysteresis area as well. The analysis leads to a strategy for the suppression of hysteresis occurring between 1:1 frequency-locked motion and quasiperiodic response. This hysteresis suppression causes the disappearance of nonlinear effects leading to a smooth transition between the quasiperiodic and the frequency-locked responses. Specifically, it appears that a rapid forcing introduces additional apparent nonlinear stiffness which can effectively suppress hysteresis in a certain range of the rapid excitation frequency. This work was motivated by the important issue of controlling and eliminating hysteresis often undesirable in mechanical systems, in general, and in application to microscale devices, especially.     

Multistability and fast-slow analysis for van der Pol–Duffing oscillator with varying exponential delay feedback factor

Time delays are many sources of complex behavior in dynamical systems. Yet its relationship with bursting dynamics needs to be further explored, particularly when the strength of feedback is a nonlinear function of delay. In this paper, we analyze the dynamics of the van der Pol–Duffing fast-slow oscillator controlled by the parametric delay feedback, where the strength of feedback control is a function exponential varying with the time delay. The system may exhibit a unique equilibrium point and three ones for the different parameters by employing the pitchfork bifurcation. Next, the stability-switches and the Hopf bifurcation curves are presented as the delay varies, which leads to the occurrence of novel bursting phenomena. Some weak resonant or non-resonant double Hopf bursting oscillations are presented due to the vanishing of real parts of two pairs of characteristic roots. Not only the magnitude of the time delay itself but also the strength of feedback control may influence the dynamical evolution process of bursting behaviors in the delayed system. Such fast-slow forms about bursting dynamics, as well as classifications about local dynamics are investigated. Furthermore, periodic and quasi-periodic bursting motions are verified in both theoretical and numerical ways.     

Duffing–van der Pol oscillator type dynamics in Murali–Lakshmanan–Chua (MLC) circuit

We have constructed a simple second-order dissipative nonautonomous circuit exhibiting ordered and chaotic behaviour. This circuit is the well known Murali–Lakshmanan–Chua(MLC) circuit but with diode based nonlinear element. For chosen circuit parameters this circuit admits familiar MLC type attractor and also Duffing–van der Pol circuit type chaotic attractors. It is interesting to note that depending upon the circuit parameters the circuit shows both period doubling route to chaos and quasiperiodic route to chaos. In our study we have constructed two-parameter bifurcation diagrams in the forcing amplitude–frequency plane, one parameter bifurcation diagrams, Lyapunov exponents, 0–1 test and phase portrait. The performance of the circuit is investigated by means of laboratory experiments, numerical integration of appropriate mathematical model and explicit analytic studies.     

Effect of fast harmonic excitation on frequency-locking in a van der Pol–Mathieu–Duffing oscillator

We study the effect of high-frequency harmonic excitation on the entrainment area of the main resonance in a van der Pol–Mathieu–Duffing oscillator. An averaging technique is used to derive a self- and parametrically driven equation governing the slow dynamic of the oscillator. The multiple scales method is then performed on the slow dynamic near the main resonance to obtain a reduced autonomous slow flow equations governing the modulation of amplitude and phase of the slow dynamic. These equations are used to determine the steady state response, bifurcation and frequency–response curves. A second multiple scales expansion is used for each of the dependent variables of the slow flow to obtain slow slow flow modulation equations. Analysis of non-trivial equilibrium of this slow slow flow provides approximation of the slow flow limit cycle corresponding to quasi-periodic motion of the slow dynamic of the original system. Results show that fast harmonic excitation can change the nonlinear characteristic spring behavior and affect significantly the entrainment region. Numerical simulations are used to confirm the analytical results.     

Projective synchronization of chaotic fractional-order energy resources demand–supply systems via linear control

A fractional-order energy resources demand–supply system is proposed. A projective synchronization scheme is proposed as an extension on the synchronization scheme of Odibat et al. (2010). The scheme is applied to achieve projective synchronization of the chaotic fractional-order energy resource demand–supply systems. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.     

Optimal control and synchronization of Lotka–Volterra model

In this paper we discuss the problem of optimal control for the steady state of Lotka–Volterra model. The conditions of the asymptotic stability of the steady state of this model are used to obtain the optimal control functions. In such study, the optimal Lyapunov function is used. The general solution of the equations of the perturbed state is obtained as a function of time. In addition, the optimal control is also applied to achieve the state synchronization of two identical Lotka–Volterra systems. Graphical and numerical simulation studies of the obtained results are presented.     

Chaos control of fractional order Rabinovich–Fabrikant system and synchronization between chaotic and chaos controlled fractional order Rabinovich–Fabrikant system

In this article the local stability of the Rabinovich–Fabrikant (R–F) chaotic system with fractional order time derivative is analyzed using fractional Routh–Hurwitz stability criterion. Feedback control method is used to control chaos in the considered fractional order system and after controlling the chaos the authors have introduced the synchronization between fractional order non-chaotic R–F system and the chaotic R–F system at various equilibrium points. The fractional derivative is described in the Caputo sense. Numerical simulation results which are carried out using Adams–Boshforth–Moulton method show that the method is effective and reliable for synchronizing the systems.     

Bifurcation analysis and chaos control of the modified Chua’s circuit system

From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.     

Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators

In this paper, we investigate the damping characteristics of two Duffing–van der Pol oscillators having damping terms described by fractional derivative and time delay respectively. The residue harmonic balance method is presented to find periodic solutions. No small parameter is assumed. Highly accurate limited cycle frequency and amplitude are captured. The results agree well with the numerical solutions for a wide range of parameters. Based on the obtained solutions, the damping effects of these two oscillators are investigated. When the system parameters are identical, the steady state responses and their stability are qualitatively different. The initial approximations are obtained by solving a few harmonic balance equations. They are improved iteratively by solving linear equations of increasing dimension. The second-order solutions accurately exhibit the dynamical phenomena when taking the fractional derivative and time delay as bifurcation parameters respectively. When damping is described by time delay, the stable steady state response is more complex because time delay takes past history into account implicitly. Numerical examples taking time delay and fractional derivative are respectively given for feature extraction and convergence study.     

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.     

On an Extension and a Refinement of Van der Corput's Inequality

By introducing a parameterα,we give an extension of Van der Corput's inequal- ity.Also a new strengthened version of it is considered.     

Convergence and permanence of a delayed Nicholson’s Blowflies model with feedback control

In this paper, we study a generalized Nicholson??s Blowflies model with feedback control and multiple time-varying delays. Under proper conditions, we employ a novel proof to establish some criteria to guarantee the global exponential convergence and permanence of this model. Moreover, we give two examples to illustrate our main results.     

On an Extension and a Refinement of Van der Corput＇s Inequality

By introducing a parameterα,we give an extension of Van der Corput's inequal- ity.Also a new strengthened version of it is considered.     

Almost periodic sequence solutions of a discrete Lotka–Volterra competitive system with feedback control

In this paper, we consider a discrete Lotka–Volterra competitive system with feedback control. Assuming that the coefficients in the system are almost periodic sequences, we obtain the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.     

Optimizing the dynamics of a two-cell DC–DC buck converter by time delayed feedback control

A study of the dynamical behavior of a two-cell DC–DC buck converter under a digital time delayed feedback control (TDFC) is presented. Various numerical simulations and dynamical aspects of this system are illustrated in the time domain and in the parameter space. Without TDFC, the system may present many undesirable behaviors such as sub-harmonics and chaotic oscillations. TDFC is able to widen the stability range of the system. Optimum values of parameters giving rise to fast response while maintaining stable periodic behavior are given in closed form. However, it is detected that in a certain region of the parameter space, the stabilized periodic orbit may coexist with a chaotic attractor. Boundary between basins of attraction are obtained by means of numerical simulations.     

Bifurcation and control of a bioeconomic model of a prey–predator system with a time delay

In this paper, we analyze the dynamical behaviour of a bioeconomic model system using differential algebraic equations. The system describes a prey–predator fishery with prey dispersal in a two-patch environment, one of which is a free fishing zone and other is a protected zone. It is observed that a singularity-induced bifurcation phenomenon appears when a variation of the economic interest of harvesting is taken into account. We have incorporated a state feedback controller to stabilize the model system in the case of positive economic interest. A discrete-type gestational delay of predators is incorporated, and its effect on the dynamical behaviour of the model is analyzed. The occurrence of Hopf bifurcation of the proposed model with positive economic profit is shown in the neighbourhood of the coexisting equilibrium point through considering the delay as a bifurcation parameter. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.