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This paper discusses a fractional-order prey–predator system with Gompertz growth of prey population in terms of the Caputo fractional derivative. The non-negativity and boundedness of the solutions of the considered model are successfully analyzed. We utilize the Mittag-Leffler function and the Laplace transform to prove the boundedness of the solutions of this model. We describe the topological categories of the fixed points of the model. It is theoretically demonstrated that under certain parametric conditions, the fractional-order prey–predator model can undergo both Neimark–Sacker and period-doubling bifurcations. The piecewise constant argument approach is invoked to discretize the considered model. We also formulate some necessary conditions under which the stability of the fixed points occurs. We find that there are two fixed points for the considered model which are semi-trivial and coexistence fixed points. These points are stable under some specific constraints. Using the bifurcation theory, we establish the Neimark–Sacker and period-doubling bifurcations under certain constraints. We also control the emergence of chaos using the OGY method. In order to guarantee the accuracy of the theoretical study, some numerical investigations are performed. In particular, we present some phase portraits for the stability and the emergence of the Neimark–Sacker and period-doubling bifurcations. The biological meaning of the given bifurcations is successfully discussed. The used techniques can be successfully employed for other models.

Systems of complex partial differential equations, which include the famous nonlinear Schrödinger, complex Ginzburg–Landau and Nagumo equations, as examples, are important from a practical point of view. These equations appear in many important fields of physics. The goal of this paper is to concentrate on this class of complex partial differential equations and study the fixed points and their stability analytically, the chaotic behavior and chaos control of their unstable periodic solutions. The presence of chaotic behavior in this class is verified by the existence of positive maximal Lyapunov exponent.The problem of chaos control is treated by applying the method of Pyragas. Some conditions on the parameters of the systems are obtained analytically under which the fixed points are stable (or unstable).

Stochastic forces or random noises have been greatly used in studying the control of chaos of random real systems, but little is reported for random complex systems. Chaotic limit cycles of a complex Duffing–Van der Pol system with a random excitation is studied. Generating chaos via adjusting the intensity of random phase is investigated. We consider the positive top Lyapunov exponent as a criterion of chaos for random dynamical systems. It is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. We demonstrate the stable behavior of deterministic system when noise intensity is zero by means of the top (local) Lyapunov exponent. Poincaré surface analysis and phase plot are used to confirm our results. Later, random noise is used to generate chaos by adjusting the noise intensity to make the top (local) Lyapunov exponent changes from a negative sign to a positive one, and the Poincaré surface analysis is also applied to verify the obtained results and excellent agreement between these results is found.

Studied in this paper is the control problem of hyperchaotic systems. By combining Takagi–Sugeno (T–S) fuzzy model with parallel distributed compensation design technique, we propose a delay-dependent control criterion via pure delayed state feedback. Because the result is expressed in terms of linear matrix inequalities (LMIs), it is quite convenient to check in practice. Based on this criterion, a procedure is provided for designing fuzzy controller for such systems. This method is a universal one for controlling continuous hyperchaotic systems. As illustrated by its application to hyperchaotic Chen's system, the controller design is quite effective.

A weak harmonic parametric excitation with random phase has been introduced to tame chaotic arrays. It has been shown that when the amplitude of random phase properly increases, two different kinds of chaotic arrays, unsynchronized and synchronized, can be controlled by the criterion of top Lyapunov exponent. The Lyapunov exponent was computed based on Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. In particular, it was found that with stronger coupling the synchronized chaotic arrays are more controllable than the unsynchronized ones. The bifurcation analysis, the spatiotemporal evolution, and the Poincaré map were carried out to confirm the results of the top Lyapunov exponent on the dynamical behavior of control stability. Excellent agreement was found between these results.

This paper studies the problem of controlling the chaotic behavior of a modified coupled dynamos system. Two different methods, feedback and non-feedback methods, are used to control chaos in the modified coupled dynamos system. Based on the Lyapunov direct method and Routh–Hurwitz criterion, the conditions suppressing chaos to unstable equilibrium points or unstable periodic orbits (limit cycles) are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the two different methods.

The resistive–capacitive–inductive-shunted (RCL-shunted) Josephson junction (RCLSJJ) shows chaotic behavior under some parameter conditions. This paper studies chaos control in the RCLSJJ based on the delayed feedback theory. Numerical results demonstrate that chaotic states in this junction can be controlled into stable periodic states by appropriately adjusting the feedback intensity and delay time.

In this paper recursive and adaptive backstepping nonlinear controllers are designed to, respectively, control and synchronize the triple-well or Φ⁶ Van der Pol oscillators (VDPOs) with both parametric and external excitations. The designed recursive backstepping nonlinear controller is capable of stabilizing the Φ⁶-VDPO at any position as well as controlling it to track any trajectory that is a smooth function of time. The designed adaptive backstepping nonlinear controller globally synchronizes two Φ⁶-VDPOs evolving from different initial conditions and its application to secure communications is computationally demonstrated. The results are all validated by numerical simulations.

A new fractional order chaotic n-scroll modified Chua circuit is introduced. It can generate n-scroll with a total order less than three. The equilibrium points are classified into two types according to the characteristics of the eigenvalues of the Jacobian matrix at the equilibrium points. To overcome some disadvantages of the traditional fractional order controller, a new fractional order control method is developed for stabilizing the system to any expected equilibrium point. Numerical examples are provided to verify the effectiveness of the proposed scheme.

We present the control and synchronization of spatiotemporal chaos in the photo-refractive ring oscillator systems with coupling technology. First, we realize the synchronization of spatiotemporal chaos in the two photorefractive ring oscillator systems via mutual coupling by choosing a suitable coupling strength. With the mutual coupling strength enlarging, the two mutual coupling photorefractive ring oscillator systems are controlled into periodic state, period number differs on account of the coupling strength and lattice coordinates. By increasing the coupling strength, the photorefractive ring oscillator is converted into period 8, subsequently it is converted into periods 4 and 2, periodic synchronization of the photorefractive ring oscillator systems is achieved at the same time. Calculation results show that period 1 is impossible by mutual coupling technology. Then, we investigate the influence of noise and parameter deviation on chaotic synchronization. We find that mutual coupling chaotic synchronization method can synchronize two chaotic systems with the weak noise and parameter deviation and has very good robustness. Given that the weak noise and parameter deviation have a slight effect on synchronization. Furthermore, we investigate two dimension control and synchronization of spatiotemporal chaos in the photorefractive ring osillator systems with coupling technology and get successful results. Mutual coupling technology is suitable in practical photorefractive ring oscillator systems.

Chaos control of a Bose–Einstein condensate (BEC) loaded into a moving optical lattice with attractive interaction is investigated on the basis of Lyapunov stability theory. Three methods are designed to control chaos in BEC. As a controller, a bias constant, periodic force, or wavelet function feedback is added to the BEC system. Numerical simulations reveal that chaotic behavior can be well controlled to achieve periodicity by regulating control parameters. Different periodic orbits are available for different control parameters only if the maximal Lyapunov exponent of the system is negative. The abundant effect of chaotic control is also demonstrated numerically. Chaos control can be realized effectively by using our proposed control strategies.

Permanent magnet synchronous motor (PMSM) is a multivariable and highly coupled nonlinear system, which is prone to chaotic behavior during actual operation. In order to suppress the chaotic behavior of the PMSM, this work designs a free-will arbitrary time adaptive controller based on the inverse step derivation, adaptive control theorem and Lyapunov stability theory. Firstly, the corresponding coordination transformations are designed according to the PMSM. Secondly, the bounded problem of the controlled system is analyzed by using the Lyapunov stability theory, and the free-will arbitrary time adaptive controller is designed by backstepping derivation. Finally, numerical simulation verifies that the designed controller quickly stabilizes the system state variables to the output signals and proves that the controller is free-will arbitrary time stable for all closed-loop signals.

This paper deals with the control of the memristor-based simplest chaotic circuit by means of only one-state controller. Three distinct control techniques, namely linear feedback control, sliding mode control, and nonlinear control, are examined for the control. Routh–Hurwitz stability criteria are used for constructing the linear feedback and sliding mode gains. Lyapunov function is used for ensuring the global asymptotic stability of the system with the nonlinear controller. Numerical simulations are demonstrated not only to validate the theoretical analyses, but also to compare the control results. They have shown that even if one-state controller is used, all the methods are effective for controlling the memristor-based simplest chaotic circuit. However, the control is observed in a better time period with the sliding mode controller.

Following the experimental realization of memristors, researchers have focused on memristor-based circuits. Chaotic circuits can be implemented easily using a memristor due to its nonvolatile and nonlinear behavior. This study presents a memristor-based four-dimensional (4D) chaotic oscillator with a line equilibria. A memristor having quadratic memductance was utilized to implement the proposed chaotic oscillator. The 4D chaotic oscillator with quartic nonlinearity was designed as a result of the quadratic memductance. In terms of communication security, random number generation and image and audio encryption, systems with quartic nonlinearity or that are higher-dimensional are better than systems that are lower-dimensional or possess quadratic/cubic nonlinearity. The performance of the proposed chaotic circuit was investigated according to properties such as phase portraits, Jacobian matrices, equilibrium points, Lyapunov exponents and bifurcation analyses. Furthermore, the proposed system is multistable and its solutions tend to appear as twin attractors when initial conditions approach their equilibria. The Lyapunov-based nonlinear controller was constructed for controlling the proposed system having a line equilibria. The effect of the initial conditions on the controlling indicators was also studied. In conclusion, by using discrete circuit elements, the proposed circuit was constructed, and its experimental results demonstrated a good agreement with the simulation results.

Based on the variable structure model reference adaptive control (VS-MRAC) theory, a new control system for the control of chaos in Lorenz system, using only the measured output variable, is designed. For the derivation of the control law, it is assumed that the parameters of the model are unknown. Moreover, it is assumed that a disturbance input is present in the system. It is shown that in the closed-loop system, the output variable tracks a given reference trajectory, and the state vector converges to the equilibrium state. Digital simulation results show that the closed-loop system has good transient behavior and robustness to the uncertainties and disturbance input.

In this paper, a time-delayed chaos control method based on repetitive learning is proposed. A general repetitive learning control structure based on the invariant manifold of the chaotic system is given. The integration of the repetitive learning control principle and the time-delayed chaos control technique enables adaptive learning of appropriate control actions from learning cycles. In contrast to the conventional repetitive learning control, no exact knowledge (analytic representation) of the target unstable periodic orbits is needed, except for the time delay constant, which can be identified via either experiments or adaptive learning. The controller effectively stabilizes the states of the continuous-time chaos on desired unstable periodic orbits. Simulations on the Duffing and Lorenz chaotic systems are provided to verify the design and analysis.

Though chaotic behaviors are exhibited in many simple nonlinear models, physical chaotic systems are much more complex and contain many types of uncertainties. This paper presents a robust adaptive neural control scheme for a class of uncertain chaotic systems in the disturbed strict-feedback form, with both unknown nonlinearities and uncertain disturbances. To cope with the two types of uncertainties, we combine backstepping methodology with adaptive neural design and nonlinear damping techniques. A smooth singularity-free adaptive neural controller is presented, where nonlinear damping terms are used to counteract the disturbances. The differentiability problem in controlling the disturbed strict-feedback system is solved without employing norm operation, which is usually used in robust control design. The proposed controllers can be applied to a large class of uncertain chaotic systems in practical situations. Simulation studies are conducted to verify the effectiveness of the scheme.

In this Letter, we study the popular parametric variation chaos control and state-feedback methodologies in chaos control, and point out for the first time that they are actually equivalent in the sense that there exist diffeomorphisms that can convert one to the other for most smooth chaotic systems. Detailed conversions are worked out for typical discrete chaotic maps (logistic, Hénon) and continuous flows (Rösller, Lorenz) for illustration. This unifies the two seemingly different approaches from the physics and the engineering communities on chaos control. This new perspective reveals some new potential applications such as chaos synchronization and normal form analysis from a unified mathematical point of view.

In this Letter a feedback linearizing adaptive control system for the control of Chua's circuits is presented. It is assumed that all the parameters of the system are unknown. Using a backstepping design procedure, an adaptive control system is designed which accomplishes trajectory control of the chosen node voltage by an independent voltage source. Simulation results are presented which show precise trajectory tracking and regulation of the state vector to the desired terminal state.

In this paper, a new method, by which any point in a chaotic attractor can be guided to any target periodic orbit, is proposed. The "Middle" periodic orbit is used to lead an initial point in a chaotic attractor to a neighborhood of the target orbit, and then controlling chaos can be achieved by the improved OGY method. The time needed in the method using "Middle" periodic orbit is less than that of the OGY method, and is inversely proportional to the square of the topological entropy of the given map. An example is used to illustrate the results.