Dynamics and Feedback Control for a New Hyperchaotic System

In this paper, a novel control method that combines passive, linear feedback, and dislocated feedback control methods is proposed and applied to the control of the four-dimensional hyperchaotic finance system which has been introduced and controlled with the linear feedback and speed feedback control methods by Yu, Cai, and Li (2012). The stability of the hyperchaotic finance system at its equilibrium points is ensured on the basis of a Lyapunov function. Computer simulations are used for verifying all the theoretical analyses visually. In the simulations, the proposed control method is also compared with the speed feedback and linear feedback control methods to observe its effectiveness and the results are discussed. DOI: http://dx.doi.org/10.5755/j01.itc.47.1.16898

In this article, a novel four dimensional autonomous nonlinear systezm called hyperchaotic Rikitake system is proposed. Basic properties of the new system are investigated and the complex dynamical behaviors, such as time series, bifurcation diagram, and Lyapunov exponents are analyzed by dynamic analysis approaches. To control the new hyperchaotic system, the delayed feedback control is introduced. Regarding the time delay as a bifurcation parameter, stability and bifurcations with respect to time delay are investigated. Conditions assuring the existence of Hopf bifurcation and the distribution of roots to the associated characteristic equation are investigated by utilizing the polynomial theorem. Besides, the Hopf bifurcation is proved to occur when the bifurcation parameter (time delay) crosses through derived critical value. Finally, numerical simulations are provided to prove the consistence with the derived theoretical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 180–193, 2016

This paper reports a new hyperchaotic system by adding an additional state variable into a three-dimensional chaotic dynamical system, studies some of its basic dynamical properties, such as the hyperchaotic attractor, Lyapunov exponents, bifurcation diagram and the hyperchaotic attractor evolving into periodic, quasi-periodic dynamical behaviours by varying parameter κ. Furthermore, effective linear feedback control method is used to suppress hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbits. Numerical simulations are presented to show these results.

The global finite-time synchronization control of a type of fourth-order hyperchaotic systems with intersecting nonlinearities was studied. Firstly, the matrix form of this kind of hyperchaotic system was given to construct the master-slave synchronization models based on the generalized linear state error feedback controller. Then, the finite-time synchronization problem was equivalent to the finite-time stability of the error system originating from the master-slave hyperchaotic systems. The global finite-time stability of the error system was proven by the finite-time stability theory, further obtaining the criterion for global finite-time synchronization and the synchronization time estimation in mathematical formula. Subsequently, the theoretical results were applied to the well-known hyperchaotic Lorenz-stenflo system. The finite-time synchronization criteria for hyperchaotic Lorenz-stenflo systems were further proven by using optimization technology, and the corresponding synchronization times were estimated. Finally, the simulation was conducted by the MATLAB software, showing that two hyperchaotic Lorenz-stenflo systems could synchronize in a finite time.

In this paper, the control of chaos with a single controller, which provides simplicity in implementation, is investigated in continuous time nonlinear Rucklidge system. For this purpose, a linear feedback controller and a passive controller are constructed and added to the Rucklidge chaotic system. Lyapunov function is used to realize that the controller ensures the global asymptotic stability of the system. Owing to the controller, Rucklidge chaotic system can be regulated to its equilibrium points. Numerical simulations of the proposed methods and local relay control method have been demonstrated, compared and discussed.

In high power applications, proper gate control strategy for insulated gate bipolar transistors (IGBTs) can increase their efficient and reliability. As to the feed-back control strategy, the key issues are the control speed and control stability. This paper proposed a new feed-back control method integrated in digital gate driver for high power IGBT. It aims for controlling the current and voltage slope suppressing the voltage overshoot and electromagnetic interference (EMI) in switching transient. The digital gate driver for large power IGBT detects the collector current and voltage in real time. Then the detected analog signal is digitally sampled and sent to the digital core on board. An algorithm similar to bang-bang control is deployed in forward control loop to adjust the gate drive resistance. The control stability and robustness can be guaranteed as a result. The whole detection and control process is accomplished in real time with only hundreds of nanoseconds. Thus, the proposed control method can optimized the IGBT switching transient with high reliability and time effectiveness. Experimental results validate the feasibility of the proposed feed-back digital control method.

Supply Chain Management (SCM) is a critical business function that involves the planning, coordination, and control of the flow of goods, information, and finances as they move from the manufacturer to the wholesaler to the retailer and finally to the end customer. SCM is a holistic approach to managing the entire process of delivering products or services to consumers. In this study, we will enhance the findings as outlined in Anne et al. (2009). While certain attributes of these systems will have been investigated, numerous aspects of these systems will still require further scrutiny. This calls for additional research studies on these systems. This paper examines a Fractional-Order Supply Chain Management (FOSCM) model utilizing the Adomian Decomposition Method (ADM) and explores qualitative aspects through an approach that addresses existence and uniqueness. By using Arzelà–Ascoli's principle, this system proves that the Caputo FOSCM model has at least one solution. Furthermore, we investigate the dynamics of the system by using the Lyapunov Exponent (LE), Bifurcation Diagram (BD), Complexity Analysis (CA) and 0–1 test. Finally, we introduce the control for FOSCM model using the Linear Feedback Control (LFC) method. We verify the correctness of our analysis by using numerical simulations.

In this work, the authors investigate the stability conditions in a fractional-order financial system using the fractional Routh-Hurwitz criteria. According to the qualitative theory, the existence and uniqueness of solutions for a class of commensurate fractional-order financial systems are investigated. A necessary condition for this system to remain chaotic is obtained. It is found that chaos exists in this system with order less than three. Furthermore, the fractional Routh-Hurwitz conditions are used to control chaos in the proposed fractional-order system to its equilibria via linear feedback control method. Then, it is shown that the fractional-order financial system is controllable just in the fractional-order case when using a specific choice of linear controllers. Numerical simulations are used to verify the analytical results.

Hyperchaos has important applications in many branches of science and engineering. In this work, we propose a new 4-D hyperchaotic system with three quadratic nonlinearities by modifying the dynamics of hyperchaotic Wang system (Wang et al. 2010). The proposed new hyperchaotic system has a unique equilibrium at the origin, which is a saddle point and unstable. Thus, the new hyperchaotic system exhibits self-excited hyperchaotic attractor. We describe qualitative properties of the new hyperchaotic system such as symmetry, Lyapunov exponents, Kaplan-Yorke dimension, etc. Furthermore, an active control method is derived for the synchronization of two identical new hyperchaotic systems. The circuit experimental results of the new hyperchaotic system show agreement with the numerical simulations.

This paper investigates the hybrid synchronization of identical hyperbolic Qi systems and hybrid synchronization of hyperchaotic Qi and Lu systems. The hyperchaotic Qi system (2008) and hyperchaotic Lu system (2006) are important models of hyperchaotic systems. Hybrid synchronization of the hyperchaotic systems is achieved through synchronization of two pairs of states and anti-synchronization of the other two pairs of states of the two hyperchaotic systems. Nonlinear control is the method used for the hybrid synchronization of identical hyperbolic Qi systems and hybrid synchronization of hyperchaotic Qi and Lu systems. Since the Lyapunov exponents are not required for these calculations, this method is effective and convenient to achieve hybrid synchronization of the two hyperchaotic systems. Numerical simulations are shown to verify the results.

In this paper, a new hyperchaotic finance system which is constructed based on a chaotic finance system by adding an additional state variable is presented. The basic dynamical behaviors of this hyperchaotic finance system are investigated, such as the equilibrium, stability, hyperchaotic attractor, Lyapunov exponents, and bifurcation analysis. Furthermore, effective speed feedback controllers and linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium points. Numerical simulations are given to illustrate and verify the results.

The synchronisation problem of Lorenz system is further investigated in this paper. On the basis of the Lyapunov stability theory, some simple and useful criteria for synchronisation to be delay-independent asymptotically stable are presented via combining drive and Delayed Feedback Control (DFC) methods. Especially, only one controller is needed to obtain synchronisation asymptotically. Theoretical analysis and numerical simulations show the effectiveness and feasibility of the proposed schemes.

In this paper, design of SMC scheme based controller is proposed for synchronization of two different hyperchaotic systems. Synchronization is achieved using the Lyapunov stability theory. A proportional integral (PI) switching surface is used to ensure the stability of the closed-loop error dynamics in sliding motion. This SMC scheme is effective and guarantees the occurrence of sliding motion and achieves synchronization of master hyperchaotic Xu and slave hyperchaotic Lorenz system. Combinations of two different hyperchaotic systems are used for the synchronization reflecting the novelty of paper. Finally, numerical simulations are performed to demonstrate the effectiveness of the proposed control strategy.

The refuge of prey species is a biological factor necessary for the coexistence of the species and hence it is yet another factor of great interest due to defensive properties of the prey against the predation. We propose a class of discrete differential algebraic prey predator system with Holling Type II functional response and prey refuge. The system is found to possess trivial, semi-trivial and interior equilibrium states existing under certain conditions. Also we examine the existence and uniqueness of the solutions of the system. Dynamical behaviour of the system is investigated through linear stability analysis of the equilibrium states. Moreover, this system undergoes bifurcation when the chosen parameter passes through a critical value, and closed invariant curves arise from a stable equilibrium state. The analytical results are strengthened with appropriate numerical examples and they demonstrate chaotic behaviours over a wild range of parameters. The computation of maximal Lyapunov exponents confirms the existence of chaos. Finally chaos control is achieved by linear feedback control and hybrid control methods.

A novel robust fuzzy controller design method is proposed to stabilize a class of chaotic (hyperchaotic) systems with uncertain parameters based on their equivalent T-S fuzzy models. In this method, the interval system theory is applied to deal with the parametric uncertainty firstly, and then a fuzzy state feedback controller is designed to stabilize the equilibrium of the uncertain chaotic (hyperchaotic) systems robustly based on Exact Linearization (EL) theory and Parallel Distributed Compensation (PDC) technique. Moreover, the sufficient criterion for the robust stability of the equilibrium of the controlled uncertain chaotic (hyperchaotic) systems is derived by Lyapunov stability theory and LMI technique. The designed controller with simple structure and rapid response can stabilize many kinds of uncertain chaotic or hyperchaotic systems. Simulation results are presented to demonstrate the effectiveness of the proposed method.