Chaos anticontrol and synchronization of three time scales brushless DC motor system

Chaos anticontrol of three time scale brushless dc motors and chaos synchronization of different order systems are studied. Nondimensional dynamic equations of three time scale brushless DC motor system are presented. Using numerical results, such as phase diagram, bifurcation diagram, and Lyapunov exponent, periodic and chaotic motions can be observed. By adding constant term, periodic square wave, the periodic triangle wave, the periodic sawtooth wave, and kx|x| term, to achieve anticontrol of chaotic or periodic systems, it is found that more chaotic phenomena of the system can be observed. Then, by coupled terms and linearization of error dynamics, we obtain the partial synchronization of two different order systems, i.e. brushless DC motor system and rate gyroscope system.     

Dynamical behaviors of the chaotic Brushless DC motors model

This article is concerned with the boundedness solutions of the brushless DC motor system. For this system, the global attractive set and positively invariant set are obtained based on generalized Lyapunov function stability theory and the extremum principle of function. Furthermore, the rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme. © 2014 Wiley Periodicals, Inc. Complexity 21: 79–85, 2016     

Adaptive fuzzy tracking control for the chaotic permanent magnet synchronous motor drive system via backstepping

An adaptive fuzzy control method is developed to suppress chaos in the permanent magnet synchronous motor drive system via backstepping technology. Fuzzy logic systems are used to approximate unknown nonlinearities and an adaptive backstepping technique is employed to construct controllers. Compared with the conventional backstepping, the designed fuzzy controllers’ structure is very simple. The simulation results indicate that the proposed control scheme can suppress the chaos of PMSM drive systems and track the reference signal successfully even under the parameter uncertainties.     

Parameters identification of chaotic system by chaotic gravitational search algorithm

In this paper, for the parameter identification problem of chaotic system, a chaotic gravitational search algorithm (CGSA) is proposed. At first, an iterative chaotic map with infinite collapses is introduced and chaotic local search (CLS) is designed, then CLS and basic gravitational search are combined in the procedure frame. The CGSA is composed of coarse gravitational search and fine chaotic local search, while chaotic search seeks the optimal solution further, based on the current best solution found by the coarse gravitational search. In order to show the effectiveness of CGSA, both offline and online parameter identifications of Lorenz system are conducted in comparative experiments, while the performances of CGSA are compared with GA, PSO and GSA. The results demonstrate the effectiveness and efficiency of CGSA in solving the problem of parameter identification of chaotic system, and the improvement to GSA has been verified.     

On a four-dimensional chaotic system

This paper reports a new four-dimensional continuous autonomous chaotic system, in which each equation in the system contains a 3-term cross product. Basic properties of the system are analyzed by means of Lyapunov exponents and bifurcation diagrams.     

On a new chaotic system

In this paper, a novel 3D nonlinear system with chaotic behavior is considered. Our proposed system is a generalized Lorenz‐like system, with no reflections about coordinates axes. Although the considered system depends on seven real parameters, its stability is completely analyzed. Also, the presence of Hopf bifurcation is pointed out. Copyright © 2014 John Wiley & Sons, Ltd.     

Generalized synchronization of continuous chaotic system

Sufficient condition for the generalized synchronization of continuous chaotic system with a kind of nonlinear transformation is derived. The method is illustrated by applications to Lorenz and Duffing chaotic systems and the simulation results demonstrate the effectiveness of the proposed theorem.     

Synchronization of a chaotic finance system

Synchronization strategies of a three-dimensional chaotic finance system are investigated in this paper. Based on Lyapunov stability theory and Routh-Hurwitz criteria, some effective controllers are designed for the global asymptotic synchronization on different conditions. When the system parameters are known, the hybrid feedback control and a method based on special matrix structure are adopted respectively, to realize the synchronization of the chaotic finance system. When the parameters are unknown, the active control is extended and introduced to realize the synchronization. Numerical simulations show the validity and feasibility of the synchronization schemes.     

Modified projective synchronization of chaotic system

A modified projective synchronization is proposed to acquire a general kind of proportional relationships between the drive and response systems. From rigorously control theory, a sufficient condition is attained for the stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, we take Lorenz system as an example for illustration and verification.     

Synchronization of the unified chaotic system

This paper studies the synchronization problem of the unified chaotic system. Three different methods, linear feedback method, nonlinear feedback method and impulsive control method are used to control synchronization of the unified chaotic systems. Based on the Lyapunov stability theory and impulsive control method, the conditions of synchronization are discussed, and they are also proved theoretically. Numerical simulations show the effectiveness of the three different methods.     

Speed feedback control of chaotic system

Based on the Lyapunov stabilization theory and matrix measure, this paper addresses the strategies of speed feedback control of chaotic system to the unsteadily equilibrium points, illustrated by a unified chaotic system and Rössler chaotic system. It is proved that the infimum of speed feedback control coefficient is less than that of displacement feedback control coefficient.     

Dynamics of a new Lorenz-like chaotic system

The present work is devoted to giving new insights into a new Lorenz-like chaotic system. The local dynamical entities, such as the number of equilibria, the stability of the hyperbolic equilibria and the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations and the local manifold character, are all analyzed when the parameters are varied in the space of parameters. The existence of homoclinic and heteroclinic orbits of the system is also rigorously studied. More exactly, for $b\ge 2a>0$ and $c>0$, we prove that the system has no homoclinic orbit but has two and only two heteroclinic orbits.     

Energy cycle for the Lorenz attractor

In this paper we identify an energetic cycle in the Lorenz-63 system through its Lie–Poisson structure. A new geometrical view of this Lorenz system is presented and sheds light on its energetic properties by recovering its Hamiltonian structure and the associated Casimir. It is shown that this approach gives notable physical insight on the dynamical behaviour of the system. A link between energetics and chaotic properties has been found and specific inequalities, involving conversion terms, have been identified. Traps and jumps between the lobes are studied and benefits of our formalism to capture predictability-energy connections are discussed.     

The parametric synchronization scheme of chaotic system

By constructing the parametric error vectors between drive system and response system, a parametric synchronization scheme of chaotic system which is different from all other schemes is proposed in this paper. Controller of the scheme is designed. The proposed scheme and controller not only realize the synchronization of the state vectors, but also synchronize the unknown response parameters to the given drive parameter as time goes to infinity. That is to say, to achieving the synchronization, we have no need to know the parameters of response system when the parameters of drive system are given. The scheme and controller are successfully applied to the Rössler and the hyperchaotic Rössler systems, corresponding numerical simulations are presented to show the validity of the proposed synchronization scheme and effectiveness of the controller.     

Chaos control of Chen chaotic dynamical system

This paper is devoted to study the problem of controlling chaos in Chen chaotic dynamical system. Two different methods of control, feedback and nonfeedback methods are used to suppress chaos to unstable equilibria or unstable periodic orbits (UPO). The Lyapunov direct method and Routh–Hurwitz criteria are used to study the conditions of the asymptotic stability of the steady states of the controlled system. Numerical simulations are presented to show these results.     

A characteristic frequency of chaotic dynamical system

A simple algorithm for determination of characteristic frequency at which the chaotic system may be synchronized by external force is proposed. This frequency can be defined without application of the synchronizing force to the investigated system, but merely from the time series at its output. The knowledge of synchronization region helps in easy quantification of chaotic motion. The characteristic frequency depends on the parameters of the dynamical system. Hence, its value can be used for the purposes of monitoring and diagnostics.     

Analysis of a 3D chaotic system

A 3D nonlinear chaotic system, called the T system, is analyzed in this paper. Horseshoe chaos is investigated via the heteroclinic Shilnikov method constructing a heteroclinic connection between the saddle equilibrium points of the system. Partially numerical computations are carried out to support the analytical results.     

Hopf bifurcation in the Lorenz-type chaotic system

In this paper, we analyze dynamical behaviors of the Lorenz-type system via the complementary-cluster energy-barrier criterion. Moreover the Hopf bifurcation of this system is also investigated by means of the first Lyapunov coefficient. As a consequence, it is proved that this system has three Hopf bifurcation points, at which these Hopf bifurcations are nondegenerate and supercritical.