Shijian Cang

Chaotic behavior and circuit implementation of a fractional-order permanent magnet synchronous motor model

Abstract A fractional-order permanent magnet synchronous motor (PMSM) model is proposed and analyzed by phase portraits, bifurcation diagrams, Lyapunov exponents and Poincare maps in this paper. Numerical simulations show that the proposed PMSM model with fractional integral of order α ∈ [ 0.94 , 1 ] is chaotic on the condition of great changes of parameter. Further, through a transfer function approximation of fractional-order operator, an experimental circuit composed of resistors, capacitors, operational amplifiers and analog multipliers is presented to realize the fractional-order PMSM model, and whose chaotic behavior is also recorded by a digital oscilloscope in experiment.

Distinguishing Lorenz and Chen Systems Based Upon Hamiltonian Energy Theory

Solving the linear first-order Partial Differential Equations (PDEs) derived from the unified Lorenz system, it is found that there is a unified Hamiltonian (energy function) for the Lorenz and Chen systems, and the unified energy function shows a hyperboloid of one sheet for the Lorenz system and an ellipsoidal surface for the Chen system in three-dimensional phase space, which can be used to explain that the Lorenz system is not equivalent to the Chen system. Using the unified energy function, we obtain two generalized Hamiltonian realizations of these two chaotic systems, respectively. Moreover, the energy function and generalized Hamiltonian realization of the Lu system and a four-dimensional hyperchaotic Lorenz-type system are also discussed.

Projective synchronisation of fractional-order memristive systems with different structures based on active control method

An active control strategy is proposed to investigate the problem of projective synchronisation of two coupled fractional-order memristive chaotic systems in this paper. Based on the Laplace transform, it is proved that the proposed active controller can realise the projective synchronisation between two fractional-order memristive systems with different structures. The phase portraits and the error curves are also used to verify the effectiveness of the proposed method.

A Strange Double-Deck Butterfly Chaotic Attractor from a Permanent Magnet Synchronous Motor with Smooth Air Gap: Numerical Analysis and Experimental Observation

A permanent magnet synchronous motor (PMSM) model with smooth air gap and an exogenous periodic input is introduced and analyzed in this paper. With a simple mathematical transformation, a new nonautonomous Lorenz-like system is derived from this PMSM model, and this new three-dimensional system can display the complicated dynamics such as the chaotic attractor and the multiperiodic orbits by adjusting the frequency and amplitude of the exogenous periodic inputs. Moreover, this new system shows a double-deck chaotic attractor that is completely different from the four-wing chaotic attractors on topological structures, although the phase portrait shapes of the new attractor and the four-wing chaotic attractors are similar. The exotic phenomenon has been well demonstrated and investigated by numerical simulations, bifurcation analysis, and electronic circuit implementation.

Adaptive Sliding Mode Controller Design for Projective Synchronization of Different Chaotic Systems with Uncertain Terms and External Bounded Disturbances

Synchronization is very useful in many science and engineering areas. In practical application, it is general that there are unknown parameters, uncertain terms, and bounded external disturbances in the response system. In this paper, an adaptive sliding mode controller is proposed to realize the projective synchronization of two different dynamical systems with fully unknown parameters, uncertain terms, and bounded external disturbances. Based on the Lyapunov stability theory, it is proven that the proposed control scheme can make two different systems (driving system and response system) be globally asymptotically synchronized. The adaptive global projective synchronization of the Lorenz system and the Lu system is taken as an illustrative example to show the effectiveness of this proposed control method.

Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Chaotic dynamics exists in many natural systems, such as weather and climate, and there are many applications in different disciplines. However, there are few research results about chaotic conservative systems especially the smooth hyperchaotic conservative system in both theory and application. This paper proposes a five-dimensional (5D) smooth autonomous hyperchaotic system with nonhyperbolic fixed points. Although the proposed system includes four linear terms and four quadratic terms, the new system shows complicated dynamics which has been proven by the theoretical analysis. Several notable properties related to conservative systems and the existence of perpetual points are investigated for the proposed system. Moreover, its conservative hyperchaotic behavior is illustrated by numerical techniques including phase portraits and Lyapunov exponents.

Projective synchronization of two fractional-order memristive systems via an active controller

An active control strategy is presented to realize the projective synchronization of two different coupled fractional-order chaotic systems in this paper. The theoretical analysis shows whether the proposed active controller can realize the projective synchronization between two coupled fractional-order memristive systems with different structures. The phase portraits and the error curves are also used to verify the effectiveness of the proposed method.

On a New Three-dimensional Chaotic System with Only One Equilibrium Based on a Series Memristive Circuit

This paper introduces several basic principles of chaos generated by memristive circuits in accordance with the characteristic of memristor, and a new three-dimensional chaotic system with only one equilibrium are further proposed based on a series memristive circuit. The numerical results show that the proposed memristive system has the common features of nonlinear system under different parameters, such as periodic orbit and chaos which are demonstrated by numerical simulations, bifurcation analysis. Most importantly, the system can be used for illustrating continuous period-double bifurcations routing to chaos.

SMC-based Projective Synchronization of Lorenz System and Chen System with Fully Unknown Parameters

In this paper, an adaptive sliding mode controller (SMC) with a parameter update law is developed to realize projective synchronization of the Lorenz system and the Chen system with fully unknown parameters. The projective synchronization includes complete synchronization and anti-phase synchronization. Moreover, it is proven that the proposed adaptive SMC can maintain the existence of sliding mode in uncertain chaotic systems based on Lyapunov stability theory. Finally, numerical simulations are presented to illustrate the effectiveness of the proposed control method.

Coexistence of multiple strange attractors governed by different initial conditions in a deterministic system

This paper presents a new four-dimension autonomous system which shows extraordinary dynamical properties . Chaotic attractor and periodic attractor or hyper-chaotic attractor and quasi-periodic attractor, which are governed by different initial conditions instead of the system parameters, can coexist in the deterministic system. These interesting phenomena are verified through numerical simulations and analyses including time series, phase portraits, Poincare maps, bifurcation diagrams, and Lyapunov exponents.