Analysis and control of a hyperchaotic system with only one nonlinear term

Abstract

In order to promote the development of chaos in nonlinear systems, and explore more convenient controllers for the engineering application, a four-dimensional nonlinear dynamic system with only one nonlinear term was constructed and its complex dynamic characteristics were analyzed, including the phase trajectory map, Lyapunov exponents, and so on. Furthermore, the recursive backstepping method was proposed to design a different controller; the hyperchaotic system was controlled to an equilibrium point and a periodic orbit. Theoretical analysis is in agreement with simulation results. The results show that the recursive backstepping control method can wipe off chaos, and make the hyperchaotic system achieve stable states. The control process is a smooth transition, and the transition time is short.

Synchronization and Antisynchronization of a Class of Chaotic Systems With Nonidentical Orders and Uncertain Parameters

“…(17) and v ¼ 1, the controllers are obtained as follows: u 1 ¼ ÀD 0:95 x 1 À À35x 1 þ 35y 1 ð Þ þÀ 5e 1 þ 5e 2 ð Þ À 5s 1 þ 0:2signðs 1 Þ ð Þ u 2 ¼ ÀD 0:95 y 1 À ðÀ7x 1 þ 28z 1 Þ þ À3e 1 þ 2e 3 ð Þ À 3sint À x 1 z 1 ð Þ À5s 2 þ 0:2signðs 2 Þ ð Þ u 3 ¼ ÀD 0:95 z 1 þ 3z 1 À x 1 y 1 À 5s 3 þ 0:2signðs 3 Þ ð Þ 8 > > > > > > < > > > > > > : (29) Obviously, Fig. 9 demonstrates that the error system converges to zero gradually.…”

“…Up to now, a variety of chaos synchronization techniques have been applied in chemistry [2], physics [3,4], medicine [5], engineering [6], information science [7,8], and so on. Inspired by its universal utilities, researchers have put forward dozens of methods such as impulse synchronization [9,10], adaptive synchronization [11,12], feedback synchronization [13,14], fuzzy synchronization [15,16], back-stepping [17], active synchronization [18,19], and sliding mode synchronization [20][21][22].…”

Chaos Synchronization of Nonlinear Fractional Discrete Dynamical Systems via Linear Control

“…Chaos synchronization has been a hot topic [17][18][19]. There are many synchronization schemes for fractional differential systems, such as synchronization via the linear control technique [20], synchronization via the adaptive sliding mode [21], projective synchronization via single sinusoidal coupling [22], hybrid chaos synchronization with a robust method [23], synchronization with activation feedback control [24], synchronization via a scalar transmitted signal [25], adaptive synchronization via a single driving variables [26], synchronization via novel active pinning controls [27].…”

Projective Synchronization of Chaotic Discrete Dynamical Systems via Linear State Error Feedback Control

“…Odibat et al [20] studied synchronization for 3-dimensional chaotic fractional-order systems via linear control. Chen et al [21,22] also did some sound work designing controllers which are less than the number of dimensions of the chaotic systems. By using linear state error feedback control technology, Xin et al [2,3,23] studied the projective synchronization for three kinds of chaotic fractional-order systems.…”