Control of chaos: Lie algebraic exact linearization approach for the Lü system

Abstract.In this article, nonlinear control of the Lü chaos is presented by transforming a Lü system in a linear controllable system by using the Lie algebraic exact linearization method. Numerically, control of chaos does not have much place in current literature unless one actually finds and displays the controller in the chaotic system. The controller for the Lü system is designed for an arbitrary output. Based on this controller, both linear and nonlinear outputs are discussed in detail. Numerical simulations are carried out for all necessary aspects of the Lü problem that justify our analytical findings. Moreover, the stabilization of the Lü system based on the linear controllable system is discussed in the form of limit cycles and Hopf bifurcation. In particular, the numerical Hopf bifurcation diagram is presented using the numerical continuation technique.

[1]  I. A. Baba,et al.  Global stability analysis of two-strain epidemic model with bilinear and non-monotone incidence rates , 2017 .

[2]  M. Bologna.  Exact analytical approach to differential equations with variable coefficients , 2015, 1505.06509.

[3]  Weiping Li,et al.  Applied Nonlinear Control , 1991 .

[4]  Jinde Cao,et al.  Adaptive control of multiple chaotic systems with unknown parameters in two different synchronization modes , 2016 .

[5]  Alistair I. Mees,et al.  Optimal control of nonlinear systems to given orbits , 1997 .

[6]  Mitul Islam,et al.  Chaos control in Shimizu Morioka system by Lie algebraic exact linearization , 2014 .

[7]  Ioannis Antoniou,et al.  Probabilistic control of chaos through small perturbations , 2000 .

[8]  Kestutis Pyragas.  Continuous control of chaos by self-controlling feedback , 1992 .

[9]  Jinhu Lu,et al.  A New Chaotic Attractor Coined , 2002, Int. J. Bifurc. Chaos.

[10]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[11]  Aria Alasty,et al.  Stabilizing periodic orbits of chaotic systems using fuzzy adaptive sliding mode control , 2008 .

[12]  Li-Qun Chen,et al.  A Modified Exact Linearization Control for Chaotic Oscillators , 1999 .

[13]  D. López-Mancilla,et al.  Output synchronization of chaotic systems under nonvanishing perturbations , 2008 .

[14]  Lin Chen,et al.  Stabilization of parameters perturbation chaotic system via adaptive backstepping technique , 2008, Appl. Math. Comput..

[15]  Jinde Cao,et al.  Active control strategy for synchronization and anti-synchronization of a fractional chaotic financial system , 2017 .

[16]  Liu Yanzhu,et al.  Control of the Lorenz chaos by the exact linearization , 1998 .

[17]  Xinghuo Yu,et al.  Variable structure control approach for controlling chaos , 1997 .