Hyperchaos, adaptive control and synchronization of a novel 5-D hyperchaotic system with three positive Lyapunov exponents and its SPICE implementation

Abstract

In this research work, a twelve-term novel 5-D hyperchaotic Lorenz system with three quadratic nonlinearities has been derived by adding a feedback control to a ten-term 4-D hyperchaotic Lorenz system with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system has the Lyapunov exponents L 1 = 0.3684, L 2 = 0.2174, L 3 = 0 and L 4 = −12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as D KY = 3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L 1 = 0.4195, L 2 = 0.2430, L 3 = 0.0145, L 4 = 0 and L 5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as D KY = 4.0159. Thus, the novel 5-D hyperchaotic Lorenz system has a maximal Lyapunov exponent (MLE), which is greater than the maximal Lyapunov exponent (MLE) of the 4-D hyperchaotic Lorenz system. The 5-D novel hyperchaotic Lorenz system has a unique equilibrium point at the origin, which is a saddle-point and hence unstable. Next, an adaptive controller is designed to stabilize the novel 5-D hyperchaotic Lorenz system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 5-D hyperchaotic Lorenz systems with unknown system parameters. Finally, an electronic circuit realization of the novel 5-D hyperchaotic Lorenz system using SPICE is described in detail to confirm the feasibility of the theoretical model.

Anti-synchronisation of identical chaotic systems via novel sliding control and its application to a novel chaotic system

“…Various control schemes have been developed to investigate the synchronisation and anti-synchronisation problems of the chaos literature such as active control method (Ho and Hung, 2002;Chen, 2005;Karthikeyan and Sundarapandian, 2014;Sarasu and Sundarapandian, 2011;Sundarapandian and Karthikeyan, 2012a;Vaidyanathan and Rajagopal, 2011;Zhang et al, 2012), adaptive control method (Abouelsoud and Mohamed, 2015;Liao and Tsai, 2000;Sundarapandian, 2012a, 2012b;Sundarapandian and Karthikeyan, 2011a, 2011b, 2012bVaidyanathan, 2012aVaidyanathan, , 2013cPakiriswamy, 2013, 2014;Vaidyanathan et al, 2014bVaidyanathan et al, , 2015d, backstepping method (Yu and Zhang, 2004;Park, 2006;Rasappan and Vaidyanathan, 2012a, 2012b, 2014Suresh and Sundarapandian, 2013;Vaidyanathan and Rasappan, 2014;Vaidyanathan et al, 2015f), sampled-data feedback control (Zhao and Lu, 2008), time-delay feedback method (Guo and Zhong, 2009), sliding mode control (Dhanalakshmi et al, 2015;Meng et al, 2014;Rhif, 2014;Shang and Wang, 2013;Sundarapandian and Sivaperumal, 2011;Vaidyanathan, 2012bVaidyanathan, , 2012cVaidyanathan and Sampath, 2012;Vaidyanathan et al, 2015g;…”

Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation

“…6-9, the complete synchronization of the identical 4-D novel hyperchaotic hyperjerk systems (52) and (53) is shown, when the adaptive control law and the parameter update law are implemented. Also, in Fig.…”

“…(52) and (53) with unknown parameters a, b and c are globally and exponentially synchronized by the adaptive control law…”

“…Since the discovery of a first 4-D hyperchaotic system by Rössler in 1979 [44], many 4-D hyperchaotic systems have been found in the literature such as hyperchaotic Lorenz system [45], hyperchaotic Lü system [46], hyperchaotic Chen system [47], hyperchaotic Wang system [48], hyperchaotic Newton-Leipnik system [49], hyperchaotic Jia system [50], hyperchaotic Vaidyanathan systems [51,52], etc. In mechanics, if the scalar x(t) represents the position of a moving object at time t, then the first derivative, x(t), represents the velocity, the second derivative,ẍ(t), represents the acceleration and the third derivative, ... x (t), represents the jerk or jolt [53].…”

Analysis, Adaptive Control and Synchronization of a Novel 4-D Hyperchaotic Hyperjerk System via Backstepping Control Method

“…Since the discovery of a first 4-D hyperchaotic system by Rössler in 1979 [52], many 4-D hyperchaotic systems have been found in the literature such as hyperchaotic Lorenz system [53], hyperchaotic Lü system [54], hyperchaotic Chen system [55], hyperchaotic Wang system [56], hyperchaotic Newton-Leipnik system [57], hyperchaotic Jia system [58], hyperchaotic Vaidyanathan systems [59,60,61,62,63,64,65,66,67,68], hyperchaotic Pham system [69], hyperchaotic Sampath system [70], etc.…”