Ordering chaos and synchronization transitions by chemical delay and coupling on scale-free neuronal networks

Chaos theory has wide applications and its importance can be seen by the voluminous publications on various applications in several branches of science, commerce and engineering. Control, tracking or regulation and synchronization of different types of chaotic systems are importance areas of research in the control literature and various methods have been adopted over the past few decades for tackling these research problems. Also, the discovery of novel chaotic and hyperchaotic systems in various applications, their qualitative properties and the control of such systems are also important research areas in chaos theory. This paper announces a novel 4-D hyperchaotic Rikitake dynamo system, which is derived by adding a state feedback control to the famous 3-D Rikitake two-disk dynamo system (1958). The frequent and irregular reversals of the Earth's magnetic field inspired a number of early studies involving electrical currents within the Earth's molten core. One of the first such models to exhibit reversals was Rikitake's two-disk dynamo system (Rikitake, 1958). This paper discusses the qualitative properties of the novel hyperchaotic Rikitake dynamo system. We note that the novel hyperchaotic Rikitake dynamo system has no equilibrium points. The Lyapunov exponents of the hyperchaotic Rikitake dynamo system are found as ! = 0.09136, ! = 0.02198, ! = 0 and ! = −2.11190. The Kaplan-Yorke fractional dimension of the novel hyperchaotic Rikitake dynamo system is found as !" = 3.05367. Next, this paper discusses control and synchronization of the novel hyperchaotic Rikitake dynamo system with unknown parameters using adaptive control method. The main results are established using Lyapunov stability theory and numerically illustrated using MATLAB. Finally, for the 4-D novel hyperchaotic system, an electronic circuit realization in SPICE has been described to confirm the feasibility of the theoretical hyperchaotic Rikitake dynamo model.

Section: Introductionmentioning

confidence: 99%

Analysis, Control, Synchronization and SPICE Implementation of a Novel 4 - D Hyperchaotic Rikitake Dynamo System without Equilibrium

Vaidyanathan¹,

Volos²,

Pham³

2015

“…The study of chaos theory has many important applications in science and engineering such as oscillators [38][39][40], lasers [41][42][43], robotics [44][45][46][47], chemical reactors [48][49][50], biology [51,52], ecology [53,54], cardiology [55], memristors [56][57][58][59][60], neural networks [61][62][63], secure communications [64][65][66][67], cryptosystems [68][69][70][71], economics [72][73][74], etc.…”

Section: Introductionmentioning

confidence: 99%

A 3 - D Novel Conservative Chaotic System and its Generalized Projective Synchronization via Adaptive Control

Vaidyanathan¹,

Pakiriswamy²

2015

This research work proposes a five-term 3-D novel conservative chaotic system with a quadratic nonlinearity and a quartic nonlinearity. The conservative chaotic systems have the important property that they are volume conserving. The Lyapunov exponents of the 3-D novel chaotic system are obtained as ! = 0.0836, ! = 0 and ! = −0.0836. Since the sum of the Lyapunov exponents is zero, the 3-D novel chaotic system is conservative. Thus, the Kaplan-Yorke dimension of the 3-D novel chaotic system is easily seen as 3.0000. The phase portraits of the novel chaotic system simulated using MATLAB depict the chaotic attractor of the novel system. This research work also discusses other qualitative properties of the system. Next, an adaptive controller is designed to achieve Generalized Projective Synchronization (GPS) of two identical novel chaotic systems with unknown system parameters. MATLAB simulations are shown to validate and demonstrate the GPS results derived in this work.

“…Chaos theory has many important applications in science and engineering such as vibration control [35][36][37], oscillators [38][39][40], lasers [41][42][43], robotics [44][45][46][47], chemical reactors [48][49][50], biology [51,52], ecology [53,54], cardiology [55], memristors [56][57][58][59], neural networks [60][61][62], secure communications [63][64][65][66], cryptosystems [67][68][69][70], network design [71,72], economics [73][74][75][76], market forecasting [77], etc.…”

Section: Introductionmentioning

confidence: 99%

A 3 - D Novel Highly Chaotic System with Four Quadratic Nonlinearities, its Adaptive Control and Anti - Synchronization with Unknown Parameters

Vaidyanathan¹

2015

This research work proposes a seven-term 3-D novel dissipative chaotic system with four quadratic nonlinearities. The Lyapunov exponents of the 3-D novel chaotic system are obtained as L 1 = 11.36204, L 2 = 0 and L 3 =-47.80208. Since the sum of the Lyapunov exponents is negative, the 3-D novel chaotic system is dissipative. Also, the Kaplan-Yorke dimension of the 3-D novel chaotic system is obtained as D KY = 2.23769. The maximal Lyapunov exponent (MLE) of the novel chaotic system is L 1 = 11.36204, which is a large value for a polynomial chaotic system. Thus, the proposed 3-D novel chaotic system is highly chaotic. The phase portraits of the novel chaotic system simulated using MATLAB depict the highly chaotic attractor of the novel system. This research work also discusses other qualitative properties of the system. Next, an adaptive controller is designed to stabilize the 3-D novel chaotic system with unknown parameters. Also, an adaptive synchronizer is designed to achieve anti-synchronization of the identical 3-D novel chaotic systems with unknown parameters. The adaptive results derived in this work are established using Lyapunov stability theory. MATLAB simulations have been shown to illustrate and validate all the main results derived in this work.