This research work investigates the global chaos synchronization of Sprott’s jerk chaotic system using backstepping control method. Sprott’s jerk system (1997) is algebraically the simplest dissipative chaotic system consisting of five terms and a quadratic nonlinearity. Sprott’s chaotic system involves only five terms and one quadratic nonlinearity, while Rossler’s chaotic system (1976) involves seven terms and one quadratic nonlinearity. This work first details the properties of the Sprott’s jerk chaotic system. The phase portraits of the Sprott’s jerk system are described. The Lyapunov exponents of the Sprott’s jerk system are obtained as L 1 = 0.0525, L 2 = 0 and L 3 = −2.0727. The Lyapunov dimension of the Sprott’s jerk system is obtained as D L = 2.0253. Next, an active backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with known parameters. The backstepping control method is a recursive procedure that links the choice of a Lyapunov function with the design of a controller and guarantees global asymptotic stability of strict-feedback chaotic systems. Finally, an adaptive backstepping controller is designed for the global chaos synchronization of identical Sprott’s jerk systems with unknown parameters. MATLAB simulations are provided to validate and demonstrate the effectiveness of the proposed active and adaptive chaos synchronization schemes for the Sprott’s jerk systems.

Backstepping Controller Design for the Global Chaos Synchronization of Sprott’s Jerk Systems

Anti-synchronization is an important type of synchronization of a pair of chaotic systems called the master and slave systems. The anti-synchronization characterizes the asymptotic vanishing of the sum of the states of the master and slave systems. In other words, anti-synchronization of master and slave system is said to occur when the states of the synchronized systems have the same absolute values but opposite signs. Anti-synchronization has applications in science and engineering. This work derives a general result for the anti-synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. Next, as an application of the main result, anti-synchronizing controller has been designed for Vaidyanathan–Madhavan chaotic systems (2013). The Lyapunov exponents of the Vaidyanathan–Madhavan chaotic system are found as $L_1 = 3.2226, L_2 = 0$ and $L_3 = -30.3406$ and the Lyapunov dimension of the novel chaotic system is found as $D_L = 2.1095$. The maximal Lyapunov exponent of the Vaidyanathan–Madhavan chaotic system is $L_1 = 3.2226$. As an application of the general result derived in this work, a sliding mode controller is derived for the anti-synchronization of the identical Vaidyanathan–Madhavan chaotic systems. MATLAB simulations have been provided to illustrate the qualitative properties of the novel 3-D chaotic system and the anti-synchronizer results for the identical novel 3-D chaotic systems.

Anti-synchronization of Identical Chaotic Systems Using Sliding Mode Control and an Application to Vaidyanathan–Madhavan Chaotic Systems

Hybrid phase synchronization is a new type of synchronization of a pair of chaotic systems called the master and slave systems. In hybrid phase synchronization, the odd numbered states of the master and slave systems are completely synchronized (CS), while their even numbered states are anti-synchronized (AS). The hybrid phase synchronization has applications in secure communications and cryptosystems. This work derives a new result for the hybrid phase synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. As an application of this general result, a sliding mode controller is derived for the hybrid phase synchronization of the identical 3-D Vaidyanathan chaotic systems (2014). MATLAB simulations have been provided to illustrate the Vaidyanathan system and the hybrid synchronizer results for the identical Vaidyanathan systems.

Hybrid Synchronization of Identical Chaotic Systems Using Sliding Mode Control and an Application to Vaidyanathan Chaotic Systems

Hyperchaotic systems are defined as chaotic systems with more than one positive Lyapunov exponent. Combined with one null Lyapunov exponent along the flow and one negative Lyapunov exponent to ensure boundedness of the solution, the minimal dimension for a continuous hyperchaotic system is four. The hyperchaotic systems are known to have important applications in secure communications and cryptosystems. First, this work describes an eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities. The qualitative properties of the novel hyperchaotic system are described in detail. The Lyapunov exponents of the system are obtained as $ L_{1} = 0.7781,L_{2} = 0.2299,L_{3} = 0 $ and $ L_{4} = - 12.5062 $. The maximal Lyapunov exponent of the system (MLE) is $ L_{1} = 0.7781 $. The Lyapunov dimension of the novel hyperchaotic system is obtained as $ D_{L} = 3.0806 $. Next, the work describes an adaptive controller design for the global chaos control of the novel hyperchaotic system. The main result for the adaptive controller design has been proved using Lyapunov stability theory. MATLAB simulations are described in detail for all the main results derived in this work for the eleven-term 4-D novel hyperchaotic system with four quadratic nonlinearities.

Analysis and Control of a 4-D Novel Hyperchaotic System

This research work describes a nine-term 3-D novel chaotic system with four quadratic nonlinearities. First, this work describes the dynamic analysis of the novel chaotic system and qualitative properties of the novel chaotic system are derived. The Lyapunov exponents of the nine-term novel chaotic system are obtained as $ L_{1} = 9.45456,\;L_{2} = 0 $ and $ L_{3} = - 30.50532 $. Since the maximal Lyapunov exponent (MLE) of the novel chaotic system is $ L_{1} = 9.45456 $, which is a high value, the novel chaotic system exhibits strong chaotic properties. Next, this work describes the adaptive control of the novel chaotic system with unknown system parameters. Also, this work describes the adaptive synchronization of the identical novel chaotic systems with unknown system parameters. The adaptive control and synchronization results are proved using Lyapunov stability theory. MATLAB simulations are given to demonstrate and validate all the main results derived in this work for the nine-term 3-D novel chaotic system.

https://www.springer.com/cda/content/document/cda_downloaddocument/9783319131313-c2.pdf?SGWID=0-0-45-1489612-p177078942

Analysis, Control and Synchronization of a Nine-Term 3-D Novel Chaotic System

Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping

In this paper, we mainly focus our attention on the global dynamical behaviour of some ubiquitous nonlinear oscillators under the presence of nonlinear dissipation. We particularly consider the parametrically driven Duffing oscillator and externally driven Helmholtz–Duffing oscillators with nonlinear dissipation term proportional to the power of velocity $$(v\left| v \right| ^{p-1})$$
 . We obtain the threshold condition for the occurrence of chaos analytically as well as numerically for all the cases $$p=$$
 1, 2, 3 and 4. We also identify the regions of 2D parameter space (consisting of external forcing amplitude and damping coefficient) corresponding to various asymptotic dynamics and analyse the effect of nonlinear damping on the overall dynamical behaviour of these nonlinear oscillators.

Dynamical behaviour of parametrically driven Duffing and externally driven Helmholtz–Duffing oscillators under nonlinear dissipation

In this communication we study the forced and quadratically damped Duffing oscillator system experimentally through an analogue circuit. For this purpose we design a circuit and observe its dynamical behaviour under various sets of control parameters and also compare the experimental results with the numerical simulation results.

/pdf/an-analogue-circuit-to-study-the-forced-and-quadratically-3w1shft35t.pdf

An analogue circuit to study the forced and quadratically damped Duffing oscillator

Abstract In this communication, we mainly focus our attention on the global dynamical behavior of forced and nonlinearly damped pendulum. We consider the nonlinear damping term proportional to the power of velocity (v|v|p−1). Analytically we obtain the threshold condition for the occurrence of homoclinic bifurcation using Melnikov technique. We carry out the bifurcation analysis to identify the route to chaos for various cases of damping (p = 1, 2 and 3) and analyze the topological shape and fractalness of phase space attractor by computing Kaplan-Yorke dimension and Correlation dimension. We identify the regions of 2D parameter space (consists of external forcing amplitude and damping coefficient) corresponding to various types of asymptotic dynamics through extensive Lyapunov exponent calculation and analyze the fragileness and globalness of chaos in the parameter space with the introduction of nonlinear damping. We also analyze the complexity of basin of attraction for all three cases (p = 1, 2 and 3) and compute the correlation dimension to quantify the fractalness of basin boundaries.

Anjali Sharma

Papers

Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping

Dynamical behaviour of parametrically driven Duffing and externally driven Helmholtz–Duffing oscillators under nonlinear dissipation

An analogue circuit to study the forced and quadratically damped Duffing oscillator

Bifurcation and Chaos in Periodically Forced and Nonlinearly Damped Pendulum