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A new two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium, its bifurcation analysis, circuit design and a control application to complete synchronization

Vaidyanathan Sundarapandian, Moroz Irene M., Sambas Aceng

Archives of Control Sciences

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2023

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Vol. 33, No. 2

277--298

EN

In this work, we present new results for a two-scroll 4-D hyperchaotic system with a unique saddle point equilibrium at the origin. The bifurcation and multi-stability analysis for the new hyperchaotic system are discussed in detail. As a control application, we develop a feedback control based on integral sliding mode control (ISMC) for the complete synchronization of a pair of two-scroll hyperchaotic systems developed in this work. Numerical simulations using Matlab are provided to illustrate the hyperchaotic phase portraits, bifurcation diagrams and synchronization results. Finally, as an electronic application, we simulate the new hyperchaotic system using Multisim for real-world implementations.

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A new multistable double-scroll 4-D hyperchaotic system with no equilibrium point, its bifurcation analysis, synchronization and circuit design

Vaidyanathan Sundarapandian, He Shaobo, Simbas Aceng

Archives of Control Sciences

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2021

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Vol. 31, No. 1

99--128

EN

In this work, we have developed a new 4-D dynamical system with hyperchaos and hidden attractor. First, by introducing a feedback input control into the 3-D Ma chaos system (2004), we obtain a new 4-D hyperchaos system with no equilibrium point. Thus, we derive a new hyperchaos system with hidden attractor. We carry out an extensive bifurcation analysis of the new hyperchaos model with respect to the three parameters. We also carry out probability density distribution analysis of the new hyperchaotic system. Interestingly, the new nonlinear hyperchaos system exhibits multistability with coexisting attractors. Next, we discuss global hyperchaos self-synchronization for the new hyperchaos system via Integral Sliding Mode Control (ISMC). As an engineering application, we realize the new 4-D hyperchaos system with an electronic circuit via MultiSim. The outputs of the MultiSim hyperchaos circuit show good match with the numerical MATLAB plots of the hyperchaos model. We also analyze the power spectral density (PSD) of the hyperchaos of the state variables using MultiSim.

3

Regular and chaotic dynamics of a 4-DOF mechanical system with dry friction

Kosińska A., Awrejcewicz J., Grzelczyk D.

Vibrations in Physical Systems

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2016

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Vol. 27

195--202

EN

In this paper the model of four degree-of-freedom mechanical sliding system with dry friction is considered. One of the components of the mentioned system rides on driving belt, which is driven at constant velocity. This model corresponds to a row of carriage laying on a guideway, which moves at constant velocity with respect to the guideway as a foundation. From a mathematical point of view the analyzed problem is governed by four second order differential equations of motion, and numerical analysis is performed in Mathematica software. Some interesting behaviors are detected and reported using Phase Portraits, Poincaré Maps and Lyapunov Exponents. Moreover, Power Spectral Densities obtained by the Fast Fourier Transform technique are reported. The presented results show different behaviors of the system, including periodic, quasi-periodic and chaotic orbits.

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Hyperchaos, adaptive control and synchronization of a novel 4-D hyperchaotic system with two quadratic nonlinearities

Vaidyanathan S.

Archives of Control Sciences

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2016

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Vol. 26, no. 4

471--495

EN

This research work announces an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. We describe the qualitative properties of the novel 4-D hyperchaotic system and illustrate their phase portraits. We show that the novel 4-D hyperchaotic system has two unstable equilibrium points. The novel 4-D hyperchaotic system has the Lyapunov exponents L1 = 3.1575, L2 = 0.3035, L3 = 0 and L4 = −33.4180. The Kaplan-Yorke dimension of this novel hyperchaotic system is found as DKY = 3.1026. Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, we deduce that the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to stabilize the novel 4-D hyperchaotic system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 4-D hyperchaotic systems with unknown system parameters. The adaptive control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work

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

Vaidyanathan S.

Archives of Control Sciences

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2016

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Vol. 26, no. 3

311--338

EN

A hyperjerk system is a dynamical system, which is modelled by an nth order ordinary differential equation with n ≥ 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n ≥ 4. In this research work, a 4-D novel hyperchaotic hyperjerk system with two nonlinearities has been proposed, and its qualitative properties have been detailed. The novel hyperjerk system has a unique equilibrium at the origin, which is a saddle-focus and unstable. The Lyapunov exponents of the novel hyperjerk system are obtained as L1 = 0.14219, L2 = 0.04605, L3 = 0 and L4 = −1.39267. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained as DKY = 3.1348. Next, an adaptive controller is designed via backstepping control method to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive controller is designed via backstepping control method to achieve global synchronization of the identical novel hyperjerk systems with three unknown parameters. MATLAB simulations are shown to illustrate all the main results derived in this research work on a novel hyperjerk system.

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

Vaidyanathan S., Volos C., Pham V.-T., Madhavan K.

Archives of Control Sciences

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2015

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Vol. 25, no. 1

135--158

EN

A hyperjerk system is a dynamical system, which is modelled by an nth order ordinary differential equation with n 4 describing the time evolution of a single scalar variable. Equivalently, using a chain of integrators, a hyperjerk system can be modelled as a system of n first order ordinary differential equations with n 4. In this research work, a 4-D novel hyperchaotic hyperjerk system has been proposed, and its qualitative properties have been detailed. The Lyapunov exponents of the novel hyperjerk system are obtained as L1 = 0:1448;L2 = 0:0328;L3 = 0 and L4 = −1:1294. The Kaplan-Yorke dimension of the novel hyperjerk system is obtained as DKY = 3:1573. Next, an adaptive backstepping controller is designed to stabilize the novel hyperjerk chaotic system with three unknown parameters. Moreover, an adaptive backstepping controller is designed to achieve global hyperchaos synchronization of the identical novel hyperjerk systems with three unknown parameters. Finally, an electronic circuit realization of the novel jerk chaotic system using SPICE is presented in detail to confirm the feasibility of the theoretical hyperjerk model.

7

Dynamics of Mechanical Sliding System with Dry Friction

Grzelczyk D., Awrejcewicz J., Kudra G.

Machine Dynamics Research

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2014

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Vol. 38, No. 3

61-70

EN

In the paper we consider a four degree of freedom model of a mechanical system with dry friction, which can explain non-regular vibrations of mechanical sliding system. Stick-slip vibrations are studied for the case, where body is riding on a driving belt as a foundation that moves at a constant velocity. From a mathematical point of view, the analyzed problem is governed by a set of nonlinear ordinary second order differential equations of motion, obtained using the Lagrangian method. Numerical analysis is carried out with the qualitative and quantitative theories of nonlinear differential equations reduced to the non-dimensional form. The behavior of the mentioned system is monitored via standard phase portraits, bifurcation diagram as well as Lyapunov exponents. Some interesting numerical results (periodic, quasi-periodic, chaotic and hyper-chaotic orbits) are obtained and reported in the paper.

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

Vaidyanathan S., Volos C., Pham V.-T.

Archives of Control Sciences

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2014

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Vol. 24, no. 4

409--446

EN

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 (Jia, 2007) with three quadratic nonlinearities. The 4-D hyperchaotic Lorenz system (Jia, 2007) has the Lyapunov exponents L1 = 0.3684,L2 = 0.2174,L3 = 0 and L4 =−12.9513, and the Kaplan-Yorke dimension of this 4-D system is found as DKY =3.0452. The 5-D novel hyperchaotic Lorenz system proposed in this work has the Lyapunov exponents L1 = 0.4195,L2 = 0.2430,L3 = 0.0145,L4 = 0 and L5 = −13.0405, and the Kaplan-Yorke dimension of this 5-D system is found as DKY =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.