A novel 3-D jerk chaotic system with three quadratic nonlinearities and its adaptive control

This paper announces an eight-term novel 3-D jerk chaotic system with three quadratic nonlinearities. The phase portraits of the novel jerk chaotic system are displayed and the qualitative properties of the jerk system are described. The novel jerk chaotic system has two equilibrium points, which are saddle-foci and unstable. The Lyapunov exponents of the novel jerk chaotic system are obtained as L1 = 0.20572,L2 = 0 and L3 = −1.20824. Since the sum of the Lyapunov exponents of the jerk chaotic system is negative, we conclude that the chaotic system is dissipative. The Kaplan-Yorke dimension of the novel jerk chaotic system is derived as DKY = 2.17026. Next, an adaptive controller is designed via backstepping control method to globally stabilize the novel jerk chaotic system with unknown parameters. Moreover, an adaptive controller is also designed via backstepping control method to achieve global chaos synchronization of the identical jerk chaotic systems with unknown 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 systems. MATLAB simulations have been depicted to illustrate the phase portraits of the novel jerk chaotic system and also the adaptive backstepping control results.

[1]  B. Crozier,et al.  Politics and the Military in Modern Spain , 1967 .

[2]  Sundarapandian Vaidyanathan,et al.  A no-equilibrium hyperchaotic system with a cubic nonlinear term , 2016 .

[3]  Sundarapandian Vaidyanathan,et al.  Hidden attractors in a chaotic system with an exponential nonlinear term , 2015 .

[4]  Sundarapandian Vaidyanathan,et al.  Adaptive Control of Rikitake Two-Disk Dynamo System , 2015 .

[5]  Sundarapandian Vaidyanathan,et al.  Global chaos synchronisation of identical chaotic systems via novel sliding mode control method and its application to Zhu system , 2015, Int. J. Model. Identif. Control..

[6]  Sundarapandian Vaidyanathan,et al.  Hybrid Synchronization of Hyperchaotic Wang-Chen and Hyperchaotic Lorenz Systems by Active Non-linear Control , 2011 .

[7]  Sundarapandian Vaidyanathan,et al.  Integral Sliding Mode Control Design for the Global Chaos Synchronization of Identical Novel Chemical Chaotic Reactor Systems , 2015 .

[8]  Sundarapandian Vaidyanathan,et al.  Synchronization of Hyperchaotic Liu System via Backstepping Control with Recursive Feedback , 2012 .

[9]  V. Sundarapandian,et al.  Analysis, control, synchronization, and circuit design of a novel chaotic system , 2012, Math. Comput. Model..

[10]  Sundarapandian Vaidyanathan,et al.  HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL , 2011 .

[11]  Sundarapandian Vaidyanathan,et al.  ANTI -SYNCHRONIZATION OF HYPERCHAOTIC XU SYSTEMS VIA SLIDING MODE CONTROL , 2012 .

[12]  Sundarapandian Vaidyanathan,et al.  Anti-synchronization of Mathieu-Van der Pol Chaotic Systems via Adaptive Control Method , 2015 .

[13]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of Rucklidge Chaotic Systems for Double Convection via Sliding Mode Control , 2015 .

[14]  Sundarapandian Vaidyanathan.  Analysis, control and synchronisation of a six-term novel chaotic system with three quadratic nonlinearities , 2014, Int. J. Model. Identif. Control..

[15]  Sundarapandian Vaidyanathan.  Global chaos synchronisation of identical Li-Wu chaotic systems via sliding mode control , 2014, Int. J. Model. Identif. Control..

[16]  J. Sprott,et al.  Some simple chaotic flows. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of the Forced Van der Pol Chaotic Oscillators via Adaptive Control Method , 2015 .

[18]  V. Sundarapan,et al.  Generalized Projective Synchronization of Two-Scroll Systems via Adaptive Control , 2012 .

[19]  Sundarapandian Vaidyanathan,et al.  A Novel Chemical Chaotic Reactor System and its Output Regulation via Integral Sliding Mode Control , 2015 .

[20]  Chongxin Liu,et al.  A new chaotic attractor , 2004 .

[21]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of Chemical Chaotic Reactors via Novel Sliding Mode Control Method , 2015 .

[22]  S. Vaidyanathan.  Analysis and Adaptive Synchronization of Two Novel Chaotic Systems with Hyperboli c Sinusoidal and Cosinusoidal Nonlinearity and Unknown Parameters , 2013 .

[23]  Sundarapandian Vaidyanathan,et al.  GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC NEWTON -LEIPNIK SYSTEMS BY SLIDING MODE CONTROL , 2011 .

[24]  Viet-Thanh Pham,et al.  Constructing a Novel No-Equilibrium Chaotic System , 2014, Int. J. Bifurc. Chaos.

[25]  Sundarapandian Vaidyanathan,et al.  Output Regulation of the Unified Chaotic System , 2011 .

[26]  Sundarapandian Vaidyanathan,et al.  Backstepping Controller Design for the Global Chaos Synchronization of Sprott's Jerk Systems , 2015, Chaos Modeling and Control Systems Design.

[27]  Sundarapandian Vaidyanathan,et al.  Adaptive Synchronization of the Identical FitzHugh-Nagumo Chaotic Neuron Models , 2015 .

[28]  S. Vaidyanathan,et al.  A 3 - D Novel Conservative Chaotic System and its Generalized Projective Synchronization via Adaptive Control , 2015 .

[29]  Sundarapandian Vaidyanathan,et al.  Adaptive Control Design for the Anti-synchronization of Novel 3-D Chemical Chaotic Reactor Systems , 2015 .

[30]  G. Tigan,et al.  Analysis of a 3D chaotic system , 2006, math/0608568.

[31]  Sundarapandian Vaidyanathan,et al.  Anti-Synchronization of 3-Cells Cellular Neural Network Attractors via Adaptive Control Method , 2015 .

[32]  Sundarapandian Vaidyanathan,et al.  Adaptive Synchronization of Chemical Chaotic Reactors , 2015 .

[33]  Sundarapandian Vaidyanathan,et al.  Analysis, Control, Synchronization and SPICE Implementation of a Novel 4 - D Hyperchaotic Rikitake Dynamo System without Equilibrium , 2015 .

[34]  Sundarapandian Vaidyanathan,et al.  Hybrid Chaos Synchronization of 3-Cells Cellular Neural Network Attractors via Adaptive Control Method , 2015 .

[35]  Sundarapandian Vaidyanathan,et al.  An Eight - Term Novel Four - Scroll Chaotic System with Cubic Nonlinearity and its Circuit Simulation , 2015 .

[36]  Sundarapandian Vaidyanathan.  Global Chaos Synchronization of Hyperchaotic Pang and Hyperchaotic Wang Systems via Adaptive Control , 2011 .

[37]  Sundarapandian Vaidyanathan,et al.  State Regulation of Rikitake Two-Disk Dynamo Chaotic System via Adaptive Control Method , 2015 .

[38]  Sundarapandian Vaidyanathan,et al.  Analysis, adaptive control and circuit simulation of a novel nonlinear finance system , 2016, Appl. Math. Comput..

[39]  Sundarapandian Vaidyanathan,et al.  A Novel Coupled Van der Pol Conservative Chaotic System and its Adaptive Control , 2015 .

[40]  Sundarapandian Vaidyanathan,et al.  Anti-Synchronization of the FitzHugh-Nagumo Chaotic Neuron Models via Adaptive Control Method , 2015 .

[41]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of Hyperchaotic Bao and Xu Systems by Active Nonlinear Control , 2011 .

[42]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of 3-Cells Cellular Neural Network Attractors via Integral Sliding Mode Control , 2015 .

[43]  M. Hénon,et al.  The applicability of the third integral of motion: Some numerical experiments , 1964 .

[44]  Sundarapandian Vaidyanathan,et al.  Analysis , Adaptive Control and Adaptive Synchronization of a Nine - Term Novel 3 - D Chaotic System with Four Quadratic Nonlinearities and its Circuit Simulation , 2015 .

[45]  Sundarapandian Vaidyanathan,et al.  Hybrid Synchronization of the Generalized Lotka-Volterra Three-Species Biological Systems via Adaptive Control , 2016 .

[46]  Julien Clinton Sprott,et al.  Simple chaotic flows with a line equilibrium , 2013 .

[47]  V. Sundarapandian,et al.  Analysis and Anti-Synchronization of a Novel Chaotic System via Active and Adaptive Controllers , 2013 .

[48]  Sundarapandian Vaidyanathan,et al.  Chaotic Attractor in a Novel Time-Delayed System with a Saturation Function , 2015 .

[49]  Sundarapandian Vaidyanathan,et al.  A 5-D hyperchaotic Rikitake dynamo system with hidden attractors , 2015 .

[50]  Sundarapandian Vaidyanathan,et al.  Analysis, adaptive control and synchronization of a novel 4-D hyperchaotic hyperjerk system and its SPICE implementation , 2015 .

[51]  Sundarapandian Vaidyanathan,et al.  Synchronization of 3-Cells Cellular Neural Network (CNN) Attractors via Adaptive Control Method , 2015 .

[52]  Sundarapandian Vaidyanathan,et al.  Memristor: A New Concept in Synchronization of Coupled Neuromorphic C ircuits , 2015 .

[53]  Sundarapandian Vaidyanathan,et al.  Analysis, properties and control of an eight-term 3-D chaotic system with an exponential nonlinearity , 2015, Int. J. Model. Identif. Control..

[54]  Sundarapandian Vaidyanathan,et al.  Synchronization Phenomena in Coupled Colpitts Circuits , 2015 .

[55]  Dequan Li,et al.  A three-scroll chaotic attractor , 2008 .

[56]  A. E. Matouk,et al.  Chaos, feedback control and synchronization of a fractional-order modified Autonomous Van der Pol–Duffing circuit , 2011 .

[57]  Sundarapandian Vaidyanathan,et al.  Anti-synchronization of four-wing chaotic systems via sliding mode control , 2012, Int. J. Autom. Comput..

[58]  Sundarapandian Vaidyanathan,et al.  Analysis, Control, and Synchronization of a 3-D Novel Jerk Chaotic System with Two Quadratic Nonlinearities , 2015 .

[59]  Sundarapandian Vaidyanathan,et al.  Adaptive Chaotic Synchronization of Enzymes-Substrates System with Ferroelectric Behaviour in Brain Waves , 2015 .

[60]  Sundarapandian Vaidyanathan,et al.  Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities , 2014 .

[61]  O. Rössler.  An equation for continuous chaos , 1976 .

[62]  Sundarapandian Vaidyanathan,et al.  Adaptive Control of a Chemical Chaotic Reactor , 2015 .

[63]  Sundarapandian Vaidyanathan,et al.  Design and SPICE implementation of a 12-term novel hyperchaotic system and its synchronisation via active control , 2015, Int. J. Model. Identif. Control..

[64]  Sundarapandian Vaidyanathan,et al.  Analysis and adaptive control of a novel 3-D conservative no-equilibrium chaotic system , 2015 .

[65]  Ioannis M. Kyprianidis,et al.  Text Encry ption Scheme Realized with a Chaotic Pseudo - Random Bit Generator , 2013 .

[66]  Guoliang Cai,et al.  Chaos Synchronization of a New Chaotic System via Nonlinear Control , 2007 .

[67]  S. Vaidyanathan.  A 3 - D Novel Highly Chaotic System with Four Quadratic Nonlinearities, its Adaptive Control and Anti - Synchronization with Unknown Parameters , 2015 .

[68]  Sundarapandian Vaidyanathan,et al.  Active Control Design for the Anti-Synchronization of Lotka- Volterra Biological Systems with Four Competitive Species , 2015 .

[69]  Sundarapandian Vaidyanathan,et al.  Random Bit Generator Based on Non-Autonomous Chaotic Systems , 2015 .

[70]  Sundarapandian Vaidyanathan,et al.  Hyperchaos, qualitative analysis, control and synchronisation of a ten-term 4-D hyperchaotic system with an exponential nonlinearity and three quadratic nonlinearities , 2015, Int. J. Model. Identif. Control..

[71]  Sundarapandian Vaidyanathan,et al.  A nalysis, Adaptive Control and Anti - Synchronization of a Six - Term Novel Jerk Chaotic System with two Exponential Nonlinearities and its Circuit Simulation , 2015 .

[72]  Sundarapandian Vaidyanathan,et al.  Analysis, Adaptive Control and Synchronization of a Seven - Term Novel 3 - D Chaotic System with Three Quadratic Nonlinearities and its Digital Implementation in LabVIEW , 2015 .

[73]  Dr.V. Sundarapandian.  Anti-Synchronization of Liu and Li Systems by Active Nonlinear Control , 2011 .

[74]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of Mathieu-Van der Pol Chaotic Systems via Adaptive Control Method , 2015 .

[75]  Sundarapandian Vaidyanathan.  Generalised projective synchronisation of novel 3-D chaotic systems with an exponential non-linearity via active and adaptive control , 2014, Int. J. Model. Identif. Control..

[76]  Sundarapandian Vaidyanathan,et al.  Sliding Mode Control of Rucklidge Chaotic System for Nonlinear Double Convection , 2015 .

[77]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of the Lotka-Volterra Biological Systems with Four Competitive Species via Active Control , 2015 .

[78]  Sundarapandian Vaidyanathan,et al.  Chaos in Neurons and Adaptive Control of Birkhoff-Shaw Strange Chaotic Attractor , 2015 .

[79]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of Hyperchaotic Pang and Hyperchaotic Wang Systems via Adaptive Control , 2011 .

[80]  Sundarapandian Vaidyanathan,et al.  A Novel Chemical Chaotic Reactor System and its Adaptive Control , 2015 .

[81]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of n-Scroll Chua Circuit and Lur’e System using Backstepping Control Design with Recursive Feedback , 2014 .

[82]  Sundarapandian Vaidyanathan,et al.  Output Regulation of the Forced Van der Pol Chaotic Oscillator via Adaptive Control Method , 2015 .

[83]  Sundarapandian Vaidyanathan,et al.  Adaptive backstepping control, synchronization and circuit simulation of a 3-D novel jerk chaotic system with two hyperbolic sinusoidal nonlinearities , 2014 .

[84]  Sundarapandian Vaidyanathan,et al.  Adaptive Backstepping Controller Design for the Anti - Synchronization of Identical WINDMI Chaotic Systems with Unknown Parameters and its SPICE Implementation , 2015 .

[85]  Sundarapandian Vaidyanathan,et al.  A Memristor-Based Hyperchaotic System with Hidden Attractors: Dynamics, Synchronization and Circuital Emulating , 2015 .

[86]  Ioannis M. Kyprianidis,et al.  Image encryption process based on chaotic synchronization phenomena , 2013, Signal Process..

[87]  Alain Arneodo,et al.  Possible new strange attractors with spiral structure , 1981 .

[88]  Sundarapandian Vaidyanathan,et al.  Global Chaos Control of 3-Cells Cellular Neural Network Attractor via Integral Sliding Mode Control , 2015 .

[89]  Sundarapandian Vaidyanathan,et al.  Output Regulation of Arneodo-Coullet Chaotic System , 2011 .

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

[91]  P. Olver.  Nonlinear Systems , 2013 .

[92]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[93]  Sundarapandian Vaidyanathan,et al.  Analysis, synchronization and circuit design of a novel butterfly attractor , 2014 .

[94]  Sundarapandian Vaidyanathan,et al.  Global Chaos Synchronization of Hyperchaotic Lorenz Systems by Sliding Mode Control , 2011 .

[95]  J. Yorke,et al.  Chaos: An Introduction to Dynamical Systems , 1997 .

[96]  Sundarapandian Vaidyanathan,et al.  ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF THREE -SCROLL CHAOTIC SYSTEMS , 2012 .

[97]  George S. Tombras,et al.  Synchronization Phenomena in Coupled Non - identical Chaotic Circuits , 2016 .

[98]  Sundarapandian Vaidyanathan.  Adaptive Synchronization of Novel 3-D Chemical Chaotic Reactor Systems , 2015 .

[99]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[100]  Sundarapandian Vaidyanathan,et al.  Synchronization of Tokamak Systems with Symmetric and Magnetically Confined Plasma via Adaptive Control , 2015 .

[101]  V. Sundarapan,et al.  Sliding Controller Design of Hybrid Synchronization of Four-Wing Chaotic Systems , 2011 .

[102]  Sundarapandian Vaidyanathan,et al.  Anti-Synchronization of Chemical Chaotic Reactors via Adaptive Control Method , 2015 .

[103]  Uchechukwu E. Vincent,et al.  Synchronization of chaos in non-identical parametrically excited systems , 2009 .