Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis, chaos control and synchronisation in its fractional-order form

In this paper, a Van der Pol–Duffing (VdPD) jerk oscillator is designed. The proposed VdPD jerk oscillator is built by converting the autonomous two-dimensional VdPD oscillator to a jerk oscillator. Dynamical behaviours of the proposed VdPD jerk oscillator are investigated analytically, numerically and analogically. The numerical results indicate that the proposed VdPD jerk oscillator displays chaotic oscillations, symmetrical bifurcations and coexisting attractors. The physical existence of the chaotic behaviour found in the proposed VdPD jerk oscillator is verified by using Orcad-PSpice software. A good qualitative agreement is shown between the numerical simulations and the PSpice results. Moreover, the fractional-order form of the proposed VdPD jerk oscillator is studied using stability theorem of fractional-order systems and numerical simulations. It is found that chaos, periodic oscillations and coexistence of attractors exist in the fractional-order form of the proposed jerk oscillator with order less than three. The effect of fractional-order derivative on controlling chaos is illustrated. It is shown that chaos control is achieved in fractional-order form of the proposed VdPD jerk oscillator only for the values of linear controller used. Finally, the problem of drive–response synchronisation of the fractional-order form of the chaotic proposed VdPD jerk oscillators is considered using active control technique.

[1]  Paul Woafo,et al.  Synchronizing modified van der Pol–Duffing oscillators with offset terms using observer design: application to secure communications , 2007 .

[2]  Xinping Guan,et al.  Formation and obstacle avoidance control for multiagent systems , 2011 .

[3]  Uchechukwu E. Vincent,et al.  Adaptive backstepping control and synchronization of a modified and chaotic Van der Pol-Duffing oscillator , 2011 .

[4]  T. Kapitaniak.  Chaos for Engineers: Theory, Applications, and Control , 2012 .

[5]  Kyandoghere Kyamakya,et al.  Regular oscillations, chaos, and multistability in a system of two coupled van der Pol oscillators: numerical and experimental studies , 2014 .

[6]  P. Woafo,et al.  Adaptive synchronization of a modified and uncertain chaotic Van der Pol-Duffing oscillator based on parameter identification , 2005 .

[7]  Hsien-Keng Chen,et al.  Implementation of the fractional-Order Chen-Lee System by Electronic Circuit , 2013, Int. J. Bifurc. Chaos.

[8]  Andrew G. Glen,et al.  APPL , 2001 .

[9]  K. O’Grady,et al.  柔軟記録媒体のための金属粒子(MP)技術の開発 , 2008 .

[10]  U. Feudel,et al.  Control of multistability , 2014 .

[11]  AYUB KHAN,et al.  Multiswitching combination–combination synchronization of chaotic systems , 2017 .

[12]  Jan Danckaert,et al.  Bursting oscillations in a 3D system with asymmetrically distributed equilibria: Mechanism, electronic implementation and fractional derivation effect , 2015 .

[13]  YangQuan Chen,et al.  Fractional order robust control for cogging effect compensation in PMSM position servo systems: Stability analysis and experiments , 2010 .

[14]  Samuel Bowong,et al.  Synchronization of uncertain chaotic systems via backstepping approach , 2004 .

[15]  Tao Wang,et al.  Chaos control and hybrid projective synchronization of several new chaotic systems , 2012, Appl. Math. Comput..

[16]  J.-M. Malasoma.  What is the simplest dissipative chaotic jerk equation which is parity invariant , 2000 .

[17]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[18]  Jiaming Liang,et al.  Stability analysis for periodic solutions of the Van der Pol–Duffing forced oscillator , 2016 .

[19]  Julien Clinton Sprott,et al.  A New Chaotic Jerk Circuit , 2011, IEEE Transactions on Circuits and Systems II: Express Briefs.

[20]  K SEBASTIAN SUDHEER,et al.  Modified function projective combination synchronization of hyperchaotic systems , 2017 .

[21]  Tomasz Kapitaniak.  Chaos for Engineers , 1998 .

[22]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

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

[24]  Hans Peter Gottlieb,et al.  What is the Simplest Jerk Function that Gives Chaos , 1996 .

[25]  Mohammad Ataei,et al.  A chattering-free sliding mode control design for uncertain chaotic systems , 2009 .

[26]  Samuel Bowong,et al.  Practical finite-time synchronization of jerk systems: Theory and experiment , 2014, Nonlinear Dynamics.

[27]  Leonardo Acho,et al.  Chaotification of the Van der Pol System Using Jerk Architecture , 2006, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[28]  A. Maccari,et al.  Vibration amplitude control for a van der Pol–Duffing oscillator with time delay , 2008 .

[29]  runden Tisch,et al.  AM , 2020, Catalysis from A to Z.

[30]  P. K. Talla,et al.  Emergence of complex dynamical behaviors in improved Colpitts oscillators: antimonotonicity, coexisting attractors, and metastable chaos , 2017 .

[31]  Reza Ghaderi,et al.  Fuzzy fractional order sliding mode controller for nonlinear systems , 2010 .

[32]  A. Jonscher.  Dielectric relaxation in solids , 1983 .

[33]  ADELEH NOURIAN,et al.  The adaptive synchronization of fractional-order Liu chaotic system with unknown parameters , 2016 .

[34]  Adv , 2019, International Journal of Pediatrics and Adolescent Medicine.

[35]  Paul Woafo,et al.  Dynamics and synchronization analysis of coupled fractional-order nonlinear electromechanical systems , 2012 .

[36]  Kehui Sun,et al.  Dynamical properties and complexity in fractional-order diffusionless Lorenz system , 2016 .

[37]  Uchechukwu E. Vincent,et al.  Synchronization of identical and non-identical 4-D chaotic systems using active control , 2008 .

[38]  YangQuan Chen,et al.  Linear Feedback Control: Analysis and Design with MATLAB , 2008 .

[39]  Paul Woafo,et al.  Bursting generation mechanism in a three-dimensional autonomous system, chaos control, and synchronization in its fractional-order form , 2014 .

[40]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[41]  Andrew Y. T. Leung,et al.  Anti-synchronization between identical and non-identical fractional-order chaotic systems using active control method , 2014 .

[42]  Elsayed Ahmed,et al.  On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems , 2006 .

[43]  Z. Njitacke Tabekoueng,et al.  Periodicity, chaos, and multiple attractors in a memristor-based Shinriki's circuit. , 2015, Chaos.

[44]  Jamal Daafouz,et al.  Adaptive synchronization of two chaotic systems consisting of modified Van der Pol–Duffing and Chua oscillators , 2005 .

[45]  Teh-Lu Liao,et al.  Optimal PID control design for synchronization of delayed discrete chaotic systems , 2008 .

[46]  Hadi Taghvafard,et al.  Phase and anti-phase synchronization of fractional order chaotic systems via active control , 2011 .

[47]  Jacques Kengne,et al.  Dynamical analysis of a simple autonomous jerk system with multiple attractors , 2016 .

[48]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[49]  J. Kengne,et al.  Experiment on Bifurcation and Chaos in Coupled Anisochronous Self-Excited Systems: Case of Two Coupled van der Pol-Duffing Oscillators , 2014 .

[50]  Wanda Szemplińska-Stupnicka,et al.  THE COEXISTENCE OF PERIODIC, ALMOST-PERIODIC AND CHAOTIC ATTRACTORS IN THE VAN DER POL-DUFFING OSCILLATOR , 1997 .

[51]  A N Pisarchik,et al.  Controlling the multistability of nonlinear systems with coexisting attractors. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  Mohammad Saleh Tavazoei,et al.  The effect of fractionality nature in differences between computer simulation and experimental results of a chaotic circuit , 2013 .