Abstract

ABSTRACTIn this paper, an autonomous Toda jerk oscillator is proposed and analysed. The autonomous Toda jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator with an exponential nonlinear term to a jerk oscillator. The existence of Hopf bifurcation is established during the stability analysis of the unique equilibrium point. For a suitable choice of the parameters, the proposed autonomous Toda jerk oscillator can generate antimonotonicity, periodic oscillations, chaotic oscillations and bubbles. By introducing two additional parameters in the proposed autonomous Toda jerk oscillator, it is possible to control partially or totally the amplitude of its signals. In addition, electronic circuit realization of the proposed Toda jerk oscillator is carried out to confirm results found during numerical simulations. The commensurate fractional-order version of the proposed autonomous chaotic Toda jerk oscillator is studied using the stability theorem of fractional-order oscillators and numerical simulations. It is found that periodic oscillations and chaos exist in the fractional-order form of the proposed Toda jerk oscillator with order less than three. Finally, combination synchronization of two fractional-order proposed autonomous chaotic Toda jerk oscillators with another fractional-order proposed autonomous chaotic Toda jerk oscillator is analysed using the nonlinear feedback control method.

Highlights

  • Chaotic oscillations can be generated in the thirdorder or higher-order autonomous nonlinear differential equations

  • By using the Routh–Hurwitz stability criterion and linear stability of the only equilibrium point, it was found that Hopf bifurcation occurs in the proposed autonomous Toda jerk oscillator

  • One-scroll and bubble chaotic attractors obtained during numerical simulations were confirmed using electronic circuit realization of the proposed autonomous Toda jerk oscillator

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Summary

Introduction

Chaotic oscillations can be generated in the thirdorder or higher-order autonomous nonlinear differential equations. Motivated by the above works, in this paper, an autonomous chaotic jerk oscillator is obtained by converting an autonomous two-dimensional Toda oscillator into a three-dimensional differential equations using the jerk architecture which belongs to the family of MO11. This memory oscillator is a simple autonomous jerk oscillator with exponential nonlinearity. In this paper, the integer and fractional autonomous Toda jerk oscillator is analysed in order to understand the dynamics of this class of jerk oscillator with exponential nonlinearity.

Description and analysis of the autonomous Toda jerk oscillator
Analytical study
Numerical study
Antimonotonicity phenomenon
Partial and total amplitude control
Electronic circuit realization of the autonomous Toda jerk oscillator
Conclusion
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