Autonomous Jerk Oscillator with Cosine Hyperbolic Nonlinearity: Analysis, FPGA Implementation, and Synchronization

Karthikeyan Rajagopal,Viet-Thanh Pham,Sifeu Takougang Kingni,Gaetan Fautso Kuiate,Victor Kamdoum Tamba

doi:10.1155/2018/7273531

Abstract

A two-parameter autonomous jerk oscillator with a cosine hyperbolic nonlinearity is proposed in this paper. Firstly, the stability of equilibrium points of proposed autonomous jerk oscillator is investigated by analyzing the characteristic equation and the existence of Hopf bifurcation is verified using one of the two parameters as a bifurcation parameter. By tuning its two parameters, various dynamical behaviors are found in the proposed autonomous jerk oscillator including periodic attractor, one-scroll chaotic attractor, and coexistence between chaotic and periodic attractors. The proposed autonomous jerk oscillator has period-doubling route to chaos with the variation of one of its parameters and reverse period-doubling route to chaos with the variation of its other parameter. The proposed autonomous jerk oscillator is modelled on Field Programmable Gate Array (FPGA) and the FPGA chip statistics and phase portraits are derived. The chaotic and coexistence of attractors generated in the proposed autonomous jerk oscillator are confirmed by FPGA implementation of the proposed autonomous jerk oscillator. A good qualitative agreement is illustrated between the numerical and FPGA results. Finally synchronization of unidirectional coupled identical proposed autonomous jerk oscillators is achieved using adaptive sliding mode control method.

Highlights

There has been a significant increase in studies of chaotic oscillators over the past decades because of their complex behaviors and their promising applications [ – ]
An elegant chaotic oscillators based on jerk equation was implemented in a circuit where the nonlinearity was provided by a single diode [ ]
Njitacke et al discovered the coexistence of multiple attractors and crisis route to chaos in a chaotic jerk circuit [ ] and a simple autonomous jerk oscillator with multiple attractors was introduced in [ ]

Summary

Introduction

There has been a significant increase in studies of chaotic oscillators over the past decades because of their complex behaviors and their promising applications [ – ]. There are increasing works on chaos in jerk oscillators [ – ]. An elegant chaotic oscillators based on jerk equation was implemented in a circuit where the nonlinearity was provided by a single diode [ ]. Njitacke et al discovered the coexistence of multiple attractors and crisis route to chaos in a chaotic jerk circuit [ ] and a simple autonomous jerk oscillator with multiple attractors was introduced in [ ]. This work revealed that the hybrid diode based jerk circuit exhibits rich dynamical behaviors including multiple coexisting self-excited attractors. For the same values of system parameters, the coexistence of four different attractors was obtained in such a memristor-based jerk circuit.

Theoretical Analysis of Proposed Autonomous Jerk Oscillator

FPGA Implementation of Proposed

Conclusion