Nonlinear dynamics, coexistence of attractors and microcontroller implementation of a modified Helmholtz Jerk oscillator

In this work, we converted a two-dimensional modified Helmholtz oscillator into a three-dimensional modified Helmholtz jerk oscillator. The study of the stability of the fixed points is made and by using the theorem of Hopf, the condition of existence of the bifurcation of Hopf is sought. By numerical simulations relating to the diagrams of the basin of parameters, attraction, bifurcation, the Lyapunov exponents and the phase portrait, the global dynamics as well as the coexistence of the attractors of the system are analyzed. This study revealed that the considered modified Jerk Helmholtz oscillator can generate Hopf bifurcation, bistable limit cycles, coexistence of chaotic and periodic attractors for appropriate choices of system parameter values. The microcontroller based implementation of the modified Jerk Helmholtz oscillator is proposed to experimentally verify the obtained analytical and numerical results. Finally, to control the amplitude of the Lyapunov attractor and exponent, we added two new parameters in the modified Helmholtz jerk oscillator.

Automated summary

This study converts a two-dimensional modified Helmholtz oscillator into a three-dimensional modified Helmholtz jerk oscillator, and investigates the stability of fixed points as well as the bifurcation of Hopf using numerical simulations. The global dynamics and the coexistence of attractors are analyzed through the basin of parameters, attraction, bifurcation, Lyapunov exponents, and phase portrait. It is found that the modified Jerk Helmholtz oscillator can generate Hopf bifurcation, bistable limit cycles, and the coexistence of chaotic and periodic attractors under appropriate system parameter values. An experimental verification of the modified Jerk Helmholtz oscillator based on microcontroller implementation is proposed. Two new parameters are added to control the amplitude of the Lyapunov attractor and exponent in the modified Helmholtz jerk oscillator.