Van der Pol equation with sine nonlinearity: dynamical behavior and real time control to a target trajectory

Abstract

In this work, the dynamical behavior and real time control of a target trajectory of a modified Van der Pol model so the potential is proportional to the term ( ) x sin n is proposed. A generalization in the case of small oscillations on the ( ) x sin n function is studied. Due to ( ) x sin n function, the system presents periodical regions of stability and unstability, a very rich dynamical behavior. Analytical investigations based on the harmonic balance method came out some specific values of the excitation frequency for which the model is subjected to a phenomenon of frequency entrainment. Also, under effect of the sine function power, chaos appears for even small value of the nonlinearity coefficient, in contrast to the classical Van der Pol oscillator. An investigation as an artificial pacemaker is done based on the real frequency of the natural pacemaker. We found that the modified Van der Pol model, like the classic Van der Pol model, can play the role of an artificial pacemaker with some approximations. Due to the complexity of the analogical sine function, an experimental study was made by real implementation of an Arduino Card based on the Runge-Kutta 4th order algorithm. The results obtained show a good correlation with the numerical results. RECEIVED

Dynamical behavior analysis of the heart system by the bifurcation structures

“…Followed by adapted controllers and attached to sensors, cardiac activity is monitored. References [ 46 , 68 ] have shown that the microcontroller-based differential equation conversion mechanism can be used to reproduce the operation of artificial pacemakers. The principle of this simulation is based on algorithms.…”

Analysis of the dynamics of new models of nonlinear systems with state variable damping and elastic coefficients

“…Microcontroller simulation is one of the simplest because it is performed using a computer in which suitable simulation software is installed. Recent work has shown that it can be used to do electronics using just a programming language ( Fonkou et al, 2021a , Fonkou et al, 2021b , Fonkou et al, 2021c ). Experimentally, it is less cumbersome and requires significant period savings for component wiring, a higher degree of integration, and lower power consumption.…”

“…A microcontroller is an integrated circuit that gathers the essential elements of a computer. Recent work has shown that microcontrollers can be used to simulate nonlinear differential equations ( Fonkou et al, 2021a , Fonkou et al, 2021b , Fonkou et al, 2021c ) by producing simple and complex electrical signals. Compared to conventional electronic systems based on separate components, they can reduce the size of the components, as well as the cost of the products.…”

Dynamical behavior analysis of the Van der Pol oscillator with sine nonlinearity subjected to non-sinusoidal periodic excitations by the bifurcation structures

“…The Van der Pol oscillator (VdP) is the starting point for cardiac modelling, thanks to the qualitative characteristics of its pulse (sinusoidal and relaxation), but also to its capacity to control the cardiac rhythm since it can be used as a corrector in the case of blood pulse signals [51]. With this in mind, Fonkou et al [52] proposed a modified model of the Van der Pol oscillator by replacing the potential x with the term ( ) sin x . n In their work, they show that their model can describe the role of the artificial pacemaker just as well as the classic model of the Van der Pol oscillator, with some approximations.…”

“…The most manipulable ideal signals are of a square, triangular or even pulsed nature, Simo and Woafo [50] show that when the Van der Pol oscillator is subjected to non-sinusoidal periodic excitations (signals of a square and triangular nature), chaos only appears for large values of the dissipation term ( ) e = 5.0 . On the basis of these observations, the question arises as to how the model proposed by Fonkou et al [52] would behave if it were excited by nonsinusoidal periodic signals of a square and triangular nature? What would be the effect of the power n on the appearance of chaotic dynamics in the case of small and large values of ( ) e e = e = 0.1and 5.0 ?…”