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

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 sin(n) (x) is proposed. A generalization in the case of small oscillations on the sinn (x) function is studied. Due to sinn (x) 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.