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

Abstract

The present work studies the dynamical behavior of the Van der Pol oscillator with sine nonlinearity oscillator subjected to the effects of non-sinusoidal excitations through the analysis of bifurcation structures. In this work, a modification of the classical Van der Pol oscillator is carried out by replacing the linear term x with the nonlinear term sinn (x) . The idea is to see the impact of the strength of this function on the appearance of chaotic dynamics for small and large values of the nonlinear dissipation term  . For this purpose, studies by numerical simulation and by analogue simulation are proposed. First, using nonlinear analysis tools, a similarity on the bifurcation sequences is observed despite a difference in the ranks of obtaining the particular behaviours and the bifurcation points. This study is confirmed by their respective maximum Lyapunov exponents. In a second step, a real implementation using microcontroller technology, which is motivated by the use of an artificial pacemaker, is carried out. For a more practical case, the excitation signal is generated by another microcontroller. The study shows a similarity with the results obtained numerically. Thirdly, in order to show that the model can be derived from mathematical modelling, an electronic simulation using OrCAD-Pspice software is also proposed. The results obtained are in qualitative agreement

On the heart rhythm analysis using a nonlinear dynamics perspective: analytical study and electronic simulation

“…FitzHugh and Nagumo also proposed an improvement on Van der Pol's model, seeking to provide a simpler and clearer description in qualitative terms [38]. Inspired by the work of Balthazar Van der Pol and Van der Mark in 1926 [39,40], there has also been a great deal of work on heart rate modeling, notably the work of Fonkou et al [41][42][43][44][45][46]. We might also mention the work of Moe et al [47].…”

Analysis of Sequences of Bases of Human DNA Using the Square Wave Method (SWM)

“…Other sources, [8], [9], [10],[11] and [12], also refer to that article. The SWM has been mentioned frequently in patents: [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89], [90], [91], [92], [93], [94], [95], [96], [97], [98], [99], [100], ...…”

“…Other sources, [8], [9], [10], [11] and [12], also refer to that article. The SWM has been mentioned frequently in patents: [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57],…”