Study of Bursting Oscillations in a Simple System with Signum Nonlinearity with Two Timescales: Theoretical Analysis and FPGA Implementation

• Published in: Circuits, systems, and signal processing systems, and signal processing, 2022-08, Vol.41 (8), p.4185-4209
• Main Authors: Simo, Herve, Tchendjeu, Achille Ecladore Tchahou, Kenmogne, Fabien
• Format: Article
• Language: English
• Subjects: Bifurcations; Bursting; Circuits and Systems; Continuity (mathematics); Electrical Engineering; Electronics and Microelectronics; Engineering; Excitation; Field programmable gate arrays; Hyperbolic functions; Instrumentation; Nonlinearity; Oscillations; Parameter sensitivity; Signal,Image and Speech Processing; Smooth boundaries; Spiking; Subsystems; Waveforms
• ISSN: 0278-081X; 1531-5878
• DOI: 10.1007/s00034-022-01982-z

In this paper, the effects of slowly varying external excitation and the sign of the strength of signum term of system with a simple signum nonlinearity function are investigated by using the fast-slow analysis method and the transformation phase portrait method. Firstly, a double-well and single-well non-smooth potential are presented. A comparison of the proposed system and the Duffing system with the same fixed points is presented. Secondly, by taking the periodic excitation as a slow-varying parameter, the non-smooth system can possess two timescales and it can be transformed into two linear autonomous systems. The phase space can be divided into two regions by the non-smooth boundaries, in which the trajectory is governed by two different subsystems, respectively. Based on the analysis of the two subsystems, the stabilities and the bifurcations of the equilibrium branches of the fast subsystem are presented when the sign of parameter b of signum term is positive and negative. The results show that the stability of the system and its bifurcations are most sensitive to the change of sign of parameter b .Thirdly, through numerical simulations, the bursting oscillations with different waveforms are observed in the non-smooth system. It is found that for b > 0 , bursting oscillations presents two types of quiescent state (QS) and spiking state (SP), respectively, while if b < 0 , bursting patterns with four type of quiescent state and spiking state which are represented by Q S ± i ( i = 1 , 2 ) and S P ± i ( i = 1 , 2 ) , respectively. Based on the analysis of the bifurcations and portrait phase, the dynamical mechanism of the bursting oscillations is analyzed. Finally, the signum function is replaced by a sharply varying continuous hyperbolic tangent function. The results show that for a small value of constant parameter n , the system exhibits bursting patterns. It is also found that the amplitude and the number of the spikes of bursting oscillation depend on the value of n . The proposed system is implemented in field programmable gate array (FPGA) using hardware/software code-sign to verify the numerical simulation, because it can be applied in embedded engineering based on bursting.