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

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Abstract

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 \(QS_{ \pm i} (i = 1,2)\) and \(SP_{ \pm i} (i = 1,2)\), 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.

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Acknowledgements

The authors thank the editor and anonymous reviewers for their valuable comments and suggestions that helped to improve the presentation of the paper. H. Simo thanks Dr. Sifeu Takougang Kingni (Department of Mechanical, Petroleum and Gas Engineering, Faculty of Mines and Petroleum Industries, University of Maroua, Cameroun) for interesting discussions. Tchahou thanks Rodrigo Alejandro Melo for interesting discussions we had.

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The authors received no specific funding for this work.

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Authors and Affiliations

  1. Department of Mechanical Engineering, the University Institute of Technology, The University of Ngaoundéré, P.O. Box 455, Ngaoundéré, Cameroon

    Herve Simo

  2. Laboratory of Modelling and Simulation in Engineering, Biomimetics and Prototypes and Twas Research Unit, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde, Cameroun

    Herve Simo & Fabien Kenmogne

  3. Department of Computer Engineering, National Higher Polytechnic Institute, University of Bamenda, P. O. Box 39, Bambili, Cameroon

    Achille Ecladore Tchahou Tchendjeu

  4. Department of Civil Engineering, Higher Teacher Training College of the Technical Education (ENSET), University of Douala, Douala, Cameroon

    Fabien Kenmogne

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  1. Herve Simo
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Correspondence to Herve Simo.

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Simo, H., Tchendjeu, A.E.T. & Kenmogne, F. Study of Bursting Oscillations in a Simple System with Signum Nonlinearity with Two Timescales: Theoretical Analysis and FPGA Implementation. Circuits Syst Signal Process 41, 4185–4209 (2022). https://doi.org/10.1007/s00034-022-01982-z

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