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Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory

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Abstract

This paper deals with problems for which it is necessary to represent the oscillations of the electromechanical seismograph about their position of equilibrium as regards the synchronous condition. The loss of stability of the system occurs through a saddle-node bifurcation, where there is a collision of the stable orbit with an unstable one. Then, global bifurcations and chaotic dynamics of an electromechanical seismograph are the aims of this study. The electrical part of the model is described by an extended force Rayleigh oscillator with Φ 6-potential, while the mechanical part is described by a damped and driven linear oscillator. By using the direct perturbation technique, we analytically obtain the general solution of the first-order equation. Through the boundedness condition of the general solution we get the famous Melnikov function predicting the onset of chaos in the case where the Φ 6-potential is three wells, which are complemented by numerical simulations by which we illustrate the bifurcation curves and the fractality of the basins of attraction. The results show that the threshold amplitude of harmonic excitation for the onset of instability will move upwards as the amplitude intensity of the ground motion increases. These results suggest that much attention should be paid to controlling the increase of the amplitude of the ground motion, especially when the harmonic excited electromechanical seismograph system as a main device is applied to some practical systems.

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Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

    M. Siewe Siewe

  2. Faculté des Sciences, Département de Physique, Laboratoire de Mécanique, Université de Yaoundé I, BP 812, Yaoundé, Cameroun

    M. Siewe Siewe & C. Tchawoua

  3. Department of Physics, Higher Teachers Training College, Annex Bambili, Université de Yaoundé I, P.O. Box 39, Bamenda, Cameroon

    S. B. Yamgoué

  4. Department of Physics, Faculty of Science, University of Buea, P.O. Box 63, Buea, Cameroon

    F. M. Moukam Kakmeni

Authors
  1. M. Siewe Siewe
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  2. S. B. Yamgoué
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  3. F. M. Moukam Kakmeni
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  4. C. Tchawoua
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Correspondence to M. Siewe Siewe.

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Siewe Siewe, M., Yamgoué, S.B., Moukam Kakmeni, F.M. et al. Chaos controlling self-sustained electromechanical seismograph system based on the Melnikov theory. Nonlinear Dyn 62, 379–389 (2010). https://doi.org/10.1007/s11071-010-9725-3

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  • DOI: https://doi.org/10.1007/s11071-010-9725-3

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