Dynamics and synchronization in feedback cyclic structures with hysteresis oscillators

A. M. Kamachkin ; D. K. Potapov ; V. V. Yevstafyeva

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 16 (2020) no. 2, pp. 186-199

Résumé

Consider the dynamics of complex systems with cyclic links. The systems have the ring structures composed of hysteresis-feedback oscillators. In one of these systems, each oscillator has an additional feedback with the next oscillator. Sufficient conditions for the existence of periodic and recurrent motions are established. The periodic motion corresponds to a synchronous process occurring in such cyclic structures. In particular cases, we obtain the conditions for synchronization, as well as the stability conditions for synchronous oscillatory processes.

Détail

Keywords: complex system dynamics, cyclic structure, oscillator, hysteresis feedback, synchronization, stability.

Bibliographie

[1] K. J. Astrom, “Oscillations in systems with relay feedback”, Adaptive Control, Filtering and Signal Processing, Springer-Verlag, New York, 1995, 1–25 | MR | Zbl

[2] C. C. Yu, Autotuning of PID controllers: relay feedback approach, Springer-Verlag, New York, 1999, 235 pp. | Zbl

[3] X. Zou, R. A. Siegel, “Modeling of oscillatory dynamics of a simple enzyme-diffusion system with hysteresis. The case of lumped permeabilities”, J. Chem. Phys., 110:4 (1999), 2267–2279 | DOI

[4] R. B. Northrop, Endogenous and exogenous regulation and control of physiological systems, Chapman Hall, London, 2000, 452 pp.

[5] B. Li, R. A. Siegel, “Global analysis of a model pulsing drug delivery oscillator based on chemomechanical feedback with hysteresis”, Chaos, 10:3 (2000), 682–690 | DOI | MR | Zbl

[6] J. Grasman, Asymptotic methods for relaxation oscillations and applications, Springer Verlag, New York, 1987, 221 pp. | MR | Zbl

[7] E. A. Jackson, Perspectives of nonlinear dynamics, v. In 2 vol., Cambridge University Press, New York, 1991, 469 pp. ; v. 2, 633 pp. | MR | Zbl

[8] S. K. Scott, Oscillations, waves and chaos in chemical kinetics, Oxford University Press, New York, 1994, 90 pp. | MR

[9] I. R. Epstein, K. Showalter, “Nonlinear chemical dynamics: Oscillations, patterns, and chaos”, J. Phys. Chem., 100:31 (1996), 13132–13147 | DOI

[10] J. W. Macki, P. Nistri, P. Zecca, “Mathematical models for hysteresis”, SIAM Review, 35:1 (1993), 94–123 | DOI | MR | Zbl

[11] S. Varigonda, T. T. Georgiou, “Dynamics of relay relaxation oscillators”, IEEE Trans. Automat. Control, 46:1 (2001), 65–77 | DOI | MR | Zbl

[12] V. I. Zubov, Dynamics of control systems, Saint Petersburg University Press, St. Petersburg, 2004, 380 pp. (In Russian)

[13] A. Yu. Aleksandrov, S. V. Sokolov, “Stability and estimations of solutions for some classes of nonlinear systems”, Proc. Middle Volga Math. Soc., 7:1 (2005), 113–123 (In Russian) | Zbl

[14] M. A. Barron, M. Sen, “Synchronization of four coupled van der Pol oscillators”, Nonlin. Dyn., 56:4 (2009), 357–367 | DOI | MR | Zbl

[15] V. V. Evstaf'eva, “On necessary conditions for existence of periodic solutions in a dynamic system with discontinuous nonlinearity and an external periodic influence”, Ufa Math. Journal, 3:2 (2011), 20–27 (In Russian)

[16] A. S. Kuznetsova, “The dynamics of coupled oscillators with a feedback”, Middle Volga Math. Soc. Journal, 13:2 (2011), 63–69 (In Russian) | MR | Zbl

[17] Z. Balanov, W. Krawcewicz, D. Rachinskii, A. Zhezherun, “Hopf bifurcation in symmetric networks of coupled oscillators with hysteresis”, J. Dyn. Differ. Equations, 24:4 (2012), 713–759 | DOI | MR | Zbl

[18] V. V. Yevstafyeva, “Existence of the unique kT-periodic solution for one class of nonlinear systems”, J. Sib. Fed. Univ. Math. Phys., 6:1 (2013), 136–142

[19] V. V. Evstaf'eva, “On existence conditions for a twopoint oscillating periodic solution in an non-autonomous relay system with a Hurwitz matrix”, Automat. Remote Control, 2015, no. 6, 42–56 (In Russian)

[20] A. M. Kamachkin, G. M. Chitrov, V. N. Shamberov, “Normal matrix forms to decomposition and control problems for manydimensional systems”, Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 13:4 (2017), 417–430 (In Russian) | MR

[21] E. Hooton, Z. Balanov, W. Krawcewicz, D. Rachinskii, “Noninvasive stabilization of periodic orbits in O$_4$-symmetrically coupled systems near a Hopf bifurcation point”, Intern. Journal Bifurcation and Chaos, 27:6 (2017), 1750087, 1–18 | DOI | MR

[22] A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva, “Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence”, Intern. Journal Robust Nonlinear Control, 27:2 (2017), 204–211 | DOI | MR | Zbl

[23] A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva, “Existence of subharmonic solutions to a hysteresis system with sinusoidal external influence”, Electron. Journal Differential Equations, 2017, no. 140, 1–10 | MR

[24] A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva, “On uniqueness and properties of periodic solution of second-order nonautonomous system with discontinuous nonlinearity”, J. Dyn. Control Syst., 23:4 (2017), 825–837 | DOI | MR | Zbl

[25] V. V. Evstaf'eva, “Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero”, Ukrainian Math. Journal, 70:8 (2018), 1085–1096 (In Russian) | MR

[26] A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva, “Existence of periodic modes in automatic control system with a three-position relay”, Intern. Journal Control, 93:4 (2020), 763–770 | DOI | MR | Zbl

[27] U. A. Rozikov, An introduction to mathematical billiards, World Scientific Publishing Co. Pte. Ltd., Singapore, 2019, 224 pp. | MR | Zbl