On functional differential equations connected to Huygens synchronization under propagation

Authors

  • Vladimir Rasvan Universitatea din Craiova

DOI:

https://doi.org/10.52846/stccj.2022.2.1.33

Keywords:

conservation laws, distortionless propagation, time delays

Abstract

The structure represented by one or several oscillators couple to a one-dimensional transmission environment (e.g. a vibrating string in the mechanical case or a lossless transmission line in the electrical case) turned to be attractive for the research in the field of complex structures and/or complex behavior. This is due to the fact that such a structure represents some generalization of various interconnection modes with lumped parameters for the oscillators.
On the other hand the lossless and distortionless propagation along transmission lines has generated several research in electrical, thermal, hydro and control engineering leading to the association of some functional differential equations to the basic initial boundary value problems.
The present research is performed at the crossroad of the aforementioned directions. We shall associate to the starting models some functional differential equations - in most cases of neutral type - and make use of the general theorems for existence and stability of forced oscillations for functional differential equations. The challenges introduced by the analyzed problems for the general theory are emphasized, together with the implication of the results for various applications.

References

Y. I. Neymark, Dynamical Systems and Controlled Processes (in Russian). Moscow: Nauka Publishing House, 1978, pp. 11–14.

J. K. Hale, “Coupled Oscillators on a Circle,” Resenhas IME-USP, vol. 1, no. 4, pp. 441–457, 1994.

——, “Diffusive coupling, dissipation and synchronization,” J. Dyn. Diff. Equ., vol. 9, no. 1, pp. 1–52, 1997. DOI: https://doi.org/10.1007/BF02219051

M. Earl and S. Strogatz, “Synchronization in oscillator networks with delayed coupling: A stability criterion,” Physical Review E, vol. 67, 036204, pp. 1–4, 2003. DOI: https://doi.org/10.1103/PhysRevE.67.036204

J. J. Fox, C. Jayaprakash, D. L. Wang, and S. R. Campbell, “Synchronization in Relaxation Oscillator Networks with Conduction Delays,” Neural Computation, vol. 13, pp. 1003–1021, 2001. DOI: https://doi.org/10.1162/08997660151134307

J. K. Hale, “Synchronization through boundary interactions,” in Advances in Time Delay Systems, ser. Lect. Notes in Comp. Sci. Eng., S. I. Niculescu and K. Gu, Eds., no. 38. Berlin Heidelberg: Springer, 2004, pp. 225–232. DOI: https://doi.org/10.1007/978-3-642-18482-6_16

A. Pikovsky, “The simplest case of chaotic wave scattering,” Chaos, vol. 3, no. 4, pp. 505–506, 1993. DOI: https://doi.org/10.1063/1.165995

S. Lepri and A. Pikovsky, “Nonreciprocal wave scattering on nonlinear string-coupled oscillators,” Chaos, vol. 24, 043119, pp. 1–9, 2014. DOI: https://doi.org/10.1063/1.4899205

G. B. Ermentrout and N. Koppell, “Frequency plateaus in a chain of weakly coupled oscillators, I,” SIAM J. Math. Anal., vol. 15, no. 2, pp. 215–237, 1984. DOI: https://doi.org/10.1137/0515019

N. Koppell and G. B. Ermentrout, “Symmetry and Phaselocking in Chains of Weakly Coupled Oscillators,” Comm. Pure Appl. Math., vol. XXXIX, pp. 623–660, 1986. DOI: https://doi.org/10.1002/cpa.3160390504

N. G. Cˇ etaev, “Stability and classical laws,” Coll. Sci. Works Kazan Aviation Inst., no. 5, pp. 3–18, 1936.

V. Abolinia and A. Myshkis, “Mixed problem for an almost linear hyperbolic system in the plane (in Russian),” Mat. Sbornik, vol. 50(92), no. 4, pp. 423–442, 1960.

K. L. Cooke and D. W. Krumme, “Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations,” Journ. Math. Anal. Appl., vol. 24, no. 2, pp. 372–387, 1968. DOI: https://doi.org/10.1016/0022-247X(68)90038-3

K. L. Cooke, “A linear mixed problem with derivative boundary conditions.” in Seminar on Differential Equations and Dynamical Systems (III)., ser. Lecture Series, D. Sweet and J. A. Yorke, Eds., no. 51. College Park: University of Maryland, 1970, pp. 11–17.

V. R˘asvan, “Augmented validation and a stabilization approach for systems with propagation,” in Systems Theory: Perspectives, Applications and Developments., ser. Systems Science Series, F. Miranda, Ed., no. 1. New York: Nova Science Publishers, 2014, pp. 125–170.

A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics (in Russian). Moscow: Nauka Publishing House, 1977, pp. 64–70.

V. A. Yakubovich, “The method of the matrix inequalities in the theory of stability of nonlinear control systems, I. Absolute stability of the forced oscillations (in Russian),” Avtom. i telemekhanika, vol. XXVIII, no. 7, pp. 1017–1029, 1964.

R. W. Brockett, “Synchronization without periodicity,” in Mathematical Systems Theory, A Volume in Honor of U. Helmke., K. H¨uper and J. Trumpf, Eds. CreateSpace, 2013, pp. 65–74.

V. R˘asvan, Absolute stability of time lag control systems (in Romanian). Bucharest: Editura Academiei, 1975.

J. K. Hale and S. V. Lunel, Introduction to Functional Differential Equations. New York: Springer, 1993. DOI: https://doi.org/10.1007/978-1-4612-4342-7

A. Halanay and V. R˘asvan, “Periodic and almost periodic solutions for a class of systems described by coupled delay-differential and difference equations,” Nonlinear Analysis:Theory,Methods & Applications, vol. 1, no. 3, pp. 197–206, 1977. DOI: https://doi.org/10.1016/0362-546X(77)90029-3

V. R˘asvan, “Synchronization with propagation – the functional differential equations,” AIP Conference Proceedings, vol. 1738, p. 210011, 2016. DOI: https://doi.org/10.1063/1.4951994

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Published

2022-06-30

How to Cite

[1]
V. Rasvan, “On functional differential equations connected to Huygens synchronization under propagation”, Syst. Theor. Control Comput. J., vol. 2, no. 1, pp. 34–43, Jun. 2022.