Abstract
This analysis is concerned with Huygens synchronization over distributed media—vibrating strings or LC transmission lines. If e.g., two oscillators with lumped parameters i.e., described by ordinary differential equations, which display self sustained oscillations, are coupled to some distributed medium, they interact in function of the structural properties of the resulting system. If this medium has infinite length then, according to the structural properties of the difference operator describing propagation, either synchronization with the external frequency or some “complex behavior” can occur. If the coupling has a finite length, again the qualitative properties are determined by the structure of the aforementioned difference operator: either the self sustained oscillations are quenched, the system approaching asymptotically a stable equilibrium (opposite of the Turing coupling of two “cells” that is lumped oscillators) or again the aforementioned “complex behavior” can occur. We state finally the conjecture that this “complex behavior” is in fact some almost periodic oscillation and not a chaotic behavior. It is worth mentioning that the two types of qualitative behavior are in connection with the physical nature of the considered systems. Specifically, the difference operator is strongly stable for electrical systems and critically stable for mechanical systems. It is this aspect that explains proneness to standard or “complex” behavior. However, if the aforementioned conjecture will not be disproved, the difference between the corresponding oscillatory behaviors will consist only in the contents of harmonics given by the Fourier series attached.
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References
Abolinia, V.E., Myshkis, A.D.: Mixed problem for an almost linear hyperbolic system in the plane (in Russian). Mat Sb. 50(92), 423–442 (1960)
Amerio, L., Prouse, G.: Almost-Periodic Functions and Functional Equations. In: The University Series in Higher Mathematics. Van Nostrand Reinhold, New York, Cincinnati, Toronto, London, Melbourne (1971)
Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Dover Publications, New York (1966)
Bellman, R.E., Cooke, K.L.: Differential difference equations. In: Mathematics in Science and Engineering, vol. 6. Academic Press, New York (1963)
Blekhman, I.I., Fradkov, A.L., Nijmeijer, H., Pogromsky, A.Y.: On self-synchronization and controlled synchronization. Syst. Control. Lett. 31, 299–306 (1997)
Boas, R.P.: Entire functions. In: Pure and Applied Mathematics, vol. 5. Academic Press, New York (1954)
Bogolyubov, N.N., Mitropol’skii, Y.A.: Asymptotic Methods in the Theory of Nonlinear Oscillations. Fizmatgiz, Moscow USSR (1961). (in Russian)
Bulgakov, B.V.: Oscillations. Gostekhizdat, Moscow USSR (1954). (in Russian)
Cesari, L.: Asymptotic behavior and stability problems in ordinary differential equations, 2nd edn. In: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 16. Springer, Berlin (1963)
Četaev, N.G.: On the stable trajectories of the dynamics (Russian). Sci. Pap. Kazan Aviat. Inst. 4(5), 3–18 (1936a)
Četaev, N.G.: Stability and the classical laws (Russian). Sci. Pap. Kazan Aviat. Inst. 4(6), 3–5 (1936b)
Cooke, K.: A linear mixed problem with derivative boundary conditions. In: Sweet, D., Yorke, J. (eds.) Seminar on Differential Equations and Dynamical Systems (III). Lecture Series, vol. 51, pp. 11–17. University of Maryland, College Park (1970)
Cooke, K.L., Krumme, D.W.: Differential-difference equations and nonlinear initial-boundary value problems for linear hyperbolic partial differential equations. J. Math. Anal. Appl. 24, 372–387 (1968)
Corduneanu, C.: Almost Periodic Functions, 2nd edn. AMS Chelsea Publishing, New York (1989)
Corduneanu, C.: Almost Periodic Oscillations and Waves. Springer, New York (2009)
Dickson, D.G.: Expansions in series of solutions of linear difference- differential and infinite order differential equations with constant coefficients. In: Memoirs of the American Mathematical Society, vol. 23. AMS Publications, Providence RI, USA (1957)
Edelman, K., Gendelman, O.V.: Dynamics of self-excited oscillators with neutral delay coupling. Nonlinear Dyn. 72(3), 683–694 (2013)
Gromova, P.S.: Stability of solutions of nonlinear equations of neutral type in an asymptotically critical case (in Russian). Matem zametki 1(6), 715–726 (1967)
Gromova, P.S., Zverkin, A.M.: About the trigonometric series whose sum is a continuous unbounded on the real axis function—solution of an equation with deviated argument (in Russian). Differ Uravn. 4(10), 1774–1784 (1968)
Halanay, A., Răsvan, V.: Applications of Liapunov methods in stability. In: Mathematics and Its Applications, vol. 245. Kluwer Academic Publishers, Dordrecht (1993)
Halanay, A.: Invariant manifolds for systems with time lags. In: Hale, J.K., LaSalle, J.P. (eds.) Differential Equations and Dynamical Systems, pp. 199–213. Academic Press, New York (1967)
Halanay, A., Răsvan, V.: Almost periodic solutions for a class of systems described by delay differential and difference equations. J. Nonlinear Anal. Theory Methods Appl. 19(1), 197–206 (1977)
Hale, J.K., Lunel, S.V.: Introduction to functional differential equations. In: Applied Mathematical Sciences, vol. 99. Springer International Edition (1993)
Hale, J.K.: Coupled oscillators on a circle. Resenhas IME-USP 1, 441–457 (1994)
Hale, J.K.: Diffusive coupling, dissipation and synchronization. J. Dyn. Differ. Equ. 9, 1–52 (1997)
Hale, J.K., Martinez-Amores, P.: Stability in neutral equations. J. Nonlinear Anal. Theory Methods Appl. 1, 161–172 (1977)
Hartman, P.: Ordinary Differential Equations. Wiley, New York (1964)
Kolesov, A.Y., Mishchenko, E.F., Rozov, H.N.: Asymptotic methods for the analysis of periodic solutions of nonlinear hyperbolic equations. In: Proceedings of the Steklov Institute of Mathematics, vol. 222. Nauka, Moscow, Russia (1998)
Kurzweil, J.: Invariant manifolds for flows. In: Hale, J.K., LaSalle, J.P. (eds.) Differential Equations and Dynamical Systems, pp. 431–468. Academic Press, New York (1967)
Langer, R.E.: On the zeros of exponential sums and integrals. Bull. Am. Math. Soc. 37, 213–239 (1931)
Lepri, S., Pikovsky, A.: Nonreciprocal wave scattering on nonlinear string-coupled oscillators. Chaos 24(043119), 1–9 (2014)
Levin, B.Y.: Zeros Distribution for the Entire Functions. Gostekhizdat, Moscow USSR (1956). (in Russian)
Liénard, A.: Étude des oscillations entretenues. Revue générale d’Électricité 23(901–912), 946–954 (1928)
Malkin, I.G.: Some Problems in the Theory of Nonlinear Oscillations. Gostekhizdat, Moscow USSR (1956). (in Russian)
Marchenko, Y.I., Rubanik, V.P.: Mutual synchronization of molecular oscillators (in russian). Izvestya VUZ USSR Radiofizika 8(4), 679–687 (1965)
Minorsky, N.: Introduction to Non-Linear Mechanics. In: Edwards, J.W. (ed.) Ann Arbor, USA (1947)
Neymark, Y.I.: Dynamical Systems and Controlled Processes. Nauka, Moscow USSR (1978)
Pikovsky, A.: The simplest case of chaotic wave scattering. Chaos 3(4), 505–506 (1993)
Pikovsky, A., Rosenblum, M.: Dynamics of globally coupled oscillators: progress and perspectives. Chaos 25(097616), 1–11 (2015)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization. A Universal Concept in Nonlinear Sciences, Cambridge University Press, Cambridge, UK (2001)
Răsvan, V., Danciu, D., Popescu, D.: On Huygens synchronization. application to van der Pol oscillators with distributed couplings. In: 18th International Carpathian Control Conference (ICCC), pp. 521–526. IEEE Publications (2017)
Răsvan, V., Danciu, D., Popescu, D.: Qualitative properties of a model of coupled drilling oscillations. In: 22nd International Conference on System Theory Control and Computing (ICSTCC), pp. 99–104. IEEE Publications (2018)
Răsvan, V., Niculescu, S.I.: Oscillations in lossless propagation models: a Liapunov–Krasovskii approach. IMA J. Math. Control. Inf. 19(1–2), 157–172 (2002)
Răsvan, V.: Augmented validation and a stabilization approach for systems with propagation. In: Miranda, F. (ed.) Systems Theory: Perspectives, Applications and Developments. Systems Science Series, vol. 1, pp. 125–170. Nova Science Publishers, New York (2014)
Răsvan, V.: Functional differential equations of lossless propagation and almost linear behavior (plenary talk). In: IFAC Proceedings Volumes. 6th Workshop on Time Delay Systems, pp. 138–150. Elsevier (2006)
Răsvan, V.: Stability theory. Concepts. Methods. Applications (in Romanian). Editura Ştiinţifică şi Enciclopedică, Bucharest, Romania (1987)
Răsvan, V.: Stable and critical cases in Huygens synchronization. Bull. Math. Soc. Sci Math. Roumanie 61(109)(4), 461–471 (2018b)
Răsvan, V.: Synchronization versus oscillation quenching. In: 10th International Symposium on Advanced Topics in Electrical Engineering (ATEE), pp. 463–468. IEEE Publications (2017)
Răsvan, V.: The stability postulate of NG Cetaev and the augmented model validation. IFAC-PapersOnLine 50(1), 7450–7455 (2018a)
Răsvan, V.: Absolute stability of a class of control systems described by functional differential equations of neutral type. In: Janssens, P., Mawhin, J., Rouche, N. (eds.) Equations Differentielles et Fonctionnelles Non Lineaires, pp. 381–396. Hermann, Paris (1973b)
Răsvan, V.: Absolute stability of a class of control systems described by coupled delay-differential and difference equations. Rev. Roumaine Sci. Techn. Serie Electrotechn. Energ. 18(2), 329–346 (1973a)
Răsvan, V.: Some results concerning the theory of electrical networks containing lossless transmission lines. Rev. Roumaine Sci. Techn. Serie Electrotechn. Energ. 19(4), 595–602 (1974)
Reissig, R., Sansone, G., Conti, R.: Qualitative Theorie nichtlinearer Differentialgleichungen. Edizioni Cremonese, Roma (1963)
Rubanik, V.P.: Delay influence over the process of synchronization of self sustained oscillations by an external periodic force (in Russian). Izvestya VUZ USSR Radiofizika 5, 561–571 (1962)
Thomson, W., Tait, P.G.: Treatise on Natural Philosophy. Oxford University Press, Oxford, UK (1867)
Tikhonov, A.N., Samarskii, A.A.: Equations of the Mathematical Physics. Nauka, Moscow USSR (1977). (in Russian)
Wiener, N.: Cybernetics, or Control and Communication in the Animal and the Machine, 2nd edn. MIT Press, Cambridge, USA (1967)
Yakubovich, V.A.: The method of the matrix inequalities in the stability theory of nonlinear controlled systems i. absolute stability of the forced oscillations (in Russian). Avtom i telemekh 25, 1017–1029 (1964)
Zverkin, A.M.: Series expansion of the solutions of linear differential difference equations i: quasi-polynomials (in russian). In: El’sgol’ts. L.E., Zverkin, A.M. (eds.) Papers of the Seminar on the theory of differential equations with deviated argument, vol. 3, pp. 3–39. University of Peoples’ Friendship, Moscow USSR (1965)
Zverkin, A.M.: Series expansion of the solutions of linear differential difference equations ii: series expansions (in russian). In: El’sgol’ts, L.E., Zverkin, A.M. (eds.) Papers of the Seminar on the theory of differential equations with deviated argument, vol. 4, pp. 3–50. University of Peoples’ Friendship, Moscow USSR (1967)
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Răsvan, V. (2020). Huygens Synchronization Over Distributed Media—Structure Versus Complex Behavior. In: Zattoni, E., Perdon, A., Conte, G. (eds) Structural Methods in the Study of Complex Systems. Lecture Notes in Control and Information Sciences, vol 482. Springer, Cham. https://doi.org/10.1007/978-3-030-18572-5_8
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