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Synchronization of weakly nonlinear oscillators with Huygens' coupling
1. Oxford Advanced Learner's Dictionary (Oxford University Press, Oxford, 1997).
2. A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators (Dover Publications, New York, 1987).
4. I. I. Blekhman, Synchronization of Dynamic Systems (Nauka, Moscow, 1971).
5. I. I. Blekhman, Synchronization in Science and Technology (ASME Press, New York, 1988).
6. C. Huygens, in Oeuvres completes de Christiaan Huygens, edited by M. Nijhoff (J. Enschedé & Fils, The Hague, 1932), Vol. 17, pp. 156–189.
8. I. G. Malkin, Some Problems in the Theory of Nonlinear Oscillations (State Publishing House of Technical and Theoretical Literature, Moscow, 1956).
9. F. C. Moon, “ Chaotic clocks: A paradigm for the evolution of noise in machines,” in IUTAM Symposium, edited by G. Rega and F. Vestroni (Springer, London, 2005).
10. A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations (John Wiley & Sons, New York, 1979).
11. H. Nijmeijer, I. Blekhman, A. Fradkov, and A. Pogromski, “ Self-synchronization and controlled synchronization,” in Proceedings of the First International Conference on Control of Oscillations and Chaos, St. Petersburg, Russia, 27-29 August 1997, pp. 36–41.
12.Because μ is considered to be small, system (1) is often called quasilinear system.
13.However, it is required that the algebraic multiplicity of these roots is equal to their geometric multiplicity.
14.An abbreviated notation is used for the determinant of a n × n matrix with components asj as , s, j = 1,…, n. The symbol is the Kronecker delta.
15.Note that this transformation is valid since the matrix of eigenvectors associated to Eq. (23) has full rank. Eigenvalue i as well as eigenvalue –i have geometric and algebraic multiplicity equal to 2.
16.Actually, Eq. (30) is satisfied for with n = 0, 1, 2,….
17.For the given parameters, condition (37) is satisfied if .
18.For the given parameters, condition (57) in Theorem 4 is satisfied if .
19.Clearly, the coupling strength may be affected by modifying m and/or m3. However, modifying m requires to modify the dynamics of the uncoupled oscillators, which is not intended in the present study. Hence, in the analysis presented in this paper, the coupling strength has been modified by means of the mass of the coupling bar m3.
20. W. T. Oud, H. Nijmeijer, and A. Y. Pogromsky, “ A study of Huijgens' synchronization: Experimental results,” in Group Coordination and Cooperative Control, Lecture Notes in Control and Information Sciences Vol. 336, edited by K. Pettersen, J. Gravdahl, and H. Nijmeijer (Springer, Berlin, 2006), pp. 191–203.
23. J. Pena-Ramirez, “ Huygens' synchronization of dynamical systems,” Ph.D. thesis (Eindhoven University of Technology, 2013).
24. J. Pena-Ramirez, R. H. B. Fey, and H. Nijmeijer, “ An experimental study on synchronization of nonlinear oscillators with Huygens' coupling,” NOLTA, IEICE 3, 128–142 (2012).
25. A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization. A Universal Concept in Nonlinear Sciences (Cambridge University Press, Cambridge, 2001).
26. B. van der Pol, “ On relaxation-oscillations,” Philos. Mag. Ser. 7 2, 978–992 (1926).
27. A. V. Roup, D. S. Bernstein, S. G. Nersesov, W. S. Haddad, and V. Chellaboina, “ Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincaré maps,” Int. J. Control 76, 1685–1698 (2003).
29. S. Strogatz, Sync. The Emerging Science of Spontaneous Order (Hyperon, New York, 2003).
31. H. Ulrichs, A. Mann, and U. Parlitz, “ Synchronization and chaotic dynamics of coupled mechanical metronomes,” Chaos 19, 043120–1043120–6 (2009).
32. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems (Springer-Verlag, Heidelberg, 1990).
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