Huygens' clocks: ‘sympathy’ and resonance

References

• Arnold, V. (1977). Loss of stability of self-oscillations close to resonance and versal deformations of equivalent vector fields. Functional Analysis and its Applications, 11, 1–10. doi: 10.1007/BF01135526
   Google Scholar
• Arnold, V. (1988). Geometrical methods in the theory of ordinary differential equations (2nd ed.). New York: Springer-Verlag.
   Google Scholar
• Arnold, V. (1989). Mathematical methods of classical mechanics (2nd ed.). New York: Springer-Verlag.
   Google Scholar
• Bennett, M., Schatz, M., Rockwood, H., & Wiesenfield, K. (2002). Huygens' clocks. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 458, 563–579. doi: 10.1098/rspa.2001.0888
   Web of Science ®Google Scholar
• Blekhman, I. (1988). Synchronization in science and technology. New York, NY: ASME Press.
   Google Scholar
• Broer, H., & Vegter, G. (1992). Bifurcational aspects of parametric resonance. In Dynamics reported. Expositions in dynamical systems (pp. 1–53). Berlin: Springer.
   Google Scholar
• Czolczynski, K., Perlikowski, P., Stefanski, A., & Kapitaniak, T. (2013). Synchronization of the self-excited pendula suspended on the vertically displacing beam. Communications in Nonlinear Science and Numerical Simulation, 18, 386–400. doi: 10.1016/j.cnsns.2012.07.007
   Web of Science ®Google Scholar
• Doedel, E., & Oldeman, B. (2012, January). Auto-07p: Continuation and bifurcation software for ordinary differential equations [Computer software manual]. Montreal, Canada.
   Google Scholar
• Fradkov, A., & Andrievsky, B. (2008). Control of wave motion in the chain of pendulums. In Proceedings of the 17th IFAC world congress (pp. 3136–3141). Seoul, Korea, July 6–11.
   Google Scholar
• Fradkov, A., & Andrievsky, B. (2009). Behavior analysis of harmonically forced chain of pendulums. In 18th IEEE international conference on control application (pp. 1563–1567). St. Petersburg, Russia, July 8–10.
   Google Scholar
• Fradkov, A., Andrievsky, B., & Boykov, K. (2012). Multipendulum mechatronic setup: Design and experiments. Mechatronics, 22, 76–82. doi: 10.1016/j.mechatronics.2011.11.006
   Web of Science ®Google Scholar
• Francke, M. (2016). Huygens' synchronization: Parametric resonance and multi-dimensional coupling (Unpublished master's thesis). Eindhoven University of Technology.
   Google Scholar
• Hoogeboom, F. (2015). Huygens' synchronization: Experiments, modeling, and stability analyses (Unpublished master's thesis). Eindhoven University of Technology.
   Google Scholar
• Hoogeboom, F., Pogromsky, A., & Nijmeijer, H. (2016). Huygens' inspired multi-pendulum setups: Experiments and stability analysis. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26, 116304. doi: 10.1063/1.4965031
   PubMed Web of Science ®Google Scholar
• Huygens, C. (1893). Correspondance 1664–1665. In Ouvres completes de Christiaan Huygens (Vol. V). The Hague: La Societe Hollandaise Sciences.
   Google Scholar
• Kapitaniak, M., Brzeski, P., Czolczynski, K., Perlikowski, P., Stefanski, A., & Kapitaniak, T. (2012). Synchronization thresholds of coupled self-excited nonidentical pendula suspended on the vertically displacing beam. Progress of Theoretical Physics, 128(6), 1141–1173. doi: 10.1143/PTP.128.1141
   Google Scholar
• Kuznetsov, N., Leonov, G., Nijmeijer, H., & Pogromsky, A. (2007). Synchronization of two metronomes. IFAC Proceedings Volumes, 40(14), 49–52. doi: 10.3182/20070829-3-RU-4912.00007
   Google Scholar
• Martens, E., Thutupalli, S., Fourriere, A., & Halltschek, O. (2013). Chimera states in mechanical oscillator networks. Proceedings of the National Academy of Sciences, 110, 10563–10567. doi: 10.1073/pnas.1302880110
   PubMed Web of Science ®Google Scholar
• Nijmeijer, H., & Ramirez, J. P. (2017). De sympathie van twee slingeruurwerken: Wat zag Huygens (NIET)? Nederlands Tijdschrift voor Natuurkunde [Dutch Journal of Physics], 36–39.
   Google Scholar
• Oud, W., Nijmeijer, H., & Pogromsky, A. (2006). A study of Huygens' synchronization: Experimental results. In Group coordination and cooperative control: Lecture notes in control and information science (pp. 191–203). Berlin: Springer-Verlag.
   Google Scholar
• Pantaleone, J. (2002). Synchronization of metronomes. American Journal of Physics, 70, 992–1000. doi: 10.1119/1.1501118
   Web of Science ®Google Scholar
• Pena Ramirez, J., & Nijmeijer, H. (2016). The Poincare method: A powerful tool for analyzing synchronization of coupled oscillators. Indagationes Mathematics, 27, 1127–1146. doi: 10.1016/j.indag.2015.11.008
   Web of Science ®Google Scholar
• Peña Ramírez, J., Aihara, K., Fey, R., & Nijmeijer, H. (2014). Further understanding of Huygens' coupled clocks: The effect of stiffness. Physica D: Nonlinear Phenomena, 270, 11–19. doi: 10.1016/j.physd.2013.12.005
   Web of Science ®Google Scholar
• Peña Ramírez, J., Olvera, L., Nijmeijer, H., & Alvarez, J. (2016). The sympathy of two pendulum clocks: Beyond Huygens' observations. Scientific Reports, 6, 23580. doi: 10.1038/srep23580
   PubMed Web of Science ®Google Scholar
• Pikovsky, A., Rosenblum, M., & Kurths, J. (2003). Synchronization: A universal concept in non-linear sciences. Cambridge: Cambridge University Press.
   Google Scholar
• Strogatz, S. (2003). SYNC: The emerging science of sponteneous order. New York, NY: Hyperion.
   Google Scholar
• Verhulst, F. (2002). Parametric and autoparametric resonance. Acta Applicandae Mathematica, 70, 231–264. doi: 10.1023/A:1013934501001
   Web of Science ®Google Scholar
• Yakubovich, V., & Starzhinskii, V. (1975). Linear differential equations with periodic coefficients. New York, NY: John Wiley and Sons.
   Google Scholar