A taxonomy for generalized synchronization between flat-coupled systems
Abstract: Generalized synchronization is plausibly the most complex form of synchronization. Previous studies have revealed the existence of weak or strong forms of generalized synchronization depending on the multi- or mono-valued nature of the mapping between the attractors of two unidirectionally-coupled systems. Generalized synchronization is here obtained by coupling two systems with a flat control law. Here, we demonstrate that the corresponding first-return maps can be topologically conjugate in some cases. Conversely, the response map can foliated while the drive map is not. We describe the corresponding types of generalized synchronization, explicitly focusing on the influence of the coupling strength when significantly different dimensions or dissipation properties characterize the coupled systems. A taxonomy of generalized synchronization based on these properties is proposed.
- J. H. Vincent, Proceedings of the Physical Society of London 32, 84 (1919).
- E. V. Appleton, Proceedings of the Cambridge Philosophical Society 21, 231 (1922).
- R. D. Huntoon and A. Weiss, Proceedings of the IRE 35, 1415 (1947).
- E. N. Lorenz, Journal of the Atmospheric Sciences 20, 130 (1963).
- O. E. Rössler, Physics Letters A 57, 397 (1976).
- J. L. Hindmarsh and R. M. Rose, Proceedings of the Royal Society of London B 221, 87 (1984).
- L. M. Pecora and T. L. Carroll, Physical Review Letters 64, 821 (1990).
- J. Lissajous, Annales de Chimie et de Physique iii 51, 146 (1857).
- W. He and J. Cao, Chaos 19, 013118 (2009).
- L. Kocarev and U. Parlitz, Physical Review Letters 76, 1816 (1996).
- W. Perruquetti and J.-P. Barbot, Chaos in automatic control (CRC Press, Boca Raton, FL, 2005).
- K. Pyragas, Physical Review E 54, R4508 (1996).
- U. Parlitz, Nonlinear Theory and Its Applications, IEICE 3, 113 (2012).
- C.-T. Lin, IEEE Transactions on Automatic Control 19, 201 (1974).
- C. Letellier and J.-P. Barbot, Chaos 31, 103114 (2021).
- Student, Biometrika 6, 1 (1908), the actual name of the author is W. S. Gosset.
- O. E. Rössler, Zeitschrift für Naturforschung A 31, 259 (1976).
- H. Bai-Lin, Elementary symbolic dynamics and chaos in dissipative systems (World Scientific Publishing, Singapore, 1989).
- C. Letellier, P. Dutertre, and O. Ménard, “Codes for topological analysis,” http://www.atomosyd.net/spip.php?article239 (2013).
- B. van der Pol, Philosophical Magazine vii, 2, 978 (1926).
- K. F. Bonhöffer, Journal of General Physiology 32, 69 (1948).
- R. FitzHugh, Biophysical Journal 1, 445 (1961).
- T. Saito, IEEE Transactions on Circuits and Systems 37, 399 (1990).
- C. Letellier and R. Gilmore, Physical Review E 63, 016206 (2001).
- J. C. Sprott, Physical Review E 50, 647 (1994).
- M. Hénon and C. Heiles, The astronomical Journal 69, 73 (1964).
- C. Letellier and L. A. Aguirre, Chaos 12, 549 (2002).
- M. C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, NY, 1990).
- Y. Ueda, in The road to chaos (Aerial Press, 1992).
- U. Parlitz and W. Lauterborn, Physics Letters A 107, 351 (1985).
- R. Miranda and E. Stone, Physics Letters A 178, 105 (1993).
- J. W. Alexander and G. B. Briggs, Annals of Mathematics ii 28, 562 (1926).
- K. Reidemeister, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 5, 7 (1927).
- L. Cao, Physica D 110, 43 (1997).
- L. Rosier, Systems & Control Letters 19, 467 (1992).
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.
