5 Normal Hyperbolicity for Non-autonomous Oscillators and Oscillator Networks
83
8. J. Eldering, Normally Hyperbolic Invariant Manifolds—The Noncompact Case (Springer,
2013)
9. B. Ermentrout, N. Kopell, Parabolic bursting in an excitable system coupled with a slow
oscillation. SIAM J. Appl. Math. 46, 233–253 (1986)
10. N. Fenichel, Persistence and smoothness of invariant manifolds for flows. Indiana U Math. J.
21, 193–226 (1971)
11. G. Froyland, K. Padberg, M.H. England, A.M. Treguier, Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98, 224503 (2007)
12. A.H. Gin, Aperiodically forced oscillators. Ph.D. Thesis, Univ Warwick, 2013
13. H. Haken, Generalised Ginzburg-Landau equations for phase transition-like phenomena in
lasers, nonlinear optics, hydrodynamics and chemical reactions. Z. Phys. B 21 105 (1975)
14. D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer, 1981)
15. M.W. Hirsch, C.C. Pugh, M. Shub, Invariant manifolds. Lecture Notes in Mathematics, vol.
583 (Springer, 1977)
16. S.P. Kuznetsov, Dynamical chaos and uniformly hyperbolic attractors: from mathematics to
physics. Phys. Uspekhi 54(2), 119–144 (2011)
17. J. Lopes Dias, Renormalisation scheme for vector fields on T 2 with a Diophantine frequency.
Nonlinearity 15, 665–679 (2002)
18. R.S. MacKay, Renormalisation in area-preserving maps, 1982 Princeton. Ph.D. Thesis, revised
version published by World Scientific Publishing Co (1992)
19. R.S. MacKay, Dynamics of networks, in: Stochastic and Spatial Structures of Dynamical
Systems, ed. by S van Strien, S. Verduyn Lunel (1996), pp. 81–104
20. T. Matsumoto, L.O. Chua, M. Komuro, The double scroll. IEEE Trans. Circ. Syst. CAS-32,
798–818 (1985)
21. C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35,
193–213 (1981)
22. A. Pikovsky, M. Rosenblum, J. Kurths. Synchronisation (CUP, 2001)
23. R.V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces. Math. USSR Sbornik 23
233 (1974)
24. O. Rössler, An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)
25. S. Schecter, The saddle-node separatrix-loop bifurcation. SIAM J. Math. Anal. 18, 1142–1156
(1987)
26. A. Shilnikov, L. Shilnikov, D. Turaev, On some mathematical topics in classical synchronization. A tutorial. Int. J .Bif. Chaos 14, 2143–2160 (2004)
27. Y.F. Suprunenko, P.T. Clemson, A. Stefanovska, Chronotaxic systems: a new class of selfsustained nonautonomous oscillators. Phys. Rev. Lett. 111, 024101 (2013)
28. A. Tehrani Safa, M. Ghaffari Saadat, M. Naraghi, Passive dynamic of the simplest walking
model: replacing ramps with stairs. Mech. Mach. Theory 42, 1314–1325 (2007)
29. J.C. Willems, The behavioral approach to open and interconnected systems. IEEE Control Syst.
Mag. (Dec 2007)
30. J.F. Yamagishi, K. Kaneko, Chaos on a high-dimensional torus. Phys Rev Res 2, 023044 (2020)
31. R. Yamapi, R.S. MacKay, Stability of synchronisation in a shift-invariant ring of mutually
coupled oscillators. Discr. Conts. Dyn .Syst. B 10, 973–96 (2008)
83
8. J. Eldering, Normally Hyperbolic Invariant Manifolds—The Noncompact Case (Springer,
2013)
9. B. Ermentrout, N. Kopell, Parabolic bursting in an excitable system coupled with a slow
oscillation. SIAM J. Appl. Math. 46, 233–253 (1986)
10. N. Fenichel, Persistence and smoothness of invariant manifolds for flows. Indiana U Math. J.
21, 193–226 (1971)
11. G. Froyland, K. Padberg, M.H. England, A.M. Treguier, Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98, 224503 (2007)
12. A.H. Gin, Aperiodically forced oscillators. Ph.D. Thesis, Univ Warwick, 2013
13. H. Haken, Generalised Ginzburg-Landau equations for phase transition-like phenomena in
lasers, nonlinear optics, hydrodynamics and chemical reactions. Z. Phys. B 21 105 (1975)
14. D. Henry, Geometric Theory of Semilinear Parabolic Equations (Springer, 1981)
15. M.W. Hirsch, C.C. Pugh, M. Shub, Invariant manifolds. Lecture Notes in Mathematics, vol.
583 (Springer, 1977)
16. S.P. Kuznetsov, Dynamical chaos and uniformly hyperbolic attractors: from mathematics to
physics. Phys. Uspekhi 54(2), 119–144 (2011)
17. J. Lopes Dias, Renormalisation scheme for vector fields on T 2 with a Diophantine frequency.
Nonlinearity 15, 665–679 (2002)
18. R.S. MacKay, Renormalisation in area-preserving maps, 1982 Princeton. Ph.D. Thesis, revised
version published by World Scientific Publishing Co (1992)
19. R.S. MacKay, Dynamics of networks, in: Stochastic and Spatial Structures of Dynamical
Systems, ed. by S van Strien, S. Verduyn Lunel (1996), pp. 81–104
20. T. Matsumoto, L.O. Chua, M. Komuro, The double scroll. IEEE Trans. Circ. Syst. CAS-32,
798–818 (1985)
21. C. Morris, H. Lecar, Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35,
193–213 (1981)
22. A. Pikovsky, M. Rosenblum, J. Kurths. Synchronisation (CUP, 2001)
23. R.V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces. Math. USSR Sbornik 23
233 (1974)
24. O. Rössler, An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)
25. S. Schecter, The saddle-node separatrix-loop bifurcation. SIAM J. Math. Anal. 18, 1142–1156
(1987)
26. A. Shilnikov, L. Shilnikov, D. Turaev, On some mathematical topics in classical synchronization. A tutorial. Int. J .Bif. Chaos 14, 2143–2160 (2004)
27. Y.F. Suprunenko, P.T. Clemson, A. Stefanovska, Chronotaxic systems: a new class of selfsustained nonautonomous oscillators. Phys. Rev. Lett. 111, 024101 (2013)
28. A. Tehrani Safa, M. Ghaffari Saadat, M. Naraghi, Passive dynamic of the simplest walking
model: replacing ramps with stairs. Mech. Mach. Theory 42, 1314–1325 (2007)
29. J.C. Willems, The behavioral approach to open and interconnected systems. IEEE Control Syst.
Mag. (Dec 2007)
30. J.F. Yamagishi, K. Kaneko, Chaos on a high-dimensional torus. Phys Rev Res 2, 023044 (2020)
31. R. Yamapi, R.S. MacKay, Stability of synchronisation in a shift-invariant ring of mutually
coupled oscillators. Discr. Conts. Dyn .Syst. B 10, 973–96 (2008)