Chaos in Fast-Slow Systems

  • Christian Kuehn
Part of the Applied Mathematical Sciences book series (AMS, volume 191)


Similar to Chapter 13, the current chapter uses the theory previously discussed in parts of this book to gain substantial insight into nonlinear dynamics.


  1. [Abr12b]
    R.V. Abramov. Suppression of chaos at slow variables by rapidly mixing fast dynamics through linear energy-preserving coupling. Comm. Math. Sci., 10(2):595–624, 2012.zbMATHMathSciNetGoogle Scholar
  2. [AG13]
    G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.Google Scholar
  3. [AK13d]
    H. Aoki and K. Kaneko. Slow stochastic switching by collective chaos of fast elements. Phys. Rev. Lett., 111(14):144102, 2013.Google Scholar
  4. [AM86]
    K. Aihara and G. Matsumoto. Chaotic oscillations and bifurcations in squid giant axons. In A. Holden, editor, Chaos, pages 257–269. Manchester University Press, 1986.Google Scholar
  5. [AS86]
    R.H. Abraham and H.B. Stewart. A chaotic blue sky catastrophe in forced relaxation oscillations. Physica D, 21(2):394–400, 1986.zbMATHMathSciNetGoogle Scholar
  6. [ASY96]
    K.T. Alligood, T.D. Sauer, and J.A. Yorke. Chaos: An Introduction to Dynamical Systems. Springer, 1996.Google Scholar
  7. [BC91]
    M. Benedicks and L. Carleson. The dynamics of the Hénon map. Annals of Mathematics, 133:73–169, 1991.zbMATHMathSciNetGoogle Scholar
  8. [BDV04]
    C. Bonatti, L.J. Díaz, and M. Viana. Dynamics Beyond Uniform Hyperbolicity. Springer, 2004.Google Scholar
  9. [BEG+03]
    K. Bold, C. Edwards, J. Guckenheimer, S. Guharay, K. Hoffman, J. Hubbard, R. Oliva, and W. Weckesser. The forced van der Pol equation II: canards in the reduced system. SIAM Journal of Applied Dynamical Systems, 2(4):570–608, 2003.zbMATHMathSciNetGoogle Scholar
  10. [BG93]
    B. Braaksma and J. Grasman. Critical dynamics of the Bonhoeffer–van der Pol equation and its chaotic response to periodic stimulation. Physica D, 68(2):265–280, 1993.zbMATHMathSciNetGoogle Scholar
  11. [BKK13]
    H.W. Broer, T.J. Kaper, and M. Krupa. Geometric desingularization of a cusp singularity in slow–fast systems with applications to Zeeman’s examples. J. Dyn. Diff. Eq., 25(4):925–958, 2013.zbMATHMathSciNetGoogle Scholar
  12. [Bog07]
    V.I. Bogachev. Measure Theory, volume 1. Springer, 2007.Google Scholar
  13. [BPPV85]
    R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani. Characterisation of intermittency in chaotic systems. J. Phys. A, 18(12):2157–2165, 1985.zbMATHMathSciNetGoogle Scholar
  14. [BR93]
    M.V. Berry and J.M. Robbins. Chaotic classical and half-classical adiabatic reactions: geometric magnetism and deterministic friction. Proc. R. Soc. A, 442(1916):659–672, 1993.MathSciNetGoogle Scholar
  15. [Bro95]
    R. Brown. Horseshoes in the measure-preserving Hénon map. Ergodic Theor. Dyn.Syst., 15(6): 1045–1060, 1995.zbMATHGoogle Scholar
  16. [BS02]
    M. Brin and G. Stuck. Introduction to Dynamical Systems. CUP, 2002.Google Scholar
  17. [BS09a]
    R. Barrio and S. Serrano. Bounds for the chaotic region in the Lorenz model. Physica D, 16(1): 1615–1624, 2009.MathSciNetGoogle Scholar
  18. [BT10]
    H. Broer and F. Takens. Dynamical Systems and Chaos. Springer, 2010.Google Scholar
  19. [CAA09]
    N. Corson and M.A. Aziz-Alaoui. Asymptotic dynamics of the slow–fast Hindmarsh–Rose neuronal system. Dyn. Contin. Discrete Impuls. Syst. Ser. B, 16(4):535–549, 2009.zbMATHMathSciNetGoogle Scholar
  20. [CAL90]
    B. Christiansen, P. Alstrøm, and M.T. Levinsen. Routes to chaos and complete phase locking in modulated relaxation oscillators. Phys. Rev. A, 42(4):1891–1900, 1990.MathSciNetGoogle Scholar
  21. [Car52]
    M.L. Cartwright. Van der Pol’s equation for relaxation oscillations. In Contributions to the Theory of Nonlinear Oscillations II, pages 3–18. Princeton University Press, 1952.Google Scholar
  22. [Car79]
    G.A. Carpenter. Bursting phenomena in excitable membranes. SIAM J. Appl. Math., 36(2):334–372, 1979.zbMATHMathSciNetGoogle Scholar
  23. [CCB90]
    C.C. Canavier, J.W. Clark, and J.H. Byrne. Routes to chaos in a model of a bursting neuron. Biophys. J., 57(6):1245–1251, 1990.Google Scholar
  24. [CEAM13]
    M. Ciszak, S. Euzzor, T. Arecchi, and R. Meucci. Experimental study of firing death in a network of chaotic FitzHugh–Nagumo neurons. Phys. Rev. E, 87:022919, 2013.Google Scholar
  25. [CFL95]
    T.R. Chay, Y.S. Fan, and Y.S. Lee. Bursting, spiking, chaos, fractals, and universality in biological rhythms. Int. J. Bif. Chaos, 5(3):595–635, 1995.zbMATHGoogle Scholar
  26. [Cha85]
    T.R. Chay. Chaos in a three-variable model of an excitable cell. Physica D, 16(2):233–242, 1985.zbMATHGoogle Scholar
  27. [CL45]
    M.L. Cartwright and J.E. Littlewood. On non-linear differential equations of second order. I. The equation \(\ddot{y} - k(1 - y^{2})\dot{y} + y = b\lambda k\cos (\lambda t + a)\), k large. J. London Math. Soc., 20:180–189, 1945.Google Scholar
  28. [CL47]
    M.L. Cartwright and J.E. Littlewood. On non-linear differential equations of second order. II. The equation \(\ddot{y} - kf(y,\dot{y}) + g(y,k) = p(t), k > 0\), f(y) ≥ 1. Ann. Math., 48(2):472–494, 1947.Google Scholar
  29. [CO00]
    S. Coombes and A.H. Osbaldestin. Period-adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator. Phys. Rev. E, 62(3):4057–4066, 2000.Google Scholar
  30. [CRO86]
    W.L. Chien, H. Rising, and J.M. Ottino. Laminar mixing and chaotic mixing in several cavity flows. J. Fluid Mech., 170(1): 355–377, 1986.Google Scholar
  31. [CS05]
    G.S. Cymbalyuk and A.L. Shilnikov. Coexistence of tonic spiking oscillations in a leech neuron model. J. Comput. Neurosci., 18(3):255–263, 2005.MathSciNetGoogle Scholar
  32. [Den94]
    B. Deng. Constructing homoclinic orbits and chaotic attractors. Int. J. Bif. Chaos, 4(4):823–841, 1994.zbMATHGoogle Scholar
  33. [Den95a]
    B. Deng. Constructing Lorenz type attractors through singular perturbations. Int. J. Bif. Chaos, 5(6):1633–1642, 1995.zbMATHGoogle Scholar
  34. [Den99]
    B. Deng. Glucose-induced period-doubling cascade in the electrical activity of pancreatic β-cells. J. Math. Biol., 38(1):21–78, 1999.zbMATHMathSciNetGoogle Scholar
  35. [Den01]
    B. Deng. Food chain chaos due to junction-fold point. Chaos, 11(3):514–525, 2001.zbMATHGoogle Scholar
  36. [Den04]
    B. Deng. Food chain chaos with canard explosion. Chaos, 14(4): 1083–1092, 2004.zbMATHMathSciNetGoogle Scholar
  37. [DFGRL02]
    E. Doedel, E. Freire, E. Gamero, and A. Rodriguez-Luis. An analytical and numerical study of a modified van der Pol oscillator. J. Sound Vibration, 256(4):755–771, 2002.zbMATHMathSciNetGoogle Scholar
  38. [DH02a]
    B. Deng and G. Hines. Food chain chaos due to Shilnikov orbit. Chaos, 12(3):533–538, 2002.zbMATHMathSciNetGoogle Scholar
  39. [DIK04]
    S. Doi, J. Inoue, and S. Kumagai. Chaotic spiking in the Hodgkin–Huxley nerve model with slow inactivation in the sodium current. J. Integr. Neurosci., 3(2):207–225, 2004.Google Scholar
  40. [DK05]
    S. Doi and S. Kumagai. Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models. J. Comput. Neurosci., 19(3):325–356, 2005.MathSciNetGoogle Scholar
  41. [dW11]
    A.S. de Wijn. Internal degrees of freedom and transport of benzene on graphite. Phys. Rev. E, 84:011610, 2011.Google Scholar
  42. [dWF09]
    A.S. de Wijn and A. Fasolino. Relating chaos to deterministic diffusion of a molecule adsorbed on a surface. J. Phys.: Condens. Matter, 21:264002, 2009.Google Scholar
  43. [dWK07]
    A.S. de Wijn and H. Kantz. Vertical chaos and horizontal diffusion in the bouncing-ball billiard. Phys. Rev. E, 75:046214, 2007.MathSciNetGoogle Scholar
  44. [EKR87]
    J.P. Eckmann, S.O. Kamphorst, and D. Ruelle. Recurrence plots of dynamical systems. Europhys. Lett., 4(9):973–977, 1987.Google Scholar
  45. [FC94]
    H. Fan and T.R. Chay. Generation of periodic and chaotic bursting in an excitable cell model. Biol. Cybernet., 71(5):417–431, 1994.zbMATHGoogle Scholar
  46. [FK82]
    S. Fraser and R. Kapral. Analysis of flow hysteresis by a one-dimensional map. Phys. Rev. A, 25(6):3223–3233, 1982.MathSciNetGoogle Scholar
  47. [FK03a]
    K. Fujimoto and K. Kaneko. Bifurcation cascade as chaotic itinerancy with multiple time scales. Chaos, 13(3):1041–1056, 2003.zbMATHMathSciNetGoogle Scholar
  48. [FK03b]
    K. Fujimoto and K. Kaneko. How fast elements can affect slow dynamics. Physica D, 180:1–16, 2003.zbMATHMathSciNetGoogle Scholar
  49. [Gal93]
    J.A. Gallas. Structure of the parameter space of the Hénon map. Phys. Rev. Lett., 70(18):2714–2717, 1993.Google Scholar
  50. [Gar09]
    C. Gardiner. Stochastic Methods. Springer, Berlin Heidelberg, Germany, 4th edition, 2009.zbMATHGoogle Scholar
  51. [GH83]
    J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, NY, 1983.zbMATHGoogle Scholar
  52. [GHW03]
    J. Guckenheimer, K. Hoffman, and W. Weckesser. The forced van der Pol equation I: the slow flow and its bifurcations. SIAM J. Appl. Dyn. Syst., 2(1):1–35, 2003.zbMATHMathSciNetGoogle Scholar
  53. [GI96]
    A. Gorodetski and Yu.S. Ilyashenko. Minimal and strange attractors. Int. J. Bifur. Chaos, 6:1177–1183, 1996.MathSciNetGoogle Scholar
  54. [GKM89]
    P. Grassbeger, H. Kantz, and U. Moenig. On the symbolic dynamics of the Hénon map. J. Phys. A, 22(24):5217–5230, 1989.MathSciNetGoogle Scholar
  55. [GM13]
    G. Gottwald and I. Melbourne. Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. A, 469:20130201, 2013.MathSciNetGoogle Scholar
  56. [GO02]
    J. Guckenheimer and R.A. Oliva. Chaos in the Hodgkin–Huxley model. SIAM J. Appl. Dyn. Syst., 1:105–114, 2002.zbMATHMathSciNetGoogle Scholar
  57. [GP83]
    P. Grassbeger and I. Procaccia. Measuring the strangeness of strange attractors. Physica D, 9(1): 189–208, 1983.MathSciNetGoogle Scholar
  58. [GR89]
    J. Grasman and J. Roerdink. Stochastic and chaotic relaxation oscillations. J. Stat. Phys., 54(3): 949–970, 1989.zbMATHMathSciNetGoogle Scholar
  59. [GT08]
    V. Gelfreich and D. Turaev. Unbounded energy growth in Hamiltonian systems with a slowly varying parameter. Comm. Math. Phys., 283(3):769–794, 2008.zbMATHMathSciNetGoogle Scholar
  60. [Guc80a]
    J. Guckenheimer. Dynamics of the van der Pol equation. IEEE Trans. Circ. Syst., 27(11):983–989, 1980.zbMATHMathSciNetGoogle Scholar
  61. [Guc80b]
    J. Guckenheimer. Symbolic dynamics and relaxation oscillations. Physica D, 1(2):227–235, 1980.zbMATHMathSciNetGoogle Scholar
  62. [Guc03]
    J. Guckenheimer. Global bifurcations of periodic orbits in the forced van der Pol equation. In H.W. Broer, B. Krauskopf, and G. Vegter, editors, Global Analysis of Dynamical Systems - Festschrift dedicated to Floris Takens, pages 1–16. Inst. of Physics Pub., 2003.Google Scholar
  63. [Guc13]
    J. Guckenheimer. The birth of chaos. In Recent Trends in Dynamical Systems, volume 35 of Proceed. Math. Stat., pages 3–24. Springer, 2013.Google Scholar
  64. [GVS05]
    J. Grasman, F. Verhulst, and S.D. Shih. The Lyapunov exponents of the van der Pol oscillator. Math. Meth. Appl. Sci., 28(10):1131–1139, 2005.zbMATHMathSciNetGoogle Scholar
  65. [GVW76]
    J. Grasman, E.J.M. Veling, and G.M. Willems. Relaxation oscillations governed by a van der Pol equation with periodic forcing term. SIAM J. Appl. Math., 31(4):667–676, 1976.zbMATHMathSciNetGoogle Scholar
  66. [GW79]
    J. Guckenheimer and R.F. Williams. Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math., 50:59–72, 1979.zbMATHMathSciNetGoogle Scholar
  67. [GWY06]
    J. Guckenheimer, M. Wechselberger, and L.-S. Young. Chaotic attractors of relaxation oscillations. Nonlinearity, 19:701–720, 2006.zbMATHMathSciNetGoogle Scholar
  68. [Hai05]
    R. Haiduc. Horseshoes in the forced van der Pol equation. PhD Thesis, Cornell University, 2005.Google Scholar
  69. [Hai09]
    R. Haiduc. Horseshoes in the forced van der Pol system. Nonlinearity, 22:213–237, 2009.zbMATHMathSciNetGoogle Scholar
  70. [Hal98]
    G. Haller. Multi-dimensional homoclinic jumping and the discretized NLS equation. Comm. Math. Phys., 193(1):1–46, 1998.zbMATHMathSciNetGoogle Scholar
  71. [HBC+91]
    J.J. Healey, D.S. Broomhead, K.A. Cliffe, R. Jones, and T. Mullin. The origins of chaos in a modified van der Pol oscillator. Physica D, 48(2):322–339, 1991.zbMATHMathSciNetGoogle Scholar
  72. [Hol80]
    P.J. Holmes. Averaging and chaotic motions in forced oscillations. SIAM J. Appl. Math., 38(1):65–80, 1980.zbMATHMathSciNetGoogle Scholar
  73. [Hol84]
    P.J. Holmes. Bifurcation sequences in horseshoe maps: infinitely many routes to chaos. Phys. Lett. A, 104(6):299–302, 1984.MathSciNetGoogle Scholar
  74. [Hol86]
    P.J. Holmes. Knotted periodic orbits in suspensions of Smale’s horseshoe: period multiplying and cabled knots. Physica D, 21(1):7–41, 1986.zbMATHMathSciNetGoogle Scholar
  75. [Hol00]
    R.A. Holmgren. A first course in discrete dynamical systems. Springer, 2000.Google Scholar
  76. [HSD03]
    M.W. Hirsch, S. Smale, and R. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, 2nd edition, 2003.Google Scholar
  77. [HW85]
    P.J. Holmes and R.F. Williams. Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences. Arch. Rat. Mech. Anal., 90(2):115–194, 1985.zbMATHMathSciNetGoogle Scholar
  78. [IL99]
    Yu. Ilyashenko and W. Li. Nonlocal Bifurcations. AMS, 1999.Google Scholar
  79. [IM94]
    M. Itoh and H. Murakami. Chaos and canards in the van der Pol equation with periodic forcing. Int. J. Bif. Chaos, 4(4):1023–1029, 1994.zbMATHMathSciNetGoogle Scholar
  80. [JGB+03]
    W. Just, K. Gelfert, N. Baba, A. Riegert, and H. Kantz. Elimination of fast chaotic degrees of freedom: on the accuracy of the Born approximation. J. Stat. Phys., 112:277–292, 2003.zbMATHMathSciNetGoogle Scholar
  81. [JKRH01]
    W. Just, H. Kantz, C. Röderbeck, and M. Helm. Stochastic modelling: replacing fast degrees of freedom by noise. J. Phys. A, 34:3199–3213, 2001.zbMATHMathSciNetGoogle Scholar
  82. [Jos98]
    K. Josic. Invariant manifolds and synchronization of coupled dynamical systems. Phys. Rev. Lett., 80(14):3053–3056, 1998.zbMATHGoogle Scholar
  83. [Jos00]
    K. Josic. Synchronization of chaotic systems and invariant manifolds. Nonlinearity, 13(4):1321–1336, 2000.zbMATHMathSciNetGoogle Scholar
  84. [KC86]
    M. Kennedy and L. Chua. Van der Pol and chaos. IEEE Trans. Circ. Syst., 33(10):974–980, 1986.MathSciNetGoogle Scholar
  85. [KC09]
    M. Kuwamura and H. Chiba. Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators. Chaos, 19:043121, 2009.MathSciNetGoogle Scholar
  86. [KG98]
    D. Kaplan and L. Glass. Understanding Nonlinear Dynamics. Springer, 1998.Google Scholar
  87. [KH95]
    A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. CUP, 1995.Google Scholar
  88. [KJB+04]
    H. Kantz, W. Just, N. Baba, K. Gelfert, and A. Riegert. Fast chaos versus white noise – entropy analysis and a Fokker–Planck model for the slow dynamics. Physica D, 187:200–213, 2004.zbMATHGoogle Scholar
  89. [KKR96]
    A.Yu. Kolesov, Yu.S. Kolesov, and N.Kh. Rozov. Chaos of the broken torus type in three-dimensional relaxation systems. J. Math. Sci., 80(1):1533–1545, 1996.zbMATHMathSciNetGoogle Scholar
  90. [KR03]
    A.Yu. Kolesov and N.Kh. Rozov. On-off intermittency in relaxation systems. Differ. Equat., 39(1): 36–45, 2003.Google Scholar
  91. [KR09]
    A.Yu. Kolesov and N.Kh. Rozov. On the definition of chaos. Russian Math. Surveys, 64(4):701–744, 2009.Google Scholar
  92. [KRS04]
    A.Yu. Kolesov, N.Kh. Rozov, and V.A. Sadovnichiy. Life on the edge of chaos. J. Math. Sci., 120(3):1372–1398, 2004.MathSciNetGoogle Scholar
  93. [KS86]
    A. Katok and J.-M. Strelcyn. Invariant Manifolds, Entropy and Billiards; Smooth Maps with Singularities, volume 1222 of Springer Lecture Notes in Math. Springer, 1986.Google Scholar
  94. [Lev49]
    N. Levinson. A second order differential equation with singular solutions. Ann. Math., 50:127–153, 1949.MathSciNetGoogle Scholar
  95. [Lev50]
    N. Levinson. Perturbations and discontinuous solutions of non-linear systems of differential equations. Acta. Math., 50:127–153, 1950.Google Scholar
  96. [Lev80]
    M. Levi. Periodically forced relaxation oscillations. In Global Theory of Dynamical Systems. Springer, 1980.Google Scholar
  97. [Lev81]
    M. Levi. Qualitative analysis of the periodically forced relaxation oscillations, volume 32 of Mem. Amer. Math. Soc. AMS, 1981.Google Scholar
  98. [Lev98]
    M. Levi. A new randomness-generating mechanism in forced relaxation oscillations. Physica D, 114(3):230–236, 1998.zbMATHMathSciNetGoogle Scholar
  99. [LG09]
    C. Letellier and J.-M. Ginoux. Development of the nonlinear dynamical systems theory from radio engineering to electronics. Int. J. Bif. Chaos, 19:2131–2163, 2009.zbMATHMathSciNetGoogle Scholar
  100. [Lit57a]
    J.E. Littlewood. On non-linear differential equations of second order: III. The equation \(\ddot{y} - k (1 - y^{2})\dot{y} + y = b\mu k\cos (\mu t+\alpha )\) for large k, and its generalizations. Acta. Math., 97:267–308, 1957.Google Scholar
  101. [Lit57b]
    J.E. Littlewood. On non-linear differential equations of second order: IV. The general equation \(\ddot{y} - kf(y)\dot{y} + g(y) = bkp(\varphi )\), \(\varphi = t + a\) for large k and its generalizations. Acta. Math., 98: 1–110, 1957.Google Scholar
  102. [LM97a]
    Y. Li and D.W. McLaughlin. Homoclinic orbits and chaos in discretized perturbed NLS systems: Part I. Homoclinic orbits. J. Nonlinear Sci., 7(3):211–269, 1997.Google Scholar
  103. [LM97b]
    Y. Li and D.W. McLaughlin. Homoclinic orbits and chaos in discretized perturbed NLS systems: Part II. Symbolic dynamics. J. Nonlinear Sci., 7(4):315–370, 1997.Google Scholar
  104. [LM13]
    P.S. Landa and P.V.E. McClintock. Nonlinear systems with fast and slow motions. Changes in the probability distribution for fast motions under the influence of slower ones. Phys. Rep., 532(1):1–26, 2013.Google Scholar
  105. [LMP05]
    S. Luzzatto, I. Melbourne, and F. Paccaut. The Lorenz attractor is mixing. Comm. Math. Phys., 260(2):393–401, 2005.zbMATHMathSciNetGoogle Scholar
  106. [Lor63]
    E.N. Lorenz. Deterministic nonperiodic flows. J. Atmosph. Sci., 20:130–141, 1963.Google Scholar
  107. [LY75]
    T.-Y. Li and J.A. Yorke. Period three implies chaos. Amer. Math. Monthly, 82(10):985–992, 1975.zbMATHMathSciNetGoogle Scholar
  108. [Med06]
    G.S. Medvedev. Transition to bursting via deterministic chaos. Phys. Rev. Lett., 97(4):048102, 2006.Google Scholar
  109. [Mil85]
    J. Milnor. On the concept of attractor. Comm. Math. Phys., 99:177–195, 1985.zbMATHMathSciNetGoogle Scholar
  110. [MKKR94]
    E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov. Asymptotic Methods in Singularly Perturbed Systems. Plenum Press, 1994.Google Scholar
  111. [Moo66]
    R.E. Moore. Interval Analysis. Prentice-Hall, 1966.Google Scholar
  112. [MS11]
    I. Melbourne and A. Stuart. A note on diffusion limits of chaotic skew product flows. Nonlinearity, 24:1361–1367, 2011.zbMATHMathSciNetGoogle Scholar
  113. [Mur94]
    J. Murdock. Some foundational issues in multiple scale theory. Applicable Analysis, 53(3):157–173, 1994.zbMATHMathSciNetGoogle Scholar
  114. [NDL+11]
    A.B. Neiman, K. Dierkes, B. Lindner, L. Han, and A.L. Shilnikov. Spontaneous voltage oscillations and response dynamics of a Hodgkin–Huxley type model of sensory hair cells. J. Math. Neurosci., 1:1–24, 2011.MathSciNetGoogle Scholar
  115. [Nis94]
    Y. Nishiura. Coexistence of infinitely many stable solutions to reaction diffusion systems in the singular limit. In Dynamics Reported, pages 25–103. Springer, 1994.Google Scholar
  116. [OT94]
    H. Okuda and I. Tsuda. A coupled chaotic system with different time scales: possible implications of observations by dynamical systems. Int. J. Bif. Chaos, 4(4):1011–1022, 1994.zbMATHGoogle Scholar
  117. [Pal84]
    K.J. Palmer. Exponential dichotomies and transversal homoclinic points. J. Differential Equat., 55: 225–256, 1984.zbMATHGoogle Scholar
  118. [Pes77]
    Ya.B. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv., 32(4):55–114, 1977.Google Scholar
  119. [PL87]
    U. Parlitz and W. Lauterborn. Period-doubling cascades and devil’s staircases of the driven van der Pol oscillator. Phys. Rev. A, 36(3):1428–1434, 1987.Google Scholar
  120. [PR81]
    A.S. Pikovsky and M.I. Rabinovich. Stochastic oscillations in dissipative systems. Physica D, 2(1): 8–24, 1981.zbMATHMathSciNetGoogle Scholar
  121. [PS89]
    C. Pugh and M. Shub. Ergodic attractors. Trans. AMS, 312:1–54, 1989.zbMATHMathSciNetGoogle Scholar
  122. [PS02]
    P.E. Phillipson and P. Schuster. Bistability of harmonically forced relaxation oscillations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12(6):1295–1307, 2002.zbMATHMathSciNetGoogle Scholar
  123. [RL88]
    S. Rajasekar and M. Lakshmanan. Period-doubling bifurcations, chaos, phase-locking and devil’s staircase in a Bonhoeffer-van der Pol oscillator. Physica D, 32:146–152, 1988.zbMATHMathSciNetGoogle Scholar
  124. [Rob03]
    Derek J.S. Robinson. An Introduction to Abstract Algebra. Walter de Gruyter, 2003.Google Scholar
  125. [Rob13]
    R.C. Robinson. An Introduction to Dynamical Systems: Continuous and Discrete. AMS, 2013.Google Scholar
  126. [Ros89]
    B. Rossetto. Geometrical structure of attractors of slow–fast dynamical systems: the double scroll chaotic oscillator. In Differential Equations, Lecture Notes in Pure and Appl. Math., pages 621–628. Dekker, 1989.Google Scholar
  127. [Ros93]
    B. Rossetto. Chua’s circuit as a slow–fast autonomous dynamical system. J. Circuits Systems Comput., 3(2):483–496, 1993.MathSciNetGoogle Scholar
  128. [RRCL00]
    S. Ramdani, B. Rossetto, L.O. Chua, and R. Lozi. Slow manifolds of some chaotic systems with applications to laser systems. Int. J. Bif. Chaos, 10(12):2729–2744, 2000.zbMATHMathSciNetGoogle Scholar
  129. [RY12]
    B. Ryals and L.-S. Young. Horseshoes of periodically kicked van der Pol oscillators. Chaos, 22:043140, 2012.Google Scholar
  130. [Sha61]
    A.N. Sharkovskii. The reducibility of a continuous function of a real variable and the structure of the stationary points of the corresponding iteration process. Dokl. Akad.Nauk SSSR, 139:1067–1070, 1961.MathSciNetGoogle Scholar
  131. [Sha64a]
    A.N. Sharkovskii. Co-existence of cycles of a continuous map of the line into itself. Ukrainskii Matematicheskii Zhurnal, 16(1):61–71, 1964. English translation: Int. J. Bif. Chaos 5(5), pp. 1263–1273, 1995.Google Scholar
  132. [Sha64b]
    A.N. Sharkovskii. Fixed points and the center of a continuousmapping of the line into itself. Dopovidi Akad. Nauk Ukr. RSR, 1964:865–868, 1964.Google Scholar
  133. [Sha65]
    A.N. Sharkovskii. On cycles and structure of a continuous mapping. Ukrainskii Matematicheskii Zhurnal, 17(3):104–111, 1965.MathSciNetGoogle Scholar
  134. [Sha66]
    A.N. Sharkovskii. The set of convergence of one-dimensional iterations. Dopovidi Akad. Nauk Ukr. RSR, 1966:866–870, 1966.Google Scholar
  135. [Shi65]
    L.P. Shilnikov. A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl., 6:163–166, 1965.Google Scholar
  136. [SK85b]
    W.M. Schaffer and M. Kot. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol., 112: 403–427, 1985.MathSciNetGoogle Scholar
  137. [Sma63]
    S. Smale. Diffeomorphisms with many periodic points. In S.S. Cairns, editor, Differential and Combinatorial Topology, pages 63–80. Princeton University Press, 1963.Google Scholar
  138. [Sma67]
    S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 289:747–817, 1967.MathSciNetGoogle Scholar
  139. [Sma00a]
    S. Smale. Finding a horseshoe on the beaches of Rio. Math. Intelligencer, 20:39–44, 2000.MathSciNetGoogle Scholar
  140. [Sma00b]
    S. Smale. Finding a horseshoe on the beaches of Rio. In R. Abraham and Y. Ueda, editors, The Chaos-Avant-Garde, pages 7–22. World Scientific, 2000.Google Scholar
  141. [Sma00c]
    S. Smale. On how I got started in Dynamical Systems 1959–1962. In R. Abraham and Y. Ueda, editors, The Chaos-Avant-Garde, pages 1–6. World Scientific, 2000.Google Scholar
  142. [SN03]
    A. Shilnikov and F.R. Nikolai. Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity. Int. J. Bif. Chaos, 13(11):3325–3340, 2003.zbMATHGoogle Scholar
  143. [Spa82]
    C. Sparrow. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, 1982.Google Scholar
  144. [SPKP13]
    M. Schmuck, M. Pradas, S. Kalliadasis, and G.A. Pavliotis. A new stochastic mode reduction strategy for dissipative systems. arXiv:1305.4135v1, pages 1–5, 2013.Google Scholar
  145. [SR03]
    A. Shilnikov and N.F. Rulkov. Origin of chaos in a two-dimensional map modeling spiking-bursting neural activity. Int. J. Bif. Chaos, 13(11):3325–3340, 2003.zbMATHMathSciNetGoogle Scholar
  146. [SSI+11]
    M. Sekikawa, K. Shimizu, N. Inaba, H. Kita, T. Endo, K. Fujimoto, T. Yoshinaga, and K. Aihara. Sudden change from chaos to oscillation death in the Bonhoeffer–van der Pol oscillator under weak periodic perturbation. Phys. Rev. E, 84:056209, 2011.Google Scholar
  147. [Str00]
    S.H. Strogatz. Nonlinear Dynamics and Chaos. Westview Press, 2000.Google Scholar
  148. [Ter91]
    D. Terman. Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math., 51(5):1418–1450, 1991.zbMATHMathSciNetGoogle Scholar
  149. [Tuc99]
    W. Tucker. The Lorenz attractor exists. C.R. Acad. Sci. Paris, 328:1197–1202, 1999.Google Scholar
  150. [vdP20]
    B. van der Pol. A theory of the amplitude of free and forced triode vibrations. Radio Review, 1: 701–710, 1920.Google Scholar
  151. [vdP26]
    B. van der Pol. On relaxation oscillations. Philosophical Magazine, 7:978–992, 1926.Google Scholar
  152. [vdPvdM27]
    B. van der Pol and J. van der Mark. Frequency demultiplication. Nature, 120:363–364, 1927.Google Scholar
  153. [vdPvdM28]
    B. van der Pol and J. van der Mark. The heartbeat considered as a relaxation oscillation, and an electrical model of the heart. Phil. Mag. Suppl., 6:763–775, 1928.Google Scholar
  154. [VM06]
    I.B. Vivancos and A.A. Minzoni. Chaotic behaviour in a singularly perturbed system. Nonlinearity, 19:1535–1551, 2006.zbMATHMathSciNetGoogle Scholar
  155. [WS12]
    Z.-L. Wang and X.-R. Shi. Chaos bursting synchronization of mismatched Hindmarsh–Rose systems via a single adaptive feedback controller. Nonlinear Dyn., 67:1817–1823, 2012.zbMATHGoogle Scholar
  156. [WY01]
    Q. Wang and L.-S. Young. Strange attractors with one direction of instability. Commun. Math. Phys., 218:1–97, 2001.zbMATHMathSciNetGoogle Scholar
  157. [WY02]
    Q. Wang and L.-S. Young. From invariant curves to strange attractors. Commun. Math. Phys., 225: 275–304, 2002.zbMATHMathSciNetGoogle Scholar
  158. [You02]
    L.-S. Young. What are SRB measures, and which dynamical systems have them? L. Stat. Phys., 108(5):733–754, 2002.zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

Personalised recommendations