Skip to main content
SpringerLink
Log in

Chaos in Fast-Slow Systems

  • Chapter
  • First Online:
Multiple Time Scale Dynamics

Part of the book series: Applied Mathematical Sciences ((AMS,volume 191))

  • 6047 Accesses

Abstract

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

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. 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.

    MATH  MathSciNet  Google Scholar 

  2. 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. H. Aoki and K. Kaneko. Slow stochastic switching by collective chaos of fast elements. Phys. Rev. Lett., 111(14):144102, 2013.

    Google Scholar 

  4. 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. R.H. Abraham and H.B. Stewart. A chaotic blue sky catastrophe in forced relaxation oscillations. Physica D, 21(2):394–400, 1986.

    MATH  MathSciNet  Google Scholar 

  6. K.T. Alligood, T.D. Sauer, and J.A. Yorke. Chaos: An Introduction to Dynamical Systems. Springer, 1996.

    Google Scholar 

  7. M. Benedicks and L. Carleson. The dynamics of the Hénon map. Annals of Mathematics, 133:73–169, 1991.

    MATH  MathSciNet  Google Scholar 

  8. C. Bonatti, L.J. Díaz, and M. Viana. Dynamics Beyond Uniform Hyperbolicity. Springer, 2004.

    Google Scholar 

  9. 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.

    MATH  MathSciNet  Google Scholar 

  10. 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.

    MATH  MathSciNet  Google Scholar 

  11. 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.

    MATH  MathSciNet  Google Scholar 

  12. V.I. Bogachev. Measure Theory, volume 1. Springer, 2007.

    Google Scholar 

  13. R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani. Characterisation of intermittency in chaotic systems. J. Phys. A, 18(12):2157–2165, 1985.

    MATH  MathSciNet  Google Scholar 

  14. 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.

    MathSciNet  Google Scholar 

  15. R. Brown. Horseshoes in the measure-preserving Hénon map. Ergodic Theor. Dyn.Syst., 15(6): 1045–1060, 1995.

    MATH  Google Scholar 

  16. M. Brin and G. Stuck. Introduction to Dynamical Systems. CUP, 2002.

    Google Scholar 

  17. R. Barrio and S. Serrano. Bounds for the chaotic region in the Lorenz model. Physica D, 16(1): 1615–1624, 2009.

    MathSciNet  Google Scholar 

  18. H. Broer and F. Takens. Dynamical Systems and Chaos. Springer, 2010.

    Google Scholar 

  19. 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.

    MATH  MathSciNet  Google Scholar 

  20. 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.

    MathSciNet  Google Scholar 

  21. 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. G.A. Carpenter. Bursting phenomena in excitable membranes. SIAM J. Appl. Math., 36(2):334–372, 1979.

    MATH  MathSciNet  Google Scholar 

  23. 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. 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. 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.

    MATH  Google Scholar 

  26. T.R. Chay. Chaos in a three-variable model of an excitable cell. Physica D, 16(2):233–242, 1985.

    MATH  Google Scholar 

  27. 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. 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. 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. 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. 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.

    MathSciNet  Google Scholar 

  32. B. Deng. Constructing homoclinic orbits and chaotic attractors. Int. J. Bif. Chaos, 4(4):823–841, 1994.

    MATH  Google Scholar 

  33. B. Deng. Constructing Lorenz type attractors through singular perturbations. Int. J. Bif. Chaos, 5(6):1633–1642, 1995.

    MATH  Google Scholar 

  34. B. Deng. Glucose-induced period-doubling cascade in the electrical activity of pancreatic β-cells. J. Math. Biol., 38(1):21–78, 1999.

    MATH  MathSciNet  Google Scholar 

  35. B. Deng. Food chain chaos due to junction-fold point. Chaos, 11(3):514–525, 2001.

    MATH  Google Scholar 

  36. B. Deng. Food chain chaos with canard explosion. Chaos, 14(4): 1083–1092, 2004.

    MATH  MathSciNet  Google Scholar 

  37. 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.

    MATH  MathSciNet  Google Scholar 

  38. B. Deng and G. Hines. Food chain chaos due to Shilnikov orbit. Chaos, 12(3):533–538, 2002.

    MATH  MathSciNet  Google Scholar 

  39. 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. 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.

    MathSciNet  Google Scholar 

  41. A.S. de Wijn. Internal degrees of freedom and transport of benzene on graphite. Phys. Rev. E, 84:011610, 2011.

    Google Scholar 

  42. 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. A.S. de Wijn and H. Kantz. Vertical chaos and horizontal diffusion in the bouncing-ball billiard. Phys. Rev. E, 75:046214, 2007.

    MathSciNet  Google Scholar 

  44. J.P. Eckmann, S.O. Kamphorst, and D. Ruelle. Recurrence plots of dynamical systems. Europhys. Lett., 4(9):973–977, 1987.

    Google Scholar 

  45. H. Fan and T.R. Chay. Generation of periodic and chaotic bursting in an excitable cell model. Biol. Cybernet., 71(5):417–431, 1994.

    MATH  Google Scholar 

  46. S. Fraser and R. Kapral. Analysis of flow hysteresis by a one-dimensional map. Phys. Rev. A, 25(6):3223–3233, 1982.

    MathSciNet  Google Scholar 

  47. K. Fujimoto and K. Kaneko. Bifurcation cascade as chaotic itinerancy with multiple time scales. Chaos, 13(3):1041–1056, 2003.

    MATH  MathSciNet  Google Scholar 

  48. K. Fujimoto and K. Kaneko. How fast elements can affect slow dynamics. Physica D, 180:1–16, 2003.

    MATH  MathSciNet  Google Scholar 

  49. J.A. Gallas. Structure of the parameter space of the Hénon map. Phys. Rev. Lett., 70(18):2714–2717, 1993.

    Google Scholar 

  50. C. Gardiner. Stochastic Methods. Springer, Berlin Heidelberg, Germany, 4th edition, 2009.

    MATH  Google Scholar 

  51. J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, NY, 1983.

    MATH  Google Scholar 

  52. 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.

    MATH  MathSciNet  Google Scholar 

  53. A. Gorodetski and Yu.S. Ilyashenko. Minimal and strange attractors. Int. J. Bifur. Chaos, 6:1177–1183, 1996.

    MathSciNet  Google Scholar 

  54. P. Grassbeger, H. Kantz, and U. Moenig. On the symbolic dynamics of the Hénon map. J. Phys. A, 22(24):5217–5230, 1989.

    MathSciNet  Google Scholar 

  55. G. Gottwald and I. Melbourne. Homogenization for deterministic maps and multiplicative noise. Proc. R. Soc. A, 469:20130201, 2013.

    MathSciNet  Google Scholar 

  56. J. Guckenheimer and R.A. Oliva. Chaos in the Hodgkin–Huxley model. SIAM J. Appl. Dyn. Syst., 1:105–114, 2002.

    MATH  MathSciNet  Google Scholar 

  57. P. Grassbeger and I. Procaccia. Measuring the strangeness of strange attractors. Physica D, 9(1): 189–208, 1983.

    MathSciNet  Google Scholar 

  58. J. Grasman and J. Roerdink. Stochastic and chaotic relaxation oscillations. J. Stat. Phys., 54(3): 949–970, 1989.

    MATH  MathSciNet  Google Scholar 

  59. V. Gelfreich and D. Turaev. Unbounded energy growth in Hamiltonian systems with a slowly varying parameter. Comm. Math. Phys., 283(3):769–794, 2008.

    MATH  MathSciNet  Google Scholar 

  60. J. Guckenheimer. Dynamics of the van der Pol equation. IEEE Trans. Circ. Syst., 27(11):983–989, 1980.

    MATH  MathSciNet  Google Scholar 

  61. J. Guckenheimer. Symbolic dynamics and relaxation oscillations. Physica D, 1(2):227–235, 1980.

    MATH  MathSciNet  Google Scholar 

  62. 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. 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. 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.

    MATH  MathSciNet  Google Scholar 

  65. 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.

    MATH  MathSciNet  Google Scholar 

  66. J. Guckenheimer and R.F. Williams. Structural stability of Lorenz attractors. Inst. Hautes Études Sci. Publ. Math., 50:59–72, 1979.

    MATH  MathSciNet  Google Scholar 

  67. J. Guckenheimer, M. Wechselberger, and L.-S. Young. Chaotic attractors of relaxation oscillations. Nonlinearity, 19:701–720, 2006.

    MATH  MathSciNet  Google Scholar 

  68. R. Haiduc. Horseshoes in the forced van der Pol equation. PhD Thesis, Cornell University, 2005.

    Google Scholar 

  69. R. Haiduc. Horseshoes in the forced van der Pol system. Nonlinearity, 22:213–237, 2009.

    MATH  MathSciNet  Google Scholar 

  70. G. Haller. Multi-dimensional homoclinic jumping and the discretized NLS equation. Comm. Math. Phys., 193(1):1–46, 1998.

    MATH  MathSciNet  Google Scholar 

  71. 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.

    MATH  MathSciNet  Google Scholar 

  72. P.J. Holmes. Averaging and chaotic motions in forced oscillations. SIAM J. Appl. Math., 38(1):65–80, 1980.

    MATH  MathSciNet  Google Scholar 

  73. P.J. Holmes. Bifurcation sequences in horseshoe maps: infinitely many routes to chaos. Phys. Lett. A, 104(6):299–302, 1984.

    MathSciNet  Google Scholar 

  74. P.J. Holmes. Knotted periodic orbits in suspensions of Smale’s horseshoe: period multiplying and cabled knots. Physica D, 21(1):7–41, 1986.

    MATH  MathSciNet  Google Scholar 

  75. R.A. Holmgren. A first course in discrete dynamical systems. Springer, 2000.

    Google Scholar 

  76. 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. 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.

    MATH  MathSciNet  Google Scholar 

  78. Yu. Ilyashenko and W. Li. Nonlocal Bifurcations. AMS, 1999.

    Google Scholar 

  79. 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.

    MATH  MathSciNet  Google Scholar 

  80. 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.

    MATH  MathSciNet  Google Scholar 

  81. 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.

    MATH  MathSciNet  Google Scholar 

  82. K. Josic. Invariant manifolds and synchronization of coupled dynamical systems. Phys. Rev. Lett., 80(14):3053–3056, 1998.

    MATH  Google Scholar 

  83. K. Josic. Synchronization of chaotic systems and invariant manifolds. Nonlinearity, 13(4):1321–1336, 2000.

    MATH  MathSciNet  Google Scholar 

  84. M. Kennedy and L. Chua. Van der Pol and chaos. IEEE Trans. Circ. Syst., 33(10):974–980, 1986.

    MathSciNet  Google Scholar 

  85. M. Kuwamura and H. Chiba. Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators. Chaos, 19:043121, 2009.

    MathSciNet  Google Scholar 

  86. D. Kaplan and L. Glass. Understanding Nonlinear Dynamics. Springer, 1998.

    Google Scholar 

  87. A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. CUP, 1995.

    Google Scholar 

  88. 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.

    MATH  Google Scholar 

  89. 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.

    MATH  MathSciNet  Google Scholar 

  90. A.Yu. Kolesov and N.Kh. Rozov. On-off intermittency in relaxation systems. Differ. Equat., 39(1): 36–45, 2003.

    Google Scholar 

  91. A.Yu. Kolesov and N.Kh. Rozov. On the definition of chaos. Russian Math. Surveys, 64(4):701–744, 2009.

    Google Scholar 

  92. A.Yu. Kolesov, N.Kh. Rozov, and V.A. Sadovnichiy. Life on the edge of chaos. J. Math. Sci., 120(3):1372–1398, 2004.

    MathSciNet  Google Scholar 

  93. 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. N. Levinson. A second order differential equation with singular solutions. Ann. Math., 50:127–153, 1949.

    MathSciNet  Google Scholar 

  95. N. Levinson. Perturbations and discontinuous solutions of non-linear systems of differential equations. Acta. Math., 50:127–153, 1950.

    Google Scholar 

  96. M. Levi. Periodically forced relaxation oscillations. In Global Theory of Dynamical Systems. Springer, 1980.

    Google Scholar 

  97. M. Levi. Qualitative analysis of the periodically forced relaxation oscillations, volume 32 of Mem. Amer. Math. Soc. AMS, 1981.

    Google Scholar 

  98. M. Levi. A new randomness-generating mechanism in forced relaxation oscillations. Physica D, 114(3):230–236, 1998.

    MATH  MathSciNet  Google Scholar 

  99. 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.

    MATH  MathSciNet  Google Scholar 

  100. 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. 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. 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. 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. 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. S. Luzzatto, I. Melbourne, and F. Paccaut. The Lorenz attractor is mixing. Comm. Math. Phys., 260(2):393–401, 2005.

    MATH  MathSciNet  Google Scholar 

  106. E.N. Lorenz. Deterministic nonperiodic flows. J. Atmosph. Sci., 20:130–141, 1963.

    Google Scholar 

  107. T.-Y. Li and J.A. Yorke. Period three implies chaos. Amer. Math. Monthly, 82(10):985–992, 1975.

    MATH  MathSciNet  Google Scholar 

  108. G.S. Medvedev. Transition to bursting via deterministic chaos. Phys. Rev. Lett., 97(4):048102, 2006.

    Google Scholar 

  109. J. Milnor. On the concept of attractor. Comm. Math. Phys., 99:177–195, 1985.

    MATH  MathSciNet  Google Scholar 

  110. 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. R.E. Moore. Interval Analysis. Prentice-Hall, 1966.

    Google Scholar 

  112. I. Melbourne and A. Stuart. A note on diffusion limits of chaotic skew product flows. Nonlinearity, 24:1361–1367, 2011.

    MATH  MathSciNet  Google Scholar 

  113. J. Murdock. Some foundational issues in multiple scale theory. Applicable Analysis, 53(3):157–173, 1994.

    MATH  MathSciNet  Google Scholar 

  114. 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.

    MathSciNet  Google Scholar 

  115. 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. 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.

    MATH  Google Scholar 

  117. K.J. Palmer. Exponential dichotomies and transversal homoclinic points. J. Differential Equat., 55: 225–256, 1984.

    MATH  Google Scholar 

  118. Ya.B. Pesin. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surv., 32(4):55–114, 1977.

    Google Scholar 

  119. 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. A.S. Pikovsky and M.I. Rabinovich. Stochastic oscillations in dissipative systems. Physica D, 2(1): 8–24, 1981.

    MATH  MathSciNet  Google Scholar 

  121. C. Pugh and M. Shub. Ergodic attractors. Trans. AMS, 312:1–54, 1989.

    MATH  MathSciNet  Google Scholar 

  122. P.E. Phillipson and P. Schuster. Bistability of harmonically forced relaxation oscillations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 12(6):1295–1307, 2002.

    MATH  MathSciNet  Google Scholar 

  123. 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.

    MATH  MathSciNet  Google Scholar 

  124. Derek J.S. Robinson. An Introduction to Abstract Algebra. Walter de Gruyter, 2003.

    Google Scholar 

  125. R.C. Robinson. An Introduction to Dynamical Systems: Continuous and Discrete. AMS, 2013.

    Google Scholar 

  126. 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. B. Rossetto. Chua’s circuit as a slow–fast autonomous dynamical system. J. Circuits Systems Comput., 3(2):483–496, 1993.

    MathSciNet  Google Scholar 

  128. 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.

    MATH  MathSciNet  Google Scholar 

  129. B. Ryals and L.-S. Young. Horseshoes of periodically kicked van der Pol oscillators. Chaos, 22:043140, 2012.

    Google Scholar 

  130. 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.

    MathSciNet  Google Scholar 

  131. 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. 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. A.N. Sharkovskii. On cycles and structure of a continuous mapping. Ukrainskii Matematicheskii Zhurnal, 17(3):104–111, 1965.

    MathSciNet  Google Scholar 

  134. A.N. Sharkovskii. The set of convergence of one-dimensional iterations. Dopovidi Akad. Nauk Ukr. RSR, 1966:866–870, 1966.

    Google Scholar 

  135. 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. W.M. Schaffer and M. Kot. Nearly one dimensional dynamics in an epidemic. J. Theor. Biol., 112: 403–427, 1985.

    MathSciNet  Google Scholar 

  137. 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. S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 289:747–817, 1967.

    MathSciNet  Google Scholar 

  139. S. Smale. Finding a horseshoe on the beaches of Rio. Math. Intelligencer, 20:39–44, 2000.

    MathSciNet  Google Scholar 

  140. 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. 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. 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.

    MATH  Google Scholar 

  143. C. Sparrow. The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer, 1982.

    Google Scholar 

  144. 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. 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.

    MATH  MathSciNet  Google Scholar 

  146. 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. S.H. Strogatz. Nonlinear Dynamics and Chaos. Westview Press, 2000.

    Google Scholar 

  148. D. Terman. Chaotic spikes arising from a model of bursting in excitable membranes. SIAM J. Appl. Math., 51(5):1418–1450, 1991.

    MATH  MathSciNet  Google Scholar 

  149. W. Tucker. The Lorenz attractor exists. C.R. Acad. Sci. Paris, 328:1197–1202, 1999.

    Google Scholar 

  150. B. van der Pol. A theory of the amplitude of free and forced triode vibrations. Radio Review, 1: 701–710, 1920.

    Google Scholar 

  151. B. van der Pol. On relaxation oscillations. Philosophical Magazine, 7:978–992, 1926.

    Google Scholar 

  152. B. van der Pol and J. van der Mark. Frequency demultiplication. Nature, 120:363–364, 1927.

    Google Scholar 

  153. 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. I.B. Vivancos and A.A. Minzoni. Chaotic behaviour in a singularly perturbed system. Nonlinearity, 19:1535–1551, 2006.

    MATH  MathSciNet  Google Scholar 

  155. 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.

    MATH  Google Scholar 

  156. Q. Wang and L.-S. Young. Strange attractors with one direction of instability. Commun. Math. Phys., 218:1–97, 2001.

    MATH  MathSciNet  Google Scholar 

  157. Q. Wang and L.-S. Young. From invariant curves to strange attractors. Commun. Math. Phys., 225: 275–304, 2002.

    MATH  MathSciNet  Google Scholar 

  158. L.-S. Young. What are SRB measures, and which dynamical systems have them? L. Stat. Phys., 108(5):733–754, 2002.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria

    Christian Kuehn

Authors
  1. Christian Kuehn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Kuehn, C. (2015). Chaos in Fast-Slow Systems. In: Multiple Time Scale Dynamics. Applied Mathematical Sciences, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-12316-5_14

Download citation

Publish with us

Policies and ethics

Search

Navigation