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


In this chapter, we begin in Section 1.1 with a practical guide to orient the reader to how the book is structured and how it can be utilized. Several notational conventions are introduced as well. Section 1.2 covers some basic terminology for systems with two time scales.


Asymptotic Analysis Asymptotic Series Notational Convention Multiple Time Scale Singular Perturbation Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [Arn73]
    V.I. Arnold. Ordinary Differential Equations. MIT Press, 1973.Google Scholar
  2. [Arn83]
    V.I. Arnold. Geometrical Methods in the Theory of Ordinary Differential Equations. Springer, New York, NY, 1983.CrossRefzbMATHGoogle Scholar
  3. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [Ber01]
    N. Berglund. Perturbation theory of dynamical systems. arXiv:math/0111178, pages 1–111, 2001.Google Scholar
  5. [Bi04]
    Q. Bi. Dynamical analysis of two coupled parametrically excited van der Pol oscillators. Int. J. Non-Linear Mech., 39(1):33–54, 2004.CrossRefzbMATHGoogle Scholar
  6. [Bir08a]
    G.D. Birkhoff. Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc., 9(4):373–395, 1908.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [Bir08b]
    G.D. Birkhoff. On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Amer. Math. Soc., 9(2):219–231, 1908.CrossRefMathSciNetGoogle Scholar
  8. [BL23]
    G.D. Birkhoff and R.E. Langer. The boundary problems and developments associated with a system of ordinary linear differential equations of the first order. Proc. Amer. Acad. Arts Sci., 58(2):51–128, 1923.CrossRefGoogle Scholar
  9. [BLM09]
    A. Buicǎ, J. Llibre, and O. Makarenkov. Asymptotic stability of periodic solutions for nonsmooth differential equations with application to the nonsmooth van der Pol oscillator. SIAM J. Math. Anal., 40(6):2478–2495, 2009.CrossRefMathSciNetGoogle Scholar
  10. [BNS04]
    V.F. Butuzov, N.N. Nefedov, and K.R. Schneider. Singularly perturbed problems in case of exchange of stabilities. J. Math. Sci., 121(1):1973–2079, 2004.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [Bra93]
    M. Braun. Differential Equations and Their Applications. Springer, 1993.Google Scholar
  12. [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
  13. [Chi10]
    C. Chicone. Ordinary Differential Equations with Applications. Texts in Applied Mathematics. Springer, 2nd edition, 2010.Google Scholar
  14. [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
  15. [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
  16. [CO99]
    J. Cronin and R.E. O’Malley, editors. Analyzing Multiscale Phenomena Using Singular Perturbation Methods, volume 56 of Proc. Symp. Appl. Math. AMS, 1999.Google Scholar
  17. [CR88]
    T. Chakraborty and R.H. Rand. The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mech., 23(5):369–376, 1988.CrossRefzbMATHMathSciNetGoogle Scholar
  18. [DS06]
    E.N. Dancer and S. Shusen. Interior peak solutions for an elliptic system of FitzHugh–Nagumo type. J Differential Equat., 229(2):654–679, 2006.CrossRefzbMATHGoogle Scholar
  19. [E11]
    W. E. Principles of Multiscale Modeling. CUP, 2011.Google Scholar
  20. [EdJ82]
    W. Eckhaus and E.M. de Jager. Theory and Applications of Singular Perturbations. Springer, 1982.Google Scholar
  21. [EE03b]
    W. E and B. Engquist. Multiscale modeling and computation. Notices of the AMS, 50(9):1063–1070, 2003.Google Scholar
  22. [ELVE04]
    W. E, X. Li, and E. Vanden-Eijnden. Some recent progress in multiscale modeling. In Multiscale Modelling and Simulation, pages 3–21. Springer, 2004.Google Scholar
  23. [EP02]
    C.H. Edwards and D.E. Penney. Calculus. Prentice Hall, 2002.Google Scholar
  24. [FDS12]
    A. Franci, G. Drion, and R. Sepulchre. An organizing center in a planar model of neuronal excitability. SIAM J. Appl. Dyn. Syst., 11(4):1698–1722, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [FDS13]
    A. Franci, G. Drion, and R. Sepulchre. Modeling neuronal bursting: singularity theory meets neurophysiology. arXiv:1305.7364, pages 1–28, 2013.Google Scholar
  26. [Fit55]
    R. FitzHugh. Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophysics, 17:257–269, 1955.CrossRefGoogle Scholar
  27. [Fit60]
    R. FitzHugh. Thresholds and plateaus in the Hodgkin–Huxley nerve equations. J. Gen. Physiol., 43:867–896, 1960.CrossRefGoogle Scholar
  28. [Fit61]
    R. FitzHugh. Impulses and physiological states in models of nerve membranes. Biophys. J., 1:445–466, 1961.CrossRefGoogle Scholar
  29. [GB10]
    O.V. Gendelman and T. Bar. Bifurcations of self-excitation regimes in a van der Pol oscillator with a nonlinear energy sink. Physica D, 239:220–229, 2010.CrossRefzbMATHMathSciNetGoogle Scholar
  30. [Gel89]
    I.M. Gel’fand. Lectures on Linear Algebra. Dover, 1989.Google Scholar
  31. [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.CrossRefzbMATHMathSciNetGoogle Scholar
  32. [Gle94]
    P. Glendinning. Stability, Instability and Chaos. CUP, 1994.Google Scholar
  33. [Gou13]
    D.A. Goussis. The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: The case of vdP relaxation-oscillations. Physica D, 248:16–32, 2013.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [Guc96]
    J. Guckenheimer. Towards a global theory of singularly perturbed systems. Progress in Nonlinear Differential Equations and Their Applications, 19:214–225, 1996.MathSciNetGoogle Scholar
  35. [Guc02]
    J. Guckenheimer. Bifurcation and degenerate decomposition in multiple time scale dynamical systems. In Nonlinear Dynamics and Chaos: Where do we go from here?, pages 1–20. Taylor and Francis, 2002.Google Scholar
  36. [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
  37. [Hab78]
    P. Habets. On relaxation oscillations in a forced van der Pol oscillator. Proc. Roy. Soc. Edinburgh A, 82(1):41–49, 1978.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [Has75]
    S.P. Hastings. Some mathematical problems from neurobiology. The American Mathematical Monthly, 82(9):881–895, 1975.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [Has76a]
    S.P. Hastings. On the existence of homoclinic and periodic orbits in the FitzHugh–Nagumo equations. Quart. J. Math. Oxford, 2(27):123–134, 1976.CrossRefMathSciNetGoogle Scholar
  40. [HB12]
    X. Han and Q. Bi. Slow passage through canard explosion and mixed-mode oscillations in the forced van der Pol’s equation. Nonlinear Dyn., 68:275–283, 2012.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [Hek10]
    G. Hek. Geometric singular perturbation theory in biological practice. J. Math. Biol., 60:347–386, 2010.CrossRefMathSciNetGoogle Scholar
  42. [HH52a]
    A.L. Hodgkin and A.F. Huxley. The components of membrane conductance in the giant axon of Loligo. J. Physiol., 116:473–496, 1952.Google Scholar
  43. [HH52b]
    A.L. Hodgkin and A.F. Huxley. Currents carried by sodium and potassium ions through the membrane of the giant axon of Loligo. J. Physiol., 116:449–472, 1952.Google Scholar
  44. [HH52c]
    A.L. Hodgkin and A.F. Huxley. Measurement of current-voltage relations in the membrane of the giant axon of Loligo. J. Physiol., 116:424–448, 1952.Google Scholar
  45. [HH52d]
    A.L. Hodgkin and A.F. Huxley. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 117:500–544, 1952.CrossRefGoogle Scholar
  46. [HR78]
    P.J. Holmes and D.A. Rand. Bifurcations of the forced van der Pol oscillator. Quarterly Appl. Math., 35:495–509, 1978.zbMATHMathSciNetGoogle Scholar
  47. [HR82]
    J.L. Hindmarsh and R.M. Rose. A model of the nerve impulse using two first-order differential equations. Nature, 296:162–164, 1982.CrossRefGoogle Scholar
  48. [HS99]
    P.-F. Hsieh and Y. Sibuya. Basic Theory of Ordinary Differential Equations. Springer, 1999.Google Scholar
  49. [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
  50. [Jän94]
    K. Jänich. Linear Algebra. Springer, 1994.Google Scholar
  51. [JF96]
    E.M. De Jager and J. Furu. The Theory of Singular Perturbations. North-Holland, 1996.Google Scholar
  52. [Kap99]
    T.J. Kaper. An introduction to geometric methods and dynamical systems theory for singular perturbation problems. In J. Cronin and R.E. O’Malley, editors, Analyzing Multiscale Phenomena Using Singular Perturbation Methods, pages 85–131. Springer, 1999.Google Scholar
  53. [KO00]
    B. Krauskopf and H.M. Osinga. Investigating torus bifurcations in the forced van der Pol oscillator. In Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems, pages 199–208. Springer, 2000.Google Scholar
  54. [Kör12]
    T.W. Körner. Vectors, Pure and Applied: A General Introduction to Linear Algebra. CUP, 2012.Google Scholar
  55. [KR89]
    M.K. Kadalbajoo and Y.N. Reddy. Asymptotic and numerical analysis of singular perturbation problems: a survey. Appl. Math. Comp., 30(3):223–259, 1989.CrossRefzbMATHMathSciNetGoogle Scholar
  56. [KS91a]
    T. Kapitaniak and W.H. Steeb. Transition to hyperchaos in coupled generalized van der Pol equations. Phys. Lett. A, 152: 33–36, 1991.CrossRefMathSciNetGoogle Scholar
  57. [KTS11]
    K. Konishi, M. Takeuchi and T. Shimizu. Design of external forces for eliminating traveling wave in a piecewise linear FitzHugh–Nagumo model. Chaos, 21:023101, 2011.CrossRefMathSciNetGoogle Scholar
  58. [Lev47]
    N. Levinson. Perturbations of discontinuous solutions of nonlinear systems of differential equations. Proc. Nat. Acad. Sci. USA, 33:214–218, 1947.CrossRefMathSciNetGoogle Scholar
  59. [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
  60. [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
  61. [Lop82]
    O. Lopes. FitzHugh–Nagumo system: boundedness and convergence to equilibrium. J. Differential Equat., 44(3):400–413, 1982.CrossRefzbMATHMathSciNetGoogle Scholar
  62. [Lou59]
    W.S. Loud. Periodic solutions of a perturbed autonomous system. Ann. Math., 70(3):490–529, 1959.CrossRefzbMATHMathSciNetGoogle Scholar
  63. [Mac01]
    A. Maccari. The response of a parametrically excited van der Pol oscillator to a time delay state feedback. Nonlinear Dynamics, 26(2):105–119, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  64. [McK70]
    H.P. McKean. Nagumo’s equation. Adv. Math., 4:209–223, 1970.CrossRefzbMATHMathSciNetGoogle Scholar
  65. [Mei07]
    J.D. Meiss. Differential Dynamical Systems. SIAM, 2007.Google Scholar
  66. [Mil06]
    P.D. Miller. Applied Asymptotic Analysis. AMS, 2006.Google Scholar
  67. [ML93]
    K. Murali and M. Lakshmanan. Transmission of signals by synchronization in a chaotic van der Pol-Duffing oscillator. Phys. Rev. E, 48(3):R1624–R1626, 1993.CrossRefGoogle Scholar
  68. [MPL93]
    R. Mettin, U. Parlitz, and W. Lauterborn. Bifurcation structure of the driven van der Pol oscillator. Int. J. Bif. Chaos, 3(6):1529–1555, 1993.CrossRefzbMATHMathSciNetGoogle Scholar
  69. [Nag39]
    M. Nagumo. Über das Verhalten der Integrale von λ y″ + f(x, y, y′, λ) = 0 für λ → 0. Proc. Phys. Math. Soc. Japan, 21:529–534, 1939.MathSciNetGoogle Scholar
  70. [Nag93]
    M. Nagumo. Mitio Nagumo Collected Papers. Springer, 1993.Google Scholar
  71. [NAY62]
    J. Nagumo, S. Arimoto, and S. Yoshizawa. An active pulse transmission line simulating nerve axon. Proc. IRE, 50:2061–2070, 1962.CrossRefGoogle Scholar
  72. [O’M68b]
    R.E. O’Malley. Topics in singular perturbations. Adv. Math., 2:365–470, 1968.CrossRefzbMATHMathSciNetGoogle Scholar
  73. [O’M01]
    R.E. O’Malley. Naive singular perturbation theory. Math. Mod. Meth. Appl. Sci., 11:119–131, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  74. [O’M08]
    R.E. O’Malley. Singularly rerturbed linear two-point boundary value problems. SIAM Rev., 50(3): 459–482, 2008.CrossRefzbMATHMathSciNetGoogle Scholar
  75. [Pon57]
    L.S. Pontryagin. Asymptotic properties of solutions with small parameters multiplying leading derivatives. Izv. AN SSSR, Ser. Matematika, 21(5):605–626, 1957.Google Scholar
  76. [PR60]
    L.S. Pontryagin and L.V. Rodygin. Approximate solution of a system of ordinary differential equations involving a small parameter in the derivatives. Soviet Math. Dokl., 1:237–240, 1960.zbMATHMathSciNetGoogle Scholar
  77. [Pug10]
    C. Pugh. Real Mathematical Analysis. Springer, 2010.Google Scholar
  78. [RGG00]
    C. Rocsoreanu, A. Georgescu, and N. Giurgiteanu. The FitzHugh–Nagumo Model - Bifurcation and Dynamics. Kluwer, 2000.Google Scholar
  79. [RH80]
    R.H. Rand and P.J. Holmes. Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. Int. J. Non-Linear Mech., 15(4):387–399, 1980.CrossRefzbMATHMathSciNetGoogle Scholar
  80. [RK83]
    J. Rinzel and J.P. Keener. Hopf bifurcation to repetitive activity in nerve. SIAM J. Appl. Math., 43(4):907–922, 1983.CrossRefzbMATHMathSciNetGoogle Scholar
  81. [Ros86]
    B. Rossetto. Trajectoires lentes des systems dynamiques lents-rapides. In Analysis and Optimization of Systems, Lecture Notes in Contr. Inform. Sci., pages 680–695. Springer, 1986.Google Scholar
  82. [RS99]
    C. Reinecke and G. Sweers. A positive solution on \(\mathbb{R}^{N}\) to a system of elliptic equations of FitzHugh–Nagumo type. J. Differential Equat., 153(2):292–312, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  83. [Rud76]
    W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.Google Scholar
  84. [Spi06]
    M. Spivak. Calculus. CUP, 2006.Google Scholar
  85. [SR82]
    D.W. Storti and R.H. Rand. Dynamics of two strongly coupled van der Pol oscillators. Int. J. Non-Linear Mech., 17(3):143–152, 1982.CrossRefMathSciNetGoogle Scholar
  86. [SR86]
    D.W. Storti and R.H. Rand. Dynamics of two strongly coupled relaxation oscillators. SIAM J. Appl. Math., 46(1):56–67, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  87. [SSSI12]
    K. Shimizu, Y. Saito, M. Sekikawa, and N. Inaba. Complex mixed-mode oscillations in a Bonhoeffer–van der Pol oscillator under weak periodic perturbation. Physica D, 241:1518–1526, 2012.CrossRefzbMATHGoogle Scholar
  88. [Ste07]
    M. Steinhauser. Computational Multiscale Modeling of Fluids and Solids: Theory and Applications. Springer, 2007.Google Scholar
  89. [Str00]
    S.H. Strogatz. Nonlinear Dynamics and Chaos. Westview Press, 2000.Google Scholar
  90. [Str09]
    G. Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, 2009.Google Scholar
  91. [Tes12]
    G. Teschl. Ordinary Differential Equations and Dynamical Systems. AMS, 2012.Google Scholar
  92. [Tik52]
    A.N. Tikhonov. Systems of differential equations containing small parameters in the derivatives. Mat. Sbornik N. S., 31:575–586, 1952.MathSciNetGoogle Scholar
  93. [TS11c]
    J.-C. Tsai and J. Sneyd. Traveling waves in the buffered FitzHugh–Nagumo model. SIAM J. Appl. Math., 71(5):1606–1636, 2011.CrossRefzbMATHMathSciNetGoogle Scholar
  94. [Vas63]
    A.B. Vasilieva. Asymptotic behaviour of solutions of certain problems for ordinary non-linear differential equations with a small parameter multiplying the highest derivatives. Russian Math. Surveys, 18:13–84, 1963.CrossRefGoogle Scholar
  95. [Vas76]
    A.B. Vasilieva. The development of the theory of ordinary differential equations with a small parameter multiplying the highest derivative during the period 1966–1976. Russian Math. Surveys, 31(6):109–131, 1976.CrossRefGoogle Scholar
  96. [Vas94]
    A.B. Vasilieva. On the development of singular perturbation theory at Moscow State University and elsewhere. SIAM Rev., 36(3):440–452, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  97. [vdP20]
    B. van der Pol. A theory of the amplitude of free and forced triode vibrations. Radio Review, 1: 701–710, 1920.Google Scholar
  98. [vdP26]
    B. van der Pol. On relaxation oscillations. Philosophical Magazine, 7:978–992, 1926.Google Scholar
  99. [vdP34]
    B. van der Pol. The nonlinear theory of electric oscillations. Proc. IRE, 22:1051–1086, 1934.CrossRefzbMATHGoogle Scholar
  100. [vdPvdM27]
    B. van der Pol and J. van der Mark. Frequency demultiplication. Nature, 120:363–364, 1927.CrossRefGoogle Scholar
  101. [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.CrossRefGoogle Scholar
  102. [Ver05a]
    F. Verhulst. Invariant manifolds in dissipative dynamical systems. Acta Appl. Math., 87(1):229–244, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  103. [Ver05b]
    F. Verhulst. Methods and Applications of Singular Perturbations: Boundary Layers and Multiple Timescale Dynamics. Springer, 2005.Google Scholar
  104. [Ver05c]
    F. Verhulst. Quenching of self-excited vibrations. J. Eng. Mech., 53(3):349–358, 2005.zbMATHMathSciNetGoogle Scholar
  105. [Ver06]
    F. Verhulst. Nonlinear Differential Equations and Dynamical Systems. Springer, 2006.Google Scholar
  106. [VV09]
    F. Veerman and F. Verhulst. Quasiperiodic phenomena in the van der Pol–Mathieu equation. J. Sound Vibr., 326(1):314–320, 2009.CrossRefGoogle Scholar
  107. [WW05]
    J. Wei and M. Winter. Clustered spots in the FitzHugh–Nagumo system. J. Differential Equat., 213(1):121–145, 2005.CrossRefzbMATHMathSciNetGoogle Scholar
  108. [XC03]
    J. Xu and K.W. Chung. Effects of time delayed position feedback on a van der Pol–Duffing oscillator. Physica D, 180(1):17–39, 2003.CrossRefzbMATHMathSciNetGoogle 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