Abstract
In this chapter, we shall just attempt to compute asymptotic expansions for fast–slow systems by more or less brute force. It is a very instructive technique: just substitute an asymptotic expansion for the solution and see what happens. In other words, where does such a substitution seem to give a good approximation, and where are modifications required?
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
F. Awiszus, J. Dehnhardt, and T. Funke. The singularly perturbed Hodgkin–Huxley equations as a tool for the analysis of repetitive nerve activity. J. Math. Biol., 28(2):177–195, 1990.
C.M. Andersen and J.F. Geer. Power series expansions for the frequency and period of the limit cycle of the van der Pol equation. SIAM J. Appl. Math., 42(3):678–693, 1982.
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.
K.K. Anand. On relaxation oscillations governed by a second order differential equation for a large parameter and with a piecewise linear function. Canad. Math. Bull., 26(1):80–91, 1983.
V.I. Arnold. Encyclopedia of Mathematical Sciences: Dynamical Systems V. Springer, 1994.
M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions. Dover, 1965.
W. Balser. Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer, 2000.
H. Bavinck and J. Grasman. The method of matched asymptotic expansions for the periodic solution of the van der Pol equation. Int. J. Nonl. Mech., 9(6):421–434, 1974.
N.N. Bogoliubov and I.A. Mitropol’skii. Asymptotic methods in the theory of non-linear oscillations. Gordon Breach Science Pub., 1961.
S. Bottani. Pulse-coupled relaxation oscillators: from biological synchronization to self-organized criticality. Phys. Rev. Lett., 74:4189–4192, 1995.
H. Bremmer. The scientific work of Balthasar van der Pol. Philips Tech. Rev., 22:36–52, 1960.
M. Brøns. An iterative method for the canard explosion in general planar systems. arXiv:1209.1109, pages 1–9, 2012.
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.
R. Curtu and B. Ermentrout. Oscillations in a refractory neural net. J. Math. Biol., 43(1):81–100, 2001.
C. Comstock and G.C. Hsiao. Singular perturbations for difference equations. Rocky Moun. J. Math., 6(4):561, 1976.
T.M. Cherry. Uniform asymptotic expansions. J. London Math. Soc., 1(2):121–130, 1949.
T.M. Cherry. Uniform asymptotic formulae for functions with transition points. Trans. Amer. Math. Soc., 68(2):224–257, 1950.
J. Chen and R.E. O’Malley. On the asymptotic solution of a two-parameter boundary value problem of chemical reactor theory. SIAM J. Appl. Math., 26(4):717–729, 1974.
S. Coombes. Phase locking in networks of synaptically coupled McKean relaxation oscillators. Physica D, 160(3):173–188, 2001.
R.T. Davis and K.T. Alfriend. Solutions to van der Pol’s equation using a perturbation method. Int. J. Non-Linear Mech., 2(2):153–162, 1967.
M.B. Dadfar and J. Geer. Resonances and power series solutions of the forced van der Pol oscillator. SIAM J. Appl. Math., 50(5):1496–1506, 1990.
M.B. Dadfar, J. Geer, and C.M. Andersen. Perturbation analysis of the limit cycle of the free van der Pol equation. SIAM J. Appl. Math., 44(5):881–895, 1984.
A.A. Dorodnitsyn. Asymptotic solutions of van der Pol’s equation. Prikl. Matem. i Mekhan., 11(3): 313–328, 1947.
B.R. Dudley and H.W. Swift. Frictional relaxation oscillations. Philosophical Magazine, 40:849–861, 1949.
W. Eckhaus and E.M. de Jager. Theory and Applications of Singular Perturbations. Springer, 1982.
S.-I. Ei and M. Mimura. Relaxation oscillations in combustion models of thermal self-ignition. J. Dyn. Diff. Eq., 4(1):191–229, 1992.
W.F. Finden. An asymptotic approximation for singular perturbations. SIAM J. Appl. Math., 43(1):107–119, 1983.
D.B. Forger and R.E. Kronauer. Reconciling mathematical models of biological clocks by averaging on approximate manifolds. SIAM J. Appl. Math., 62(4):1281–1296, 2002.
S.J. Fraser. Double perturbation series in the differential equations of enzyme kinetics. J. Chem. Phys., 109(2):411–423, 1998.
E.D. Gilles, G. Eigenberger, and W. Ruppel. Relaxation oscillations in chemical reactors. AIChE J., 24(5):912–920, 1978.
J. Guckenheimer, K. Hoffman, and W. Weckesser. Bifurcations of relaxation oscillations near folded saddles. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15(11):3411–3421, 2005.
A.D. MacGillivray. On the leading term of the outer asymptotic expansion of van der Pol’s equation. SIAM J. Appl. Math., 43(6):1221–1239, 1983.
A.D. MacGillivray. Justification of matching with the transition expansion of van der Pol’s equation. SIAM J. Math. Anal., 21(1):221–240, 1990.
J. Grasman and M.J.W. Jansen. Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology. J. Math. Biol., 7(2):171–197, 1979.
E.V. Grigor’eva and S.A. Kashchenko. Relaxation oscillations in a system of equations describing the operation of a solid-state laser with a nonlinear element of delaying action. Differential Equations, 27(5):506–512, 1991.
J.-M. Ginoux and C. Letellier. Van der Pol and the history of relaxation oscillations: toward the emergence of a concept. Chaos, 22:023120, 2012.
J. Grasman, H. Nijmeijer, and E.J.M. Veling. Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator. Physica D, 13(1):195–210, 1984.
D.L. González and O. Piro. Global bifurcations and phase portrait of an analytically solvable nonlinear oscillator: relaxation oscillations and saddle-node collisions. Phys. Rev. A, 36(9):4402–4410, 1987.
J. Grasman. Relaxation oscillations of a van der Pol equation with large critical forcing term. Quart. Appl. Math., 38:9–16, 1980.
J. Grasman. The mathematical modeling of entrained biological oscillators. Bull. Math. Biol., 46(3):407–422, 1984.
J. Grasman. Asymptotic Methods for Relaxation Oscillations and Applications. Springer, 1987.
J. Guckenheimer. Bifurcations of relaxation oscillations. In Normal Forms, Bifurcations and Finiteness Problems in Differential Equations, volume 137 of NATO Sci. Ser. II Math. Phys. Chem., pages 295–316. Springer, 2004.
J. Grasman and E.J.M. Veling. An asymptotic formula for the period of a Volterra-Lotka system. Math. Biosci., 18(1):185–189, 1973.
S.P. Hastings. Formal relaxation oscillations for a model of a catalytic particle. Quart. Appl. Math., 41(4):395–405, 1983.
S.J. Hogan. Relaxation oscillations in a system with a piecewise smooth drag coefficient. J. Sound Vibration, 263(2):467–471, 2003.
M.H. Holmes. Introduction to Perturbation Methods. Springer, 1995.
F.A. Howes. Effective characterization of the asymptotic behavior of solutions of singularly perturbed boundary value problems. SIAM J. Appl. Math., 30(2):296–306, 1976.
F.A. Howes. Singular perturbations and differential inequalities, volume 5 of Memoirs of the Amer. Math. Soc. AMS, 1976.
F.A. Howes. Boundary-interior layer interactions in nonlinear singular perturbation theory, volume 203 of Memoirs of the Amer. Math. Soc. AMS, 1978.
F.A. Howes. An improved boundary layer estimate for a singularly perturbed initial value problem. Math. Zeitschr., 165(2):135–142, 1979.
S.-B. Hsu and J. Shi. Relaxation oscillation profile of limit cycle in predator–prey system. Discr. Cont. Dyn. Syst. B, 11(4):893–911, 2009.
C. Hunter and M. Tajdari. Singular complex periodic solutions of van der Pol’s equation. SIAM J. Appl. Math., 50(6):1764–1779, 1990.
E. Izhikevich. Phase equations for relaxation oscillators. SIAM J. Appl. Math., 60(5):1789–1805, 2000.
E.M. De Jager and J. Furu. The Theory of Singular Perturbations. North-Holland, 1996.
W.A. Harris Jr. Singular perturbations of two-point boundary problems for systems of ordinary differential equations. Arch. Rat. Mech. Anal., 5(1):212–225, 1960.
W.A. Harris Jr. Singular perturbations of a boundary value problem for a nonlinear system of differential equations. Duke Math. J., 29(3):429–445, 1962.
G. Karreman. Some types of relaxation oscillations as models of all-or-none phenomena. Bull. Math. Biophys., 11(4):311–318, 1949.
A.Yu. Kolesov and Yu.S. Kolesov. Relaxation oscillations in mathematical models of ecology. Proc. Steklov Inst. Math., 199(1):1–126, 1995.
A. Kuznetsov, M. Kærn, and N. Kopell. Synchrony in a population of hysteresis-based genetic oscillators. SIAM J. Appl. Math., 65(2):392–425, 2004.
A.Yu. Kolesov and E.F. Mishchenko. Existence and stability of the relaxation torus. Russ. Math. Surv., 44(3):204–205, 1989.
A.Yu. Kolesov. Specific relaxation cycles of systems of Lotka–Volterra type. Math. USSR-Izvestiya, 38(3):503–523, 1992.
L.I. Kononenko. The influence of the integral manifold shape on the onset of relaxation oscillations. J. Appl. Ind. Math., 2(4):508–512, 2008.
N. Kopell and D. Somers. Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J. Math. Biol., 33(3):261–280, 1995.
B. Krauskopf, W.A. van der Graaf, and D. Lenstra. Bifurcations of relaxation oscillations in an optically injected diode laser. Quantum Semiclass. Optics, 9(5):797–809, 1997.
J. LaSalle. Relaxation oscillations. Quart. Appl. Math., 7:1–19, 1949.
S.A. Lomov and A.G. Eliseev. Asymptotic integration of singularly perturbed problems. Russ. Math. Surv., 43(3):1–63, 1988.
S. Lefschetz. Differential Equations: Geometric Theory. Interscience Publishers, 1957.
S.A. Lomov. Introduction to the General Theory of Singular Perturbations. AMS, 1992.
C.C. Lin and A.L. Rabenstein. On the asymptotic solutions of a class of ordinary differential equations of the fourth order: I. Existence of regular formal solutions. Trans. Amer. Math. Soc., 94(1):24–57, 1960.
P. Lundberg and L. Rahm. A nonlinear convective system with oscillatory behaviour for certain parameter regimes. J. Fluid Mech., 139:237–260, 1984.
N. Levinson and O.K. Smith. A general equation for relaxation oscillations. Duke Math. J., 9(2): 382–403, 1942.
J. Lorenz and R. Sanders. Second order nonlinear singular perturbation problems with boundary conditions of mixed type. SIAM J. Math. Anal., 17(3):580–594, 1986.
E. Lee and D. Terman. Stable antiphase oscillations in a network of electrically coupled model neurons. SIAM J. Appl. Dyn. Syst., 12(1):1–27, 2013.
C.R. Laing, Y. Zou, B. Smith, and I.G. Kevrekidis. Managing heterogeneity in the study of neural oscillator dynamics. J. Math. Neurosci., 2:5, 2012.
B.D. MacMillan. Asymptotic methods for systems of differential equations in which some variables have very short response times. SIAM J. Appl. Math., 16(4):704–722, 1968.
P.D. Miller. Applied Asymptotic Analysis. AMS, 2006.
N. Minorsky. Introduction to Non-Linear Mechanics. Topological Methods. Analytical Methods. Non-Linear Resonance. Relaxation Oscillations. Ann Arbor [Mich.]: J.W. Edwards, 1947.
N. Minorsky. Nonlinear Oscillations. Van Nostrand, 1962.
E.F. Mishchenko. Asymptotic theory of relaxation oscillations described by systems of second order. Mat. Sb. N.S. (in Russian), 44(86):457–480, 1958.
E.F. Mishchenko. Asymptotic calculation of periodic solutions of systems of differential equations containing small parameters in the derivatives. AMS Transl. Ser., 2(18):199–230, 1961.
E.F. Mishchenko and A.Yu. Kolesov. Asymptotical theory of relaxation oscillations. Proc. Steklov Inst. Math., 197:1–93, 1993.
E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov. Asymptotic Methods in Singularly Perturbed Systems. Plenum Press, 1994.
R. Mankin, T. Laasa, E. Soika, and A. Ainsaar. Noise-controlled slow–fast oscillations in predator–prey models with the Beddington functional response. Eur. Phys. J. B, 59:259–269, 2007.
E.F. Mishchenko and N.Kh. Rozov. Differential Equations with Small Parameters and Relaxation Oscillations (translated from Russian). Plenum Press, 1980.
P.A. Markowich and C.A. Ringhofer. Singular perturbation problems with a singularity of the second kind. SIAM J. Math. Anal., 14(5):897–914, 1983.
K. Nipp. An extension of Tikhonov’s theorem in singular perturbations for the planar case. Z. Angew. Math. Phys., 34(3):277–290, 1983.
R.E. O’Malley and J.E. Flaherty. Analytical and numerical methods for nonlinear singular singularly-perturbed initial value problems. SIAM J. Appl. Math., 38(2):225–248, 1980.
R.E. O’Malley. A boundary value problem for certain nonlinear second order differential equations with a small parameter. Arch. Rat. Mech. Anal., 29(1):66–74, 1968.
R.E. O’Malley. Singular perturbations of a boundary value problem for a system of nonlinear differential equations. J. Differential Equat., 8:431–447, 1970.
R.E. O’Malley. Phase-plane solutions to some singular perturbation problems. J. Math. Anal. Appl., 54(2):449–466, 1976.
R.E. O’Malley. On singular singularly-perturbed initial value problems. Applicable Analysis, 8(1): 71–81, 1978.
R.E. O’Malley. A singular singularly-perturbed linear boundary value problem. SIAM J. Math. Anal., 10(4):695–708, 1979.
Y. Pomeau and M. Le Berre. Critical speed-up vs critical slow-down: a new kind of relaxation oscillation with application to stick-slip phenomena. arXiv:1107.3331, pages 1–8, 2011.
S.S. Pul’kin and N.H. Rozov. The asymptotic theory of relaxation oscillations in systems with one degree of freedom. I. Calculation of the phase trajectories. Vestnik Moskov. Univ. Ser. I Mat. Meh. (in Russian), 1964(2):70–82, 1964.
D. Quinn, B. Gladman, P. Nicholson, and R. Rand. Relaxation oscillations in tidally evolving satellites. Celestial Mech. Dynam. Astronom., 67(2):111–130, 1997.
A.L. Rabenstein. Asymptotic solutions of u iv +λ 2(zu″ +α u′ +β u) = 0 for large | λ | . Arch. Rat. Mech. Anal., 1(1):418–435, 1957.
N.H. Rozov. Asymptotic calculation of nearly discontinuous solutions of a second-order system of differential equations. Dokl. Akad. Nauk SSSR (in Russian), 145:38–40, 1962.
N.H. Rozov. On the asymptotic theory of relaxation oscillations in systems with one degree of freedom. II. Calculation of the period of the limit cycle. Vestnik Moskov. Univ. Ser. I Mat. Meh. (in Russian), 1964(3):56–65, 1964.
P.F. Rowast and A.I. Selverston. Modeling the gastric mill central pattern generator of the lobster with a relaxation-oscillator network. J. Neurophysiol., 70(3):1030–1053, 1993.
A. Rasmussen, J. Wyller and J.O. Vik. Relaxation oscillations in spruce-budworm interactions. Nonlinear Anal. Real World Appl., 12:304–319, 2011.
Y. Sibuya. Asymptotic solutions of initial value problems of ordinary differential equations with a small parameter in the derivative, I. Arch. Rat. Mech. Anal., 14:304–311, 1963.
D. Somers and N. Kopell. Rapid synchronization through fast threshold modulation. Biol. Cybern., 68(5):393–407, 1993.
D. Somers and N. Kopell. Waves and synchrony in networks of oscillators of relaxation and non-relaxation type. Physica D, 89(1):169–183, 1995.
I. Siekmann and H. Malchow. Local collapses in the Truscott–Brindley model. Math. Model. Nat. Phenom., 3(4):114–130, 2008.
D.R. Smith. Singular-Perturbation Theory: An Introduction with Applications. CUP, 1985.
R. Singh and S. Sinha. Spatiotemporal order, disorder, and propagating defects in homogeneous system of relaxation oscillators. Phys. Rev. E, 87:012907, 2013.
F.J. Solis and C. Yebra. Modeling the pursuit in natural systems: a relaxed oscillation approach. Math. Comput. Modelling, 52(7):956–961, 2010.
D. Terman and E. Lee. Partial synchronization in a network of neural oscillators. SIAM J. Appl. Math., 57(1):252–293, 1997.
D. Terman, E. Lee, J. Rinzel, and T. Bem. Stability of anti-phase and in-phase locking by electrical coupling but not fast inhibition alone. SIAM J. Appl. Dyn. Syst., 10(3):1127–1153, 2011.
W.C. Troy. Bifurcation phenomena in FitzHugh’s nerve conduction equations. J. Math. Anal. Appl., 54(3):678–690, 1976.
J.J. Tyson. Relaxation oscillations in the revised Oregonator. J. Chem. Phys., 80(12):6079–6082, 1984.
W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Dover, 2002.
A.J. Wesselink. Stellar variability and relaxation oscillations. Astrophys. J., 89:659–668, 1939.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Kuehn, C. (2015). Direct Asymptotic Methods. In: Multiple Time Scale Dynamics. Applied Mathematical Sciences, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-12316-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-12316-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12315-8
Online ISBN: 978-3-319-12316-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)