Skip to main content

Direct Asymptotic Methods

  • Chapter
  • First Online:
Multiple Time Scale Dynamics

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

  • 6014 Accesses

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?

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

Access this chapter

eBook
USD 16.99
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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  2. 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.

    Article  MATH  MathSciNet  Google Scholar 

  3. 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 

  4. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. V.I. Arnold. Encyclopedia of Mathematical Sciences: Dynamical Systems V. Springer, 1994.

    Google Scholar 

  6. M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions. Dover, 1965.

    Google Scholar 

  7. W. Balser. Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer, 2000.

    Google Scholar 

  8. 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.

    Article  MATH  Google Scholar 

  9. N.N. Bogoliubov and I.A. Mitropol’skii. Asymptotic methods in the theory of non-linear oscillations. Gordon Breach Science Pub., 1961.

    Google Scholar 

  10. S. Bottani. Pulse-coupled relaxation oscillators: from biological synchronization to self-organized criticality. Phys. Rev. Lett., 74:4189–4192, 1995.

    Article  Google Scholar 

  11. H. Bremmer. The scientific work of Balthasar van der Pol. Philips Tech. Rev., 22:36–52, 1960.

    MathSciNet  Google Scholar 

  12. M. Brøns. An iterative method for the canard explosion in general planar systems. arXiv:1209.1109, pages 1–9, 2012.

    Google Scholar 

  13. 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 

  14. R. Curtu and B. Ermentrout. Oscillations in a refractory neural net. J. Math. Biol., 43(1):81–100, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  15. C. Comstock and G.C. Hsiao. Singular perturbations for difference equations. Rocky Moun. J. Math., 6(4):561, 1976.

    Google Scholar 

  16. T.M. Cherry. Uniform asymptotic expansions. J. London Math. Soc., 1(2):121–130, 1949.

    Article  MathSciNet  Google Scholar 

  17. T.M. Cherry. Uniform asymptotic formulae for functions with transition points. Trans. Amer. Math. Soc., 68(2):224–257, 1950.

    Article  MATH  MathSciNet  Google Scholar 

  18. 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.

    Article  MATH  MathSciNet  Google Scholar 

  19. S. Coombes. Phase locking in networks of synaptically coupled McKean relaxation oscillators. Physica D, 160(3):173–188, 2001.

    Article  MATH  MathSciNet  Google Scholar 

  20. 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.

    Article  MATH  MathSciNet  Google Scholar 

  21. 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.

    Article  MATH  MathSciNet  Google Scholar 

  22. 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.

    Article  MATH  MathSciNet  Google Scholar 

  23. A.A. Dorodnitsyn. Asymptotic solutions of van der Pol’s equation. Prikl. Matem. i Mekhan., 11(3): 313–328, 1947.

    MATH  MathSciNet  Google Scholar 

  24. B.R. Dudley and H.W. Swift. Frictional relaxation oscillations. Philosophical Magazine, 40:849–861, 1949.

    Article  MATH  Google Scholar 

  25. W. Eckhaus and E.M. de Jager. Theory and Applications of Singular Perturbations. Springer, 1982.

    Google Scholar 

  26. S.-I. Ei and M. Mimura. Relaxation oscillations in combustion models of thermal self-ignition. J. Dyn. Diff. Eq., 4(1):191–229, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  27. W.F. Finden. An asymptotic approximation for singular perturbations. SIAM J. Appl. Math., 43(1):107–119, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  28. 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.

    Article  MATH  MathSciNet  Google Scholar 

  29. S.J. Fraser. Double perturbation series in the differential equations of enzyme kinetics. J. Chem. Phys., 109(2):411–423, 1998.

    Article  Google Scholar 

  30. E.D. Gilles, G. Eigenberger, and W. Ruppel. Relaxation oscillations in chemical reactors. AIChE J., 24(5):912–920, 1978.

    Article  MathSciNet  Google Scholar 

  31. 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.

    Article  MATH  MathSciNet  Google Scholar 

  32. 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.

    Article  MATH  MathSciNet  Google Scholar 

  33. 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.

    Article  MATH  MathSciNet  Google Scholar 

  34. 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.

    Article  MATH  MathSciNet  Google Scholar 

  35. 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.

    MATH  MathSciNet  Google Scholar 

  36. 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.

    Article  Google Scholar 

  37. 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.

    Article  MATH  MathSciNet  Google Scholar 

  38. 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.

    Article  MathSciNet  Google Scholar 

  39. J. Grasman. Relaxation oscillations of a van der Pol equation with large critical forcing term. Quart. Appl. Math., 38:9–16, 1980.

    MATH  MathSciNet  Google Scholar 

  40. J. Grasman. The mathematical modeling of entrained biological oscillators. Bull. Math. Biol., 46(3):407–422, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. Grasman. Asymptotic Methods for Relaxation Oscillations and Applications. Springer, 1987.

    Google Scholar 

  42. 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.

    Google Scholar 

  43. 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.

    Article  MATH  Google Scholar 

  44. S.P. Hastings. Formal relaxation oscillations for a model of a catalytic particle. Quart. Appl. Math., 41(4):395–405, 1983.

    MathSciNet  Google Scholar 

  45. S.J. Hogan. Relaxation oscillations in a system with a piecewise smooth drag coefficient. J. Sound Vibration, 263(2):467–471, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  46. M.H. Holmes. Introduction to Perturbation Methods. Springer, 1995.

    Google Scholar 

  47. 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.

    Article  MATH  MathSciNet  Google Scholar 

  48. F.A. Howes. Singular perturbations and differential inequalities, volume 5 of Memoirs of the Amer. Math. Soc. AMS, 1976.

    Google Scholar 

  49. F.A. Howes. Boundary-interior layer interactions in nonlinear singular perturbation theory, volume 203 of Memoirs of the Amer. Math. Soc. AMS, 1978.

    Google Scholar 

  50. F.A. Howes. An improved boundary layer estimate for a singularly perturbed initial value problem. Math. Zeitschr., 165(2):135–142, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  51. 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.

    Article  MATH  MathSciNet  Google Scholar 

  52. C. Hunter and M. Tajdari. Singular complex periodic solutions of van der Pol’s equation. SIAM J. Appl. Math., 50(6):1764–1779, 1990.

    Article  MATH  MathSciNet  Google Scholar 

  53. E. Izhikevich. Phase equations for relaxation oscillators. SIAM J. Appl. Math., 60(5):1789–1805, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  54. E.M. De Jager and J. Furu. The Theory of Singular Perturbations. North-Holland, 1996.

    Google Scholar 

  55. 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.

    Article  MATH  Google Scholar 

  56. 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.

    Article  MATH  MathSciNet  Google Scholar 

  57. G. Karreman. Some types of relaxation oscillations as models of all-or-none phenomena. Bull. Math. Biophys., 11(4):311–318, 1949.

    Article  MathSciNet  Google Scholar 

  58. A.Yu. Kolesov and Yu.S. Kolesov. Relaxation oscillations in mathematical models of ecology. Proc. Steklov Inst. Math., 199(1):1–126, 1995.

    Google Scholar 

  59. 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.

    Article  MATH  MathSciNet  Google Scholar 

  60. A.Yu. Kolesov and E.F. Mishchenko. Existence and stability of the relaxation torus. Russ. Math. Surv., 44(3):204–205, 1989.

    Google Scholar 

  61. A.Yu. Kolesov. Specific relaxation cycles of systems of Lotka–Volterra type. Math. USSR-Izvestiya, 38(3):503–523, 1992.

    Article  MathSciNet  Google Scholar 

  62. 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.

    Article  MathSciNet  Google Scholar 

  63. N. Kopell and D. Somers. Anti-phase solutions in relaxation oscillators coupled through excitatory interactions. J. Math. Biol., 33(3):261–280, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  64. 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.

    Article  MathSciNet  Google Scholar 

  65. J. LaSalle. Relaxation oscillations. Quart. Appl. Math., 7:1–19, 1949.

    MathSciNet  Google Scholar 

  66. S.A. Lomov and A.G. Eliseev. Asymptotic integration of singularly perturbed problems. Russ. Math. Surv., 43(3):1–63, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  67. S. Lefschetz. Differential Equations: Geometric Theory. Interscience Publishers, 1957.

    Google Scholar 

  68. S.A. Lomov. Introduction to the General Theory of Singular Perturbations. AMS, 1992.

    Google Scholar 

  69. 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.

    Google Scholar 

  70. P. Lundberg and L. Rahm. A nonlinear convective system with oscillatory behaviour for certain parameter regimes. J. Fluid Mech., 139:237–260, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  71. N. Levinson and O.K. Smith. A general equation for relaxation oscillations. Duke Math. J., 9(2): 382–403, 1942.

    Article  MATH  MathSciNet  Google Scholar 

  72. 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.

    Article  MATH  MathSciNet  Google Scholar 

  73. 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.

    Article  MATH  MathSciNet  Google Scholar 

  74. 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.

    Article  MathSciNet  Google Scholar 

  75. 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.

    Article  MathSciNet  Google Scholar 

  76. P.D. Miller. Applied Asymptotic Analysis. AMS, 2006.

    Google Scholar 

  77. N. Minorsky. Introduction to Non-Linear Mechanics. Topological Methods. Analytical Methods. Non-Linear Resonance. Relaxation Oscillations. Ann Arbor [Mich.]: J.W. Edwards, 1947.

    Google Scholar 

  78. N. Minorsky. Nonlinear Oscillations. Van Nostrand, 1962.

    Google Scholar 

  79. 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.

    Google Scholar 

  80. 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.

    Google Scholar 

  81. E.F. Mishchenko and A.Yu. Kolesov. Asymptotical theory of relaxation oscillations. Proc. Steklov Inst. Math., 197:1–93, 1993.

    MathSciNet  Google Scholar 

  82. E.F. Mishchenko, Yu.S. Kolesov, A.Yu. Kolesov, and N.Kh. Rozov. Asymptotic Methods in Singularly Perturbed Systems. Plenum Press, 1994.

    Google Scholar 

  83. 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.

    Article  Google Scholar 

  84. E.F. Mishchenko and N.Kh. Rozov. Differential Equations with Small Parameters and Relaxation Oscillations (translated from Russian). Plenum Press, 1980.

    Google Scholar 

  85. 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.

    Article  MATH  MathSciNet  Google Scholar 

  86. K. Nipp. An extension of Tikhonov’s theorem in singular perturbations for the planar case. Z. Angew. Math. Phys., 34(3):277–290, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  87. 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.

    Article  MATH  MathSciNet  Google Scholar 

  88. 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.

    MATH  MathSciNet  Google Scholar 

  89. 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.

    Article  MATH  MathSciNet  Google Scholar 

  90. R.E. O’Malley. Phase-plane solutions to some singular perturbation problems. J. Math. Anal. Appl., 54(2):449–466, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  91. R.E. O’Malley. On singular singularly-perturbed initial value problems. Applicable Analysis, 8(1): 71–81, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  92. R.E. O’Malley. A singular singularly-perturbed linear boundary value problem. SIAM J. Math. Anal., 10(4):695–708, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  93. 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.

    Google Scholar 

  94. 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.

    Google Scholar 

  95. D. Quinn, B. Gladman, P. Nicholson, and R. Rand. Relaxation oscillations in tidally evolving satellites. Celestial Mech. Dynam. Astronom., 67(2):111–130, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  96. 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.

    Article  MathSciNet  Google Scholar 

  97. 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.

    Google Scholar 

  98. 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.

    Google Scholar 

  99. 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.

    Google Scholar 

  100. A. Rasmussen, J. Wyller and J.O. Vik. Relaxation oscillations in spruce-budworm interactions. Nonlinear Anal. Real World Appl., 12:304–319, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  101. 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.

    Article  MATH  MathSciNet  Google Scholar 

  102. D. Somers and N. Kopell. Rapid synchronization through fast threshold modulation. Biol. Cybern., 68(5):393–407, 1993.

    Article  Google Scholar 

  103. 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.

    Article  MATH  MathSciNet  Google Scholar 

  104. I. Siekmann and H. Malchow. Local collapses in the Truscott–Brindley model. Math. Model. Nat. Phenom., 3(4):114–130, 2008.

    Article  MathSciNet  Google Scholar 

  105. D.R. Smith. Singular-Perturbation Theory: An Introduction with Applications. CUP, 1985.

    Google Scholar 

  106. R. Singh and S. Sinha. Spatiotemporal order, disorder, and propagating defects in homogeneous system of relaxation oscillators. Phys. Rev. E, 87:012907, 2013.

    Article  Google Scholar 

  107. F.J. Solis and C. Yebra. Modeling the pursuit in natural systems: a relaxed oscillation approach. Math. Comput. Modelling, 52(7):956–961, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  108. D. Terman and E. Lee. Partial synchronization in a network of neural oscillators. SIAM J. Appl. Math., 57(1):252–293, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  109. 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.

    Article  MATH  MathSciNet  Google Scholar 

  110. W.C. Troy. Bifurcation phenomena in FitzHugh’s nerve conduction equations. J. Math. Anal. Appl., 54(3):678–690, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  111. J.J. Tyson. Relaxation oscillations in the revised Oregonator. J. Chem. Phys., 80(12):6079–6082, 1984.

    Article  MathSciNet  Google Scholar 

  112. W. Wasow. Asymptotic Expansions for Ordinary Differential Equations. Dover, 2002.

    Google Scholar 

  113. A.J. Wesselink. Stellar variability and relaxation oscillations. Astrophys. J., 89:659–668, 1939.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints 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

Publish with us

Policies and ethics