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Multiple Time Scale Dynamics

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

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Abstract

In this chapter we touch on several application areas in which time scale separation arises naturally. As you can guess from the rather diverse set of section headings, it is quite reasonable to conjecture that most quantitative sciences that employ mathematical modeling may eventually encounter various multiscale problems. Each section centers on one or two key examples in which one can clearly identify the time scale separation parameter as well as apply many of the methods discussed in this book.

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Bibliography

  1. P. Auger and E. Benoit. A prey-predator model in a multi-patch environment with different time scales. J. Biol. Syst., 1(2):187–197, 1993.

    Google Scholar 

  2. O.E. Akman, D.S. Broomhead, R.V. Abadi, and R.A. Clement. Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system. J. Math. Biol., 51(6):661–694, 2005.

    MATH  MathSciNet  Google Scholar 

  3. J.E. Avron and A. Elgart. Adiabatic theorem without a gap condition. Commun. Math. Phys., 203: 445–463, 1999.

    MATH  MathSciNet  Google Scholar 

  4. J.E. Avron, M. Fraas, G.M. Graf, and P. Grech. Adiabatic theorems for generators of contracting evolutions. Commun. Math. Phys., 314:163–191, 2012.

    MATH  MathSciNet  Google Scholar 

  5. Z. Artstein and V. Gaitsgory. Linear-quadratic tracking of coupled slow and fast targets. Math. Control Signals Systems, 10: 1–30, 1997.

    MATH  MathSciNet  Google Scholar 

  6. 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 

  7. V.I. Arnold, V.V. Kozlov, and A.I. Neishstadt. Mathematical Aspects of Classical and Celestial Mechanics. Springer, 3rd edition, 2006.

    Google Scholar 

  8. B. Avramovic, P.V. Kokotovic, J.R. Winkleman, and J.H. Chow. Area decomposition for electromechanical models of power systems. Automatica, 16(6):637–648, 1980.

    MATH  Google Scholar 

  9. K. Al-Naimee, F. Marino, M. Ciszak, R. Meucci, and F.T. Arecchi. Chaotic spiking and incomplete homoclinic scenarios in semiconductor lasers with optoelectric feedback. New Journal of Physics, 11:073022, 2009.

    Google Scholar 

  10. P. Auger and D. Pontier. Fast game dynamics coupled to slow population dynamics: a single population with hawk–dove strategies. Aggregation and emergence in population dynamics. Math. Comput. Modelling, 27(4):81–88, 1998.

    Google Scholar 

  11. P. Auger and D. Pontier. Fast game theory coupled to slow population dynamics: the case of domestic cat populations. Math. Biosci., 148:65–82, 1998.

    MATH  MathSciNet  Google Scholar 

  12. Y. Ando and M. Suzuki. Control of active suspension systems using the singular perturbation method. Contr. Eng. Prac., 4(3):287–293, 1996.

    Google Scholar 

  13. S. Ahn, B.H. Smith, A. Borisyuk, and D. Terman. Analyzing neuronal networks using discrete-time dynamics. Physica D, 239(9):515–528, 2010.

    MATH  MathSciNet  Google Scholar 

  14. D. Anderson, A. Tenzer, G. Barlev, M. Girvan, T.M. Antonsen, and E. Ott. Multiscale dynamics in communities of phase oscillators. Chaos, 22(1):013102, 2012.

    Google Scholar 

  15. D. Barkley. A model for fast computer simulation of waves in excitable media. Physica D, 49:61–70, 1991.

    Google Scholar 

  16. D. Barkley. Linear stability analysis of rotating spiral waves in excitable media. Phys. Rev. Lett., 68(13):2090–2093, 1992.

    Google Scholar 

  17. D. Barkley. Fast simulation of waves in three-dimensional excitable media. Int. J. Bif. Chaos, 7(11):2529–2545, 1997.

    MATH  MathSciNet  Google Scholar 

  18. G.K. Batchelor. An Introduction to Fluid Dynamics. CUP, 1967.

    Google Scholar 

  19. C.G. Diniz Behn and V. Booth. A fast–slow analysis of the dynamics of REM sleep. SIAM J. Appl. Dyn. Syst., 11(1):212–242, 2012.

    MATH  MathSciNet  Google Scholar 

  20. A. Barrat, M. Barthélemy, and A. Vespignani. Dynamical Processes on Complex Networks. CUP, 2008.

    Google Scholar 

  21. M. Burger, L. Caffarelli, P. Markowich, and M.-T. Wolfram. On a Boltzmann type price formation model. Proc. R. Soc. A, 469:1–21, 2013.

    MathSciNet  Google Scholar 

  22. T.J. Burns, R.W. Davis, and E.F. Moore. A perturbation study of particle dynamics in a plane wake flow. J. Fluid Mech., 384(1):1–26, 1999.

    MATH  MathSciNet  Google Scholar 

  23. P.G. Bergmann. Introduction to the Theory of Relativity. Prentice Hall, 1960.

    Google Scholar 

  24. A.J. Bernoff. Spiral wave solutions for reaction–diffusion equations in a fast reaction/slow diffusion limit. Phsica D, 53:125–150, 1991.

    MATH  MathSciNet  Google Scholar 

  25. M. Berg. Viscoelastic continuum model of nonpolar solvation. 1. Implications for multiple time scales in liquid dynamics. J. Phys. Chem., 102:17–30, 1998.

    Google Scholar 

  26. M. Born and V. Fock. Beweis des Adiabatensatzes. Z. Phys., 51:165–169, 1928.

    MATH  Google Scholar 

  27. B. Brighi, A. Fruchard, and T. Sari. On the Blasius problem. Adv. Differential Equat., 13(5):509–600, 2008.

    MATH  MathSciNet  Google Scholar 

  28. V.S. Buslaev and E.A. Grinina. Remarks on the quantum adiabatic theorem. St. Petersburg Math. J., 16(4):639–649, 2005.

    MATH  MathSciNet  Google Scholar 

  29. V. Bykov, I. Goldfarb, V. Gol’dshtein, and J.B. Greenberg. Thermal explosion in a hot gas mixture with fuel droplets: a two reactant model. Combust. Theor. Model., 6(2):339–359, 2002.

    Google Scholar 

  30. O. Brandman, J.E. Ferrell Jr, R. Li, and T. Meyer. Interlinked fast and slow positive feedback loops drive reliable cell decisions. Science, 310:496–498, 2005.

    MATH  MathSciNet  Google Scholar 

  31. G.L. Browning and H.-O. Kreiss. Analysis of periodic updating for systems with multiple timescales. H. Atmos. Sci., 53(2):335–348, 1996.

    MathSciNet  Google Scholar 

  32. A. Birzu and K. Krischer. Resonance tongues in a system of globally coupled FitzHugh–Nagumo oscillators with time-periodic coupling strength. Chaos, 20:043114, 2010.

    Google Scholar 

  33. M. Brøns and R. Kaasen. Canards and mixed-mode oscillations in a forest pest model. Theor. Popul. Biol., 77:238–242, 2010.

    Google Scholar 

  34. J.A. Biello and A.J. Majda. A new multiscale model for the Madden Julian oscillation. J. Atmosph. Sci., 62:1694–1720, 2005.

    MathSciNet  Google Scholar 

  35. J.A. Biello and A.J. Majda. Intraseasonal multi-scale moist dynamics of the tropical tropospheres. Commun. Math. Sci., 8(2):519–540, 2010.

    MATH  MathSciNet  Google Scholar 

  36. E. Brown, J. Moehlis, and P. Holmes. On the phase reduction and response dynamics of neural oscillator populations. Neural Comput., 16(4):673–715, 2004.

    MATH  Google Scholar 

  37. C.L. Buckley and T. Nowotny. Multiscale model of an inhibitory network shows optimal properties near bifurcation. Phys. Rev. Lett., 106:238109, 2011.

    Google Scholar 

  38. C.M. Bender and S.A. Orszag. Asymptotic Methods and Perturbation Theory. Springer, 1999.

    Google Scholar 

  39. F. Bornemann. Homogenization in Time of Singularly Perturbed Mechanical Systems. Springer, 1998.

    Google Scholar 

  40. J. Banasiak, E.K. Phongi, and M. Lachowicz. A singularly perturbed SIS model with age structure. Math. Biosci. Eng., 10(3):499–521, 2013.

    MATH  MathSciNet  Google Scholar 

  41. J. Best, C. Park, D. Terman, and C. Wilson. Transitions between irregular and rhythmic firing patterns in excitatory-inhibitory neuronal networks. J. Comput. Neurosci., 23(2):217–235, 2007.

    MathSciNet  Google Scholar 

  42. S. Bornholdt and T. Rohlf. Topological evolution of dynamical networks: global criticality from local dynamics. Phys. Rev. Lett., 84(26):6114–6117, 2000.

    Google Scholar 

  43. F.P. Bretherton. Slow viscous motion round a cylinder in a simple shear. J. Fluid Mech., 12:591–613, 1962.

    MATH  MathSciNet  Google Scholar 

  44. R. Breban. Role of environmental persistence in pathogen transmission: a mathematical modeling approach. J. Math. Biol., 66(3):535–546, 2013.

    MATH  MathSciNet  Google Scholar 

  45. M. Brøns. Relaxation oscillations and canards in a nonlinear model of discontinuous plastic deformation in metals at very low temperature. Proc. R. Soc. A, 461:2289–2302, 2005.

    Google Scholar 

  46. S.V. Bogatyrev and V.A. Sobolev. Separating the rapid and slow motions in the problems of the dynamics of systems of rigid bodies and gyroscopes. J. Appl. Math. Mech., 52(1):41–48, 1988.

    MathSciNet  Google Scholar 

  47. H. Boudjellaba and T. Sari. Stability loss delay in harvesting competing populations. J. Differential Equat., 152(2):394–408, 1999.

    MATH  MathSciNet  Google Scholar 

  48. S. Bornholdt and H.G. Schuster, editor. Handbook of Graphs and Networks. Wiley, 2003.

    Google Scholar 

  49. V.N. Biktashev and R. Suckley. Non-Tikhonov asymptotic properties of cardiac excitability. Phys. Rev. Lett., 93(16):169103, 2004.

    Google Scholar 

  50. C.S. Bretherton, J. Uchida, and P.N. Blossey. Slow manifolds and multiple equilibria in stratocumulus-capped boundary layers. J. Adv. Model. Earth Syst., 2(14):1–20, 2010.

    Google Scholar 

  51. A. Bovier and S.-D. Wang. Multi-time scales in adaptive dynamics: microscopic interpretation of a trait substitution tree model. arXiv:1207.4690v1, pages 1–23, 2012.

    Google Scholar 

  52. A. Bovier and S.-D. Wang. Trait substitution trees on two time scales analysis. arXiv:1304.4640v1, pages 1–28, 2013.

    Google Scholar 

  53. A.J. Calise. Singular perturbation analysis of the atmospheric orbital plane change problem. J. Astronaut. Sci., 36:35–43, 1988.

    Google Scholar 

  54. G.F. Carrier. Singular perturbation theory and geophysics. SIAM Rev., 12(2):175–193, 1970.

    Google Scholar 

  55. C.Cotter. Data assimilation on the exponentially accurate slow manifold. Phil. Trans. R. Soc. A, 317:(20120300), 2013.

    Google Scholar 

  56. S. Conti, A. DeSimone, G. Dolzmann, S. Müller, and F. Otto. Multiscale modeling of materials - the role of analysis. In Trends in Nonlinear Analysis, pages 375–408. Springer, 2003.

    Google Scholar 

  57. K. Christensen, R. Donangelo, B. Koiller, and K. Sneppen. Evolution of random networks. Phys. Rev. Lett., 81(11):2380–2383, 1998.

    Google Scholar 

  58. M.H. Cortez and S.P. Ellner. Understanding rapid evolution in predator–prey interactions using the theory of fast–slow systems. Am. Nat., 176(5):109–127, 2010.

    Google Scholar 

  59. J.H. Chow, R. Galarza, P. Accari, and W.W. Prince. Inertial and slow coherency aggregation algorithms for power system dynamic model reduction. IEEE Trans. Power Syst., 10(2):680–685, 1995.

    Google Scholar 

  60. M. Costa, A.L. Goldberger, and C.K. Peng. Multiscale entropy analysis of complex physiologic time series. Phys. Rev. Lett., 89(6):068102, 2002.

    Google Scholar 

  61. A. Chen. Modeling a synthetic biological chaotic system: relaxation oscillators coupled by quorum sensing. Nonlinear Dyn., 63:711–718, 2011.

    Google Scholar 

  62. J. Cisternas, P. Holmes, I.G. Kevrekidis, and X. Li. CO oxidation on thin Pt crystals: temperature slaving and the derivation of lumped models. J. Chem. Phys., 118:3312, 2003.

    Google Scholar 

  63. F. Castella, J.-P. Hoffbeck, and Y. Lagadeuc. A reduced model for spatially structured predator–prey systems with fast spatial migrations and slow demographic evolutions. Asymptot. Anal., 61(3):125–175, 2009.

    MATH  MathSciNet  Google Scholar 

  64. J.H. Chow, editor. Time-Scale Modeling of Dynamic Networks with Applications to Power Systems, volume 46 of Lect. Notes Contr. Infor. Sci. Springer, 1982.

    Google Scholar 

  65. J.H. Chow. Aggregation properties of linearized two-time-scale power networks. IEEE Trans. Circ. Syst., 38(7):720–730, 1991.

    Google Scholar 

  66. J.H. Chow and P.V. Kokotovic. Time scale modeling of sparse dynamic networks. IEEE Trans. Aut. Contr., 30(8):714–722, 1985.

    MATH  Google Scholar 

  67. W.W. Chow, S.W. Koch, and M. Sargent. Semiconductor Laser Physics. Springer, 1994.

    Google Scholar 

  68. E.J. Collins and D.S. Leslie. Convergent multiple-timescales reinforcement learning algorithms in normal form games. Ann. Appl. Probab., 13(4):1231–1251, 2003.

    MATH  MathSciNet  Google Scholar 

  69. K. Christensen and N.R. Moloney. Complexity and Criticality. Imperial College Press, 2005.

    Google Scholar 

  70. J. Cousteix and J. Mauss. Asymptotic Analysis and Boundary Layers. Springer, 2007.

    Google Scholar 

  71. A.-S. Crépin. Using fast and slow processes to manage resources with thresholds. Environ. Resource Econ., 36(2):191–213, 2007.

    Google Scholar 

  72. F. Clément and A. Vidal. Foliation-based parameter tuning in a model of the GnRH pulse and surge generator. SIAM J. Appl. Dyn. Syst., 8(4):1591–1631, 2009.

    MATH  MathSciNet  Google Scholar 

  73. J.H. Chow, J.R. Winkelman, M.A. Pai, and P.W. Sauer. Singular perturbation analysis of large-scale power systems. Int. J. Elec. Power Ener. Syst., 12(2):117–126, 1990.

    Google Scholar 

  74. M.J.H. Dantas. Quenching in a class of singularly perturbed mechanical systems. Int. J. Non-Linear Mech., 50:48–57, 2013.

    Google Scholar 

  75. M. Dam, M. Brøns, J.J. Rasmussen, V. Naulin, and G. Xu. Bifurcation analysis and dimension reduction of a predator–prey model for the LH transition. Physics of Plasmas, 20:102302, 2013.

    Google Scholar 

  76. C. DuBois, J. Farnham, E. Aaron, and A. Radunskaya. A multiple time-scale computational model of a tumor and its micro environment. Math. Biosci. Eng., 10:121–150, 2013.

    MATH  MathSciNet  Google Scholar 

  77. F. Dercole, R. Ferrière, A. Gragnani, and S. Rinaldi. Coevolution of slow–fast populations: evolutionary sliding, evolutionary pseudo-equilibria and complex Red Queen dynamics. Proc. R. Soc. B, 273:983–990, 2006.

    Google Scholar 

  78. A. Doelman, R.A. Gardner, and T.J. Kaper. Stability analysis of singular patterns in the 1D Gray-Scott model: a matched asymptotics approach. Physica D, 122(1):1–36, 1998.

    MATH  MathSciNet  Google Scholar 

  79. A. Doelman, R.A. Gardner, and T.J. Kaper. Large stable pulse solutions in reaction–diffusion equations. Indiana Univ. Math. J., 50(1):443–507, 2001.

    MATH  MathSciNet  Google Scholar 

  80. J.L.A. Dubbeldam and B. Krauskopf. Self-pulsations of lasers with saturable absorber: dynamics and bifurcations. Optics Communications, 159:325–338, 1999.

    Google Scholar 

  81. J.L.A. Dubbeldam, B. Krauskopf, and D. Lenstra. Excitability and coherence resonance in lasers with saturable absorber. Phys. Rev. E, 3(60):6580–6588, 1999.

    Google Scholar 

  82. A. Doelman, T.J. Kaper, and H. van der Ploeg. Spatially periodic and aperiodic multi-pulse patterns in the one-dimensional Gierer–Meinhardt equation. Meth. Appl. Anal., 8(3):387–414, 2001.

    MATH  Google Scholar 

  83. A. Doelman, T.J. Kaper, and P.A. Zegeling. Pattern formation in the one-dimensional Gray–Scott model. Nonlinearity, 10(2):523–563, 1997.

    MATH  MathSciNet  Google Scholar 

  84. U. Dieckmann and R. Law. The dynamical theory of coevolution: a derivation from stochastic ecological processes. J. Math. Biol., 34:579–612, 1996.

    MATH  MathSciNet  Google Scholar 

  85. L. DeVille and E. Lerman. Dynamics on networks I. Modular continuous-time systems. J. Euro. Math. Soc., pages 1–59, 2013. to appear.

    Google Scholar 

  86. B. D’Andréa-Novel, G. Campion, and G. Bastin. Control of wheeled mobile robots not satisfying ideal velocity constraints: a singular perturbation approach. Int. J. Robust Nonl. Contr., 5(4):243–267, 1995.

    MATH  Google Scholar 

  87. J. Duan, C. Pötzsche, and S. Siegmund. Slow integral manifolds for Lagrangian fluid dynamics in unsteady geophysical flows. Physica D, 233(1):73–82, 2007.

    MATH  MathSciNet  Google Scholar 

  88. F. Dercole and S. Rinaldi. Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications. Princeton University Press, 2008.

    Google Scholar 

  89. J. Drover, J. Rubin, J. Su, and B. Ermentrout. Analysis of a canard mechanism by which excitatory synaptic coupling can synchronize neurons at low firing frequencies. SIAM J. Appl. Math., 65(1):69–92, 2004.

    MATH  MathSciNet  Google Scholar 

  90. I. Dvorak and J. Siska. Analysis of metabolic systems with complex slow and fast dynamics. Bull. Math. Biol., 51(2):255–274, 1989.

    MATH  Google Scholar 

  91. M.J. Dauphine-Tanguy, P. Borne, and M. Lebrun. Order reduction of multi-time scale systems using bond graphs, the reciprocal system and the singular perturbation method. J. Frank. Inst., 319:157–171, 1985.

    Google Scholar 

  92. R. Durrett. Random Graph Dynamics. CUP, 2010.

    Google Scholar 

  93. A. Doelman and H. van der Ploeg. Homoclinic stripe patterns. SIAM J. Appl. Dyn. Syst., 1(1):65–104, 2002.

    MATH  MathSciNet  Google Scholar 

  94. M.J. Donovan, P. Wenner, N. Chub, J. Tabak, and J. Rinzel. Mechanisms of spontaneous activity in the developing spinal cord and their relevance to locomotion. Ann. New York Acad. Sci., 860(1):130–141, 1998.

    Google Scholar 

  95. M. Enculescu, A. Gholami, and M. Falcke. Dynamic regimes and bifurcations in a model of actin-based motility. Phys. Rev. E, 78(3):031915, 2007.

    Google Scholar 

  96. T. Erneux, P. Gavrielides, P. Peterson, and M.P. Sharma. Dynamics of passively Q-switched microchip lasers. IEEE J. Quant. Electr., 35:1247–1256, 1999.

    Google Scholar 

  97. E. Endres and H.K. Jenssen. Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier–Stokes shocks. Discr. Cont. Dyn. Sys., 27(1):133–169, 2010.

    MATH  MathSciNet  Google Scholar 

  98. E. Endres, H.K. Jenssen, and M. Milliams. Symmetric Euler and Navier–Stokes shocks in stationary barotropic flow on a bounded domain. J. Differential Equat., 245(10):3025–3067, 2008.

    MATH  Google Scholar 

  99. E. Endres, H.K. Jenssen, and M. Milliams. Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier–Stokes shocks. Dynamical Systems, 27(1):133–169, 2010.

    MATH  MathSciNet  Google Scholar 

  100. Y. Estrin and L. Kubin. Criterion for thermomechanical instability of low temperature plastic deformation. Scripta Metallurgica, 14:1359–1364, 1980.

    Google Scholar 

  101. G.B. Ermentrout and N. Kopell. Symmetry and phaselocking in chains of weakly coupled oscillators. Comm. Pure Appl. Math., 39(5):623–660, 1986.

    MATH  MathSciNet  Google Scholar 

  102. H. Erzgräber, B. Krauskopf, and D. Lenstra. Bifurcation analysis of a semiconductor laser with filtered optical feedback. SIAM J. Appl. Dyn. Syst., 6(1):1–28, 2007.

    MATH  MathSciNet  Google Scholar 

  103. A. Erisir, D. Lau, B. Rudy, and C.S. Leonard. Function of specific K + channels in sustained high-frequency firing of fast-spiking interneurons. J. Neurophysiol., 82:2476–2489, 1999.

    Google Scholar 

  104. T. Erneux and P. Mandel. Temporal aspects of absorptive optical bistability. Phys. Rev. A, 28(2): 896–909, 1983.

    Google Scholar 

  105. T. Erneux, P. Peterson, and A. Gavrielides. The pulse shape of a passively Q-switched microchip laser. Eur. Phys. J. D, 10(3):423–431, 2000.

    Google Scholar 

  106. G.B. Ermentrout. Reduction of conductance-based models with slow synapses to neural nets. Neural Comput., 6(4):679–695, 1994.

    Google Scholar 

  107. G.B. Ermentrout. Type I membranes, phase resetting curves, and synchrony. Neural Comput., 8(5):979–1001, 1996.

    Google Scholar 

  108. G.B. Ermentrout. Linearization of FI curves by adaptation. Neural Comput., 10(7):1721–1729, 1998.

    Google Scholar 

  109. T. Erneux. Q-switching bifurcation in a laser with a saturable absorber. J. Opt. Soc. Amer. B Opt. Phys., 5:1063–1069, 1988.

    Google Scholar 

  110. G.B. Ermentrout and D.H. Terman. Mathematical Foundations of Neuroscience. Springer, 2010.

    Google Scholar 

  111. T. Erneux, E.A. Viktorov, and P. Mandel. Time scales and relaxation dynamics in quantum-dot lasers. Phys. Rev. A, 76(2):023819, 2007.

    Google Scholar 

  112. A.C. Fowler and P.D. Howell. Intermittency in the transition to turbulence. SIAM J. Appl. Math., 63(4):1184–1207, 2003.

    MATH  MathSciNet  Google Scholar 

  113. A. Fasano, M.A. Herrero, and M.R. Rodrigo. Slow and fast invasion waves in a model of acid-mediated tumour growth. Math. Biosci., 220:45–56, 2009.

    MATH  MathSciNet  Google Scholar 

  114. R. FitzHugh. Thresholds and plateaus in the Hodgkin–Huxley nerve equations. J. Gen. Physiol., 43:867–896, 1960.

    Google Scholar 

  115. R.J. Field and R.M. Noyes. Oscillations in chemical systems IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys., 60:1877–1884, 1974.

    Google Scholar 

  116. B.D. Fulcher, A.J.K. Phillips, and P.A. Robinson. Modeling the impact of impulsive stimuli on sleep-wake dynamics. Phys. Rev. E, 78(5):051920, 2008.

    Google Scholar 

  117. O. De Feo and S. Rinaldi. Singular homoclinic bifurcations in tritrophic food chains. Math. Biosci., 148:7–20, 1998.

    MATH  MathSciNet  Google Scholar 

  118. Z. Feng, D.L. Smith, F. Ellise McKenzie, and S.A. Levin. Coupling ecology and evolution: malaria and the S-gene across time scales. Math. Biosci., 189(1):1–19, 2004.

    MATH  MathSciNet  Google Scholar 

  119. Z. Feng, Y. Yi, and H. Zhu. Fast and slow dynamics of malaria and the S-gene frequency. J. Dyn. Diff. Eq., 16(4):869–896, 2004.

    MATH  MathSciNet  Google Scholar 

  120. M. Giona, A. Adrover, F. Creta, and M. Valorani. Slow manifold structure in explosive kinetics. 2. Extension to higher dimensional systems. J. Phys. Chem. A, 110(50):13463–13474, 2006.

    Google Scholar 

  121. T. Gross and B. Blasius. Adaptive coevolutionary networks: a review. Journal of the Royal Society – Interface, 5:259–271, 2008.

    Google Scholar 

  122. I.T. Georgiou, A.K. Bajaj, and M. Corless. Slow and fast invariant manifolds, and normal modes in a two degree-of-freedom structural dynamical system with multiple equilibrium states. Int. J. Non-Linear Mech., 33(2):275–300, 1998.

    MATH  MathSciNet  Google Scholar 

  123. C. Germay, N. Van de Wouw, H. Nijmeijer, and R. Sepulchre. Nonlinear drillstring dynamics analysis. SIAM J. Appl. Dyn. Syst., 8(2):527–553, 2009.

    MATH  MathSciNet  Google Scholar 

  124. J. Guckenheimer and S. Ellner. Dynamic Models in Biology. Princeton University Press, 2006.

    Google Scholar 

  125. I. Georgiou. On the global geometric structure of the dynamics of the elastic pendulum. Nonlinear Dyn., 18(1):51–68, 1999.

    MATH  MathSciNet  Google Scholar 

  126. I. Georgiou. Advanced proper orthogonal decomposition tools: using reduced order models to identify normal modes of vibration and slow invariant manifolds in the dynamics of planar nonlinear rods. Nonlinear Dyn., 41(1):69–110, 2005.

    MATH  MathSciNet  Google Scholar 

  127. I. Goldfarb, V. Gol’dshtein, G. Kuzmenko, and S. Sazhin. Thermal radiation effect on thermal explosion in gas containing fuel droplets. Combust. Theor. Model., 3(4):769–787, 1999.

    Google Scholar 

  128. I. Goldfarb, V. Gol’dshtein, and A. Zinoviev. Delayed thermal explosion in porous media: method of invariant manifolds. IMA J. Appl. Math., 67(3):263–280, 2002.

    Google Scholar 

  129. R.M. Ghigliazza and P. Holmes. A minimal model of a central pattern generator and motoneurons for insect locomotion. SIAM J. Appl. Dyn. Syst., 3(4):671–700, 2004.

    MATH  MathSciNet  Google Scholar 

  130. R.M. Ghigliazza and P. Holmes. Minimal models of bursting neurons: how multiple currents, conductances, and timescales affect bifurcation diagrams. SIAM J. Appl. Dyn. Syst., 3(4):636–670, 2004.

    MATH  MathSciNet  Google Scholar 

  131. K.M. Grimsrud and R. Huffaker. Solving multidimensional bioeconomic problems with singular-perturbation reduction methods: Application to managing pest resistance to pesticidal crops. J. Environ. Econ. Manag., 51(3):336–353, 2006.

    MATH  Google Scholar 

  132. M. Grace and M.-T. Hütt. Predictability of spatio-temporal patterns in a lattice of coupled FitzHugh–Nagumo oscillators. J. R. Soc. Interface, 10:20121016, 2013.

    Google Scholar 

  133. M. Ghisi. Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation. Adv. Differential Equat., 17(1):1–36, 2012.

    MATH  MathSciNet  Google Scholar 

  134. E.A. Gaffney, J.K. Heath and M.Z. Kwiatkowska. A mass action model of a Fibroblast Growth Factor signaling pathway and its simplification. Bull. Math. Biol., 70(8):2229–2263, 2008.

    MATH  MathSciNet  Google Scholar 

  135. H. Gingold. An asymptotic decomposition method applied to multi-turning point problems. SIAM J. Math. Anal., 16(1):7–27, 1985.

    MATH  MathSciNet  Google Scholar 

  136. A. Goldbeter and D.E. Koshland. An amplified sensitivity arising from covalent modification in biological systems. Proc. Natl. Acad. Sci USA, 78:6840–6844, 1981.

    MathSciNet  Google Scholar 

  137. W. Gerstner and W. Kistler. Spiking Neuron Models. Cambridge University Press, 2002.

    Google Scholar 

  138. T. Götz, A. Klar, N. Marheineke, and R. Wegener. A stochastic model and associated Fokker–Planck equation for the fiber lay-down process in nonwoven production processes. SIAM J. Appl. Math., 67(6):1704–1717, 2007.

    MATH  MathSciNet  Google Scholar 

  139. J. Guckenheimer and I.S. Labouriau. Bifurcations of the Hodgkin and Huxley equations; a new twist. Bull. Math. Biol., 55:937–952, 1993.

    MATH  Google Scholar 

  140. V.Y. Glizer. Asymptotic solution of zero-sum linear-quadratic differential game with cheap control for minimizer. Nonl. Diff. Eq. Appl., 7(2):213–258, 2000.

    MathSciNet  Google Scholar 

  141. A. Gierer and H. Meinhardt. A theory of biological pattern formation. Kybernetic, 12:30–39, 1972.

    Google Scholar 

  142. L. Giomi, L. Mahadevan, B. Chakraborty, and M.F. Hagan. Banding, excitability and chaos in active nematic suspensions. Nonlinearity, 25(8):2245–2269, 2012.

    MATH  MathSciNet  Google Scholar 

  143. J.E. Gough, H.I. Nurdin, and S. Wildfeuer. Commutativity of the adiabatic elimination limit of fast oscillatory components and the instantaneous feedback limit in quantum feedback networks. J. Math. Phys., 51:123518, 2010.

    MathSciNet  Google Scholar 

  144. A. Goldbeter. A minimal cascade model for the mitotic oscillator involving cyclin and cdc2 kinase. Proc. Natl. Acad. Sci USA, 88:9107–9111, 1991.

    Google Scholar 

  145. A. Goldbeter. Biochemical Oscillations and Cellular Rhythms. CUP, 1997.

    Google Scholar 

  146. G. Gottwald, M. Oliver, and N. Tecu. Long-time accuracy for approximate slow manifolds in a finite dimensional model of balance. J. Nonlinear Sci., 17:283–307, 2007.

    MATH  MathSciNet  Google Scholar 

  147. D. Golomb and J. Rinzel. Dynamics of globally coupled inhibitory neurons with heterogeneity. Phys. Rev. E, 48(6):4810, 1993.

    Google Scholar 

  148. D. Golomb and J. Rinzel. Clustering in globally coupled inhibitory neurons. Physica D, 72(3):259–282, 1994.

    MATH  Google Scholar 

  149. I. Georgiou and I.B. Schwartz. Dynamics of large scale coupled structural/mechanical systems: a singular perturbation/proper orthogonal decomposition approach. SIAM J. Appl. Math., 59(4): 1178–1207, 1999.

    MATH  MathSciNet  Google Scholar 

  150. F. Ghorbel and M.W. Spong. Integral manifolds of singularly perturbed systems with application to rigid-link flexible-joint multibody systems. Int. J. Non-Linear Mech., 35(1):133–155, 2000.

    MATH  MathSciNet  Google Scholar 

  151. I. Georgiou and I.B. Schwartz. Invariant manifolds, nonclassical normal modes, and proper orthogonal modes in the dynamics of the flexible spherical pendulum. Nonlinear Dyn., 25(1):3–31, 2001.

    MATH  MathSciNet  Google Scholar 

  152. T. Gross and H. Sayama, editors. Adaptive Networks: Theory, Models and Applications. Springer, 2009.

    Google Scholar 

  153. I. Gucwa and P. Szmolyan. Geometric singular perturbation analysis of an autocatalator model. Discr. Cont. Dyn. Syst. S, 2(4):783–806, 2009.

    MATH  MathSciNet  Google Scholar 

  154. J. Guckenheimer and C. Scheper. A geometric model for mixed-mode oscillations in a chemical system. SIAM J. Appl. Dyn. Sys., 10(1):92–128, 2011.

    MATH  MathSciNet  Google Scholar 

  155. J. Gunawardena. A linear framework for time-scale separation in nonlinear biochemical systems. PLoS ONE, 7:e36321, 2012.

    Google Scholar 

  156. O.V. Gendelman, A.F. Vakakis, L.A. Bergman, and D.M. McFarland. Asymptotic analysis of passive nonlinear suppression of aeroelastic instabilities of a rigid wing in subsonic flow. SIAM J. Appl. Math., 70(5):1655–1677, 2009.

    MathSciNet  Google Scholar 

  157. M.N. Galtier and G. Wainrib. Multiscale analysis of slow–fast neuronal learning models with noise. J. Math. Neurosci., 2:13, 2012.

    MathSciNet  Google Scholar 

  158. S. Gil and D.H. Zanette. Coevolution of agents and networks: opinion spreading and community disconnection. Phys. Lett. A, 356:89–94, 2006.

    MATH  Google Scholar 

  159. V. Gol’dshtein, A. Zinoviev, V. Sobolev, and E. Shchepakina. Criterion for thermal explosion with reactant consumption in a dusty gas. Proc. R. Soc. London A, 542(1952):2013–2119, 1996.

    Google Scholar 

  160. B. Hassard. Bifurcation of periodic solutions of the Hodgkin–Huxley model for the squid giant axon. J. Theor. Biol., 71(3): 401–420, 1978.

    MathSciNet  Google Scholar 

  161. A. Hastings. Timescales, dynamics, and ecological understanding. Ecology, 91:3471–3480, 2010.

    Google Scholar 

  162. W. Hauck. Kinks and rotations in long Josephson junctions. Math. Meth. Appl. Sci., 24(15):1189–1217, 2001.

    MATH  MathSciNet  Google Scholar 

  163. G. Hek. Geometric singular perturbation theory in biological practice. J. Math. Biol., 60:347–386, 2010.

    MathSciNet  Google Scholar 

  164. P.J. Holmes, R.J. Full, D. Koditschek, and J. Guckenheimer. The dynamics of legged locomotion: models, analyses, and challenges. SIAM Rev., 48(2):207–304, 2006.

    MATH  MathSciNet  Google Scholar 

  165. A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis. Localized synchronization in two coupled nonidentical semiconductor lasers. Phys. Rev. Lett., 78(25):4745–4748, 1997.

    Google Scholar 

  166. 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.

    Google Scholar 

  167. R. Huffaker and R. Hotchkiss. Economic dynamics of reservoir sedimentation management: optimal control with singularly perturbed equations of motion. J. Econ. Dyn. Contr., 30(12):2553–2575, 2006.

    MATH  MathSciNet  Google Scholar 

  168. J.M. Hong, C.-H. Hsu, and W. Liu. Inviscid and viscous stationary waves of gas flow through contracting–expanding nozzles. J. Differential Equat., 248(1):50–76, 2010.

    MATH  MathSciNet  Google Scholar 

  169. J.M. Hong, C.-H. Hsu, and W. Liu. Viscous standing asymptotic states of isentropic compressible flows through a nozzle. Arch. Rat. Mech. Anal., 196(2):575–597, 2010.

    MATH  MathSciNet  Google Scholar 

  170. F.C. Hoppenstaedt and E.M. Izhikevich. Weakly Connected Neural Networks. Springer, 1997.

    Google Scholar 

  171. S.-Y. Ha, S. Jung, and M. Slemrod. Fast–slow dynamics of planar particle models for flocking and swarming. J. Differen. Equat., 252:2563–2579, 2012.

    MATH  MathSciNet  Google Scholar 

  172. C.J. Honey, R. Kötter, M. Breakspear, and O. Sporns. Network structure of cerebral cortex shapes functional connectivity on multiple time scales. Proc. Natl. Acad. Sci., 104(24):10240–10245, 2007.

    Google Scholar 

  173. E. Harvey, V. Kirk, H.M. Osinga, J. Sneyd, and M. Wechselberger. Understanding anomalous delays in a model of intracellular calcium dynamics. Chaos, 20:045104, 2010.

    MathSciNet  Google Scholar 

  174. E. Harvey, V. Kirk, M. Wechselberger, and J. Sneyd. Multiple time scales, mixed-mode oscillations and canards in models of intracellular calcium dynamics. J. Nonlinear Sci., 21:639–683, 2011.

    MATH  Google Scholar 

  175. M.Y. Hussaini and W.D. Lakin. Existence and non-uniqueness of similarity solutions of a boundary-layer problem. Quarterly. J. Mech. Appl. Math., 39:15–24, 1986.

    MATH  MathSciNet  Google Scholar 

  176. M.Y. Hussaini, W.D. Lakin, and A. Nachman. On similarity solutions of a boundary layer problem with an upstream moving wall. SIAM J. Appl. Math., 47(4):699–709, 1987.

    MATH  MathSciNet  Google Scholar 

  177. S.P. Hastings and J.B. McLeod. On the periodic solutions of a forced second-order equation. J. Nonlinear Sci., 1(2):225–245, 1991.

    MATH  MathSciNet  Google Scholar 

  178. H. Hu, M. Martina, and P. Jonas. Fast-spiking hippocampal interneurons dendritic mechanisms underlying rapid synaptic activation of fast-spiking hippocampus interneurons. Science, 327:52–58, 2010.

    Google Scholar 

  179. F.C. Hoppenstaedt. A slow selection analysis of two locus, two allele traits. Theor. Popul. Biol., 9: 68–81, 1976.

    Google Scholar 

  180. M. Higuera, J. Porter, and E. Knobloch. Faraday waves, streaming flow, and relaxation oscillations in nearly circular containers. Chaos, 18(1):015104, 2008.

    Google Scholar 

  181. M. El Hajji and A. Rapaport. Practical coexistence of two species in the chemostat - a slow–fast characterization. Math. Biosci., 218(1):33–39, 2009.

    MATH  MathSciNet  Google Scholar 

  182. H. Hofmann and S.R. Sanders. Speed-sensorless vector torque control of induction machines using a two-time-scale approach. IEEE Trans. Ind. Appl., 34:169–177, 1998.

    Google Scholar 

  183. A. Huber and P. Szmolyan. Geometric singular perturbation analysis of the Yamada model. SIAM J. Applied Dynamical Systems, 4(3):607–648, 2005.

    MATH  MathSciNet  Google Scholar 

  184. G. Haller and T. Sapsis. Where do inertial particles go in fluid flows? Physica D, 237(5):573–583, 2008.

    MATH  MathSciNet  Google Scholar 

  185. S.-Y. Ha and M. Slemrod. Flocking dynamics of singularly perturbed oscillator chain and the Cucker-Smale system. J. Dyn. Diff. Eq., 22:325–330, 2010.

    MATH  MathSciNet  Google Scholar 

  186. S.-Y. Ha and M. Slemrod. A fast–slow dynamical systems theory for the Kuramoto type phase model. J. Differential Equat., 251(10):2685–2695, 2011.

    MATH  MathSciNet  Google Scholar 

  187. Y. Huang. How do cross-migration models arise? Math. Biosci., 195(2):127–140, 2005.

    MATH  MathSciNet  Google Scholar 

  188. E. Izhikevich and F. Hoppensteadt. Slowly coupled oscillators: phase dynamics and synchronization. SIAM J. Appl. Math., 63(6):1935–1953, 2003.

    MATH  MathSciNet  Google Scholar 

  189. G. Iñiguez, J. Kertész, K.K. Kaski, and R.A. Barrio. Opinion and community formation in coevolving networks. Phys. Rev. E, 80(6):066119, 2009.

    Google Scholar 

  190. G. Iñiguez, J. Kertész, K.K. Kaski, and R.A. Barrio. Phase change in an opion-dynamics model with separation of time scales. Phys. Rev. E, 83:016111, 2011.

    MathSciNet  Google Scholar 

  191. A.P. Itin, A.I. Neishtadt, and A.A. Vasiliev. Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave. Physica D, 141(3):281–296, 2000.

    MATH  MathSciNet  Google Scholar 

  192. D. Iron and M.J. Ward. The dynamics of multispike solutions to the one-dimensional Gierer–Meinhardt model. SIAM J. Appl. Math., 62(6):1924–1951, 2002.

    MATH  MathSciNet  Google Scholar 

  193. D. Iron, M.J. Ward, and J. Wei. The stability of spike solutions to the one-dimensional Gierer–Meinhardt model. Physica D, 150(1):25–62, 2001.

    MATH  MathSciNet  Google Scholar 

  194. E. Izhikevich. Simple model of spiking neurons. IEEE Trans. Neural Netw., 14(6):1569–1572, 2003.

    MathSciNet  Google Scholar 

  195. E. Izhikevich. Dynamical Systems in Neuroscience. MIT Press, 2007.

    Google Scholar 

  196. G. Jongen, J. Anemüller, D. Bollé, A.C.C. Coolen, and C. Pérez-Vicente. Coupled dynamics of fast spins and slow exchange interactions in the XY spin glass. J. Phys. A, 34(19):3957–3984, 2001.

    MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  198. A. Joye, H. Kunz, and C.E. Pfister. Exponential decay and geometric aspect of transition probabilities in the adiabatic limit. Ann. Phys., 208(2):299–332, 1991.

    MATH  MathSciNet  Google Scholar 

  199. C.K.R.T. Jones. Geometric singular perturbation theory. In Dynamical Systems (Montecatini Terme, 1994), volume 1609 of Lect. Notes Math., pages 44–118. Springer, 1995.

    Google Scholar 

  200. A. Joye. Proof of the Landau-Zener formula. Asymp. Anal., 9(3):209–258, 1994.

    MATH  MathSciNet  Google Scholar 

  201. A. Joye and C.E. Pfister. Exponentially small adiabatic invariant for the Schrödinger equation. Comm. Math. Phys., 140(1):15–41, 1991.

    MATH  MathSciNet  Google Scholar 

  202. A. Joye and C.E. Pfister. Full asymptotic expansion of transition probabilities in the adiabatic limit. J. Phys. A, 24(4):753, 1991.

    Google Scholar 

  203. A. Joye and C.E. Pfister. Superadiabatic evolution and adiabatic transition probability between two nondegenerate levels isolated in the spectrum. J. Math. Phys., 34(2):454, 1993.

    Google Scholar 

  204. J. Kumar and G. Ananthakrishna. Multi-scale modeling approach to acoustic emission during plastic deformation. Phys. Rev. Lett., 106:106001, 2011.

    Google Scholar 

  205. P.V. Kokotovic, B. Avramovic, J.H. Chow, and J.R. Winkelman. Coherency based decomposition and aggregation. Automatica, 18:47–56, 1982.

    MATH  Google Scholar 

  206. T. Kato. On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Japan, 5:435–439, 1950.

    Google Scholar 

  207. A.I. Khibnik, Y. Braimanc, T.A.B. Kennedy, and K. Wiesenfeld. Phase model analysis of two lasers with injected field. Physica D, 111(1):295–310, 1998.

    MATH  Google Scholar 

  208. J. Kevorkian and J.D. Cole. Multiple Scale and Singular Perturbation Methods. Springer, 1996.

    Google Scholar 

  209. N. Kopell and G.B. Ermentrout. Phase transitions and other phenomena in chains of coupled oscillators. SIAM J. Appl. Math., 50(4):1014–1052, 1990.

    MATH  MathSciNet  Google Scholar 

  210. G. Kozyreff and T. Erneux. Singular Hopf bifurcation to strongly pulsating oscillations in lasers containing a saturable absorber. Euro. J. Appl. Math., 14:407–420, 2003.

    MATH  MathSciNet  Google Scholar 

  211. H.J. Kelley. Flight path optimization with multiple time scales. J. Aircraft, 8(4):238–240, 1971.

    Google Scholar 

  212. H.K. Khalil. Multimodel design of a Nash strategy. J. Optim. Theor. Appl., 31(4):553–564, 1980.

    MATH  MathSciNet  Google Scholar 

  213. H.K. Khalil and P.V. Kokotovic. Feedback and well-posedness of singularly perturbed Nash games. IEEE Trans. Aut. Contr., 24(5):699–708, 1979.

    MATH  MathSciNet  Google Scholar 

  214. K. Khorasani and P.V. Kokotovic. Feedback linearization of a flexible manipulator near its rigid body manifold. Syst. Control Lett., 6(3):187–192, 1985.

    MATH  MathSciNet  Google Scholar 

  215. P. Kokotovic, H.K. Khalil, and J. O’Reilly. Singular Perturbation Methods in Control: Analysis and Design. SIAM, 1999.

    Google Scholar 

  216. R. Klein and A.J. Majda. Systematic multiscale models for deep convection on mesoscales. Theor. Comp. Fluid Dyn., 20:525–551, 2006.

    MathSciNet  Google Scholar 

  217. T. Kolokolnikov, M. Nizette, T. Erneux, N. Joly, and S. Bielawski. The Q-switching instability in passively mode-locked lasers. Physica D, 219(1):13–21, 2006.

    MATH  MathSciNet  Google Scholar 

  218. N. Kopell. We got rhythm: Dynamical systems of the nervous system. Notices of the AMS, 47(1):6–16, 2000.

    MATH  MathSciNet  Google Scholar 

  219. B.W. Kooi, J.C. Poggiale, P. Auger, and S.A.L.M. Kooijman. Aggregation methods in food chains with nutrient recycling. Ecol. Model., 157(1):69–86, 2002.

    Google Scholar 

  220. V. Kelptsyn, O. Romaskevich, and I. Schurov. Josephson effect and slow–fast systems. Nanostructures. Math. Phys. Model., 8(1):31–46, 2013.

    Google Scholar 

  221. E. Keller and S. Segel. Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol., 26:399–415, 1970.

    MATH  Google Scholar 

  222. M. Kunze and H. Spohn. Adiabatic limit for the Maxwell-Lorentz equations. Ann. Henri Poincaré, 1(4):625–653, 2000.

    MATH  MathSciNet  Google Scholar 

  223. M. Kunze and H. Spohn. Post-Coulombian dynamics at order c-3. J. Nonlinear Sci., 11(5):321–396, 2001.

    MATH  MathSciNet  Google Scholar 

  224. M. Kunze and H. Spohn. Radiation reaction and center manifolds. SIAM J. Math. Anal., 32(1):30–53, 2006.

    MathSciNet  Google Scholar 

  225. J. Keener and J. Sneyd. Mathematical Physiology 1: Cellular Physiology. Springer, 2008.

    Google Scholar 

  226. J. Keener and J. Sneyd. Mathematical Physiology 2: Systems Physiology. Springer, 2008.

    Google Scholar 

  227. I. Kosiuk and P. Szmolyan. A new type of relaxation oscillations in a model of the mitotic oscillator. preprint, 2013.

    Google Scholar 

  228. M. Krupa, M. Schagerl, A. Steindl, P. Szmolyan, and H. Troger. Relative equilibria of tethered satellite systems and their stability for very stiff tethers. Dynamical Systems, 16(3):253–278, 2001.

    MATH  MathSciNet  Google Scholar 

  229. C. Kuehn. Time-scale and noise optimality in self-organized critical adaptive networks. Phys. Rev. E, 85(2):026103, 2012.

    Google Scholar 

  230. C. Kuehn. Normal hyperbolicity and unbounded critical manifolds. Nonlinearity, 27(6):1351–1366, 2014.

    MATH  MathSciNet  Google Scholar 

  231. K.U. Kristiansen, P.L. Uldall, and R.M. Roberts. Numerical modelling of elastic space tethers. Celestial Mech. Dynam. Astronom., 113(2):235–254, 2012.

    MathSciNet  Google Scholar 

  232. M. Krupa, A. Vidal, and F. Clément. A network model of the periodic synchronization process in the dynamics of calcium concentration in GnRH neurons. J. Math. Neurosci., 3(4):1–40, 2013.

    MathSciNet  Google Scholar 

  233. T. Kolokolnikov, M. J. Ward, and J. Wei. The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: the pulse-splitting regime. Physica D, 202(3):258–293, 2005.

    MATH  MathSciNet  Google Scholar 

  234. T. Kolokolnikov, M.J. Ward, and J. Wei. The existence and stability of spike equilibria in the one-dimensional Gray–Scott model: the low feed rate regime. Stud. Appl. Math., 115(1):21–71, 2005.

    MATH  MathSciNet  Google Scholar 

  235. M. Lahutte-Auboin, R. Costalat, J.-P. Francoise, and R. Guillevin. Dip and buffering in a fast–slow system associated to brain lactate kinetics. arXiv:1308.0486v1, pages 1–11, 2013.

    Google Scholar 

  236. W.D. Lakin and P. Van Den Driessche. Time scales in population biology. SIAM J. Appl. Math., 32(3):694–705, 1977.

    MATH  Google Scholar 

  237. C.F. Lee. Singular perturbation analysis of a reduced model for collective motion: a renormalization group approach. Phys. Rev. E, 83:031127, 2011.

    Google Scholar 

  238. B.N. Lundstrom, M. Famulare, L.B. Sorensen, W.J. Spain, and A.L. Fairhall. Sensitivity of firing rate to input fluctuations depends on time scale separation between fast and slow variables in single neurons. J. Comput. Neurosci., 27(2):277–290, 2009.

    MathSciNet  Google Scholar 

  239. M. Linkerhand and C. Gros. Self-organized stochastic tipping in slow–fast dynamical systems. Math. Mech. Complex Syst., pages 1–16, 2013. to appear.

    Google Scholar 

  240. D. Ludwig, D.D. Jones, and C.S. Holling. Qualitative analysis of insect outbreak systems: The spruce budworm and forest. J. Animal Ecol., 47(1):315–332, 1978.

    Google Scholar 

  241. J. Lin and F.L. Lewis. Two-time scale fuzzy logic controller of flexible link robot arm. Fuzzy Sets Syst., 139:125–149, 2003.

    MATH  MathSciNet  Google Scholar 

  242. Y. Li, H. Qian, and Y. Yi. Nonlineaar oscillations and multiscale dynamics in a closed chemical reaction system. J. Dyn. Diff. Eq., 22:491–507, 2010.

    MATH  MathSciNet  Google Scholar 

  243. Y.X. Li and J. Rinzel. Equations for InsP 3 receptor-mediated \([Ca^{2+}]_{i}\) oscillations derived from a detailed kinetic model: a Hodgkin–Huxley like formalism. J. Theor. Biol., 166(4): 461–473, 1994.

    Google Scholar 

  244. K.W. Lee and S.N. Singh. Bifurcation of orbits and synchrony in inferior olive neurons. J. Math. Biol., 65:465–491, 2012.

    MATH  MathSciNet  Google Scholar 

  245. H.K.H. Lentz, T. Selhorst, and I.M. Sokolov. Unfolding accessibility provides a macroscopic approach to temporal networks. Phys. Rev. Lett., 110:118701, 2013.

    Google Scholar 

  246. K. Henzler-Wildman M. Lei, V. Thai, J. Kerns, M. Karplus, and D. Kern. A hierarchy of timescales in protein dynamics is linked to enzyme catalysis. Nature, 450:913–918, 2007.

    Google Scholar 

  247. A.M. Levine, G.H.M. Van Tartwijk, D. Lenstra, and T. Erneux. Diode lasers with optical feedback: stability of the maximum gain mode. Phys. Rev. A, 52(5):3436–3439, 1995.

    Google Scholar 

  248. T. Li, E. Vanden-Eijnden, P. Zhang, and W. E. Stochastic models of polymeric fluids at small Deborah number. J. Non-Newtonian Fluid Mech., 121:117–125, 2004.

    Google Scholar 

  249. A.J. Majda. New multiscale models and self-similarity in tropical convection. J. Atmos. Sci., 64(4):1393–1404, 2007.

    MathSciNet  Google Scholar 

  250. A.M. Mandel, M. Akke, and A.G. Palmer. Dynamics of ribonuclease H: temperature dependence of motions on multiple time scales. Biochem., 35(50):16009–16023, 1996.

    Google Scholar 

  251. P.A. Markowich. The Stationary Semiconductor Device Equations. Springer, 1986.

    Google Scholar 

  252. A.J. Majda and J.A. Biello. A multiscale model for tropical intraseasonal oscillations. Proc. Nat. Acad. Sci. USA, 101:4736–4741, 2004.

    MATH  MathSciNet  Google Scholar 

  253. M.R. Maxey and S. Corrsin. Gravitational settling of aerosol particles in randomly oriented cellular flow fields. J. Atmos. Sci., 43(11):1112–1134, 1986.

    Google Scholar 

  254. C. McCluskey. Lyapunov functions for tuberculosis models with fast and slow progression. Math. Biosci. Eng., 3(4):603–614, 2006.

    MATH  MathSciNet  Google Scholar 

  255. R.E. Meyer. A view of the triple deck. SIAM J. Appl. Math., 43(4):639–663, 1983.

    MATH  MathSciNet  Google Scholar 

  256. I. Mirzaev and J. Gunawardena. Laplacian dynamics on general graphs. Bull. Math. Biol., 75(11): 2118–2149, 2013.

    MATH  MathSciNet  Google Scholar 

  257. J.A.J. Metz, S.A.H. Geritz, G. Meszéna, F.J.A. Jacobs, and J.S. van Heerwaarden. Adaptive dynamics: a geometrical study of the consequences of nearly faithful reproduction. In S.J. van Strien and S.M. Verduyn Lunel, editors, Stochastic and Spatial Structures of Dynamical Systems, pages 183–231. North-Holland, 1991.

    Google Scholar 

  258. A. Mitrofanova. Efficient systems biology algorithms for biological networks over multiple time-scales: from evolutionary to regulatory time. PhD thesis, Courant Institute of Mathematical Sciences, NYU, New York, USA, 2009.

    Google Scholar 

  259. A.J. Majda and R. Klein. Systematic multi-scale models for the tropics. J. Atmosph. Sci., 60:393–408, 2003.

    Google Scholar 

  260. M.M. McCarthy and N. Kopell. The effect of propofol anesthesia on rebound spiking. SIAM J. Appl. Dyn. Syst., 11(4):1674–1697, 2012.

    MATH  Google Scholar 

  261. D. McMillen, N. Kopell, J. Hasty, and J.J. Collins. Synchronizing genetic relaxation oscillators by intercell signaling. Proc. Natl. Acad. Sci. USA, 99(2):679–684, 2002.

    Google Scholar 

  262. M. Moallem, K. Khorasani, and R.V. Patel. An integral manifold approach for tip-position tracking of flexible multi-link manipulators. IEEE Trans. Robot. Aut., 13(6):823–837, 1997.

    Google Scholar 

  263. F. Marino and F. Marin. Coexisting attractors and chaotic canard explosions in a slow–fast optomechanical system. Phys. Rev. E, 87(5):052906, 2013.

    Google Scholar 

  264. J. Mitry, M. McCarthy, N. Kopell, and M. Wechselberger. Excitable neurons, firing threshold manifolds and canards. J. Math. Neurosci., 3:12, 2013.

    MathSciNet  Google Scholar 

  265. S.K. Mazmuder, A.H. Nayfeh, and D. Boroyevich. Theoretical and experimental investigation of the fast-and slow-scale instabilities of a DC-DC converter. IEEE Trans. Power Electron., 16(2):201–216, 2001.

    Google Scholar 

  266. R. Moeckel. Transition tori in the five-body problem. J. Differential Equat., 129(2):290–314, 1996.

    MATH  MathSciNet  Google Scholar 

  267. A. Milik and A. Prskawetz. Slow-fast dynamics in a model of population and resource growth. Math. Popul. Stud., 6(2):155–169, 1996.

    MATH  Google Scholar 

  268. M. Marvá, J.-C. Poggiale, and R. Bravo de la Parra. Reduction of slow–fast periodic systems with applications to population dynamics models. Math. Models Methods Appl. Sci., 22(10):1250025, 2012.

    Google Scholar 

  269. A. Milik, A. Prskawetz, G. Feichtinger, and W.C. Sanderson. Slow-fast dynamics in Wonderland. Environ. Model. Assessm., 1(1):3–17, 1996.

    Google Scholar 

  270. P.A. Markowich and C.A. Ringhofer. A singularly perturbed boundary value problem modelling a semiconductor device. SIAM J. Appl. Math., 44:213–256, 1984.

    MathSciNet  Google Scholar 

  271. K.L. Maki and Y. Renardy. The dynamics of a viscoelastic fluid which displays thixotropic yield stress behavior. J. Non-Newtonian Fluid Mech., 181:30–50, 2012.

    Google Scholar 

  272. P.A. Markowich, C.A. Ringhofer, and C. Schmeiser. Semiconductor Equations. Springer, 1990.

    Google Scholar 

  273. J.A.C. Martins, N.V. Rebrova, and V.A. Sobolev. On the (in)-stability of quasi-static paths of smooth systems: definitions and sufficient conditions. Math. Meth. Appl. Sci., 29(6):741–750, 2006.

    MATH  MathSciNet  Google Scholar 

  274. P.A. Markowich and C. Schmeiser. Uniform asymptotic representation of solutions of the basic semiconductor-device equations. IMA J. Appl. Math., 36(1):43–57, 1986.

    MATH  MathSciNet  Google Scholar 

  275. P.A. Markowich and P. Szmolyan. A system of convection-diffusion equations with small diffusion-coefficient arising in semiconductor physics. J. Differential Equat., 81(2):234–254, 1989.

    MATH  MathSciNet  Google Scholar 

  276. A. Milik and P. Szmolyan. Multiple time scales and canards in a chemical oscillator. In C.K.R.T. Jones, editor, Multiple Time Scale Dynamical Systems, volume 122, pages 117–140. Springer, 2001.

    Google Scholar 

  277. M. Marvá, E. Sánchez, R. Bravo de la Parra, and L. Sanz. Reduction of slow–fast discrete models coupling migration and demography. J. Theoret. Biol., 258(3):371–379, 2009.

    MathSciNet  Google Scholar 

  278. A. Milik, P. Szmolyan, H. Loeffelmann, and E. Groeller. Geometry of mixed-mode oscillations in the 3-d autocatalator. Int. J. Bif. Chaos, 8(3):505–519, 1998.

    MATH  Google Scholar 

  279. T.V. Martins and R. Toral. Synchronisation induced by repulsive interactions in a system of van der Pol oscillators. Prog. Theor. Phys., 126(3):353–368, 2011.

    MATH  Google Scholar 

  280. J.D. Murray. On the role of myoglobin in muscle respiration. J. Theor. Biol., 47(1):115–126, 1974.

    Google Scholar 

  281. J.D. Murray. Mathematical Biology I: An Introduction. Springer, 3rd edition, 2002.

    Google Scholar 

  282. J.D. Murray. Mathematical Biology II: Spatial Models and Biomedical Applications. Springer, 3rd edition, 2003.

    Google Scholar 

  283. A. Mustafin. Two mutually loss-coupled lasers featuring astable multivibrator. Physica D, 218(2):167–176, 2006.

    MATH  MathSciNet  Google Scholar 

  284. S. Namachchivaya. Spindle speed variation for the suppression of regenerative chatter. J. Nonl. Sci., 13(3):265–288, 2003.

    MATH  MathSciNet  Google Scholar 

  285. A.H. Nayfeh. Nonlinear stability of a liquid jet. Phys. Fluids, 13:841, 1970.

    MATH  Google Scholar 

  286. W.H. Nesse, A. Borisyuk, and P.C. Bressloff. Fluctuation-driven rhythmogenesis in an excitatory neuronal network with slow adaptation. J. Comput. Neurosci., 25(2):317–333, 2008.

    MathSciNet  Google Scholar 

  287. M.E.J. Newman, A.-L. Barabási, and D.J. Watts. The Structure and Dynamics of Networks. Princeton University Press, 2003.

    Google Scholar 

  288. D.S. Naidu and A.J. Calise. Singular perturbations and time scales in guidance and control of aerospace systems: a survey. J. Guid. Contr. Dyn., 24(6):1057–1078, 2001.

    Google Scholar 

  289. W.I. Newman and M. Efroimsky. The method of variation of constants and multiple time scales in orbital mechanics. Chaos, 13(2):476–485, 2003.

    MATH  MathSciNet  Google Scholar 

  290. M.E.J. Newman. The structure and function of complex networks. SIAM Review, 45:167–256, 2003.

    MATH  MathSciNet  Google Scholar 

  291. M.E.J. Newman. Networks - An Introduction. OUP, 2011.

    Google Scholar 

  292. T. Nguyen-Huu, R. Bravo de la Parra, and P. Auger. Approximate aggregation of linear discrete models with two time scales: re-scaling slow processes to the fast scale. J. Difference Equ. Appl., 17(4):621–635, 2011.

    MATH  MathSciNet  Google Scholar 

  293. F. Noé, I. Horenko, C. Schütte, and J.C. Smith. Hierarchical analysis of conformational dynamics in biomolecules: transition networks of metastable states. J. Chem. Phys., 126:(155101), 2007.

    Google Scholar 

  294. J. Nowacki,, S.H. Mazlan, H.M. Osinga, and K.T. Tsaneva-Atanasova. The role of large-conductance Calcium-activated K + (BK) channels in shaping bursting oscillations of a somatotroph cell model. Physica D, 239(9):485–493, 2010.

    MATH  MathSciNet  Google Scholar 

  295. E. Neumann and A. Pikovsky. Slow-fast dynamics in Josephson junctions. Eur. Phys. J. B, 34:293–303, 2003.

    Google Scholar 

  296. J. Nawrath, M.C. Romano, M. Thiel, I. Kiss, M. Wickramasinghe, J. Timmer, J. Kurths, and B. Schelter. Distinguishing direct from indirect interactions in oscillatory networks with multiple time scales. Phys. Rev. Lett., 104:038701, 2010.

    Google Scholar 

  297. A.I. Neishtadt and V.V. Sidorenko. Wisdom system: dynamics in the adiabatic approximation. Celestial Mechanics and Dynamical Astronomy, 90(3):307–330, 2004.

    MATH  MathSciNet  Google Scholar 

  298. P. Noble and S. Travadel. Non-persistence of roll-waves under viscous perturbations. Discr. Cont. Dyn. Syst. B, 1(1):61–70, 2001.

    MATH  MathSciNet  Google Scholar 

  299. E. Olbrich, J.C. Claussen, and P. Achermann. The multiple time scales of sleep dynamics as a challenge for modelling the sleeping brain. Philos. Trans. R. Soc. Lond. Ser. A, 369(1952):3884–3901, 2011.

    MATH  MathSciNet  Google Scholar 

  300. R.E. O’Malley. Singular perturbation theory: a viscous flow out of Göttingen. Ann. Rev. Fluid Mech., 42:1–17, 2010.

    MathSciNet  Google Scholar 

  301. M. Oh and V. Mateev. Non-weak inhibition and phase resetting at negative values of phase in cells with fast–slow dynamics at hyperpolarized potentials. J. Comput. Neurosci., 31:31–42, 2011.

    MathSciNet  Google Scholar 

  302. L. Øyehaug, E. Plathe, and S.W. Omholt. Targeted reduction of complex models with time scale hierarchy - a case study. Math. Biosci., 185(2):123–152, 2003.

    MathSciNet  Google Scholar 

  303. R.E. O’Malley and C. Schmeiser. The asymptotic solution of a semiconductor device problem involving reverse bias. SIAM J. Appl. Math., 50(2):504–520, 1990.

    MATH  MathSciNet  Google Scholar 

  304. O.A. Oleinik and V.N. Samokhin. Mathematical Models in Boundary Layer Theory. Chapman & Hall, 1999.

    Google Scholar 

  305. J.-C. Poggiale, P. Auger, F. Cordoleani, and T. Nguyen-Huu. Study of a virus-bacteria interaction model in a chemostat: application of geometric singular perturbation theory. Phil. Trans. R. Soc. A, 367:4685–4697, 2009.

    MATH  MathSciNet  Google Scholar 

  306. R.L. Panton. Incompressible Flow. Wiley, 2005.

    Google Scholar 

  307. B. Porter and A. Bradshaw. Singular perturbation methods in the design of tracking systems incorporating inner-loop compensators and high-gain error-actuated controllers. Int. J. Syst. Sci., 12(10):1193–1205, 1981.

    MATH  MathSciNet  Google Scholar 

  308. Z. Pan and T. Basar. H ∞-optimal control for singularly perturbed systems. Part I: Perfect state measurements. Automatica, 29(2):401–423, 1993.

    Google Scholar 

  309. P. Poletti, B. Caprile, M. Ajelli, A. Pugliese, and S. Merler. Spontaneous behavioural changes in response to epidemics. J. Theor. Biol., 260(1):31–40, 2009.

    MathSciNet  Google Scholar 

  310. R.W. Penney, A.C.C. Coolen, and D. Sherrington. Coupled dynamics of fast spins and slow interactions in neural networks and spin systems. J. Phys. A, 26(15):3681–3695, 1993.

    MathSciNet  Google Scholar 

  311. L.M. Perko. Application of singular perturbation theory to the restricted three body problem. Rocky Mount. J. Math., 6(4):675, 1976.

    Google Scholar 

  312. A. Panfilov and P. Hogeweg. Spiral breakup in a modified FitzHugh–Nagumo model. Phys. Lett. A, 176(5):295–299, 1993.

    Google Scholar 

  313. P.F. Pinsky. Synchrony and clustering in an excitatory neural network model with intrinsic relaxation kinetics. SIAM J. Appl. Math., 55(1):220–241, 1995.

    MATH  MathSciNet  Google Scholar 

  314. A.V. Panfilov and J.P. Keener. Re-entry in three-dimensional Fitzhugh-Nagumo medium with rotational anisotropy. Physica D, 84(3):545–552, 1995.

    Google Scholar 

  315. G. Peponides, P. Kokotovic, and J. Chow. Singular perturbations and time scales in nonlinear models of power systems. IEEE Trans. Circ. Syst., 29(11):758–767, 1982.

    MATH  MathSciNet  Google Scholar 

  316. P. Pierobon, J. Miné-Hattab, G. Cappello, J.-L. Viovy, and M. Cosentino Lagomarsino. Separation of time scales in a one-dimensional directed nucleation-growth process. Phys. Rev. E, 82:061904, 2010.

    Google Scholar 

  317. L. Prandtl. Über Flüssigkeiten bei sehr kleiner Reibung. In Verh. III - International Math. Kongress, pages 484–491. Teubner, 1905.

    Google Scholar 

  318. V. Petrov, S.K. Scott, and K. Showalter. Mixed-mode oscillations in chemical systems. J. Chem. Phys., 97(9):6191–6198, 1992.

    Google Scholar 

  319. D.E. Postnov, O.V. Sosnovtseva, P. Scherbakov, and E. Mosekilde. Multimode dynamics in a network with resource mediated coupling. Chaos, 18, 2008.

    Google Scholar 

  320. D.A. Rakhlin. Enhanced diffusion in smoothly modulated superlattices. Phy. Rev. E, 63(1):011112, 2000.

    Google Scholar 

  321. M.J. Rempe, J. Best, and D. Terman. A mathematical model of the sleep/wake cycle. J. Math. Biol., 60:615–644, 2010.

    MATH  MathSciNet  Google Scholar 

  322. V. Rajagopalan, S. Chakraborty, and A. Ray. Estimation of slowly varying parameters in nonlinear systems via symbolic dynamic filtering. Signal Processing, 88:339–348, 2008.

    MATH  Google Scholar 

  323. J. Rankin, M. Desroches, B. Krauskopf, and M. Lowenberg. Canard cycles in aircraft ground dynamics. Nonlin. Dyn., 66(4):681–688, 2011.

    Google Scholar 

  324. A. García Cantú Ros, J.-S. Mc Ewen, and P. Gaspard. Effect of ultrafast diffusion on adsorption, desorption, and reaction processes over heterogeneous surfaces. Phys. Rev. E, 83:021604, 2011.

    Google Scholar 

  325. S. Reich. Multiple time-scales in classical and quantum-classical molecular dynamics. J. Comput. Phys., 151:49–73, 1999.

    MATH  MathSciNet  Google Scholar 

  326. S. Rinaldi and A. Gragnani. Destabilizing factors in slow–fast systems. Ecol. Model., 180:445–460, 2004.

    Google Scholar 

  327. S. Revzen and J.M. Guckenheimer. Finding the dimension of slow dynamics in a rhythmic system. J. R. Soc. Interface, 9:957–971, 2012.

    Google Scholar 

  328. J. Rinzel and G. Huguet. Nonlinear dynamics of neuronal excitability, oscillations, and coincidence detection. Comm. Pure Appl. Math., 66(9):1464–1494, 2013.

    MATH  MathSciNet  Google Scholar 

  329. M. Raghib, N.A. Hill, and U. Dieckmann. A multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics. J. Math. Biol., 62:605–653, 2011.

    MATH  MathSciNet  Google Scholar 

  330. S. Rinaldi. Laura and Petrarch: an intriguing case of cyclical love dynamics. SIAM J. Appl. Math., 58(4):1205–1221, 1994.

    MathSciNet  Google Scholar 

  331. S. Rinaldi. Synchrony in slow–fast metacommunities. Int. J. Bif. Chaos, 19(7):2447–2453, 2009.

    MATH  MathSciNet  Google Scholar 

  332. R.J. Field, E. Körös and R.M. Noyes. Oscillations in chemical systems II. Thorough analysis of temporal oscillations in the Ce −BrO3-malonic acid system. J. Am. Chem. Soc., 94:8649–8664, 1972.

    Google Scholar 

  333. J. Rubin, C.K.R.T. Jones, and M. Maxey. Settling and asymptotic motion of aerosol particles in a cellular flow field. J. Nonlinear Sci., 5:337–358, 1995.

    MATH  MathSciNet  Google Scholar 

  334. V. Rottschäfer and T.J. Kaper. Blowup in the nonlinear Schrödinger equation near critical dimension. J. Math. Anal. Appl., 268:517–549, 2002.

    MATH  MathSciNet  Google Scholar 

  335. V. Rottschäfer and T.J. Kaper. Geometric theory for multi-bump, self-similar, blowup solutions of the cubic nonlinear Schrödinger equation. Nonlinearity, 16:929–961, 2003.

    MATH  MathSciNet  Google Scholar 

  336. H.G. Rotstein and R. Kuske. Localized and asynchronous patterns via canards in coupled calcium oscillators. Physica D, 215:46–61, 2006.

    MATH  MathSciNet  Google Scholar 

  337. E. Reznik, T. Kaper, and D. Segré. The dynamics of hybrid metabolic-genetic oscillators. Chaos, 23(1):013132, 2013.

    Google Scholar 

  338. H.G. Rotstein, N. Kopell, A.M. Zhabotinsky, and I.R. Epstein. A canard mechanism for localization in systems of globally coupled oscillators. SIAM J. Appl. Math., 63(6):1998–2019, 2003.

    MATH  MathSciNet  Google Scholar 

  339. M.L. Rosenzweig and R.H. MacArthur. Graphical representation and stability conditions of predator–prey interactions. American Naturalist, 97:209–223, 1963.

    Google Scholar 

  340. S. Rinaldi and S. Muratori. Slow-fast limit cycles in predator–prey models. Ecol. Model., 61:287–308, 1992.

    Google Scholar 

  341. J.J. Rubin, J.E. Rubin, and G.B. Ermentrout. Analysis of synchronization in a slowly changing environment: how slow coupling becomes fast weak coupling. Phys. Rev. Lett., 110(20):204101, 2013.

    Google Scholar 

  342. P. Reddy and P. Sannuti. Optimal control of a coupled-core nuclear reactor by a singular perturbation method. IEEE Trans- Aut. Contr., 20(6):766–769, 1975.

    Google Scholar 

  343. S. Rinaldi and M. Scheffer. Geometric analysis of ecological models with slow and fast processes. Ecosystems, 3:507–521, 2000.

    Google Scholar 

  344. J.E. Rubin and D. Terman. Analysis of clustered firing patterns in synaptically coupled networks of oscillators. J. Math. Biol., 41:6, 2000.

    MathSciNet  Google Scholar 

  345. J.E. Rubin and D. Terman. Geometric singular perturbation analysis of neuronal dynamics. In B. Fiedler, editor, Handbook of Dynamical Systems 2, pages 93–146. Elsevier, 2002.

    Google Scholar 

  346. J.E. Rubin and D. Terman. Synchronized activity and loss of synchrony among heterogeneous conditional oscillators. SIAM J. Appl. Dyn. Syst., 1:1, 2002.

    MathSciNet  Google Scholar 

  347. M.I. Rabinovich, P. Varona, A.I. Selverston, and H.D. Abarbanel. Dynamical principles in neuroscience. Rev. Mod. Phys., 78(4):1213–1265, 2006.

    Google Scholar 

  348. J. Rubin and M. Wechselberger. Giant squid - hidden canard: the 3D geometry of the Hodgin-Huxley model. Biological Cybernetics, 97(1), 2007.

    Google Scholar 

  349. H.G. Rotstein and H. Wu. Swing, release, and escape mechanisms contribute to the generation of phase-locked cluster patterns in a globally coupled FitzHugh–Nagumo model. Phys. Rev. E, 86:066207, 2012.

    Google Scholar 

  350. V.R. Saksena and T. Basar. A multimodel approach to stochastic team problems. Automatica, 18(6):713–720, 1982.

    MATH  MathSciNet  Google Scholar 

  351. C. Schütte and F.A. Bornemann. On the singular limit of the quantum-classical molecular dynamics model. SIAM J. Appl. Math., 59(4):1208–1224, 1999.

    MATH  MathSciNet  Google Scholar 

  352. R. Suckley and V. Biktashev. Comparison of asymptotics of heart and nerve excitability. Phys. Rev. E, 68:011902, 2003.

    MathSciNet  Google Scholar 

  353. R.D. Simitev and V.N. Biktashev. Asymptotics of conduction velocity restitution in models of electrical excitation in the heart. Bull. Math. Biol., 73(1):72–115, 2011.

    MATH  MathSciNet  Google Scholar 

  354. V.R. Saksena and J.B. Cruz. Nash strategies in decentralized control of multiparameter singularly perturbed large scale systems. Large Scale Syst., 2:219–234, 1981.

    MATH  MathSciNet  Google Scholar 

  355. V.R. Saksena and J.B. Cruz. A multimodel approach to stochastic Nash games. Automatica, 18(3): 295–305, 1982.

    MATH  MathSciNet  Google Scholar 

  356. M.A. Salman and J.B. Cruz. Team-optimal closed-loop Stackelberg strategies for systems with slow and fast modes. Int. J. Contr., 37(6):1401–1416, 1983.

    MATH  MathSciNet  Google Scholar 

  357. V.R. Saksena and J.B. Cruz. Robust Nash strategies for a class of non-linear singularly perturbed problems. Int. J. Contr., 39(2):293–310, 1984.

    MATH  MathSciNet  Google Scholar 

  358. B. Song, C. Castillo-Chavez, and J.P. Aparicio. Tuberculosis models with fast and slow dynamics: the role of close and casual contacts. Math. Biosci., 180(1):187–205, 2002.

    MATH  MathSciNet  Google Scholar 

  359. C. Schmeiser. Finite deformations of thin beams, asymptotic analysis by singular perturbation methods. IMA J. Appl. Math., 34(2):155–164, 1985.

    MATH  MathSciNet  Google Scholar 

  360. C. Schmeiser. On strongly reverse biased semiconductor diodes. SIAM J. Appl. Math., 49(6):1734–1748, 1989.

    MATH  MathSciNet  Google Scholar 

  361. B. Sicardy and V. Dubois. Co-orbital motion with slowly varying parameters. Celestial Mech. Dynam. Astronom., 86(4):321–350, 2003.

    MATH  MathSciNet  Google Scholar 

  362. C. Sueur and G. Dauphin-Tanguy. Bond graph approach to multi-time scale systems analysis. J. Frank. Inst., 328(5):1005–1026, 1991.

    MATH  MathSciNet  Google Scholar 

  363. S.S. Sazhin, G. Feng, M.R. Heikal, I. Goldfarb, V. Sol’dshtein, and G. Kuzmenko. Thermal ignition analysis of a monodisperse spray with radiation. Combustion and Flame, 124(4):684–701, 2001.

    Google Scholar 

  364. H. Schlichting and K. Gersten. Boundary-Layer Theory. Springer, 2000.

    Google Scholar 

  365. L. Sacerdote and M.T. Giraudo. Stochastic integrate and fire models: a review on mathematical methods and their applications. In Stochastic Biomathematical Models, pages 99–148. Springer, 2013.

    Google Scholar 

  366. A. Surana and G. Haller. Ghost manifolds in slow–fast systems, with applications to unsteady fluid flow separation. Physica D, 237(10):1507–1529, 2008.

    MATH  MathSciNet  Google Scholar 

  367. W.E. Sherwood. Phase response in networks of bursting neurons: modeling central pattern generators. PhD thesis, Cornell University, Ithaca, USA, 2008.

    Google Scholar 

  368. C. Soria-Hoyo, F. Pontiga, and A. Castellanos. A PIC based procedure for the integration of multiple time scale problems in gas discharge physics. J. Comput. Phys., 228(4):1017–1029, 2009.

    MATH  MathSciNet  Google Scholar 

  369. Y. Shen, Z. Hou, and H. Xin. Transition to burst synchronization in coupled neuron networks. Phys. Rev. E, 77:031920, 2008.

    Google Scholar 

  370. W.T. Silfvast. Laser Fundamentals. CUP, 2004.

    Google Scholar 

  371. M.W. Spong, K. Khorasani, and P.V. Kokotovic. An integral manifold approach to the feedback control of flexible joint robots. IEEE J. Robot. Autom., 3(4):291–300, 1987.

    Google Scholar 

  372. I. Sainz, A.B. Klimov, and L. Roa. Quantum phase transitions in an effective Hamiltonian: fast and slow systems. J. Phys. A, 41:355301, 2008.

    MathSciNet  Google Scholar 

  373. D. Soudry and R. Meir. The neuron’s response at extended timescales. arXiv:1301.2631, pages 1–5, 2013.

    Google Scholar 

  374. M. Souza. Multiscale analysis for a vector-borne epidemic model. J. Math. Biol., pages 1–14, 2013. accepted, to appear.

    Google Scholar 

  375. H. Spohn. The critical manifold of the Lorentz-Dirac equation. Europhys. Lett., 50(3):287, 2000.

    Google Scholar 

  376. L.B. Shaw and I.B. Schwartz. Fluctuating epidemics on adaptive networks. Phys. Rev. E, 77:(066101), 2008.

    Google Scholar 

  377. R.I. Saye and J.A. Sethian. Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams. Science, 340:720–724, 2013.

    MathSciNet  Google Scholar 

  378. I. Surovtsova, N. Simus, T. Lorenz, A. König, S. Sahle, and U. Kummer. Accessible methods for the dynamic time-scale decomposition of biochemical systems. Bioinformatics, 25(21):2816–2823, 2009.

    Google Scholar 

  379. H. Stommel. Trajectories of small bodies sinking slowly through convection cells. J. Mar. Res., 8: 24–29, 1949.

    Google Scholar 

  380. S.H. Strogatz. Exploring complex networks. Nature, 410:268–276, 2001.

    Google Scholar 

  381. N.N. Subbotina. Asymptotic properties of minimax solutions of Isaacs-Bellman equations in differential games with fast and slow motions. J. Appl. Math. Mech., 60(6):883–890, 1996.

    MATH  MathSciNet  Google Scholar 

  382. D. Terman, S. Ahn, X. Wang, and W. Just. Reducing neuronal networks to discrete dynamics. Physica D, 237(3):324–338, 2008.

    MATH  MathSciNet  Google Scholar 

  383. J.J. Tyson, K.C. Chen, and B. Novak. Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. Current Opinion in Cell Biology, 15:221–231, 2003.

    Google Scholar 

  384. M. Thomson and J. Gunawardena. Unlimited multistability in multisite phosphorylation systems. Nature, 460:274–277, 2009.

    Google Scholar 

  385. W.C. Troy. The bifurcation of periodic solutions in the Hodgkin–Huxley equations. Quart. Appl. Math., 36:73–83, 1978.

    MATH  MathSciNet  Google Scholar 

  386. H.C. Tseng and D.D. Siljak. A learning scheme for dynamic neural networks: equilibrium manifold and connective stability. Neural Networks, 8(6):853–864, 1995.

    Google Scholar 

  387. J.-C. Tsai. Do calcium buffers always slow down the propagation of calcium waves? J. Math. Biol., pages 1–46, 2012. to appear.

    Google Scholar 

  388. Y. Tang, J.L. Stephenson, and H.G. Othmer. Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics. Biophys. J., 70:246–263, 1996.

    Google Scholar 

  389. J.M. Tuwankotta. Widely separated frequencies in coupled oscillators with energy-preserving quadratic nonlinearity. Physica D, 182(1):125–149, 2003.

    MATH  MathSciNet  Google Scholar 

  390. D. Terman and D.L. Wang. Global competition and local cooperation in a network of neural oscillators. Physica D, 81:148–176, 1995.

    MATH  MathSciNet  Google Scholar 

  391. N. Ulanovsky, L. Las, D. Farkas, and I. Nelken. Multiple time scales of adaptation in auditory cortex neurons. J. Neurosci., 24(46):10440–10453, 2004.

    Google Scholar 

  392. F. Verhulst and T. Bakri. The dynamics of slow manifolds. J. Indones. Math. Soc., 13:73–90, 2007.

    MATH  MathSciNet  Google Scholar 

  393. M. van Dyke. Perturbation Methods in Fluid Mechanics. Academic Press, 1964.

    Google Scholar 

  394. H. van der Ploeg and A. Doelman. Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction–diffusion equations. Indiana Univ. Math. J., 54(5):1219–1301, 2005.

    MATH  MathSciNet  Google Scholar 

  395. G. van der Sande, J. Danckaert, I. Weretennicoff, and T. Erneux. Rate equations for vertical-cavity surface-emitting lasers. Phys. Rev. A, 67(1):013809, 2003.

    Google Scholar 

  396. M.J. Ward. Metastable bubble solutions for the Allen–Cahn equation with mass conservation. SIAM J. Appl. Math., 56(5):1247–1279, 1996.

    MATH  MathSciNet  Google Scholar 

  397. H.Y. Wu and S.M. Baer. Analysis of an excitable dendritic spine with an activity-dependent stem conductance. J. Math. Biol., 36(6):569–592, 1998.

    MATH  Google Scholar 

  398. J.A. White, M.I. Banks, R.A. Pearce, and N.J. Kopell. Networks of interneurons with fast and slow γ-aminobutyric acid type A (GABA A ) kinetics provide substrate for mixed gamma-theta rhythm. Proc. Natl. Acad. Sci. USA, 97(14):8128–8133, 2000.

    Google Scholar 

  399. J.R. Winkelman, J.H. Chow, J.J. Allemong, and P.V. Kokotovic. Multi-time-scale analysis of a power system. Automatica, 16:35–43, 1980.

    MATH  MathSciNet  Google Scholar 

  400. J.R. Winkelman, J.H. Chow, B.C. Bowler, B. Avramovic, and P.V. Kokotovic. An analysis of interarea dynamics of multi-machine systems. IEEE Trans. Power Appar. Syst., 2:754–763, 1981.

    Google Scholar 

  401. S. Wieczorek. Stochastic bifurcation in noise-driven lasers and Hopf oscillators. Phys. Rev. E, 79(3):036209, 2009.

    Google Scholar 

  402. J.P. Wilber. Invariant manifolds describing the dynamics of a hyperbolic–parabolic equation from nonlinear viscoelasticity. Dynamical Systems, 21(4):465–489, 2006.

    MATH  MathSciNet  Google Scholar 

  403. M.W. Walser and C.H. Keitel. Geometric and algebraic approach to classical dynamics of a particle with spin. Lett. Math. Phys., 55(1):63–70, 2001.

    MATH  MathSciNet  Google Scholar 

  404. Y.-F. Wang, M. Khan, and H.A. van den Berg. Interaction of fast and slow dynamics in endocrine control systems with an application to β-cell dynamics. Math. Biosci., 235(1):8–18, 2012.

    MATH  MathSciNet  Google Scholar 

  405. B.P. Wood and J.R. Miller. Linked selected and neutral loci in heterogeneous environments. J. Math. Biol., 53(6):939–975, 2006.

    MATH  MathSciNet  Google Scholar 

  406. M.J. Ward, D. McInerney, P. Houston, D. Gavaghan, and P. Maini. The dynamics and pinning of a spike for a reaction–diffusion system. SIAM J. Appl. Math., 62(4):1297–1328, 2002.

    MATH  MathSciNet  Google Scholar 

  407. D. Wirosoetisno and T.G. Shepherd. Averaging, slaving and balance dynamics in a simple atmospheric model. Physica D, 141:37–53, 2000.

    MATH  MathSciNet  Google Scholar 

  408. M.J. Ward and J. Wei. Hopf bifurcation and oscillatory instabilities of spike solutions for the one-dimensional Gierer–Meinhardt model. J. Nonlinear Sci., 13(2):209–264, 2003.

    MATH  MathSciNet  Google Scholar 

  409. M. Yamada. A theoretical analysis of self-sustained pulsation phenomena in narrow-stripe semiconductor lasers. IEEE J. Quant. Electr., 29:1330–1336, 1993.

    Google Scholar 

  410. N. Yu, Y.X. Li, and R. Kuske. A computational study of spike time reliability in two cases of threshold dynamics. J. Math. Neurosci., 3:11, 2013.

    MathSciNet  Google Scholar 

  411. P. Yanguas, J.F. Palacián, J.F. Meyer, and H.S. Dumas. Periodic solutions in Hamiltonian systems, averaging, and the lunar problem. SIAM J. Appl. Dyn. Syst., 7(2):311–340, 2008.

    MATH  MathSciNet  Google Scholar 

  412. P. Yordanov, S. Tyanova, M.-T. Hütt, and A. Lesne. Asymmetric transition and time-scale separation in interlinked positive feedback loops. Int. J. Bif. Chaos, 21(7):1895–1905, 2011.

    MATH  Google Scholar 

  413. A. Zagaris and A. Doelman. Emergence of steady and oscillatory localized structures in a phytoplankton-nutrient model. Nonlinearity, 24(12):3437–3486, 2011.

    MATH  MathSciNet  Google Scholar 

  414. A. Zagaris, A. Doelman, N.N. Pham Thi, and B.P. Sommeijer. Blooming in a non-local, coupled phytoplankton-nutrient model. SIAM J. Appl. Math., 69(4):1174–1204, 2009.

    MATH  MathSciNet  Google Scholar 

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Kuehn, C. (2015). Applications. In: Multiple Time Scale Dynamics. Applied Mathematical Sciences, vol 191. Springer, Cham. https://doi.org/10.1007/978-3-319-12316-5_20

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