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

Abstract

This chapter collects various topics that did not fit immediately within the main flow of the book. Nevertheless, they have been included here due to their general importance and interaction with fast–slow systems.

Bibliography

  1. [ACVV13]
    M. Avendano-Camacho, J.A. Vallejo, and Yu. Vorobiev. Higher order corrections to adiabatic invariants of generalized slow–fast Hamiltonian systems. arXiv:1305.3974v1, pages 1–22, 2013.Google Scholar
  2. [AG13]
    G.G. Avalos and N.B. Gallegos. Quasi-steady state model determination for systems with singular perturbations modelled by bond graphs. Math. Computer Mod. Dyn. Syst., pages 1–21, 2013. to appear.Google Scholar
  3. [AKN06]
    V.I. Arnold, V.V. Kozlov, and A.I. Neishstadt. Mathematical Aspects of Classical and Celestial Mechanics. Springer, 3rd edition, 2006.Google Scholar
  4. [AMN+03]
    R.B. Alley, J. Marotzke, W.D. Nordhaus, J.T. Overpeck, D.M. Peteet, R.A. Pielke Jr., R.T. Pierrehumbert, P.B. Rhines, T.F. Stocker, L.D. Talley, and J.M. Wallace. Abrupt climate change. Science, 299:2005–2010, 2003.Google Scholar
  5. [And92]
    J.L. Anderson. Multiple time scale methods for adiabatic systems. Amer. J. Phys., 60:923–927, 1992.zbMATHMathSciNetGoogle Scholar
  6. [ANZ11]
    A.V. Artemyev, A.I. Neishtadt, and L.M. Zelenyi. Jumps of adiabatic invariant at the separatrix of a degenerate saddle point. Chaos, 21:043120, 2011.Google Scholar
  7. [ANZV10]
    A.V. Artemyev, A.I. Neishtadt, L.M. Zelenyi, and D.L. Vainchtein. Adiabatic description of capture into resonance and surfatron acceleration of charged particles by electromagnetic waves. Chaos, 20:043128, 2010.MathSciNetGoogle Scholar
  8. [AP98a]
    P. Auger and J.-C. Poggiale. Aggregation and emergence in systems of ordinary differential equations. Math. Comput. Model., 27(4):1–21, 1998.zbMATHMathSciNetGoogle Scholar
  9. [Arn97]
    V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer, 1997.Google Scholar
  10. [Art04]
    Z. Artstein. Distributional convergence in planar dynamics and singular perturbations. J. Differential Equations, 201:250–286, 2004.zbMATHMathSciNetGoogle Scholar
  11. [AWVC12]
    P. Ashwin, S. Wieczorek, R. Vitolo, and P. Cox. Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system. Phil. Trans. R. Soc. A, 370:1166–1184, 2012.Google Scholar
  12. [BC94a]
    D. Bainov and V. Covachev. Impulsive differential equations with a small parameter. World Scientific, 1994.Google Scholar
  13. [BCP87]
    K.E. Brenan, S.L. Campbell, and L.R. Petzhold. Numerical solution of initial-value problems in differential-algebraic equations. SIAM, 1987.Google Scholar
  14. [BdCdS10]
    C.A. Buzzi, T. de Carvalho, and P.R. da Silva. Canard cycles and Poincaré index of non-smooth vector fields on the plane. arXiv:1002:4169v2, pages 1–20, 2010.Google Scholar
  15. [BdST05]
    C.A. Buzzi, P.R. da Silva, and M.A. Teixeira. Singular perturbation problems for time-reversible systems. Proc. Amer. Math. Soc., 133(11):3323–3331, 2005.zbMATHMathSciNetGoogle Scholar
  16. [BdST06]
    C.A. Buzzi, P.R. da Silva, and M.A. Teixeira. A singular approach to discontinuous vector fields on the plane. J. Diff. Eq., 231:633–655, 2006.zbMATHGoogle Scholar
  17. [BdST12]
    C.A. Buzzi, P.R. da Silva, and M.A. Teixeira. Slow-fast systems on algebraic varieties bordering piecewise-smooth dynamical systems. Bull. Sci. Math., 136(4):444–462, 2012.zbMATHMathSciNetGoogle Scholar
  18. [Ber84]
    M.V. Berry. Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. A, 392(1802):45–57, 1984.zbMATHGoogle Scholar
  19. [Ber85]
    M.V. Berry. Classical adiabatic angles and quantal adiabatic phase. J. Phys. A, 18(1):15, 1985.Google Scholar
  20. [Ber87]
    M.V. Berry. Quantum phase corrections from adiabatic iteration. Proc. R. Soc. A, 414(1846):31–46, 1987.Google Scholar
  21. [BF13]
    F. Battelli and M. Fečkan. Fast-slow dynamical approximation of forced impact systems near periodic solutions. Bound. Value Probl., 2013:71, 2013.Google Scholar
  22. [BG02d]
    N. Berglund and B. Gentz. Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations. Stoch. Dyn., 2:327–356, 2002.zbMATHMathSciNetGoogle Scholar
  23. [BG08b]
    N. Brännström and V. Gelfreich. Drift of slow variables in slow–fast Hamiltonian systems. Physica D, 237:2913–2921, 2008.zbMATHMathSciNetGoogle Scholar
  24. [BGS97]
    F. Broner, G.H. Goldsztein, and S.H. Strogatz. Dynamical hysteresis without static hysteresis: scaling laws and asymptotic expansions. SIAM J. Appl. Math., 57(4):1163–1187, 1997.zbMATHMathSciNetGoogle Scholar
  25. [Bil10]
    A. Billoire. Distribution of timescales in the Sherrington–Kirkpatrick model. J. Stat. Mech., 11:P11034, 2010.Google Scholar
  26. [BK91]
    D.L. Bosley and J. Kevorkian. Sustained resonance in very slowly varying oscillatory Hamiltonian systems. SIAM J. Appl. Math., 51(2):439–471, 1991.zbMATHMathSciNetGoogle Scholar
  27. [BK92]
    D.L. Bosley and J. Kevorkian. Adiabatic invariance and transient resonance in very slowly varying oscillatory Hamiltonian systems. SIAM J. Appl. Math., 52(2):494–527, 1992.zbMATHMathSciNetGoogle Scholar
  28. [Bla91]
    F. Blais. Asymptotic expansions of rivers. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 181–189. Springer, 1991.Google Scholar
  29. [BLH+09]
    J. Bakke, Ø. Lie, E. Heegaard, T. Dokken, G.H. Haug, H.H. Birks, P. Dulski, and T. Nilsen. Rapid oceanic and atmospheric changes during the Younger Dryas cold period. Nature Geosci., 2:202–205, 2009.Google Scholar
  30. [BM72]
    M.V. Berry and K.E. Mount. Semiclassical approximations in wave mechanics. Rep. Prog. Phys., 35(1):315, 1972.Google Scholar
  31. [BMN10]
    M. Bobieński, P. Mardesić, and D. Novikov. Pseudo-abelian integrals on slow–fast Darboux systems. arXiv:1007.2001v1, pages 1–11, 2010.Google Scholar
  32. [Bor04a]
    A.V. Borovskikh. Investigation of relaxation oscillations by constructive nonstandard analysis. I. Differ. Equ., 40(3):309–317, 2004.Google Scholar
  33. [Bor04b]
    A.V. Borovskikh. Investigation of relaxation oscillations by constructive nonstandard analysis. II. Differ. Equ., 40(4):491–501, 2004.Google Scholar
  34. [BS96]
    M. Brokate and J. Sprekels. Hysteresis and Phase Transitions. Springer, 1996.Google Scholar
  35. [BS98]
    F.A. Bornemann and C. Schütte. A mathematical investigation of the Car–Parrinello method. Numer. Math., 78(3):359–376, 1998.zbMATHMathSciNetGoogle Scholar
  36. [BSG10]
    N. Brännström, E. De Simone, and V. Gelfreich. Geometric shadowing in slow–fast Hamiltonian systems. Nonlinearity, 23:1169, 2010.zbMATHMathSciNetGoogle Scholar
  37. [BYB10]
    M. Benbachir, K. Yadi, and R. Bebbouchi. Slow and fast systems with Hamiltonian reduced problems. Electr. J. Diff. Eq., 2010(6):1–19, 2010.MathSciNetGoogle Scholar
  38. [Cal93]
    J.-L. Callot. Champs lents-rapides complexes à une dimension lente. Ann. Sci. École Norm. Sup., 26(2):149–173, 1993.zbMATHMathSciNetGoogle Scholar
  39. [Cam77]
    S.L. Campbell. Linear systems of differential equations with singular coefficients. SIAM J. Math. Anal., 8(6):1057–1066, 1977.zbMATHMathSciNetGoogle Scholar
  40. [Cam80b]
    S.L. Campbell. Singular Systems of Differential Equations I. Pitman, 1980.Google Scholar
  41. [CB06]
    S.R. Carpenter and W.A. Brock. Rising variance: a leading indicator of ecological transition. Ecology Letters, 9:311–318, 2006.Google Scholar
  42. [CBT+11]
    M. Chertkov, S. Backhaus, K. Turzisyn, V. Chernyak, and V. Lebedev. Voltage collapse and ODE approach to power flows: analysis of a feeder line with static disorder in consumption/production. arXiv:1106.5003v1, pages 1–8, 2011.Google Scholar
  43. [CCB+01]
    J.S. Clark, S.R. Carpenter, M. Barber, S. Collins, A. Dobson, J.A. Foley, D.M. Lodge, M. Pascual, R. Pielke Jr., W. Pizer, C. Pringle, W.V. Reid, K. A. Rose, O. Sala, W.H. Schlesinger, D.H. Wall, and D. Wear. Ecological forecasts: an emerging imperative. Science, 293:657–660, 2001.Google Scholar
  44. [CCP+11]
    S.R. Carpenter, J.J. Cole, M.L. Pace, R. Batt, W.A. Brock, T. Cline, J. Coloso, J.R. Hodgson, J.F. Kitchell, D.A. Seekell, L. Smith, and B. Weidel. Early warning signs of regime shifts: a whole-ecosystem experiment. Science, 332:1079–1082, 2011.Google Scholar
  45. [Ces94]
    P. Cessi. A simple box model of stochastically forced thermohaline circulation. J. Phys. Oceanogr., 24:1911–1920, 1994.Google Scholar
  46. [CET86]
    J.R. Cary, D.F. Escande, and J.L. Tennyson. Adiabatic-invariant change due to separatrix crossing. Phys. Rev. A, 34(5):4256–4275, 1986.Google Scholar
  47. [CG95a]
    S.L. Campbell and C.W. Gear. The index of general nonlinear DAEs. Numer. Math., 72:173–196, 1995.zbMATHMathSciNetGoogle Scholar
  48. [CG95b]
    S.L. Campbell and E. Griepentrog. Solvability of general differential algebraic equations. SIAM J. Sci. Comput., 16(2):257–270, 1995.zbMATHMathSciNetGoogle Scholar
  49. [Chu92]
    C.K. Chui. An Introduction to Wavelets. Academic Press, 1992.Google Scholar
  50. [CJR76]
    S.L. Campbell, C.D. Meyer Jr, and N.J. Rose. Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math., 31(3):411–425, 1976.zbMATHMathSciNetGoogle Scholar
  51. [CKK+07]
    A.R. Champneys, V. Kirk, E. Knobloch, B.E. Oldeman, and J. Sneyd. When Shil’nikov meets Hopf in excitable systems. SIAM J. Appl. Dyn. Syst., 6(4):663–693, 2007.zbMATHMathSciNetGoogle Scholar
  52. [CL91]
    S.L. Campbell and B. Leimkuhler. Differentiation of constraints in differential-algebraic equations. J. Struct. Mech., 19:19–39, 1991.MathSciNetGoogle Scholar
  53. [CM95]
    S.L. Campbell and E. Moore. Constraint preserving integrators for general nonlinear higher index DAEs. Numer. Math., 69(4):383–399, 1995.zbMATHMathSciNetGoogle Scholar
  54. [CP83]
    S.L. Campbell and L.R. Petzhold. Canonical forms and solvable singular systems of differential equations. SIAM J. Algebraic Discr. Meth., 4(4):517–521, 1983.zbMATHGoogle Scholar
  55. [CSHD12]
    E. Cotilla-Sanchez, P. Hines, and C.M. Danforth. Predicting critical transitions from time series synchrophasor data. IEEE Trans. Smart Grid, 3(4):1832–1840, 2012.Google Scholar
  56. [Dau92]
    I. Daubechies. Ten Lectures on Wavelets. SIAM, 1992.Google Scholar
  57. [dBBCK08]
    M. di Bernardo, C.J. Budd, A.R. Champneys, and P. Kowalczyk. Piecewise-smooth Dynamical Systems, volume 163 of Applied Mathematical Sciences. Springer, 2008.Google Scholar
  58. [DD91]
    F. Diener and M. Diener. Maximal delay. In E. Benoît, editor, Dynamic Bifurcations, volume 1493 of Lecture Notes in Mathematics, pages 71–86. Springer, 1991.Google Scholar
  59. [DD95]
    F. Diener and M. Diener. Nonstandard Analysis in Practice. Springer, 1995.Google Scholar
  60. [DFD+10]
    R. Donangelo, H. Fort, V. Dakos, M. Scheffer, and E.H. Van Nes. Early warnings for catastrophic shifts in ecosystems: comparison between spatial and temporal indicators. Int. J. Bif. Chaos, 20(2):315–321, 2010.Google Scholar
  61. [DG10]
    J.M. Drake and B.D. Griffen. Early warning signals of extinction in deteriorating environments. Nature, 467:456–459, 2010.Google Scholar
  62. [DH11]
    A. Delshams and G. Huguet. A geometric mechanism of diffusion: rigorous verification in a priori unstable Hamiltonian systems. J. Differential Equat., 250(5):2601–2623, 2011.zbMATHMathSciNetGoogle Scholar
  63. [Die84]
    M. Diener. The canard unchained or how fast/slow dynamical systems bifurcate. The Mathematical Intelligencer, 6:38–48, 1984.zbMATHMathSciNetGoogle Scholar
  64. [DJ10]
    P.D. Ditlevsen and S.J. Johnsen. Tipping points: early warning and wishful thinking. Geophys. Res. Lett., 37:19703, 2010.Google Scholar
  65. [DJ11a]
    M. Desroches and M. Jeffrey. Canards and curvature: nonsmooth approximation by pinching. Nonlinearity, 24:1655, 2011.zbMATHMathSciNetGoogle Scholar
  66. [DKR+11]
    V. Dakos, S. Kéfi, M. Rietkerk, E.H. van Nes, and M. Scheffer. Slowing down in spatially patterned systems at the brink of collapse. Am. Nat., 177(6):153–166, 2011.Google Scholar
  67. [DL11]
    D. Dolgopyat and C. Liverani. Energy transfer in fast–slow Hamiltonian systems. Comm. Math. Phys., 308:201–225, 2011.zbMATHMathSciNetGoogle Scholar
  68. [DSvN+08]
    V. Dakos, M. Scheffer, E.H. van Nes, V. Brovkin, V. Petoukhov, and H. Held. Slowing down as an early warning signal for abrupt climate change. Proc. Natl. Acad. Sci. USA, 105(38):14308–14312, 2008.Google Scholar
  69. [DvND+09]
    V. Dakos, E.H. van Nes, R. Donangelo, H. Fort, and M. Scheffer. Spatial correlation as leading indicator of catastrophic shifts. Theor. Ecol., 3(3):163–174, 2009.Google Scholar
  70. [EM07]
    W. E and P. Ming. Cauchy-Born rule and the stability of crystalline solids: static problems. Arch. Rat. Mech. Anal., 183(2):241–297, 2007.Google Scholar
  71. [Fri02b]
    E. Fridman. Singularly perturbed analysis of chattering in relay control systems. IEEE Trans. Aut. Contr., 47(12):2079–2084, 2002.MathSciNetGoogle Scholar
  72. [FS80]
    N. Farber and J. Shinar. Approximate solution of singularly perturbed nonlinear pursuit-evasion games. J. Optim. Theor. Appl., 32:39–73, 1980.zbMATHMathSciNetGoogle Scholar
  73. [Gar59]
    C.S. Gardner. Adiabatic invariants of periodic classical systems. Phys. Rev., 2(115):791–794, 1959.Google Scholar
  74. [Ger12]
    M. Gerdts. Optimal control of ODEs and DAEs. de Gruyter, 2012.Google Scholar
  75. [GFZ11]
    I. Gholami, A. Fiege, and A. Zippelius. Slow dynamics and precursors of the glass transition in granular fluids. Phys. Rev. E, 84:031305, 2011.Google Scholar
  76. [GGM00]
    G. Gallavotti, G. Gentile, and V. Mastropietro. Hamilton-Jacobi equation, heteroclinic chains and Arnol’d diffusion in three time scale systems. Nonlinearity, 13(2):323, 2000.Google Scholar
  77. [GJ08]
    V. Guttal and C. Jayaprakash. Changing skewness: an early warning signal of regime shifts in ecosystems. Ecology Letters, 11:450–460, 2008.Google Scholar
  78. [GJ09]
    V. Guttal and C. Jayaprakash. Spatial variance and spatial skewness: leading indicators of regime shifts in spatial ecological systems. Theor. Ecol., 2:3–12, 2009.Google Scholar
  79. [GK09a]
    J. Guckenheimer and C. Kuehn. Computing slow manifolds of saddle-type. SIAM J. Appl. Dyn. Syst., 8(3):854–879, 2009.zbMATHMathSciNetGoogle Scholar
  80. [GK09b]
    J. Guckenheimer and C. Kuehn. Homoclinic orbits of the FitzHugh–Nagumo equation: The singular limit. Discr. Cont. Dyn. Syst. S, 2(4):851–872, 2009.zbMATHMathSciNetGoogle Scholar
  81. [GK10b]
    J. Guckenheimer and C. Kuehn. Homoclinic orbits of the FitzHugh–Nagumo equation: Bifurcations in the full system. SIAM J. Appl. Dyn. Syst., 9:138–153, 2010.zbMATHMathSciNetGoogle Scholar
  82. [GL02]
    V. Gelfreich and L. Lerman. Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system. Nonlinearity, 15(2):447–457, 2002.zbMATHMathSciNetGoogle Scholar
  83. [GL03]
    V. Gelfreich and L. Lerman. Long-periodic orbits and invariant tori in a singularly perturbed Hamiltonian system. Physica D, 176(3):125–146, 2003.zbMATHMathSciNetGoogle Scholar
  84. [GM08]
    B. Gershgorin and A. Majda. A nonlinear test model for filtering slow–fast systems. Commun. Math. Sci., 6(3):611–649, 2008.zbMATHMathSciNetGoogle Scholar
  85. [GM10]
    B. Gershgorin and A. Majda. Filtering a nonlinear slow–fast system with strong fast forcing. Commun. Math. Sci., 8(1):67–92, 2010.zbMATHMathSciNetGoogle Scholar
  86. [Gol98]
    R. Goldblatt. Lectures on the Hyperreals. Springer, 1998.Google Scholar
  87. [GP84]
    C.W. Gear and L.R. Petzhold. ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal., 21(4):716–728, 1984.zbMATHMathSciNetGoogle Scholar
  88. [GRKT12]
    V. Gelfreich, V. Rom-Kedar, and D. Turaev. Fermi acceleration and adiabatic invariants for non-autonomous billiards. Chaos, 22(3):033116, 2012.Google Scholar
  89. [GS12b]
    M. Guardi and T.M. Seara. Exponentially and non-exponentially small splitting of separatrices for the pendulum with a fast meromorphic perturbation. Nonlinearity, 25:1367–1412, 2012.MathSciNetGoogle Scholar
  90. [GSW09]
    D. Givon, P. Stinis, and J. Weare. Variance reduction for particle filters of systems with time scale separation. IEEE Trans. Signal Proc., 57(2):424–435, 2009.MathSciNetGoogle Scholar
  91. [Hal95]
    G. Haller. Diffusion at intersecting resonances in Hamiltonian systems. Phys. Lett. A, 200(1):34–42, 1995.zbMATHMathSciNetGoogle Scholar
  92. [Han95]
    M. Hanke. Regularizations of differential-algebraic equations revisited. Math. Nachr., 174:159–183, 1995.zbMATHMathSciNetGoogle Scholar
  93. [Hen82]
    J. Henrard. Capture into resonance: an extension of the use of adiabatic invariants. Celest. Mech., 27(1):3–22, 1982.zbMATHMathSciNetGoogle Scholar
  94. [Hen93]
    J. Henrard. The adiabatic invariant in classical mechanics. In Dynamics Reported Vol. 2, pages 117–235. Springer, 1993.Google Scholar
  95. [HHvNS11]
    M. Hirota, M. Holmgren, E.H. van Nes, and M. Scheffer. Global resilience of tropical forest and savanna to critical transitions. Science, 334:232–235, 2011.Google Scholar
  96. [HW10]
    A. Hastings and D.B. Wysham. Regime shifts in ecological systems can occur with no warning. Ecol. Lett., 13:464–472, 2010.Google Scholar
  97. [ILNV02]
    A.P. Itin, R. De La Llave, A.I. Neishtadt, and A.A. Vasiliev. Transport in a slowly perturbed convective cell flow. Chaos, 12(4):1043–1053, 2002.zbMATHMathSciNetGoogle Scholar
  98. [IN12]
    A.P. Itin and A.I. Neishtadt. Fermi acceleration in time-dependent rectangular billiards due to multiple passages through resonances. Chaos, 22(2):026119, 2012.Google Scholar
  99. [IVKS07]
    A.P. Itin, A.A. Vasiliev, G. Krishna, and S.Watanabe. Change in the adiabatic invariant in a nonlinear two-mode model of Feshbach resonance passage. Physica D, 232:108–115, 2007.zbMATHGoogle Scholar
  100. [JCdBS10]
    M.R. Jeffrey, A.R. Champneys, M. di Bernardo, and S.W. Shaw. Catastrophic sliding bifurcations and onset of oscillations in a superconducting resonator. Phys. Rev. E, 81:016213, 2010.Google Scholar
  101. [Jon95]
    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
  102. [KG11]
    P. Kowalczyk and P. Glendinning. Boundary-equilibrium bifurcations in piecewise-smooth slow–fast systems. Chaos, 21: 023126, 2011.MathSciNetGoogle Scholar
  103. [KG13]
    C. Kuehn and T. Gross. Nonlocal generalized models of predator–prey systems. Discr. Cont. Dyn. Syst. B, 18(3): 693–720, 2013.zbMATHMathSciNetGoogle Scholar
  104. [KM94]
    P. Kunkel and V. Mehrmann. Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math., 56(3):225–251, 1994.zbMATHMathSciNetGoogle Scholar
  105. [KM98]
    P. Kunkel and V. Mehrmann. Regular solutions of nonlinear differential-algebraic equations and their numerical determination. Numer. Math., 79(4):581–600, 1998.zbMATHMathSciNetGoogle Scholar
  106. [KM06b]
    P. Kunkel and V. Mehrmann. Differential-Algebraic Equations. European Mathematical Society, 2006.Google Scholar
  107. [KMR14]
    C. Kuehn, E.A. Martens, and D. Romero. Critical transitions in social network activity. J. Complex Networks, 2(2):141–152, 2014. see also arXiv:1307.8250.Google Scholar
  108. [Kno92]
    M. Knorrenschild. Differential/algebraic equations as stiff ordinary differential equations. SIAM J. Numer. Anal., 29(6):1694–1715, 1992.zbMATHMathSciNetGoogle Scholar
  109. [KO96]
    L.V. Kalachev and R.E. O’Malley. The regularization of linear differential-algebraic equations. SIAM J. Math. Anal., 27(1):258–273, 1996.zbMATHMathSciNetGoogle Scholar
  110. [Kon09]
    L.I. Kononenko. Relaxation oscillations and canard solutions in singular systems on a plane. J. Appl. Ind. Math., 4(2):194–199, 2009.MathSciNetGoogle Scholar
  111. [KP89]
    M.A. Krasnosel’skii and A.V. Pokrovskii. Systems with Hysteresis. Springer, 1989.Google Scholar
  112. [KRA+07]
    S. Kéfi, M. Rietkerk, C.L. Alados, Y. Peyo, V.P. Papanastasis, A. El Aich, and P.C. de Ruiter. Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems. Nature, 449:213–217, 2007.Google Scholar
  113. [Kre96]
    P. Krejčí. Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo, 1996.zbMATHGoogle Scholar
  114. [Kre05]
    P. Krejčí. Hysteresis, convexity and dissipation in hyperbolic equations. J. Phys.: Conf. Ser., 22(1):103–123, 2005.Google Scholar
  115. [KSG13]
    C. Kuehn, S. Siegmund, and T. Gross. On the analysis of evolution equations via generalized models. IMA J. Appl. Math., 78(5):1051–1077, 2013.zbMATHMathSciNetGoogle Scholar
  116. [KSS97]
    M. Krupa, B. Sandstede, and P. Szmolyan. Fast and slow waves in the FitzHugh–Nagumo equation. J. Differential Equat., 133:49–97, 1997.zbMATHMathSciNetGoogle Scholar
  117. [Kue09]
    C. Kuehn. Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes. J. Phys. A: Math. and Theor., 42(4):045101, 2009.Google Scholar
  118. [Kue11a]
    C. Kuehn. A mathematical framework for critical transitions: bifurcations, fast–slow systems and stochastic dynamics. Physica D, 240(12):1020–1035, 2011.zbMATHGoogle Scholar
  119. [Kue13a]
    C. Kuehn. A mathematical framework for critical transitions: normal forms, variance and applications. J. Nonlinear Sci., 23(3):457–510, 2013.zbMATHMathSciNetGoogle Scholar
  120. [Kue13b]
    C. Kuehn. Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts. Theor. Ecol., 6(3):295–308, 2013.Google Scholar
  121. [Kuz04]
    Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, New York, NY, 3rd edition, 2004.zbMATHGoogle Scholar
  122. [KW91]
    T.J. Kaper and S. Wiggins. Lobe area in adiabatic Hamiltonian systems. Physica D, 51(1):205–212, 1991.zbMATHMathSciNetGoogle Scholar
  123. [LdST07]
    J. Llibre, P.R. da Silva, and M.A. Teixeira. Regularization of discontinuous vector fields on via singular perturbation. J. Dyn. Diff. Eq., 19(2):309–331, 2007.zbMATHGoogle Scholar
  124. [LdST08]
    J. Llibre, P.R. da Silva, and M.A. Teixeira. Sliding vector fields via slow–fast systems. Bull. Belg. Math. Soc., 15(5):851–869, 2008.zbMATHMathSciNetGoogle Scholar
  125. [LdST09]
    J. Llibre, P.R. da Silva, and M.A. Teixeira. Study of singularities in nonsmooth dynamical systems via singular perturbation. SIAM J. Appl. Dyn. Syst., 8(1):508–526, 2009.zbMATHMathSciNetGoogle Scholar
  126. [Lee06]
    J.M. Lee. Introduction to Smooth Manifolds. Springer, 2006.Google Scholar
  127. [LG05]
    L. Lerman and V. Gelfreich. Fast-slow Hamiltonian dynamics near a ghost separatrix loop. J. Math. Sci., 126:1445–1466, 2005.zbMATHMathSciNetGoogle Scholar
  128. [LG12]
    S.J. Lade and T. Gross. Early warning signals for critical transitions: a generalized modeling approach. PLoS Comp. Biol., 8:e1002360–6, 2012.Google Scholar
  129. [LHK+08]
    T.M. Lenton, H. Held, E. Kriegler, J.W. Hall, W. Lucht, S. Rahmstorf, and H.J. Schellnhuber. Tipping elements in the Earth’s climate system. Proc. Natl. Acad. Sci. USA, 105(6):1786–1793, 2008.zbMATHGoogle Scholar
  130. [LK85b]
    D. Luse and H. Khalil. Frequency domain results for systems with slow and fast dynamics. IEEE Trans. Aut. Contr., 30(12):1171–1179, 1985.zbMATHMathSciNetGoogle Scholar
  131. [LL07]
    V.N. Livina and T.M. Lenton. A modified method for detecting incipient bifurcations in a dynamical system. Geophysical Research Letters, 34:L03712, 2007.Google Scholar
  132. [LMT13]
    R. Lamour, R. März, and C. Tischendorf. Differential-Algebraic Equations: A Projector Based Analysis. Springer, 2013.Google Scholar
  133. [LN04]
    R.I. Leine and H. Nijmeijer. Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, 2004.Google Scholar
  134. [LST98]
    C. Lobry, T. Sari, and S. Touhami. On Tykhonov’s theorem for convergence of solutions of slow and fast systems. Electron. J. Differential Equat., 19:1–22, 1998.MathSciNetGoogle Scholar
  135. [MAEL07]
    F. Mormann, R.G. Andrzejak, C.E. Elger, and K. Lehnertz. Seizure prediction: the long and winding road. Brain, 130:314–333, 2007.Google Scholar
  136. [May03]
    I.D. Mayergoyz. Mathematical Models of Hysteresis and their Applications. Academic Press, 2003.Google Scholar
  137. [Mea05]
    K.D. Mease. Multiple time-scales in nonlinear flight mechanics: diagnosis and modeling. Appl. Math. Comput., 164(2):627–648, 2005.zbMATHMathSciNetGoogle Scholar
  138. [MEvdD13]
    A. Machina, R. Edwards, and P. van den Driessche. Singular dynamics in gene network models. SIAM J. Appl. Dyn. Syst., 12(1):95–125, 2013.zbMATHMathSciNetGoogle Scholar
  139. [MG12]
    L. Mitchell and G.A. Gottwald. On finite-size Lyapunov exponents in multiscale systems. Chaos, 22(2):023115, 2012.Google Scholar
  140. [MK12b]
    C. Meisel and C. Kuehn. On spatial and temporal multilevel dynamics and scaling effects in epileptic seizures. PLoS ONE, 7(2):e30371, 2012.Google Scholar
  141. [MLS08]
    R. May, S.A. Levin, and G. Sugihara. Ecology for bankers. Nature, 451:893–895, 2008.Google Scholar
  142. [MOPS05]
    M.P. Mortell, R.E. O’Malley, A. Pokrovskii, and V. Sobolev. Singular Perturbations and Hysteresis. SIAM, 2005.Google Scholar
  143. [MST03]
    P.E. McSharry, L.A. Smith, and L. Tarassenko. Prediction of epileptic seizures. Nature Med., 9: 241–242, 2003.Google Scholar
  144. [MTA08]
    K.D. Mease, U. Topcu, and E. Aykutlug. Characterizing two-timescale nonlinear dynamics using finite-time Lyapunov exponents and vectors. arXiv:0807.0239v1, pages 1–38, 2008.Google Scholar
  145. [NA12]
    A.I. Neishtadt and A.V. Artemyev. Destruction of adiabatic invariance for billiards in a strong nonuniform magnetic field. Phys. Rev. Lett., 108:064102, 2012.Google Scholar
  146. [Nei76]
    A.I. Neishtadt. Passage through a separatrix in a resonance problem with a slowly-varying parameter. J. Appl. Math. Mech., 39(4):594–605, 1976.MathSciNetGoogle Scholar
  147. [Nei81]
    A.I. Neishtadt. On the accuracy of conservation of the adiabatic invariant. J. Appl. Math. Mech., 45(1):58–63, 1981.MathSciNetGoogle Scholar
  148. [Nei87a]
    A.I. Neishtadt. On the change in the adiabatic invariant on crossing a separatrix in systems with two degrees of freedom. J. Appl. Math. Mech., 51(5):586–592, 1987.MathSciNetGoogle Scholar
  149. [Nei91]
    A.I. Neishtadt. Probability phenomena due to separatrix crossing. Chaos, 1(1):42–48, 1991.zbMATHMathSciNetGoogle Scholar
  150. [Nei00]
    A.I. Neishtadt. On the accuracy of persistence of adiabatic invariant in single-frequency systems. Reg. Chaotic. Dyn., 5(2):213–218, 2000.zbMATHMathSciNetGoogle Scholar
  151. [Net68]
    A.V. Netushil. Nonlinear element of stop type. Avtomat. Telemech., 7:175–179, 1968. (in Russian).Google Scholar
  152. [Net70]
    A.V. Netushil. Self-oscillations in systems with negative hysteresis. In Proc. 5th International Conference on Nonlinear Oscillations, volume 4, pages 393–396. Izdanie Inst. Mat. Akad. Nauk Ukrain., 1970. (in Russian).Google Scholar
  153. [NS12]
    A.I. Neishtadt and T. Su. On phenomenon of scattering on resonances associated with discretisation of systems with fast rotating phase. Regular & Chaotic Dyn., 17(3):359–366, 2012.zbMATHMathSciNetGoogle Scholar
  154. [NS13a]
    A.I. Neishtadt and T. Su. On asymptotic description of passage through a resonance in quasilinear Hamiltonian systems. SIAM J. Appl. Dyn. Syst., 12(3):1436–1473, 2013.zbMATHMathSciNetGoogle Scholar
  155. [NST97]
    A.I. Neishtadt, V.V. Sidorenko, and D.V. Treschev. Stable periodic motions in the problem on passage through a separatrix. Chaos, 7(1):2–11, 1997.zbMATHMathSciNetGoogle Scholar
  156. [NSTV08]
    A. Neishtadt, C. Simo, D.V. Treschev, and A. Vasiliev. Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow–fast systems. Discr. Contin. Dyn. Syst. B, 10(2):621–650, 2008.zbMATHMathSciNetGoogle Scholar
  157. [NV99]
    A.I. Neishtadt and A.A. Vasiliev. Change of the adiabatic invariant at a separatrix in a volume-preserving 3D system. Nonlinearity, 12(2):303, 1999.Google Scholar
  158. [NV05]
    A.I. Neishtadt and A.A. Vasiliev. Phase change between separatrix crossings in slow–fast Hamiltonian systems. Nonlinearity, 18(3):1393–1406, 2005.zbMATHMathSciNetGoogle Scholar
  159. [NV06]
    A.I. Neishtadt and A.A. Vasiliev. Destruction of adiabatic invariance at resonances in slow–fast Hamiltonian systems. Nucl. Instr. Meth. Phys. Res. A, 561:158–165, 2006.Google Scholar
  160. [OK94]
    R.E. O’Malley and L.V. Kalachev. Regularization of nonlinear differential-algebraic equations. SIAM J. Math. Anal., 25(2):615–629, 1994.zbMATHMathSciNetGoogle Scholar
  161. [OK96]
    R.E. O’Malley and L.V. Kalachev. The regularization of linear differential-algebraic equations. SIAM J. Math. Anal., 27(1):258–273, 1996.zbMATHMathSciNetGoogle Scholar
  162. [Pro03]
    M. Procesi. Exponentially small splitting and Arnold diffusion for multiple time scale systems. Rev. Math. Phys., 15(4):339–386, 2003.zbMATHMathSciNetGoogle Scholar
  163. [PW00]
    D.B. Percival and A.T. Walden. Wavelet Methods for Time Series Analysis. CUP, 2000.Google Scholar
  164. [RCG12]
    H.G. Rotstein, S. Coombes, and A.M. Gheorghe. Canard-like explosion of limit cycles in two-dimensional piecewise-linear models of FitzHugh–Nagumo type. SIAM J. Appl. Dyn. Syst., 11(1): 135–180, 2012.zbMATHMathSciNetGoogle Scholar
  165. [RDdRvdK04]
    M. Rietkerk, S.C. Dekker, P. de Ruiter, and J. van de Koppel. Self-organized patchiness and catastrophic shifts in ecosystems. Science, 305(2):1926–1929, 2004.Google Scholar
  166. [Rei00]
    S. Reich. Smoothed Langevin dynamics of highly oscillatory systems. Physica D, 138:210–224, 2000.zbMATHMathSciNetGoogle Scholar
  167. [Ria08]
    R. Riaza. Differential-Algebraic Systems. Analytical Aspects and Circuit Applications. World Scientific, 2008.Google Scholar
  168. [Ric11]
    M.P. Richardson. New observations may inform seizure models: very fast and very slow oscillations. Prog. Biophys. Molec. Biol., 105:5–13, 2011.Google Scholar
  169. [RJM95]
    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.zbMATHMathSciNetGoogle Scholar
  170. [RR00]
    P.J. Rabier and W.C. Rheinboldt. Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint. SIAM, 2000.Google Scholar
  171. [RT12]
    J.E. Rubin and D. Terman. Explicit maps to predict activation order in multi-phase rhythms of a coupled cell network. J. Math. Neurosci., 2:4, 2012.MathSciNetGoogle Scholar
  172. [SBB+09]
    M. Scheffer, J. Bascompte, W.A. Brock, V. Brovkhin, S.R. Carpenter, V. Dakos, H. Held, E.H. van Nes, M. Rietkerk, and G. Sugihara. Early-warning signals for critical transitions. Nature, 461:53–59, 2009.Google Scholar
  173. [SC03]
    M. Scheffer and S.R. Carpenter. Catastrophic regime shifts in ecosystems: linking theory to observation. TRENDS in Ecol. and Evol., 18(12):648–656, 2003.Google Scholar
  174. [SCF+01]
    M. Scheffer, S.R. Carpenter, J.A. Foley, C. Folke, and B. Walker. Catastrophic shifts in ecosystems. Nature, 413:591–596, 2001.Google Scholar
  175. [Sch09a]
    M. Scheffer. Critical Transitions in Nature and Society. Princeton University Press, 2009.Google Scholar
  176. [SF84]
    J. Shinar and N. Farber. Horizontal variable-speed interception game solved by forced singular perturbation technique. J. Optim. Theor. Appl., 42(4):603–636, 1984.zbMATHMathSciNetGoogle Scholar
  177. [SJLS05]
    S.N. Simic, K.H. Johansson, J. Lygeros, and S. Sastry. Towards a geometric theory of hybrid systems. Dynamics of Continuous, Discrete and Impulsive Systems, Series B, 12(5):649–687, 2005.zbMATHMathSciNetGoogle Scholar
  178. [SK10]
    J. Sieber and P. Kowalczyk. Small-scale instabilities in dynamical systems with sliding. Physica D, 239(1):44–57, 2010.zbMATHMathSciNetGoogle Scholar
  179. [SL13]
    A. Ture Savadkoohi and C.-H. Lamarque. Dynamics of coupled Dahl-type and nonsmooth systems at different scales of time. Int. J. Bif. Chaos, 23(7):1350114, 2013.Google Scholar
  180. [SS10]
    S. Schecter and C. Sourdis. Heteroclinic orbits in slow–fast Hamiltonian systems with slow manifold bifurcations. J. Dyn. Diff. Equat., 22:629–655, 2010.zbMATHMathSciNetGoogle Scholar
  181. [ST96]
    J. Sotomayor and M.A. Teixeira. Regularization of discontinuous vector fields. In International Conference on Differential Equations (Lisboa), pages 207–223. World Scientific, 1996.Google Scholar
  182. [Sto49]
    H. Stommel. Trajectories of small bodies sinking slowly through convection cells. J. Mar. Res., 8: 24–29, 1949.Google Scholar
  183. [Sto61]
    H. Stommel. Thermohaline convection with two stable regimes of flow. Tellus, 13:224–230, 1961.Google Scholar
  184. [Su12]
    T. Su. On the accuracy of conservation of adiabatic invariants in slow–fast Hamiltonian systems. Reg. Chaotic Dyn., 17(1):54–62, 2012.zbMATHGoogle Scholar
  185. [SY04]
    T. Sari and K. Yadi. On Pontryagin–Rodygin’s theorem for convergence of solutions of slow and fast systems. Electron. J. Differential Equat., 139:1–17, 2004.MathSciNetGoogle Scholar
  186. [SZ02]
    J. Shatah and C. Zeng. Periodic solutions for Hamiltonian systems under strong constraining forces. J. Differential Equat., 186(2):572–585, 2002.zbMATHMathSciNetGoogle Scholar
  187. [Tak76]
    F. Takens. Constrained equations; a study of implicit differential equations and their discontinuous solutions. In P. Hilton, editor, Structural Stability, the Theory of Catastrophes, and Applications in the Sciences, volume 525 of Lecture Notes in Mathematics, pages 143–234. Springer, 1976.Google Scholar
  188. [TB91a]
    M. Tuckerman and B.J. Berne. Stochastic molecular dynamics in systems with multiple time scales and memory friction. J. Chem. Phys., 95:4389, 1991.Google Scholar
  189. [TB91b]
    M. Tuckerman and B.J. Berne. Molecular dynamics in systems with multiple time scales: systems with stiff and soft degrees of freedom and with short and long range forces. J. Chem. Phys., 95:8362, 1991.Google Scholar
  190. [TBM91]
    M. Tuckerman, B.J. Berne, and G.J. Martyna. Molecular dynamics algorithm for multiple time scales: systems with long range forces. J. Chem. Phys., 94:6811, 1991.Google Scholar
  191. [TBM94]
    M. Tuckerman, B.J. Berne, and G.J. Martyna. Reversible multiple time scale molecular dynamics. J. Chem. Phys., 97(3):1990–2001, 1994.Google Scholar
  192. [TBR91]
    M. Tuckerman, B.J. Berne, and A. Rossi. Molecular dynamics algorithm for multiple time scales: systems with disparate masses. J. Chem. Phys., 94:1465, 1991.Google Scholar
  193. [TCE86]
    J.L. Tennyson, J.R. Cary, and D.F. Escande. Change of the adiabatic invariant due to separatrix crossing. Phys. Rev. Lett., 56(20):2117–2120, 1986.MathSciNetGoogle Scholar
  194. [TdC11]
    D.J. Tonon and T. de Carvalho. Generic bifurcations of planar Filippov systems via geometric singular perturbations. Bull. Belg. Math. Soc., 18(5):861–881, 2011.zbMATHMathSciNetGoogle Scholar
  195. [TdS12]
    M.A. Teixeira and P.R. da Silva. Regularization and singular perturbation techniques for non-smooth systems. Physica D, 241(22):1948–1955, 2012.MathSciNetGoogle Scholar
  196. [TMB90]
    M. Tuckerman, G.J. Martyna, and B.J. Berne. Molecular dynamics algorithm for condensed systems with multiple time scales. J. Chem. Phys., 93:1287, 1990.Google Scholar
  197. [TOM10]
    M. Tao, H. Owhadi, and J.E. Marsden. Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging. Multiscale Model. Simul., 8(4):1269–1324, 2010.zbMATHMathSciNetGoogle Scholar
  198. [TOM11]
    M. Tao, H. Owhadi, and J.E. Marsden. From efficient symplectic exponentiation of matrices to symplectic integration of high-dimensional Hamiltonian systems with slowly varying quadratic stiff potentials. Appl. Math. Res. Express, 2:242–280, 2011.MathSciNetGoogle Scholar
  199. [TOP96]
    E.B. Tadmor, M. Ortiz, and R. Phillips. Quasicontinuum analysis of defects in solids. Phil. Mag., 73(6):1529–1563, 1996.Google Scholar
  200. [TP94a]
    M.E. Tuckerman and M. Parrinello. Integrating the Car-Parrinello equations. I. Basic integration techniques. J. Chem. Phys., 101:1302–1315, 1994.Google Scholar
  201. [TP94b]
    M.E. Tuckerman and M. Parrinello. Integrating the Car-Parrinello equations. II. Multiple time scale techniques. J. Chem. Phys., 101:1316–1329, 1994.Google Scholar
  202. [Tre04]
    D. Treschev. Evolution of slow variables in a priori unstable Hamiltonian systems. Nonlinearity, 17(5):1803, 2004.Google Scholar
  203. [TS10]
    J.M.T. Thompson and J. Sieber. Predicting climate tipping points. In B. Launder and M. Thompson, editors, Geo-Engineering Climate Change: Environmental Necessity or Pandora’s Box, pages 50–83. CUP, 2010.Google Scholar
  204. [TS11a]
    J.M.T. Thompson and J. Sieber. Climate tipping as a noisy bifurcation: a predictive technique. IMA J. Appl. Math., 76(1):27–46, 2011.zbMATHMathSciNetGoogle Scholar
  205. [TS11b]
    J.M.T. Thompson and J. Sieber. Predicting climate tipping as a noisy bifurcation: a review. Int. J. Bif. Chaos, 21(2):399–423, 2011.zbMATHMathSciNetGoogle Scholar
  206. [TZKS12]
    J.-C. Tsai, W. Zhang, V. Kirk, and J. Sneyd. Traveling waves in a simplified model of calcium dynamics. SIAM J. Appl. Dyn. Syst., 11(4):1149–1199, 2012.zbMATHMathSciNetGoogle Scholar
  207. [vdB87]
    I. van den Berg. Nonstandard Asymptotic Analysis, volume 1249 of Lecture Notes in Mathematics. Springer, 1987.Google Scholar
  208. [VFD+12]
    A.J. Veraart, E.J. Faassen, V. Dakos, E.H. van Nes, M. Lurling, and M. Scheffer. Recovery rates reflect distance to a tipping point in a living system. Nature, 481:357–359, 2012.Google Scholar
  209. [VGVC07]
    M. Venkadesan, J. Guckenheimer, and F.J. Valero-Cuevas. Manipulating the edge of instability. J. Biomech., 40:1653–1661, 2007.Google Scholar
  210. [Vis94]
    A. Visintin. Differential Models of Hysteresis. Springer, 1994.Google Scholar
  211. [VNAZ12]
    A. Vasiliev, A. Neishtadt, A. Artemyev, and L. Zelenyi. Jump of the adiabatic invariant at a separatrix crossing: degenerate cases. Physica D, 241:566–573, 2012.zbMATHGoogle Scholar
  212. [VNM06]
    D.L. Vainchstein, A.I. Neishtadt, and I. Mezic. On passage through resonances in volume-preserving systems. Chaos, 16(4):043123, 2006.Google Scholar
  213. [vNS07]
    E.H. van Nes and M. Scheffer. Slow recovery from perturbations as generic indicator of a nearby catastrophic shift. Am. Nat., 169(6):738–747, 2007.Google Scholar
  214. [VWM+05]
    J.G. Venegas, T. Winkler, G. Musch, M.F. Vidal Melo, D. Layfield, N. Tgavalekos, A.J. Fischman, R.J. Callahan, G. Bellani, and R.S. Harris. Self-organized patchiness in asthma as a prelude to catastrophic shifts. Nature, 434:777–782, 2005.Google Scholar
  215. [Wal01]
    D.F. Walnut. An Introduction to Wavelet Analysis. Birkhäuser, 2001.Google Scholar
  216. [WALC11]
    S. Wieczorek, P. Ashwin, C.M. Luke, and P.M. Cox. Excitability in ramped systems: the compost-bomb instability. Proc. R. Soc. A, 467:1243–1269, 2011.zbMATHMathSciNetGoogle Scholar
  217. [Wie85]
    K. Wiesenfeld. Noisy precursors of nonlinear instabilities. J. Stat. Phys., 38(5):1071–1097, 1985.MathSciNetGoogle Scholar
  218. [Woj97]
    P. Wojtaszczyk. A Mathematical Introduction to Wavelets, volume 37 of LMS Student Texts. CUP, 1997.Google Scholar
  219. [WS01]
    H. Wang and Y. Song. Regularization methods for solving differential-algebraic equations. Appl. Math. Comput., 119(2):283–296, 2001.zbMATHMathSciNetGoogle Scholar
  220. [WS06b]
    L. Wang and E.D. Sontag. Almost global convergence in singular perturbations of strongly monotone systems. Lect. Notes Contr. Inf. Sci., 341:415, 2006.MathSciNetGoogle Scholar
  221. [WS08]
    L. Wang and E.D. Sontag. Singularly perturbed monotone systems and an application to double phosphorylation cycles. J. Nonlinear Sci., 18(5):527–550, 2008.zbMATHMathSciNetGoogle Scholar
  222. [YE06]
    J.Z. Yang and W. E. Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: derivation of continuum models from atomistic models. Phys. Rev. B, 74(18):184110, 2006.Google Scholar
  223. [ZFM+97]
    M.G. Zimmermann, S.O. Firle, M.A. Matiello, M. Hildebrand, M. Eiswirth, M. Bär, A.K. Bangia, and I.G. Kevrekidis. Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction–diffusion model. Physica D, 110(1):92–104, 1997.zbMATHMathSciNetGoogle Scholar
  224. [ZS84]
    A.K. Zvonkin and M.A. Shubin. Non-standard analysis and singular perturbations of ordinary differential equations. Russ. Math. Surveys, 39(2):69–131, 1984.zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Kuehn
    • 1
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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