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2015 | OriginalPaper | Buchkapitel

19. Other Topics

verfasst von : Christian Kuehn

Erschienen in: Multiple Time Scale Dynamics

Verlag: Springer International Publishing

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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.

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Zurück zum Zitat D. Dolgopyat and C. Liverani. Energy transfer in fast–slow Hamiltonian systems. Comm. Math. Phys., 308:201–225, 2011.MATHMathSciNet D. Dolgopyat and C. Liverani. Energy transfer in fast–slow Hamiltonian systems. Comm. Math. Phys., 308:201–225, 2011.MATHMathSciNet
[DSvN+08]
Zurück zum Zitat 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. 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.
[DvND+09]
Zurück zum Zitat 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. 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.
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Zurück zum Zitat 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. 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.
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Zurück zum Zitat E. Fridman. Singularly perturbed analysis of chattering in relay control systems. IEEE Trans. Aut. Contr., 47(12):2079–2084, 2002.MathSciNet E. Fridman. Singularly perturbed analysis of chattering in relay control systems. IEEE Trans. Aut. Contr., 47(12):2079–2084, 2002.MathSciNet
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Zurück zum Zitat N. Farber and J. Shinar. Approximate solution of singularly perturbed nonlinear pursuit-evasion games. J. Optim. Theor. Appl., 32:39–73, 1980.MATHMathSciNet N. Farber and J. Shinar. Approximate solution of singularly perturbed nonlinear pursuit-evasion games. J. Optim. Theor. Appl., 32:39–73, 1980.MATHMathSciNet
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Zurück zum Zitat C.S. Gardner. Adiabatic invariants of periodic classical systems. Phys. Rev., 2(115):791–794, 1959. C.S. Gardner. Adiabatic invariants of periodic classical systems. Phys. Rev., 2(115):791–794, 1959.
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Zurück zum Zitat M. Gerdts. Optimal control of ODEs and DAEs. de Gruyter, 2012. M. Gerdts. Optimal control of ODEs and DAEs. de Gruyter, 2012.
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Zurück zum Zitat I. Gholami, A. Fiege, and A. Zippelius. Slow dynamics and precursors of the glass transition in granular fluids. Phys. Rev. E, 84:031305, 2011. I. Gholami, A. Fiege, and A. Zippelius. Slow dynamics and precursors of the glass transition in granular fluids. Phys. Rev. E, 84:031305, 2011.
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Zurück zum Zitat 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. 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.
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Zurück zum Zitat V. Guttal and C. Jayaprakash. Changing skewness: an early warning signal of regime shifts in ecosystems. Ecology Letters, 11:450–460, 2008. V. Guttal and C. Jayaprakash. Changing skewness: an early warning signal of regime shifts in ecosystems. Ecology Letters, 11:450–460, 2008.
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Zurück zum Zitat 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. 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.
[GK09a]
Zurück zum Zitat J. Guckenheimer and C. Kuehn. Computing slow manifolds of saddle-type. SIAM J. Appl. Dyn. Syst., 8(3):854–879, 2009.MATHMathSciNet J. Guckenheimer and C. Kuehn. Computing slow manifolds of saddle-type. SIAM J. Appl. Dyn. Syst., 8(3):854–879, 2009.MATHMathSciNet
[GK09b]
Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
[GK10b]
Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
[GL02]
Zurück zum Zitat V. Gelfreich and L. Lerman. Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system. Nonlinearity, 15(2):447–457, 2002.MATHMathSciNet V. Gelfreich and L. Lerman. Almost invariant elliptic manifold in a singularly perturbed Hamiltonian system. Nonlinearity, 15(2):447–457, 2002.MATHMathSciNet
[GL03]
Zurück zum Zitat V. Gelfreich and L. Lerman. Long-periodic orbits and invariant tori in a singularly perturbed Hamiltonian system. Physica D, 176(3):125–146, 2003.MATHMathSciNet V. Gelfreich and L. Lerman. Long-periodic orbits and invariant tori in a singularly perturbed Hamiltonian system. Physica D, 176(3):125–146, 2003.MATHMathSciNet
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Zurück zum Zitat B. Gershgorin and A. Majda. A nonlinear test model for filtering slow–fast systems. Commun. Math. Sci., 6(3):611–649, 2008.MATHMathSciNet B. Gershgorin and A. Majda. A nonlinear test model for filtering slow–fast systems. Commun. Math. Sci., 6(3):611–649, 2008.MATHMathSciNet
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Zurück zum Zitat B. Gershgorin and A. Majda. Filtering a nonlinear slow–fast system with strong fast forcing. Commun. Math. Sci., 8(1):67–92, 2010.MATHMathSciNet B. Gershgorin and A. Majda. Filtering a nonlinear slow–fast system with strong fast forcing. Commun. Math. Sci., 8(1):67–92, 2010.MATHMathSciNet
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Zurück zum Zitat R. Goldblatt. Lectures on the Hyperreals. Springer, 1998. R. Goldblatt. Lectures on the Hyperreals. Springer, 1998.
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Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
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Zurück zum Zitat V. Gelfreich, V. Rom-Kedar, and D. Turaev. Fermi acceleration and adiabatic invariants for non-autonomous billiards. Chaos, 22(3):033116, 2012. V. Gelfreich, V. Rom-Kedar, and D. Turaev. Fermi acceleration and adiabatic invariants for non-autonomous billiards. Chaos, 22(3):033116, 2012.
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Zurück zum Zitat 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.MathSciNet 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.MathSciNet
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Zurück zum Zitat 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.MathSciNet 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.MathSciNet
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Zurück zum Zitat G. Haller. Diffusion at intersecting resonances in Hamiltonian systems. Phys. Lett. A, 200(1):34–42, 1995.MATHMathSciNet G. Haller. Diffusion at intersecting resonances in Hamiltonian systems. Phys. Lett. A, 200(1):34–42, 1995.MATHMathSciNet
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Zurück zum Zitat M. Hanke. Regularizations of differential-algebraic equations revisited. Math. Nachr., 174:159–183, 1995.MATHMathSciNet M. Hanke. Regularizations of differential-algebraic equations revisited. Math. Nachr., 174:159–183, 1995.MATHMathSciNet
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Zurück zum Zitat J. Henrard. Capture into resonance: an extension of the use of adiabatic invariants. Celest. Mech., 27(1):3–22, 1982.MATHMathSciNet J. Henrard. Capture into resonance: an extension of the use of adiabatic invariants. Celest. Mech., 27(1):3–22, 1982.MATHMathSciNet
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Zurück zum Zitat J. Henrard. The adiabatic invariant in classical mechanics. In Dynamics Reported Vol. 2, pages 117–235. Springer, 1993. J. Henrard. The adiabatic invariant in classical mechanics. In Dynamics Reported Vol. 2, pages 117–235. Springer, 1993.
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Zurück zum Zitat 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. 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.
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Zurück zum Zitat A. Hastings and D.B. Wysham. Regime shifts in ecological systems can occur with no warning. Ecol. Lett., 13:464–472, 2010. A. Hastings and D.B. Wysham. Regime shifts in ecological systems can occur with no warning. Ecol. Lett., 13:464–472, 2010.
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Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
[IN12]
Zurück zum Zitat 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. 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.
[IVKS07]
Zurück zum Zitat 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.MATH 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.MATH
[JCdBS10]
Zurück zum Zitat 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. 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.
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Zurück zum Zitat 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. 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.
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Zurück zum Zitat P. Kowalczyk and P. Glendinning. Boundary-equilibrium bifurcations in piecewise-smooth slow–fast systems. Chaos, 21: 023126, 2011.MathSciNet P. Kowalczyk and P. Glendinning. Boundary-equilibrium bifurcations in piecewise-smooth slow–fast systems. Chaos, 21: 023126, 2011.MathSciNet
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Zurück zum Zitat C. Kuehn and T. Gross. Nonlocal generalized models of predator–prey systems. Discr. Cont. Dyn. Syst. B, 18(3): 693–720, 2013.MATHMathSciNet C. Kuehn and T. Gross. Nonlocal generalized models of predator–prey systems. Discr. Cont. Dyn. Syst. B, 18(3): 693–720, 2013.MATHMathSciNet
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Zurück zum Zitat P. Kunkel and V. Mehrmann. Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math., 56(3):225–251, 1994.MATHMathSciNet P. Kunkel and V. Mehrmann. Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math., 56(3):225–251, 1994.MATHMathSciNet
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Zurück zum Zitat P. Kunkel and V. Mehrmann. Regular solutions of nonlinear differential-algebraic equations and their numerical determination. Numer. Math., 79(4):581–600, 1998.MATHMathSciNet P. Kunkel and V. Mehrmann. Regular solutions of nonlinear differential-algebraic equations and their numerical determination. Numer. Math., 79(4):581–600, 1998.MATHMathSciNet
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Zurück zum Zitat P. Kunkel and V. Mehrmann. Differential-Algebraic Equations. European Mathematical Society, 2006. P. Kunkel and V. Mehrmann. Differential-Algebraic Equations. European Mathematical Society, 2006.
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Zurück zum Zitat 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. 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.
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Zurück zum Zitat M. Knorrenschild. Differential/algebraic equations as stiff ordinary differential equations. SIAM J. Numer. Anal., 29(6):1694–1715, 1992.MATHMathSciNet M. Knorrenschild. Differential/algebraic equations as stiff ordinary differential equations. SIAM J. Numer. Anal., 29(6):1694–1715, 1992.MATHMathSciNet
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Zurück zum Zitat L.V. Kalachev and R.E. O’Malley. The regularization of linear differential-algebraic equations. SIAM J. Math. Anal., 27(1):258–273, 1996.MATHMathSciNet L.V. Kalachev and R.E. O’Malley. The regularization of linear differential-algebraic equations. SIAM J. Math. Anal., 27(1):258–273, 1996.MATHMathSciNet
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Zurück zum Zitat L.I. Kononenko. Relaxation oscillations and canard solutions in singular systems on a plane. J. Appl. Ind. Math., 4(2):194–199, 2009.MathSciNet L.I. Kononenko. Relaxation oscillations and canard solutions in singular systems on a plane. J. Appl. Ind. Math., 4(2):194–199, 2009.MathSciNet
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Zurück zum Zitat M.A. Krasnosel’skii and A.V. Pokrovskii. Systems with Hysteresis. Springer, 1989. M.A. Krasnosel’skii and A.V. Pokrovskii. Systems with Hysteresis. Springer, 1989.
[KRA+07]
Zurück zum Zitat 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. 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.
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Zurück zum Zitat P. Krejčí. Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo, 1996.MATH P. Krejčí. Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo, 1996.MATH
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Zurück zum Zitat P. Krejčí. Hysteresis, convexity and dissipation in hyperbolic equations. J. Phys.: Conf. Ser., 22(1):103–123, 2005. P. Krejčí. Hysteresis, convexity and dissipation in hyperbolic equations. J. Phys.: Conf. Ser., 22(1):103–123, 2005.
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Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
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Zurück zum Zitat M. Krupa, B. Sandstede, and P. Szmolyan. Fast and slow waves in the FitzHugh–Nagumo equation. J. Differential Equat., 133:49–97, 1997.MATHMathSciNet M. Krupa, B. Sandstede, and P. Szmolyan. Fast and slow waves in the FitzHugh–Nagumo equation. J. Differential Equat., 133:49–97, 1997.MATHMathSciNet
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Zurück zum Zitat C. Kuehn. Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes. J. Phys. A: Math. and Theor., 42(4):045101, 2009. C. Kuehn. Scaling of saddle-node bifurcations: degeneracies and rapid quantitative changes. J. Phys. A: Math. and Theor., 42(4):045101, 2009.
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Zurück zum Zitat C. Kuehn. A mathematical framework for critical transitions: bifurcations, fast–slow systems and stochastic dynamics. Physica D, 240(12):1020–1035, 2011.MATH C. Kuehn. A mathematical framework for critical transitions: bifurcations, fast–slow systems and stochastic dynamics. Physica D, 240(12):1020–1035, 2011.MATH
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Zurück zum Zitat C. Kuehn. A mathematical framework for critical transitions: normal forms, variance and applications. J. Nonlinear Sci., 23(3):457–510, 2013.MATHMathSciNet C. Kuehn. A mathematical framework for critical transitions: normal forms, variance and applications. J. Nonlinear Sci., 23(3):457–510, 2013.MATHMathSciNet
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Zurück zum Zitat C. Kuehn. Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts. Theor. Ecol., 6(3):295–308, 2013. C. Kuehn. Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts. Theor. Ecol., 6(3):295–308, 2013.
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Zurück zum Zitat Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, New York, NY, 3rd edition, 2004.MATH Yu.A. Kuznetsov. Elements of Applied Bifurcation Theory. Springer, New York, NY, 3rd edition, 2004.MATH
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Zurück zum Zitat T.J. Kaper and S. Wiggins. Lobe area in adiabatic Hamiltonian systems. Physica D, 51(1):205–212, 1991.MATHMathSciNet T.J. Kaper and S. Wiggins. Lobe area in adiabatic Hamiltonian systems. Physica D, 51(1):205–212, 1991.MATHMathSciNet
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Zurück zum Zitat 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.MATH 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.MATH
[LdST08]
Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
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Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
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Zurück zum Zitat J.M. Lee. Introduction to Smooth Manifolds. Springer, 2006. J.M. Lee. Introduction to Smooth Manifolds. Springer, 2006.
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Zurück zum Zitat L. Lerman and V. Gelfreich. Fast-slow Hamiltonian dynamics near a ghost separatrix loop. J. Math. Sci., 126:1445–1466, 2005.MATHMathSciNet L. Lerman and V. Gelfreich. Fast-slow Hamiltonian dynamics near a ghost separatrix loop. J. Math. Sci., 126:1445–1466, 2005.MATHMathSciNet
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Zurück zum Zitat S.J. Lade and T. Gross. Early warning signals for critical transitions: a generalized modeling approach. PLoS Comp. Biol., 8:e1002360–6, 2012. S.J. Lade and T. Gross. Early warning signals for critical transitions: a generalized modeling approach. PLoS Comp. Biol., 8:e1002360–6, 2012.
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Zurück zum Zitat 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.MATH 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.MATH
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Zurück zum Zitat D. Luse and H. Khalil. Frequency domain results for systems with slow and fast dynamics. IEEE Trans. Aut. Contr., 30(12):1171–1179, 1985.MATHMathSciNet D. Luse and H. Khalil. Frequency domain results for systems with slow and fast dynamics. IEEE Trans. Aut. Contr., 30(12):1171–1179, 1985.MATHMathSciNet
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Zurück zum Zitat V.N. Livina and T.M. Lenton. A modified method for detecting incipient bifurcations in a dynamical system. Geophysical Research Letters, 34:L03712, 2007. V.N. Livina and T.M. Lenton. A modified method for detecting incipient bifurcations in a dynamical system. Geophysical Research Letters, 34:L03712, 2007.
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Zurück zum Zitat R. Lamour, R. März, and C. Tischendorf. Differential-Algebraic Equations: A Projector Based Analysis. Springer, 2013. R. Lamour, R. März, and C. Tischendorf. Differential-Algebraic Equations: A Projector Based Analysis. Springer, 2013.
[LN04]
Zurück zum Zitat R.I. Leine and H. Nijmeijer. Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, 2004. R.I. Leine and H. Nijmeijer. Dynamics and Bifurcations of Non-Smooth Mechanical Systems. Springer, 2004.
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Zurück zum Zitat 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.MathSciNet 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.MathSciNet
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Zurück zum Zitat F. Mormann, R.G. Andrzejak, C.E. Elger, and K. Lehnertz. Seizure prediction: the long and winding road. Brain, 130:314–333, 2007. F. Mormann, R.G. Andrzejak, C.E. Elger, and K. Lehnertz. Seizure prediction: the long and winding road. Brain, 130:314–333, 2007.
[May03]
Zurück zum Zitat I.D. Mayergoyz. Mathematical Models of Hysteresis and their Applications. Academic Press, 2003. I.D. Mayergoyz. Mathematical Models of Hysteresis and their Applications. Academic Press, 2003.
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Zurück zum Zitat K.D. Mease. Multiple time-scales in nonlinear flight mechanics: diagnosis and modeling. Appl. Math. Comput., 164(2):627–648, 2005.MATHMathSciNet K.D. Mease. Multiple time-scales in nonlinear flight mechanics: diagnosis and modeling. Appl. Math. Comput., 164(2):627–648, 2005.MATHMathSciNet
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Zurück zum Zitat 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.MATHMathSciNet 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.MATHMathSciNet
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Zurück zum Zitat L. Mitchell and G.A. Gottwald. On finite-size Lyapunov exponents in multiscale systems. Chaos, 22(2):023115, 2012. L. Mitchell and G.A. Gottwald. On finite-size Lyapunov exponents in multiscale systems. Chaos, 22(2):023115, 2012.
[MK12b]
Zurück zum Zitat C. Meisel and C. Kuehn. On spatial and temporal multilevel dynamics and scaling effects in epileptic seizures. PLoS ONE, 7(2):e30371, 2012. C. Meisel and C. Kuehn. On spatial and temporal multilevel dynamics and scaling effects in epileptic seizures. PLoS ONE, 7(2):e30371, 2012.
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[VWM+05]
Zurück zum Zitat 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. 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.
[Wal01]
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[WALC11]
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[ZFM+97]
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[ZS84]
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Metadaten
Titel
Other Topics
verfasst von
Christian Kuehn
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-12316-5_19