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The Poincaré Map of Randomly Perturbed Periodic Motion

Abstract

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations.

In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.

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Notes

  1. Lemma 2.1 in Freidlin and Wentzell (1998) is stated for a system of the form (2.5) with P(x)=id. However, the argument used in the proof of this lemma applies to systems with P(x) satisfying the assumptions in Sect. 2.1 after a suitable modification of the action functional.

References

  • Anderson, T.W.: The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Am. Math. Soc. 6, 170–176 (1955)

    Article  MATH  Google Scholar 

  • Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillations. Dover, New York (1987)

    Google Scholar 

  • Appleby, J.A., Rodkina, A., Roeger, L.-I.W.: Stability of a limit cycle for a planar system with stochastic perturbations. Funct. Differ. Equ. 16(1–2), 11–28 (2009)

    MathSciNet  Google Scholar 

  • Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd edn. Springer, New York (1988)

    Book  Google Scholar 

  • Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  • Arnold, L., Imkeller, P., Sri Namachchivaya, N.: The asymptotic stability of a noisy non-linear oscillator. J. Sound Vib. 269, 1003–1029 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  • Benzi, R., Sutera, A., Vulpiani, A.: The mechanism of stochastic resonance. J. Phys. A, Math. Gen. 14, L453–L457 (1981)

    MathSciNet  Article  Google Scholar 

  • Berglund, N., Gentz, B.: Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach. Springer, Berlin (2006)

    Google Scholar 

  • Berglund, N., Landon, D.: Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh–Nagumo model. arXiv:1105.1278 (2011)

  • Blagoveshchenskii, Yu.N.: Diffusion processes depending on small parameter. Theory Probab. Appl. 7, 130–146 (1962)

    Article  Google Scholar 

  • Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)

    Book  MATH  Google Scholar 

  • Day, M.V.: On the exponential law in the small parameter exit problem. Stochastics 8, 297–323 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  • DeVille, L., Namachchivaya, N.S., Rapti, Z.: Stability of a stochastic two-dimensional non-Hamiltonian system. SIAM J. Appl. Math. 71(4), 1458–1475 (2013)

    Article  Google Scholar 

  • Doi, S., Kumagai, S.: Generation of very slow neuronal rhythms and chaos near the Hopf bifurcation in single neuron models. J. Comput. Neurosci. 19, 325–356 (2005)

    MathSciNet  Article  Google Scholar 

  • Freidlin, M.I.: On stable oscillations and equilibriums induced by small noise. J. Stat. Phys. 103(1–2), 283–300 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  • Freidlin, M.I., Wentzell, A.D.: Random Perturbations of Dynamical Systems, 2nd edn. Springer, New York (1998)

    Book  MATH  Google Scholar 

  • Friedman, A.: Stochastic Differential Equations and Applications. Dover, New York (2006)

    MATH  Google Scholar 

  • Furstenberg, H., Kesten, H.: Products of random matrices. Ann. Math. Stat. 31, 457–469 (1960)

    MathSciNet  Article  MATH  Google Scholar 

  • Goldobin, D.S., Pikovsky, A.: Synchronization and desynchronization of self-sustained oscillators by common noise. Phys. Rev. E 71, 045201 (2005)

    MathSciNet  Article  Google Scholar 

  • Gutkin, B.S., Ermentrout, G.B.: Dynamics of membrane excitability determine interspike interval variability: a link between spike generation mechanisms and cortical spike train statistics. Neural Comput. 10(5), 1047–1065 (1998)

    Article  Google Scholar 

  • Hale, J.K.: Oscillations in Nonlinear Systems. McGraw-Hill, New York (1963)

    MATH  Google Scholar 

  • Hale, J.K.: Ordinary Differential Equations, 2nd edn. Krieger, Melbourne (1980)

    MATH  Google Scholar 

  • Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Rockville (1980)

    Book  Google Scholar 

  • Hitczenko, P., Medvedev, G.S.: Bursting oscillations induced by small noise. SIAM J. Appl. Math. 69(5), 1359–1392 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  • Horn, R.A., Johnson, C.A.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  • Izhikevich, E.M.: Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting. MIT Press, Cambridge (2007)

    Google Scholar 

  • Kesten, H.: Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  • Kwapień, S.: A remark on the median and the expectation of convex functions of Gaussian vectors. In: Probability in Banach Spaces IX, pp. 271–272. Birkhäuser, Basel (1994)

    Chapter  Google Scholar 

  • Ledoux, M., Talagrand, M.: Probability in Banach Spaces. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  • Lim, S., Rinzel, J.: Noise-induced transitions in slow wave neuronal dynamics. J. Comput. Neurosci. 28(1), 1–17 (2010)

    MathSciNet  Article  Google Scholar 

  • Malkin, I.G.: The Theory of Stability of Motion, 2nd edn. Editorial, Moscow (2004a). (in Russian)

    Google Scholar 

  • Malkin, I.G.: Methods of Lyapunov and Poincaré in the Theory of Nonlinear Oscillations, 2nd edn. Editorial, Moscow (2004b). (in Russian)

    Google Scholar 

  • Mao, X.: Stochastic stabilization and destabilization. Syst. Control Lett. 23, 279–290 (1994)

    Article  MATH  Google Scholar 

  • Medvedev, G.S.: Transition to bursting via deterministic chaos. Phys. Rev. Lett. 97, 048102 (2006)

    Article  Google Scholar 

  • Medvedev, G.S.: Synchronization of coupled limit cycles. J. Nonlinear Sci. 21(3), 441–464 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  • Medvedev, G.S., Yoo, Y.: Chaos at the border of criticality. Chaos 18, 033105 (2008)

    MathSciNet  Article  Google Scholar 

  • Medvedev, G.S., Zhuravytska, S.: Shaping bursting by electrical coupling and noise. Biol. Cybern. 106(2), 67–88 (2012a)

    MathSciNet  Article  MATH  Google Scholar 

  • Medvedev, G.S., Zhuravytska, S.: The geometry of spontaneous spiking in neuronal networks. J. Nonlinear Sci. 22(5), 689–725 (2012b)

    MathSciNet  Article  MATH  Google Scholar 

  • Muratov, C.B., Vanden-Eijnden, E.: Noise-induced mixed-mode oscillations in a relaxation oscillator near the onset of a limit cycle. Chaos 18, 015111 (2008)

    MathSciNet  Article  Google Scholar 

  • Muratov, C.B., Vanden-Eijnden, E., E, W.: Self-induced stochastic resonance in excitable systems. Physica D 210, 227–240 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  • Øksendal, B.: Stochastic Differential Equations, 6th edn. Springer, Berlin (2003)

    Book  Google Scholar 

  • Pontryagin, L.S., Andronov, A.A., Vitt, A.A.: O statitisticheskom rassmotrenii dinamicheskikh sistem. Zh. Èksp. Teor. Fiz. 3(3), 165–180 (1933). (in Russian)

    Google Scholar 

  • Shilnikov, L.P., Shilnokov, A.L., Turaev, D.V., Chua, L.O.: Methods of Qualitative Theory in Nonlinear Dynamics, Part I. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  • Skorokhod, A.V.: Asymptotic Methods in the Theory of Stochastic Differential Equations. AMS, Providence (1989)

    MATH  Google Scholar 

  • Skorokhod, A.V., Hoppensteadt, F.C., Salehi, H.: Random Perturbation Methods. Springer, New York (2002)

    MATH  Google Scholar 

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Acknowledgements

This work was partially supported by a grant from the Simons Foundation (grant No. 208766 to PH) and an NSF grant (DMS 1109367 to GM).

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Correspondence to Georgi S. Medvedev.

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Communicated by P. Newton.

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Hitczenko, P., Medvedev, G.S. The Poincaré Map of Randomly Perturbed Periodic Motion. J Nonlinear Sci 23, 835–861 (2013). https://doi.org/10.1007/s00332-013-9170-9

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  • DOI: https://doi.org/10.1007/s00332-013-9170-9

Keywords

  • Random perturbations
  • Limit cycle
  • First return map
  • Mixed-mode oscillations
  • Bursting

Mathematics Subject Classification

  • 34C15
  • 60H10
  • 92B25