doi: 10.3934/dcdsb.2021148
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Stochastic perturbation of a cubic anharmonic oscillator

1. 

Dipartimento di Scienze Statistiche Paolo Fortunati, Università di Bologna, via Belle Arti 41, Bologna, Italy

* Corresponding author

Received  May 2020 Revised  January 2021 Early access May 2021

We perturb with an additive noise the Hamiltonian system associated to a cubic anharmonic oscillator. This gives rise to a system of stochastic differential equations with quadratic drift and degenerate diffusion matrix. Firstly, we show that such systems possess explosive solutions for certain initial conditions. Then, we carry a small noise expansion's analysis of the stochastic system which is assumed to start from initial conditions that guarantee the existence of a periodic solution for the unperturbed equation. We then investigate the probabilistic properties of the sequence of coefficients which turn out to be the unique strong solutions of stochastic perturbations of the well-known Lamé's equation. We also obtain explicit expressions of these in terms of Jacobi elliptic functions. Furthermore, we prove, in the case of Brownian noise, a lower bound for the probability that the truncated expansion stays close to the solution of the deterministic problem. Lastly, when the noise is bounded, we provide conditions for the almost sure convergence of the global expansion.

Citation: Enrico Bernardi, Alberto Lanconelli. Stochastic perturbation of a cubic anharmonic oscillator. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021148
References:
[1]

S. AlbeverioA. Hilbert and E. Zehnder, Hamiltonian systems with a stochastic force: Nonlinear versus linear, and a Girsanov formula, Stochastics Stochastics Rep., 39 (1992), 159-188.  doi: 10.1080/17442509208833772.  Google Scholar

[2]

S. AlbeverioA. Hilbert and V. Kolokoltsov, Estimates uniform in time for the transition probability of diffusions with small drift and for stochastically perturbed Newton equations, J. Theoret. Probab., 12 (1999), 293-300.  doi: 10.1023/A:1021665708716.  Google Scholar

[3]

J. A. D. ApplebyX. Rodkina and A. Maoand, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[4]

F. M. Arscott, Periodic Differential Equations, The Macmillan Company, New York, 1964.  Google Scholar

[5]

F. M. Arscott and I. M.Khabaza, Table of Lamé's Polynomials, A Pergamon Press, Oxford, London, New York, Paris, 1962.  Google Scholar

[6]

E. Bernardi and T. Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness, J. d'Analyse Math., 105 (2008), 197-240.  doi: 10.1007/s11854-008-0035-3.  Google Scholar

[7]

E. Bernardi and T. Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 well-posedness, Kyoto J. Math., 51 (2011), 767-810.  doi: 10.1215/21562261-1424857.  Google Scholar

[8]

E. Delabaere and D. T. Trinh, Spectral analysis of the complex cubic oscillator, J. Phys. A: Math. Gen., 33 (2000), 8771-8796.  doi: 10.1088/0305-4470/33/48/314.  Google Scholar

[9]

E. M. Ferreira and J. Sesma, Global solution of the cubic oscillator, J. of Phys. A: Math., 47 (2014), 415306. Google Scholar

[10]

C. W. Gardiner, Handbook of Stochastic Methods, 2$^{nd}$ edition, Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 1985.  Google Scholar

[11]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7$^{th}$ edition, Elsevier, 2007. Google Scholar

[12]

L. Hörmander, The Cauchy problem for differential equations with double characteristics, Journal D'Analyse Mathématique, 32 (1977), 118-196.  doi: 10.1007/BF02803578.  Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam, New York, Oxford, Kodansha, 1981.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[15]

R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[16]

L. Markus and A. Weerasinghe, Stochastic oscillators, J. Differential Equations, 71 (1988), 288-314.  doi: 10.1016/0022-0396(88)90029-0.  Google Scholar

[17]

L. Markus and A. Weerasinghe, Stochastic nonlinear oscillators, Adv. in Appl. Probab., 25 (1993), 649-666.  doi: 10.2307/1427528.  Google Scholar

[18]

T. Nishitani, A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators, Kyoto J. Math., 55 (2015), 281-297.  doi: 10.1215/21562261-2871758.  Google Scholar

[19] W. J. Olver Frank and W. Lozier Daniel, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.   Google Scholar
[20]

W. P. Reinhardt and P. L. Walker, Jacobian elliptic functions, in Digital Library of Mathematical Functions. Available from: http://dlmf.nist.gov/22. Google Scholar

[21]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3$^{rd}$ edition, Grundlehren der Mathematischen Wissenschaften, 293, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[22]

H. Volker, Four remarks on eigenvalues of Lamé's equation, Analysis and Applications, 2 (2004), 161-175.  doi: 10.1142/S0219530504000023.  Google Scholar

[23]

H. Volkmer, Lamé functions, in Digital Library of Mathematical Functions. Available from: http://dlmf.nist.gov/29. Google Scholar

[24]

E. Weinan, T. Li and E. Vanden-Eijnden, Applied Stochastic Analysis, Graduate Studies in Mathematics, 199, American Mathematical Society, 2019. doi: 10.1090/gsm/199.  Google Scholar

[25]

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations With Periodic Coefficients Vol.1, John Wiley & Sons, New York, 1975.  Google Scholar

show all references

References:
[1]

S. AlbeverioA. Hilbert and E. Zehnder, Hamiltonian systems with a stochastic force: Nonlinear versus linear, and a Girsanov formula, Stochastics Stochastics Rep., 39 (1992), 159-188.  doi: 10.1080/17442509208833772.  Google Scholar

[2]

S. AlbeverioA. Hilbert and V. Kolokoltsov, Estimates uniform in time for the transition probability of diffusions with small drift and for stochastically perturbed Newton equations, J. Theoret. Probab., 12 (1999), 293-300.  doi: 10.1023/A:1021665708716.  Google Scholar

[3]

J. A. D. ApplebyX. Rodkina and A. Maoand, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control, 53 (2008), 683-691.  doi: 10.1109/TAC.2008.919255.  Google Scholar

[4]

F. M. Arscott, Periodic Differential Equations, The Macmillan Company, New York, 1964.  Google Scholar

[5]

F. M. Arscott and I. M.Khabaza, Table of Lamé's Polynomials, A Pergamon Press, Oxford, London, New York, Paris, 1962.  Google Scholar

[6]

E. Bernardi and T. Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 5 well-posedness, J. d'Analyse Math., 105 (2008), 197-240.  doi: 10.1007/s11854-008-0035-3.  Google Scholar

[7]

E. Bernardi and T. Nishitani, On the Cauchy problem for non-effectively hyperbolic operators, the Gevrey 4 well-posedness, Kyoto J. Math., 51 (2011), 767-810.  doi: 10.1215/21562261-1424857.  Google Scholar

[8]

E. Delabaere and D. T. Trinh, Spectral analysis of the complex cubic oscillator, J. Phys. A: Math. Gen., 33 (2000), 8771-8796.  doi: 10.1088/0305-4470/33/48/314.  Google Scholar

[9]

E. M. Ferreira and J. Sesma, Global solution of the cubic oscillator, J. of Phys. A: Math., 47 (2014), 415306. Google Scholar

[10]

C. W. Gardiner, Handbook of Stochastic Methods, 2$^{nd}$ edition, Springer Series in Synergetics, 13, Springer-Verlag, Berlin, 1985.  Google Scholar

[11]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7$^{th}$ edition, Elsevier, 2007. Google Scholar

[12]

L. Hörmander, The Cauchy problem for differential equations with double characteristics, Journal D'Analyse Mathématique, 32 (1977), 118-196.  doi: 10.1007/BF02803578.  Google Scholar

[13]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North Holland, Amsterdam, New York, Oxford, Kodansha, 1981.  Google Scholar

[14]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.  Google Scholar

[15]

R. Khasminskii, Stochastic Stability of Differential Equations, 2$^{nd}$ edition, Springer-Verlag, Berlin, 2012. doi: 10.1007/978-3-642-23280-0.  Google Scholar

[16]

L. Markus and A. Weerasinghe, Stochastic oscillators, J. Differential Equations, 71 (1988), 288-314.  doi: 10.1016/0022-0396(88)90029-0.  Google Scholar

[17]

L. Markus and A. Weerasinghe, Stochastic nonlinear oscillators, Adv. in Appl. Probab., 25 (1993), 649-666.  doi: 10.2307/1427528.  Google Scholar

[18]

T. Nishitani, A simple proof of the existence of tangent bicharacteristics for noneffectively hyperbolic operators, Kyoto J. Math., 55 (2015), 281-297.  doi: 10.1215/21562261-2871758.  Google Scholar

[19] W. J. Olver Frank and W. Lozier Daniel, NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.   Google Scholar
[20]

W. P. Reinhardt and P. L. Walker, Jacobian elliptic functions, in Digital Library of Mathematical Functions. Available from: http://dlmf.nist.gov/22. Google Scholar

[21]

D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3$^{rd}$ edition, Grundlehren der Mathematischen Wissenschaften, 293, Springer-Verlag, Berlin, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar

[22]

H. Volker, Four remarks on eigenvalues of Lamé's equation, Analysis and Applications, 2 (2004), 161-175.  doi: 10.1142/S0219530504000023.  Google Scholar

[23]

H. Volkmer, Lamé functions, in Digital Library of Mathematical Functions. Available from: http://dlmf.nist.gov/29. Google Scholar

[24]

E. Weinan, T. Li and E. Vanden-Eijnden, Applied Stochastic Analysis, Graduate Studies in Mathematics, 199, American Mathematical Society, 2019. doi: 10.1090/gsm/199.  Google Scholar

[25]

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations With Periodic Coefficients Vol.1, John Wiley & Sons, New York, 1975.  Google Scholar

Figure 1.  Energy surface $ \xi^{2}/2 - ( x^{3}/3 -x) = 0 $ of system (13)
Figure 2.  graph of $ \text{cn}(x, q) $
Figure 3.  graph of the solution of system (13)
Figure 4.  Explosive solution for the system (12) with $ x(0) = 0 $
Figure 5.  Graph of Hamiltonian with $ c = -1, a = 1 $
Figure 6.  Graph of (22) with $ c = -1, a = 1 $
Figure 7.  Graph of $ H(y, \eta) = 0 $
Figure 8.  Graph of $ u_1 $ with $ q = 2/\sqrt{5} $
Figure 9.  Graph of $ u_{2} $ with $ q = 2/\sqrt{5} $
Figure 10.  Graph of $ \mu(q) $ with $ q \in (0, 1) $
Figure 11.  Graph of three paths of process (32) with $ \{Z(t)\}_{t\geq 0} $ being a one dimensional standard Brownian motion
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