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May  2022, 27(5): 2563-2585. doi: 10.3934/dcdsb.2021148

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 Published  May 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (5) : 2563-2585. 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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[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.

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

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

[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.

[22]

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

[23]

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

[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.

[25]

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

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.

[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.

[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.

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[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.

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

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

[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.

[22]

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

[23]

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

[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.

[25]

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

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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