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Analysis of Critical Phenomena in a Dynamic System Under the Influence of Random Perturbations

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Abstract

The paper is devoted to the study of stochastic models of an electrochemical reaction with a perturbation described by a generalized white-noise random process. Noise-induced transitions are analyzed, the influence of external perturbations on limit cycles is examined, and the sensitivity of the cycle to noise was found. The dependence of the threshold value of the noise intensity on the control parameter of the system is established. The critical value of the noise intensity at which small-amplitude oscillations turn into mixed-type oscillations is obtained. The critical value of noise corresponding to the transition from canard trajectories to relaxation oscillations in the model is found. It is shown that an increase in the intensity of random perturbations can lead to significant changes of oscillation modes of the model up to their destruction.

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References

  1. I. A. Bashkirtseva, “Stochastic sensitivity analysis: theory and numerical algorithms,” IOP Conf. Ser. Mater. Sci. Eng., 192, 012024 (2017).

    Article  Google Scholar 

  2. I. A. Bashkirtceva and T. V. Perevalova, “Analysis of stochastic attractors at bifurcation rest point-cycle,” Avtomat. Telemekh., No. 10, 53-69 (2007).

  3. I. A. Bashkirtseva and L. B. Ryashko, “Sensitivity analysis of the stochastically and periodically forced brusselator,” Phys. A., 278, 126–139 (2000).

    Article  Google Scholar 

  4. I. A. Bashkirtseva and L. B. Ryashko, “Stochastic sensitivity analysis of noise-induced excitement in a preypredator plankton system,” Front. Life Sci., 5, 141-148 (2011).

    Article  Google Scholar 

  5. N. Berglund, B. Gentz, and C. Kuehn, “Hunting french ducks in a noisy environment,” J. Differ. Equations., 252 (9), 4786–4841 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. F. Berthier, J. P. Diard, and S. Nugues, “On the nature of the spontaneous oscillations observed for the Koper–Sluyters electrocatalitic reaction,” J. Electroanal. Chem., 436 (1), 35–42 (1997).

    Article  Google Scholar 

  7. H. E. De Swart, and J. Grasman, “Effect of stochastic perturbations on a low-order spectral model of the atmospheric circulation,” Tellus., 39A, 10–24 (1987).

    Article  Google Scholar 

  8. N. M. Firstova, “Study of critical phenomena in a model of an electrochemical reactor,” Vestn. Samar. Univ. Estestvennonauch. Ser., 110, No. 9/2, 221–226 (2013).

    Google Scholar 

  9. N. Firstova and E. Shchepakina, “Conditions for the critical phenomena in a dynamic model of an electrocatalytic reaction,” J. Phys. Conf. Ser., 811, 012002 (2017).

    Article  MathSciNet  Google Scholar 

  10. N. Firstova and E. Shchepakina, “Modelling of critical conditions for an electrochemical reactor model,” Proc. Eng., 201, 495–502 (2017).

    Article  Google Scholar 

  11. N. Firstova and E. Shchepakina, “Study of oscillatory processes in the one model of electrochemical reactor,” CEUR Workshop Proc., 1638, 731–741 (2016).

    Google Scholar 

  12. E. S. Golodova and E. A. Shchepakina, “Simulation of safe combustion processes with maximal temperature,” Mat. Model., 20, No. 5, 55–68 (2008).

    MATH  Google Scholar 

  13. J. Grasman, “Asymptotic analysis of nonlinear systems with small stochastic perturbations,” Math. Comput. Sim., 31, 41–54 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  14. M. T. M. Koper and J. H. Sluyters, “Instabilities and oscillations in simple models of electrocatalytic surface reactions,” J. Electroanal. Chem., 371 (1), 149 (1994).

    Article  Google Scholar 

  15. E. A. Shchepakina, “Safety conditions for the ignition of a flammable liquid in a porous insulating material,” Sib. Zh. Industr. Mat., 5, No. 3 (11), 162-169 (2002).

  16. E. A. Shchepakina, “Singular perturbations in modeling of safe combustion processes,” Mat. Model., 15, No. 8, 113–117 (2003).

    MATH  Google Scholar 

  17. E. A. Shchepakina, “Singularly perturbed models of combustion processes in in multiphase media,” Sib. Zh. Industr. Mat., 6, No. 4 (16), 142–157 (2003).

  18. E. Shchepakina, “Black swans and canards in self-ignition problem,” Nonlinear Anal. Real World Appl., 4, 45–50 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  19. E. Shchepakina, “Critical phenomena in a model of fuel’s heating in a porous medium,” CEUR Workshop Proc., 1490, 179-189 (2015).

    Google Scholar 

  20. V. A. Sobolev and E. A. Shchepakina, Model Reduction and Critical Phenomena in Macrokinetics [in Russian], Fizmatlit, M. (2010).

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Authors and Affiliations

  1. Samara National Research University, Samara, Russia

    N. M. Firstova

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  1. N. M. Firstova
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Correspondence to N. M. Firstova.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 175, Proceedings of the XVII All-Russian Youth School-Conference “Lobachevsky Readings-2018,” November 23-28, 2018, Kazan. Part 1, 2020.

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Firstova, N.M. Analysis of Critical Phenomena in a Dynamic System Under the Influence of Random Perturbations. J Math Sci 272, 783–790 (2023). https://doi.org/10.1007/s10958-023-06472-4

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