Русская версия
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

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Analytical and Numerical Estimates of the Fractal Dimension of Forced Quasiperiodic Oscillations in Control Systems

Author(s):

Mikhail M. Anikushin

Saint-Petersburg State University,
The Faculty of Mathematics and Mechanics,
Universitetsky prospekt, 28,
198504, Peterhof, St. Petersburg, Russia
PhD. Student

demolishka@gmail.com

Volker Reitmann

Saint-Petersburg State University,
The Faculty of Mathematics and Mechanics,
Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia \
professor of the Department of Applied Cybernetics
Prof. Dr.

vreitmann@aol.com

Andrey O. Romanov

Saint-Petersburg State University,
The Faculty of Mathematics and Mechanics,
Universitetsky prospekt, 28, 198504, Peterhof, St. Petersburg, Russia
PhD. Student

romanov.andrey.twai@gmail.com

Abstract:

We consider a class of nonlinear feedback control systems with monotone nonlinearities and several stationary states. If the system is under an almost periodic perturbation, one can obtain conditions for existence of almost periodic oscillations. Our purpose is to estimate the fractal dimension of the trajectory closure of forced almost periodic oscillations obtained by the mentioned way. We show that within the result of I. M. Burkin and V. A. Yakubovich, which extends the result of M. A. Krasnoselskii et al. on the existence of exactly two almost periodic solutions (the stable one and the unstable one) in the case of two stationary states, it is possible to obtain some estimates of the fractal dimension. This estimate depends on some properties of Diophantine approximations for the frequencies of the almost periodic perturbation. We also apply a similar approach to study almost periodic oscillations in the perturbed Chua circuit, where the unperturbed system has three stationary states. We provide some analytical upper estimates of the fractal dimension and some numerical simulations confirming that upper estimates provided can be exact.

Keywords

References:

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  2. Burkin I. M., Yakubovich V. A. Frequency conditions of existence of two almost periodic solutions in a nonlinear control system. Siberian Mathematical Journal, 16(5), 699-705 (1975)
  3. Samoilenko A. M. Elements of the mathematical theory of multi-frequency oscillations, Springer Science & Business Media, (2012)
  4. Khinchin A. I. Continued fractions. P. Noordhoff, (1963)
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  12. Kuznetsov N. V., Leonov G. A., Reitmann V. Attractor dimension estimates for dynamical systems: theory and computation. Switzerland: Springer International Publishing AG (2019)
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  29. Kuznetsov N. V., Leonov G. A., Reitmann V. Attractor dimension estimates for dynamical systems: theory and computation. Switzerland: Springer International Publishing AG (2019)
  30. Levitan B. M., Zhikov V. V. Almost periodic functions and differential equations. CUP Archive (1982)
  31. Naito K. Dimension estimate of almost periodic attractors by simultaneous Diophantine approximation. J. Differ. Equ., 141(1), 179-200 (1997)
  32. Reitmann V. Ü ber die Beschrä nktheit der Lö sungen diskreter nichtstationä rer Phasensysteme. Zeitschrift fü r Analysis und ihre Anwendungen, 1(1), 83-93 (1982)
  33. Stankevich N. V., Kuznetsov N. V., Leonov G. A., and Chua L. O. Scenario of the birth of hidden attractors in the Chua circuit. International Journal of Bifurcation and Chaos, 27(12), 1-18 (2017)
  34. Suresh K., Prasad A., and Thamilmaran K. Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua's oscillator, Phys. Lett. A, 377(8), 612-621 (2013)
  35. Zygmund A. Trigonometric series. Cambridge University Press (2002)
  36. Anikushin M. M., On the Liouville phenomenon in estimates of fractal dimensions of forced quasi-periodic oscillations. Vestnik St. Petersb. Univ. Math. , 52(3) (2019)
  37. Burkin I. M., Yakubovich V. A. Frequency conditions of existence of two almost periodic solutions in a nonlinear control system. Siberian Mathematical Journal, 16(5), 699-705 (1975)
  38. Samoilenko A. M. Elements of the mathematical theory of multi-frequency oscillations, Springer Science & Business Media, (2012)
  39. Khinchin A. I. Continued fractions. P. Noordhoff, (1963)
  40. Yakubovich V. A. A frequency theorem in control theory. Siberian Mathematical Journal, 14(2), 265-289 (1973)
  41. Anikushin M. M., On the Smith reduction theorem for almost periodic ODEs satisfying the squeezing property. Rus. J. Nonlin. Dyn. , 15(1), 97-108 (2019)
  42. Anikushin M. M. Dimensional aspects of almost periodic dynamics. In [12] (2019)
  43. Burkin I. M. The method of transfer to the derivative space: 40 years of evolution. Differential Equations and Control Processes (Differencialnie Uravnenia i Protsesy Upravlenia), 72(3) (2015)
  44. Cartwright M. L. Almost periodic differential equations and almost periodic flows. J. Differ. Equ., 5(1), 167-181 (1969)
  45. Fink A. M. Almost periodic differential equations. Springer (2006)
  46. Krasnoselś kii M. A., Burd V. S., Kolesov Yu. S. Nonlinear almost periodic oscillations, John Wiley & Sons (1973)
  47. Kuznetsov N. V., Leonov G. A., Reitmann V. Attractor dimension estimates for dynamical systems: theory and computation. Switzerland: Springer International Publishing AG (2019)
  48. Levitan B. M., Zhikov V. V. Almost periodic functions and differential equations. CUP Archive (1982)
  49. Naito K. Dimension estimate of almost periodic attractors by simultaneous Diophantine approximation. J. Differ. Equ., 141(1), 179-200 (1997)
  50. Reitmann V. Ü ber die Beschrä nktheit der Lö sungen diskreter nichtstationä rer Phasensysteme. Zeitschrift fü r Analysis und ihre Anwendungen, 1(1), 83-93 (1982)
  51. Stankevich N. V., Kuznetsov N. V., Leonov G. A., and Chua L. O. Scenario of the birth of hidden attractors in the Chua circuit. International Journal of Bifurcation and Chaos, 27(12), 1-18 (2017)
  52. Suresh K., Prasad A., and Thamilmaran K. Birth of strange nonchaotic attractors through formation and merging of bubbles in a quasiperiodically forced Chua's oscillator, Phys. Lett. A, 377(8), 612-621 (2013)
  53. Zygmund A. Trigonometric series. Cambridge University Press (2002)

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