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On the effect of noise on invariant tori and quasi-periodic bifurcations of different dimensions

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  • Kuznetsov, Alexander
  • Sedova, Yuliya

Abstract

A map derived by discretization of an ensemble consisting of five non-identical in control parameter van der Pol oscillators by replacing time derivatives with finite differences is studied. In this map, with a sufficiently large frequency detuning of the oscillators, a cascade of bifurcations leading to the birth of invariant tori with increasing dimensions is observed, that corresponds to the first steps of the Landau-Hopf scenario. The influence of the discretization parameter value on the observed structure is discussed. Using this model, the effect of noise on quasi-periodic Hopf bifurcations and invariant tori of varying dimensions is studied. The presence of several Lyapunov exponents is taken into account and corresponding one-parameter and two-parameter illustrations are given. The cases of noise effect on “the weakest” and “the strongest” oscillators in an ensemble are examined and compared. The possibility of alternative chaotization of tori and their stabilization by noise is demonstrated. Illustrations of such stabilization are given.

Suggested Citation

  • Kuznetsov, Alexander & Sedova, Yuliya, 2026. "On the effect of noise on invariant tori and quasi-periodic bifurcations of different dimensions," Chaos, Solitons & Fractals, Elsevier, vol. 207(C).
    • Handle: RePEc:eee:chsofr:v:207:y:2026:i:c:s096007792600202x
    DOI: 10.1016/j.chaos.2026.118061
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