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Quasi-periodic resonances and the Landau-Hopf scenario
Russian version
Kuznetsov A. P. 1, Sedova Yu. V. 1
1Saratov Branch, Kotel’nikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Saratov, Russia
Email: apkuz@rambler.ru, sedovayv@yandex.ru
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The effect of resonances on a cascade of quasi-periodic bifurcations, the sequence of which occur in accordance with the Landau-Hopf scenario, is examined using an ensemble of discrete van der Pol - Duffing oscillators. With small frequency detunings of the oscillators, tongues of quasi-periodic modes emerge, analogous to Arnold tongues, and in the region of the highest frequency oscillations. With a large frequency detuning, the general structure of regimes transformation in accordance with Landau-Hopf scenario remains, but the quasi-periodic Hopf bifurcation in the cascade can be replaced by a saddle-node bifurcation of torus. Narrow resonance regions based on tori of different dimensions are also observed. At high values of the Duffing oscillator-like nonlinear parameter, resonances can destroy high-dimensional tori in the Landau-Hopf cascade. Keywords: quasi-periodicity, resonance, Landau-Hopf scenario, Lyapunov exponents, bifurcations.
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