Квазипериодическая динамика

The paper discusses the construction of convenient and informative three-dimensional mappings demonstrating the existence of 2-tori and 3-tori. The first mapping is obtained by discretizing the continuous time system – a generator of quasi-periodic oscillations. The second is obtained via discretization of the Lorentz-84 climate model. The third mapping was proposed in the theory of quasi-periodic bifurcations by Simo, Broer, Vitolo.

A new four-dimensional model with quasi-periodic dynamics is suggested. The torus attractor originates via the saddle-node bifurcation, which may be regarded as a member of a bifurcation family embracing different types of blue sky catastrophes.

The problem of the dynamics of three coupled self­oscillators and three coupled periodically driven self­oscillators is discussed, in the last case only one of the oscillators is directly exited by the external fore. The regions of complete synchronization, two­, three­and four­frequency tori and chaos are revealed. Three typical situations of synchronization of three self­oscillators by the external driving are found.

The driven auto-oscillatory system with the delayed modulation of driving amplitude was investigated. It was shown that synchronous regime destructs in different ways at small and large modulation amplitudes. The changes in the «driving amplitude–driving frequency» plane were revealed.

We considered the discrete map with quasi-periodic dynamics in the wide band of the parameters and investigated the structure of the parameter plane of two coupled maps. We revealed the doublings of 3D-tori, the systems of 2D-tori and synchronization tongues and the resonance web. Also we revealed the attractors with complex structure and the largest Lyapunov exponent close to zero.