Quasi-periodic dynamics

The problem of the dynamics of three coupled self-oscillators and three coupled periodically driven self-oscillators 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 four-frequency tori and chaos are revealed. Three typical situations of synchronization of three self-oscillators by the external driving are found. First situation refers to the mode locking of autonomous oscillators.

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. Also the torus birth through the Neimark-Sacker bifurcation occurs in some other region of the parameter space. 

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.