Maps with quasi-periodicity of different dimension and quasi-periodic bifurcations

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. The necessity of discussing such mappings is connected with the possibility for them of a quasi-periodicity of different dimensions, as well as quasi-periodic bifurcations, i.e. bifurcations of invariant tori. This issue has not yet been adequately covered both in scientific and educational literature. The main method of investigation is the construction of Lyapunov exponents charts. Charts are obtained by numerical methods. On such charts regions of periodic modes, two-frequency quasi-periodicity, three-frequency quasi-periodicity, and chaos are marked by different colors. Illustrations of dynamics in the form of phase portraits are also presented. Specific features and classification features of quasi-periodic bifurcations - bifurcations of invariant tori - are discussed. Quasi-periodic bifurcations are analyzed using graphs of Lyapunov exponents and bifurcation trees. The difference between the quasi-periodic Hopf bifurcation and the saddlenode bifurcation of invariant tori is discussed. The dependence of the picture on the parameter - the discretization step - is discussed. At small values of this parameter, the picture is close to the traditional system of Arnold’s tongues, which, however, are now observed on the basis of two-frequency regimes and are immersed in a three-frequency region. The new moment is the appearance of regions of periodic high-order resonances built into these languages. As the sampling parameter increases, the picture changes. Tongues with characteristic cuspoidbases are replaced by bands of two-frequency modes with built-in transverse bands of periodic resonances, from which, in turn, a new system of fan-like tongues of two-frequency modes departs. The phase portraits inside languages change from multi-turn curves to a system of isolated ovals. Thus, it is shown that the picture associated with the quasi-periodic Hopf bifurcation is quite complex and requires three parameters for its analysis. The cases of different mappings are compared. It is shown that the «torus-mapping» most fully describes the range of essential phenomena in systems with quasi-periodicity of different dimensions.