Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems (Q2388059)

Recently, by the authors it was shown that the passage to chaos in a wide class of three-dimensional autonomous nonlinear dissipative systems of ODEs (Lorenz, Rössler, Chua systems et al.) fits the following scenario: the passage to chaos begins with a cascade of Feigenbaum period doubling bifurcations and then continues by a subharmonic cascade of Sharkovskii bifurcations and, if the system has a saddle-focus separatrix loop, by a homoclinic cascade of bifurcations of stable cycles converging to a homoclinic contour. It was also shown that this scenario is based on a bifurcation, after which an originally stable singular limit cycle with complex but not complex-conjugate Floquet exponents becomes a singular saddle cycle and generates a stable cycle of double period. It was proved that each of the appearing attractors lies on a two-dimensional surface that is the closure of a two-dimensional invariant unstable manifold (a separatrix surface) of a singular saddle cycle and that chaotic dynamics is due to the phase shift between trajectories forming a separatrix surface. In this case, the original singular cycle corresponds to the zero rotor-type singular point of a two-dimensional nonautonomous system. The rotation of trajectories of the two-dimensional nonautonomous system around the rotor naturally induces a continuous self-mapping (with multi-valued inverse) of a segment containing the rotor. Here, two-dimensional nonautonomous systems of ODE with rotor-type singular points are considered. A singular point of a two-dimensional nonautonomous system with periodic linear part that has complex Floquet exponents with equal imaginary parts and different real parts is called a rotor. For such systems, a canonical form is obtained and their application to the description of the passage to chaos in three-dimensional autonomous dissipative systems is analyzed. The analytic results are justified by examples and numerical results.