On hidden attractors of nonlinear systems of differential equations with an infinite number of singular points

The work carries out an analytical and numerical analysis of the bifurcations of cycles of two systems of equations, which, according to the authors of the systems, contain both an infinite number of unstable singular points and “hidden” chaotic attractors. It is shown that the transition to chaos in systems occurs, as in any other nonlinear chaotic systems of differential equations, in accordance with the universal bifurcation scenario of Feigenbaum-Sharkovsky-Magnitskii. In this case, due to the absence of homoclinic and heteroclinic separatrix contours, incomplete FShM bifurcation cascades are realized in the systems, ending with a complete subharmonic cascade and an incomplete homoclinic cascade of bifurcations. It has been proven that in both systems the so-called “hidden” attractors are in fact complex singular attractors of systems in the sense of the FShM theory.