Hidden and Singular Attractors in Nonlinear Systems of Differential Equations

The concept of a “hidden” attractor is widely used in the modern literature on autonomous nonlinear chaotic systems of ordinary differential equations in the case when the system has an chaotic attractor and together with it has either no singular points or one or more stable singular points. It is shown in the paper that any hidden attractor of nonlinear system of differential equations is one of the singular attractors of the system in sense of the Feigenbaum-Sharkovsky-Magnitskii (FShM) universal bifurcation theory. That is any hidden irregular (chaotic) attractor is born during the implementation of FShM bifurcation scenario and is bounded non-periodic trajectory in phase space of finite or infinite dimension, which is the limit of the cycles of some cascade of Feigenbaum period-doubling bifurcations and contains in any of its neighborhood an infinite number of unstable periodic orbits.