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Concept
Hidden attractor
Summary
In the bifurcation theory, a bounded oscillation that is born without loss of stability of stationary set is called a hidden oscillation. In nonlinear control theory, the birth of a hidden oscillation in a time-invariant control system with bounded states means crossing a boundary, in the domain of the parameters, where local stability of the stationary states implies global stability (see, e.g. Kalman's conjecture). If a hidden oscillation (or a set of such hidden oscillations filling a compact subset of the phase space of the dynamical system) attracts all nearby oscillations, then it is called a hidden attractor. For a dynamical system with a unique equilibrium point that is globally attractive, the birth of a hidden attractor corresponds to a qualitative change in behaviour from monostability to bi-stability. In the general case, a dynamical system may turn out to be multistable and have coexisting local attractors in the phase space. While trivial attractors, i.e. stable equilibrium points, can be easily found analytically or numerically, the search of periodic and chaotic attractors can turn out to be a challenging problem (see, e.g. the second part of Hilbert's 16th problem).
To identify a local attractor in a physical or numerical experiment, one needs to choose an initial system’s state in attractor’s basin of attraction and observe how the system’s state, starting from this initial state, after a transient process, visualizes the attractor. The classification of attractors as being hidden or self-excited reflects the difficulties of revealing basins of attraction and searching for the local attractors in the phase space.
Definition.
An attractor is called a hidden attractor if its basin of attraction does not intersect with a certain open neighbourhood of equilibrium points; otherwise it is called a self-excited attractor.
The classification of attractors as being hidden or self-excited was introduced by G. Leonov and N. Kuznetsov in connection with the discovery of the hidden Chua attractor
for the first time in 2009 year.
Official source
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