Hidden attractor

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).

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.

For a self-excited attractor, its basin of attraction is connected with an unstable equilibrium and, therefore, the self-excited attractors can be found numerically by a standard computational procedure in which after a transient process, a trajectory, starting in a neighbourhood of an unstable equilibrium, is attracted to the state of oscillation and then traces it (see, e.g. self-oscillation process).

Classical attractors in Van der Pol, Beluosov–Zhabotinsky, Rössler, Chua, Hénon dynamical systems are self-excited.

Chaotic self-excited attractor (green domain) in Chua's system . Trajectories with initial data in neighborhoods of two saddle points (blue) and zero equilibrium point (orange) tend (green) to attractor.
Chaotic hidden attractor (green domain) in Chua's system . Trajectories with initial data in neighborhoods of two saddle points (blue) tend (red arrow) to infinity or tend (black arrow) to stable zero equilibrium point (orange).
Two hidden chaotic attractors and one hidden periodic attractor coexist with two trivial attractors in Chua circuit (from the IJBC cover [ 7 ] )
Afraimovich Award granted to N. Kuznetsov for The theory of hidden oscillations and stability of dynamical systems (2021)