Hidden oscillation
Encyclopedia
From computation point of view, in nonlinear dynamical systems periodic oscillations and chaotic attractors
(a neighborhood of which is their attraction domain) can be regarded as self-exciting attractors or hidden attractors.
Here it is essential to consider numerical localization procedure in forward and backward time, since computation in backward time may localize unstable oscillation.
, Rössler
, Chua
and many others are self-exiting attractors and can be obtained numerically, with relative ease, by standard computational procedure.
(see, e.g., results on the second part of Hilbert's 16th problem
).
Other examples of hidden oscillations are counterexamples to Aizerman's
and Kalman's conjectures on absolute stability in automatic control theory (where unique stable equilibrium point and attracting periodic solution coexist) which can be constructed for system dimension not less than three and four correspondingly.
In 2010, for the first time, a chaotic hidden attractor was discovered
in Chua's circuit
which is described by three-dimensional dynamical system.
In multi-dimensional case the integration of trajectories with random initial data is unlikely to serve localization of hidden attractor, since a basin of attraction may be highly small and the attractor dimension itself may be much less than the dimension of the considered system.
Therefore for numerical localization of hidden attractors in multi-dimensional space it is necessary to develop special analytical-numerical computational procedures
, which allow to step away from equilibria (by an analytical method) and to choose initial data in an attraction domain of the hidden oscillation (which does not contain neighborhoods of equilibria) and then to perform trajectory computation there.
Attractor
An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...
(a neighborhood of which is their attraction domain) can be regarded as self-exciting attractors or hidden attractors.
- Self-exciting attractors can be localized numerically by standard computational procedure, in which after transient process a trajectory, started from a point of unstable manifold in a small neighborhood of unstable equilibrium, reaches an attractor and computes it.
- Hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria, and therefore hidden attractor cannot be localized by standard computational procedure.
Here it is essential to consider numerical localization procedure in forward and backward time, since computation in backward time may localize unstable oscillation.
Self-exciting attractor localization
Classical attractors in well-known dynamical systems of Van der Pol, Beluosov–Zhabotinsky, LorenzLorenz attractor
The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow, noted for its lemniscate shape...
, Rössler
Rössler attractor
The Rössler attractor is the attractor for the Rössler system, a system of three non-linear ordinary differential equations. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor...
, Chua
Chua's circuit
Chua's circuit is a simple electronic circuit that exhibits classic chaos theory behavior. It was introduced in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time...
and many others are self-exiting attractors and can be obtained numerically, with relative ease, by standard computational procedure.
Hidden attractor localization
Simplest examples of hidden oscillations are internal nested limit cycles in two-dimensional systems. Here for investigation of hidden oscillations can be elaborated effective analytical methods(see, e.g., results on the second part of Hilbert's 16th problem
Hilbert's sixteenth problem
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, together with the other 22 problems....
).
Other examples of hidden oscillations are counterexamples to Aizerman's
Aizerman's conjecture
In nonlinear control, Aizerman's conjecture states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of the sector. This conjecture was proven false but led to the circle criterion and Popov criterion....
and Kalman's conjectures on absolute stability in automatic control theory (where unique stable equilibrium point and attracting periodic solution coexist) which can be constructed for system dimension not less than three and four correspondingly.
In 2010, for the first time, a chaotic hidden attractor was discovered
in Chua's circuit
Chua's circuit
Chua's circuit is a simple electronic circuit that exhibits classic chaos theory behavior. It was introduced in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time...
which is described by three-dimensional dynamical system.
In multi-dimensional case the integration of trajectories with random initial data is unlikely to serve localization of hidden attractor, since a basin of attraction may be highly small and the attractor dimension itself may be much less than the dimension of the considered system.
Therefore for numerical localization of hidden attractors in multi-dimensional space it is necessary to develop special analytical-numerical computational procedures
, which allow to step away from equilibria (by an analytical method) and to choose initial data in an attraction domain of the hidden oscillation (which does not contain neighborhoods of equilibria) and then to perform trajectory computation there.