Hidden
attractors and hidden
oscillations in dynamical systems
|
Aizerman problem & Kalman problem (Describing
function method, harmonic balance, and
absolute stability) |
Keywords: hidden
oscillation, hidden attractor, periodic solution, limit cycle, absolute
stability, 16th Hilbert problem, absolute stability, Aizerman’s conjecture, Kalman’s
conjecture, describing function method (DFM), harmonic balance
During initial
establishment and development of theory of nonlinear oscillations in the first
half of twentieth century a main attention has been given to analysis and
synthesis of oscillating systems for which the solution of problems of
existence of oscillating solutions was not too difficult. The structure itself
of many systems was such that they have oscillating solutions, the existence of
which was "almost obvious". The arising in these systems periodic solutions
were well seen by numerical analysis when numerical procedure of integrating
the trajectories allowed one to pass from small neighborhood of equilibrium to
periodic trajectory.
Further there came
to light so called hidden oscillations and hidden attractors – oscillations, which
are "small" (and, therefore, are difficult for numerical analysis) or
are not "connected" with equilibrium (i.e. when the creation of
numerical procedure of integration of trajectories for the passage from
equilibrium to periodic solution is impossible because the neighbourhoods
of equilibria do not belong to such attractor).
For the first time
the problem of finding hidden oscillations had been stated by D. Hilbert in
1900 (Hilbert's 16th problem) for two-dimensional polynomial systems. For a
more than century history, in the framework of the solution of this problem the
numerous theoretical and numerical results were obtained. However the problem
is still far from being resolved even for the simple classes of systems. Thus,
the problem of finding quadratic systems, in which there exists a limit cycle,
is nontrivial. In 40-50s of the 20th century A.N. Kolmogorov
became the initiator of a few hundreds of computational experiments, in the
result of which the limit two-dimensional quadratic systems would been found.
The result was absolutely unexpected: in not a single experiment a limit cycle
was found [Arnold V.I., Experimental Mathematics, 2005], though it is known
that quadratic systems with limit cycles form open domains in the space of
coefficients and, therefore, for a random choice of polynomial coefficients,
the probability of hitting in these sets is positive.
Further the problem
of analysis of hidden oscillations arose in applied problems of automatic
control. In the process of investigation, connected with Aizerman's
(1949) and Kalman's (1957) conjectures, it was stated
that the differential equations of systems of automatic control, which satisfy
generalized Routh-Hurwitz stability criterion, can
also have hidden periodic regimes.
At present the new
analytic-numerical approaches to investigation, of hidden oscillations in
dynamical systems, were developed based on the development of numerical
methods, computers, and applied bifurcation theory, which suggests revisiting
and revising early ideas on the application of the small parameter method and
the harmonic linearization.
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