Analytical
investigation of small limit cycles and symbolic computations of Lyapunov values
(Lyapunov quantities, Poincare-Lyapunov constants, Lyapunov coefficients). The development of
methods of computation and analysis of Lyapunov quantities (or focus values,
etc.) was greatly encouraged by as purely mathematical problems (investigation
of stability in critical case of two purely imaginary roots of the first
approximation system, Hilbert's 16th problem, cyclicity problem, and
distinguishing between center and focus) as the applied problems (investigation
of boundaries of domain of stability and excitation of oscillations).
For example, the
method of N.N. Bautin [1], based on zero
Lyapunov quantities disturbance, permits one to obtain analytical conditions of
existence in the neighborhood of equilibrium of small limit cycles (so-called
local Hilbert's 16th problem [2]) in terms of
coefficients of expanding of the right-hand side of system. For
two-dimensional quadratic systems, this technique makes it possible to
construct three small limit cycles around one of equilibrium (weak focus) [1].
Also, by this technique it is possible to obtain the estimate from below of the
number of limit cycles for two-dimensional systems with the right-hand side,
which are polynomials of n-th degree (see, for
example, the surveys [3-5]). In this case a main problem is the obtaining of
independent zeros in the expressions for Lyapunov quantities [6] (so-called
basis of Bautin's ideal, based on polynomial
expressions of Lyapunov quantities in terms of system coefficients – for
quadratic systems, the number of independent zeros is equal to three).
In the engineering
mechanics an important question on the behavior of dynamical system for
parameter values close to the boundary of domain of stability is related with
the computation of Lyapunov quantities. Following the work of N.N. Bautin [7], one differs "safe" and "dangerous"
boundaries, a small violation of which implies a small (invertible) or
noninvertible changes of system state. Such changes correspond, for example, to
scenario of "soft" and "hard" excitation of oscillations,
considered by A.A. Andronov [8]. In the case of
two complex conjugate eigenvalues of linear part of two-dimensional system in
the neighborhood of stationary point, in crossing the boundary of stability
domain from negative values of real parts of roots to the positive ones, we
have the following: if the first Lyapunov quantity is negative, then there
occurs a unique single stable limit and for inverse change of parameter this
cycle "retracts" to the point, what corresponds to a "safe"
boundary. On the contrary, if the first Lyapunov quantity is positive, then for
small variations the trajectory can deviate infinitely far from equilibrium,
what corresponds to "dangerous" boundary.
At present, there
exist a few methods to calculate Lyapunov quantities and their computer
realizations, which permit one to obtain the Lyapunov quantities in the form of
symbolic expressions, depending on the expansion coefficients of the right-hand
parts of equations of system. These methods are differentiated by complexity of
realization of algorithms, a space, in which numerical computations are made,
and the compactness of the obtained symbolic expressions [3-6, 9-13].
The first method of
finding Lyapunov quantities was suggested in
the works [14] and [15]. This method is based on the sequential construction of
Lyapunov functions based on the first integral of linear part of system.
Further there were developed different methods of computing Lyapunov
quantities, using the reduction of system to normal form (however in realizing
these methods there arises the problem of nonuniqueness
of a process of constructing a normal form of system) [5, 13, 16].
Another approach to
numerical computation of Lyapunov quantities is related with the obtaining of
approximations of system solution. For example, in the work [15] it is used the
passage to the polar coordinates and the procedure of sequential construction
of solution approximations.
In the works (see Publications)
it was suggested a new direct method for
computations of Lyapunov quantities, based on the construction of
approximations of solution (in the form of finite sum with respect to degrees
of initial data) in the original Euclidian coordinates and in a time domain
(which is an essential requirement in engineering calculations). The advantage
of this method is ideological simplicity and visualization. This approach can
also be applied to the solution of problem of determination of isochronous
center since it allows one to find the approximation of return time of
trajectory to depend upon initial data [17].
Often, for
simplification of the algorithm of numerical computations and finite
expressions of Lyapunov quantities it is used different modifications of the
above-mentioned methods, related with the transformation of system to complex
variables [5, 13, 18, 19].
For example, by the
modification of the method of construction of Lyapunov function, in 1968 for
complex domain it was developed, apparently, the first computer program of
numerical computations of Lyapunov quantities [18].
Note that while
symbolic expressions of the first and second Lyapunov quantities of general
systems were computed by N.N. Bautin [1] and
N. Serebraykoba [20], respectively, in 40-50s of
last century, the numerical computation of expressions for subsequent Lyapunov
quantities became possible only much later, with the occurrence of power
computing aids. In 2008-2009, using a packet of symbolic computations was first
obtained the expression for the third Lyapunov quantity in general form in
terms of coefficients of expansion of the right-hand parts of equations of
system (see Publications). This expression occupies more than 4 pages
(21617 symbols).
Thus, the problem
of creation of effective methods for analysis and numerical computations of
Lyapunov quantities is still far from being resolved and require further
investigations.
Visualization and numerical localization of large limit
cycles.
At the present time
there exist different methods for "construction" of limit cycles (the
cycles, appearing from critical point, center, and homoclinic or heteroclinic
orbits and from infinity). 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 of visualization of limit cycles is still
far from being resolved even for the simple classes of systems.
For the study of
bifurcations of limit cycles there exist different methods such as the
investigation of Poincare mapping (for investigation of so-called small limit
cycles, which appear from degenerated focus: Bautin
bifurcation or degenerated Andronov-Hopf bifurcation;
for the cycles obtained from homoclinic trajectory of saddle: Bogdanov-Takens bifurcation; for the cycles obtained by the
disturbance of center), the investigation of Poincare-Mel'nikov
and Abel integrals, and the averaging method (for the cycles obtained by
disturbance of Hamilton systems).Main results on the largest number of limit
cycles for the classes of systems are obtained for systems with symmetry [5].
However the
"small" parameters, used for numerical construction of limit cycles on
the basis of the above methods often makes the task of numerical analysis of
large limit cycles difficult, especially in the case of a few embedded limit
cycles. Note also that the effects of "flattering" [21], discovered
in polynomial systems, also make numerical procedures of cycle’s localization
difficult. The appearance of modern computers permits one to use numerical
simulation of complicated nonlinear dynamical systems and to obtain new
information on the structure of their trajectories. However the possibilities
of "simple" approach, based on the construction of trajectories by
numerical integration of the considered differential equations, turned out to
be highly limited. V.I. Arnol’d writes [22]:
"To estimate the number of limit
cycles of square vector fields on plane, A.N. Kolmogorov had distributed several hundreds of such fields (with randomly chosen
coefficients of quadratic expressions) among a few hundreds of students of
Mechanics and Mathematics Faculty of Moscow State University as a mathematical
practice. Each student had to find the number of limit cycles of a field. The
result of this experiment was absolutely unexpected: not a single field had a
limit cycle! It is known that a limit cycle persists under a small change of
field coefficients. Therefore, the systems with one, two, three (and even, as
has become known later, four) limit cycles form an open set in the space of
coefficients, and so for a random choice of polynomial coefficients, the
probability of hitting in it is positive. The fact that this did not occur
suggests that the above-mentioned probabilities are, apparently, small."
Publications
- G.A. Leonov, N.V. Kuznetsov, Hidden
attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden
chaotic attractor in Chua circuits, International Journal of Bifurcation
and Chaos, 23(1), 2013, art. no. 1330002 [DOI]
- N.V. Kuznetsov, O.A. Kuznetsova, G.A. Leonov, Visualization
of four normal size limit cycles in two-dimensional polynomial quadratic
system, Differential equations and Dynamical
systems, 21(1-2), 2013, pp. 29-34 [DOI]
- G.A. Leonov, N.V. Kuznetsov, and E.V. Kudryashova, A Direct
Method for Calculating Lyapunov Quantities of Two-Dimensional Dynamical
Systems, Proceedings of the Steklov
Institute of Mathematics, Volume 272, Supplement 1, 2011, pp.
119-127 [DOI]
- G.A. Leonov, N.V. Kuznetsov, Limit Cycles
of Quadratic Systems with a Perturbed Weak Focus of Order 3 and a Saddle
Equilibrium at Infinity, Doklady
Mathematics, 2010, Vol. 82, No. 2, 693-696 [DOI]
- G.A. Leonov, N.V. Kuznetsov, and E.V. Kudryashova, Cycles of
Two-Dimensional Systems: Computer Calculations, Proofs, and Experiments,
Vestnik St. Petersburg University. Mathematics, 2008, Vol. 41, No. 3, pp.
216-250 (Allerton Press, USA)
- Kuznetsov N.V., Leonov G.A., Lyapunov
quantities, limit cycles and strange behavior of trajectories in
two-dimensional quadratic systems, Journal of Vibroengineering, Vol. 10, Iss.
4, 2008, pp. 460–467
- Kuznetsov N.V., Leonov G.A., Computation of Lyapunov quantities,
Proceedings of the 6th EUROMECH Nonlinear Dynamics Conference, 2008, IPACS
Electronic Library (http://lib.physcon.ru/?item=1802), pp. 1-10
- Leonov G.A., Kuznetsov N.V., Computation of
the first Lyapunov quantity for the second-order dynamical system,
IFAC Proceedings Volumes
(IFAC-PapersOnline) 3 (1), 2007, pp. 87-89 (doi:10.3182/20070829-3-RU-4912.00014)
- Kuznetsov N.V., Stability and
Oscillations of Dynamical Systems: Theory and Applications.
Jyväskylä University Printing House, 2008 (ISBN:
978-951-39-3428-6)
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House, 2009, pp. 83-147 (978-951-39-3666-2)
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- G.A. Leonov,
N.V. Kuznetsov, Limit Cycles
of Quadratic Systems with a Perturbed Weak Focus of Order 3 and a Saddle
Equilibrium at Infinity, Doklady
Mathematics, 2010, Vol. 82, No. 2, 693-696 [DOI]