=Paper=
{{Paper
|id=Vol-2556/paper19
|storemode=property
|title=Finding and Visualizing of Limit Cycles (short paper)
|pdfUrl=https://ceur-ws.org/Vol-2556/paper19.pdf
|volume=Vol-2556
|authors=Yuri K. Dem’yanovich,Aleksey A. Fefelov
}}
==Finding and Visualizing of Limit Cycles (short paper)==
https://ceur-ws.org/Vol-2556/paper19.pdf
Finding and Visualizing of Limit Cycles
Yuri K.Dem'yanovich Aleksey A.Fefelov
Department of Electrical Departmen of Mathematics
Emperor Alexander I and Mechanics
St. Petersburg State St. Petersburg State University
Transport University Saint Petersburg, Russia
Saint Petersburg, Russia fefaleksey@mail.ru
Yuri.Demjanovich@gmail.com
Abstract of various devices reduced to the definition of steady
states for a certain system of differential equations.
The article reflects a study aimed at using Inadequate investigation of the sustainability can lead
a parallel computing system for automated to catastrophic consequen\-ces for designed devices
retrieval and visualization of cycles of a and apparatuses. Hopping from a planned state of
quadratic system of two differential equations. resistance to an unplanned condition repeatedly led to
The study was conducted in the seven- the destruction of bridges and structures, railway
dimensional space of parameters --- system accidents, the total destruction of aircraft, etc.
coefficients and initial data of the Cauchy Let a pair of functions x (t), y (t) be a solution to
problem. It is very important for sustainable the Cauchy problem for a system of two differential
operation of transport systems. To implement equations of the form
the calculations, supercomputers of Moscow x'=P(x,y) (1)
State University were remotely used. y'=Q(x,y), (2)
Visualization of the results carried out x(0)=x0,y(0)=y0. (3)
on Hewlett Packard personal computers. The The behavior of the trajectory of the point
developed software model is applicable to (x (t), y (t)) is important in the plane (x, y) with
weaning and visualization of cycles for increasing parameter t ( t usually represents the time
different systems of two differential equations. of the corresponding physical system). The mentioned
trajectories are called phase trajectories, and the plane
1. Introduction (x, y) is the phase plane. Steady state defined by
stability points and limit cycles (the word "cycle"
A.Poincare examined the geometric pattern means a closed curve). Limit cycles (attractors) are
solutions of the differential equation. In 1900, characterized by the fact that they are approached
D.Hilbert set the task of research limit cycles (wind on them) by phase trajectories (i.e. they are
(attractors) in two-dimensional quadratic systems attraction cycles). Repulsion points and cycles may
(see [Per01], [Li03], [Leo14], [Ruz15], [Leo15], also exist: phase trajectories are wound from them.
[Leo17]). In the fifties A. N. Kolmogorov suggested Note that between two nested limit cycles (they
estimating the number of these cycles. correspond steady states) there is always a cycle
In his book [Arn05] V. I. Arnold wrote: "To repulsion (it corresponds to an unstable state).
estimate the number of limit cycles quadratic vector The aforementioned tragedies show an urgent need
fields on the plane, A. N. Kolmogorov distributed to develop reliable methods of finding attraction cycles
several hundred such fields (with random selected and repulsion cycles.
coefficients of polynomials of the second degree) In the second half of the twentieth century, a large
to several hundred students at the mechanical and number of theoretical papers in which the existence of
mathematical Faculty of Moscow State University as a limit cycles and the boundaries of the parameters are
mathematical workshop. Each student had to find the indicated, where they should be searched (we skip the
number of limit cycles of their field.The results of this review of these works).
experiment were completely unexpected: there were no Unfortunately, in the vast majority of cases, there
limit cycles at all." are no analytical methods (formulas) for determining
It is well known that the definition of sustainability these cycles. In view of this, for finding cycles have
Copyright c by the paper's authors. Use permitted under Creative become widely used in modern computing methods
Commons License Attribution 4.0 International (CC BY 4.0). In: A. and computers.
Khomonenko, B. Sokolov, K. Ivanova (eds.): Selected Papers of the
Models and Methods of Information Systems Research
Workshop, St. Petersburg, Russia, 4-5 Dec. 2019, published at
http://ceur-ws.org
108
The exception of unplanned states of transport showed significant advantage of this method (relative
devices, and bridge and tunnel structures come down to computational speed) compared to the high-
to finding all attraction and repulsion cycles for phase precision Gear's method used by other researchers. In
trajectories of the Cauchy problem this work, the cycles of attraction and repulsion cycles
x'=x*x+x*y+y, (4) are automatically determined.
y'=a*x*x+b*x*y+c*y*y+alpha*x+beta*y (5) Note that between every two nested attractors there
x(0)=x0, y(0)=y0. (6) is repulsion cycle (the cycle of unstable equilibrium).
The selection of the seven parameters appearing The definition of the location of these cycles is very
here a, b, c, alpha,beta,x0,y0 should exclude device important in calculating stability in the case of
jump from a planned attractor to an unplanned one. designing mechanisms and structures (unstable
Such a jump may cause the device to malfunction and equilibrium cycle unsafe for designed devices).
even to its complete destruction. For reliable results in Initial testing of algorithms and programs was
points of the selected region of the seven-dimensional conducted in uniprocessor mode, and then with
parameter space all cycles (attractive and repulsive) parallelization emulation with an MPI interface on a
have to be defined, and they have to be presented with laptop and on a parallel cluster. After that, work was
a video monitor. Problem (4) - (6) is a special case of a carried out remotely on supercomputers "Chebyshev"
more general problem (1) - (3). and "Lomonosov-1" of Supercomputer Research
The problem of finding and visualizing cycles was Computing Center of Moscow State University. The
solved, thanks to the use of modern computing tools most interesting calculation results in the latter case
and high-speed computing systems. To solve this were automatically saved on the supercomputer,
problem, Professor G.A. Leonov formed a group then sent and autonomously visualized on the HP
with researchers who conducted a series of numerical 27-p251ur All-in-One and on the HP Pavilon x360
experiments using various methods on computers of Convertable Notebook (Figures 1 -- 3 show the results
various types. of some visualizations).
Due to difficulties in processing seven-dimensional The results can be used in calculating and
parameter spaces to these studies, the authors of this designing various devices, as well as for
work were also involved in the organization of parallel checking reliability of created designs. The simple
computing on a super\-computer. modification of algorithms and programs allows you
This article reflects a study aimed at using to use the program in the case of solving similar
a parallel computing system for automated retrieval problems for other autonomous systems of differential
and visualization of cycles of a quadratic system of equations.
two differential equations in the seven-dimensional
space of parameters. The parameters are the 3 Results
coefficients systems and initial data of the Cauchy
problem for the mentioned system.
When implementing calculations, supercompu- 3.1 First Series of Values Parameter beta
ters "Chebyshev" and "Lomonosov-1" of Super-
computer Research Computing Center of Moscow
State University are remotely used. The visualization Search for limit cycles (attractors) for a set of
of the obtained cycles is carried out on HP 27-p251 parameters a=-10.0, b=2.7, c=0.4, alpha=-473.5,
ur All-in-One and HP Pavilon x360 Convertable beta=0.003-epsilon, epsilon=s*0.0000000001,
Notebook PC. The developed software model is s=0,1,…, 1000, in each of these options led to 3
applicable to the finding and visualization of cycles attractors (the case epsilon=0 with a gradual expansion
for different systems of two differential equations. of the study area, see Fig. 1 -- 3).
3.2 Second Series of Values Parameter beta
2 Methods and Algorithms
To solve problem (3) - (4), the authors use the Runge- When searching for limit cycles (attractors) in another
Kutta method of fourth order precision with the set parameters, namely, for a=-10.0, b=2.7, c=0.4,
automatic choice of step. A numerical experiment
alpha=-173.5,beta=0.004+ 0.0001*s, s=0,1,…,9950.
109
In each of the options listed, exactly one attractor space of parameters. The developed software model
appeared. is obviously applicable to weaning and visualization
of cycles for different systems of two differential
equations.
3.3 Pair (b, c) gets 32 Million Values
Here we consider 32 million of the pairs of parameter (b,c)
according to the next formulas
a=(b-1)*(b-1)/(4*(c-1)+1), b=2.1+0.0001*s, s=0,1,…,8000,
c=0.5+0.0001*p, p=0,1,…,4000, alpha=a*(2+b)/(b*c-1)
+|a*(2+b)/( b*c-1)|/2, beta=0.
Here, in each variant, three attractors appeared, but for
some parameters (and for small perturbation of the
parameter \beta) a fourth attractor arose. The
occurrence of the fourth attractor cannot be considered
reliable because rounding errors are occurring when
floating point is used. Therefore we will not discuss
the fourth attractor. Figure 1: Visualization of wound trajectories
3.4 Wavelet Decomposition
To speed up data transfer was considered a first-order
wavelet decomposition with the following parameters
a=-10.0, b=2.7, c=0.4, alpha=-173.5,beta=0.003.
1000 Cauchy problems were solved with the initial data
(x_0, y_0) = (j, 0) ; here j = 1,2, \ldots, 1000. The resulting
sets of values were saved and then divided into the main and
wavelet streams (the main stream turned out to be about 2
times less source). Next, to another computer via ssh-
protocol source and main streams are transferred. Let T0
be the transmission time of the original flow, let T1 be the
transmission time of the main flow, and let k be their
ratio, i.e. k = T0 / T1. In the described numerical experiment,
the coefficient k turned out to be 1.92. Thus this indicates Figure 2: Expansion of the area. More wound
savings significant resource when the wavelet trajectories
decomposition is used.
4 Conclusion
This study showed that the problem of finding and
visualization of cycles in multidimensional (in seven-
dimensional) space of parameters can be solved using
modern computational algorithms and high-speed
parallel computing systems. In particular, the
application of the Runge-Kutta method proved to be
very effective. The usage of a parallel computing
system for automated search and visualization of
cycles quadratic system of two differential equations
gives very precise results in the seven-dimensional Figure 3: Visualization of three cycles
110
5 Acknowledgments
This work was supported by Botan Investments and [Kuz15] Kuznetsov, N.V., Leonov, G.A., Shumafov, M.M.:
Supercomputer Research Computing Center of A Short Survey on Pyragas Time-delay Feedback
Stabilization and Odd Number Limitation. In:
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Science\-Direct, 48-11, IFAC-PapersOnLine, 2015. P.
706-709.
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