STABILITY ANALYSIS OF RIGID SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
Keywords:
Lyapunov stability, analysis of the stability of rigid systems, solution of rigid systems
Abstract
A method for analyzing stability in the sense of Lyapunov for systems of ordinary differential
equations is proposed. The method is based on stability criteria in the form of necessary and sufficient
conditions obtained on the basis of vector-matrix transformations of difference numerical
integration schemes. The varieties of criteria in multiplicative, additive and matrix form are presented.
The design of the criteria implies the possibility of their programmatic realization. To increase
the reliability of the stability analysis, the approximations of the solution included in the
construction of the criteria are based on piecewise interpolation approximation by Lagrange polynomials
converted to a form with numerical coefficients. A programming and numerical experiment
is carried out to analyze the stability of the Belousov-Jabotinsky periodic reaction model,
which belongs to the class of rigid systems, under given initial conditions. The analysis is carried
out on the basis of the presented criteria and the results of the program clearly determine the nature
of the stability in real time. Based on the results of the experiment, it can be argued that replacing
the difference approximations of the solution with piecewise interpolation approximations
increases the reliability of the stability analysis, reduces the study time, and makes it possible to
determine the asymptotic properties of the solution. In general, the proposed approach is an alternative
to the methods of the qualitative theory of differential equations and makes it possible to
reliably determine the stability of rigid systems of ordinary differential equations in real time.
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varying piecewise interpolation solution of the Cauchy problem for ordinary differential equations
with iterative refinement], Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki
[Computational Mathematics and Mathematical Physics Journal], 2017, Vol. 57, No. 10,
pp. 1616-1634.
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sistem [Computer-oriented stability analysis of solutions of differential systems], Sovremennye
naukoemkie tekhnologii [Modern high technologies], 2020, No. 4, pp. 42-63].
18. Bulanov S.G. Computer analysis of differential systems stability based on linearization and
matrix multiplicative criteria, Journal of Physics: Conf. Series, 2021, 1902 012101.
19. Khayrer E., Vanner G. Reshenie obyknovennykh differentsial'nykh uravneniy. Zhestkie i
differentsial'no-algebraicheskie zadachi [Solving ordinary differential equations. Rigid and differential-
algebraic problems]. Moscow: Mir, 1999, 685 p.
20. Doban A., Lazar M. Computation of Lyapunov functions for nonlinear differential equations
via a Yoshizawa-type construction, 10th IFAC Symp. on Nonlinear Control Systems NOLCOS:
IFAC-PapersOnLine, 2016, pp. 29-34.
21. Zhaolu T., Chuanqing G. A numerical algorithm for Lyapunov equations, J. Appl. Math.
Comput, 2008, Vol. 202, No. 1, pp. 44-53.
22. Xiao-Lin L., Yao-Lin J. Numerical algorithm for constructing Lyapunov functions of polynomial
differential systems, J. Appl. Math. Comput, 2009, Vol. 29, No. 1-2, pp. 247-262.
nelineynykh dinamicheskikh sistem pri postoyanno deystvuyushchikh vozmushcheniyakh
[tability of nonlinear dynamical system motion under constantly acting perturbations],
Nauchno-tekhnicheskiy vestnik informatsionnykh tekhnologiy, mekhaniki i optiki [Scientific
and technical journal of information technologies, mechanics and optics], 2019, Vol. 19, No. 2,
pp. 216-221.
2. Mironov V.V., Mitrokhin Yu.S. Tekhnologicheskiy podkhod k issledovaniyu ustoychivosti
dinamicheskikh sistem: prikladnye voprosy [Constructive approach to the research of dynamic
systems stability: applied problems], Vestnik RGRTU [Vestnik of RSREU]. 2017, No. 59,
pp. 127-135.
3. Aleksandrov A.Yu., Zhabko A.P., Kosov A.A. Analiz ustoychivosti i stabilizatsiya nelineynykh
sistem na osnove dekompozitsii [Analysis of stability and stabilization of nonlinear systems
via decomposition], Sibirskiy matematicheskiy zhurnal [Siberian mathematical journal]. 2015,
Vol. 56, No. 6, pp. 1215-1233.
4. Hammarling S.J. Numerical solution of the stable, non-negative definite Lyapunov equation,
IMA J. of Num. Analysis, 1982, Vol. 2, No. 3, pp. 303-323.
5. Luyckx L., Loccufier M., Noldus E. Computational methods in nonlinear stability analysis:
stability boundary calculations, J. Comput. Appl. Math, 2004, Vol. 168, No. 12, pp. 289-297.
6. Giesl P., Hafstein S. Computation of Lyapunov functions for nonlinear discrete time systems
by linear programming, J. Difference Equ. Appl, 2014, Vol. 20, No. 4, pp. 610-640.
7. Olgac N., Sipahi R. A practical method for analyzing the stability of neutral type LTI-time
delayed systems, Automatica, 2004, Vol. 40, No. 5, pp. 847-853.
8. Hafstein S. A constructive converse Lyapunov theorem on asymptotic stability for nonlinear
autonomous ordinary differential equations, Dynamical Systems, 2005, Vol. 20, pp. 281-299.
9. Romm Ya.E. Komp'yuterno-orientirovannyy analiz ustoychivosti na osnove rekurrentnykh
preobrazovaniy raznostnykh resheniy obyknovennykh differentsial'nykh uravneniy [Computeroriented
stability analysis based on recurrent transformation of difference solutions of ordinary
differential equations], Kibernetika i sistemnyy analiz [Cybernetics and Systems Analysis],
2015, Vol. 51, No. 3, pp. 107-124.
10. Bulanov S.G. Analiz ustoychivosti sistem lineynykh differentsial'nykh uravneniy na osnove
preobrazovaniya raznostnykh skhem [Stability analysis of systems of linear differential equations
based on transformation of difference schemes], Mekhatronika, avtomatizatsiya,
upravlenie [Mekhatronika, Avtomatizatsiya, Upravlenie], 2019, Vol. 20, No. 9, pp. 542-549.
11. Romm Ya.E., Bulanov S.G. Programmnye kriterii ustoychivosti resheniya zadachi Koshi dlya
sistem lineynykh differentsial'nykh uravneniy na osnove raznostnykh skhem chislennogo
integrirovaniya [Program stability criteria for the solution of the Cauchy problem for systems
of linear differential equations based on difference numerical integration schemes], Izvestiya
vuzov. Severo-Kavkazskiy region. Tekhnicheskie nauki. Spetsial'nyy vypusk «Matematicheskoe
modelirovanie i komp'yuternye tekhnologii» [University news. North-caucasian region. Technical
sciences series. Special issue «Mathematical Modeling and Computer Technologies»],
2004, pp. 75-80.
12. Romm Ya.E., Bulanov S.G. Chislennyy eksperiment po komp'yuternomu analizu ustoychivosti
resheniy obyknovennykh differentsial'nykh uravneniy na osnove kriteriev matrichnogo vida
[Computer analysis of the stability of systems of linear differential equations with nonlinear
addition], Dep. v VINITI [Dep. in VINITI], 14.08.17, No. 89, 20 p.
13. Romm Ya.E., Bulanov S.G. Komp'yuternyy analiz ustoychivosti sistem lineynykh
differentsial'nykh uravneniy s nelineynoy dobavkoy [Computer analysis of the stability of systems
of linear differential equations with nonlinear addition], Dep. v VINITI [Dep. in VINITI],
11.03.10, No. 147, 33 p.
14. Romm Ya.E. Modelirovanie ustoychivosti po Lyapunovu na osnove preobrazovaniy
raznostnykh skhem resheniy obyknovennykh differentsial'nykh uravneniy [Modeling of stability
according to Lyapunov based on difference schemes transformations for solutions of ordinary
differential equations], Matematicheskoe modelirovanie [Mathematical Modeling], 2008,
Vol. 20, No. 12, pp. 105-118.
15. Romm Ya.E., Dzhanunts G.A. Kusochnaya interpolyatsiya funktsiy, proizvodnykh i integralov
s prilozheniem k resheniyu obyknovennykh differentsial'nykh uravneniy [Piecewise interpolation
of functions, derivatives and integrals with application to the solution of ordinary differential
equations], Sovremennye naukoemkie tekhnologii [Modern high technologies], 2020, No.
12 (part 2), pp. 291-316.
16. Dzhanunts G.A., Romm Ya.E. Var'iruemoe kusochno-interpolyatsionnoe reshenie zadachi
Koshi dlya obyknovennykh differentsial'nykh uravneniy s iteratsionnym utochneniem [The
varying piecewise interpolation solution of the Cauchy problem for ordinary differential equations
with iterative refinement], Zhurnal vychislitel'noy matematiki i matematicheskoy fiziki
[Computational Mathematics and Mathematical Physics Journal], 2017, Vol. 57, No. 10,
pp. 1616-1634.
17. Romm Ya.E. Komp'yuterno-orientirovannyy analiz ustoychivosti resheniy differentsial'nykh
sistem [Computer-oriented stability analysis of solutions of differential systems], Sovremennye
naukoemkie tekhnologii [Modern high technologies], 2020, No. 4, pp. 42-63].
18. Bulanov S.G. Computer analysis of differential systems stability based on linearization and
matrix multiplicative criteria, Journal of Physics: Conf. Series, 2021, 1902 012101.
19. Khayrer E., Vanner G. Reshenie obyknovennykh differentsial'nykh uravneniy. Zhestkie i
differentsial'no-algebraicheskie zadachi [Solving ordinary differential equations. Rigid and differential-
algebraic problems]. Moscow: Mir, 1999, 685 p.
20. Doban A., Lazar M. Computation of Lyapunov functions for nonlinear differential equations
via a Yoshizawa-type construction, 10th IFAC Symp. on Nonlinear Control Systems NOLCOS:
IFAC-PapersOnLine, 2016, pp. 29-34.
21. Zhaolu T., Chuanqing G. A numerical algorithm for Lyapunov equations, J. Appl. Math.
Comput, 2008, Vol. 202, No. 1, pp. 44-53.
22. Xiao-Lin L., Yao-Lin J. Numerical algorithm for constructing Lyapunov functions of polynomial
differential systems, J. Appl. Math. Comput, 2009, Vol. 29, No. 1-2, pp. 247-262.
Published
2021-08-11
Issue
Section
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