Effective analytical-numerical methods for the study of regular and chaotic oscillations in dynamical systems
Julkaistu sarjassa
JYU dissertationsTekijät
Päivämäärä
2019Tekijänoikeudet
© The Author & University of Jyväskylä
This dissertation examines the difficulties in analyzing the onset of oscillations in
the process of loss of stability in various nonlinear dynamical systems. The study
of the onset of oscillations originated with the discovery of periodic regimes in
automatic control systems, as well as with the discovery of chaos associated with
attempts to explain a laminar fluid flow becoming turbulent. One of the first
methods revealing and analyzing stability of periodic oscillations applied to automatic
control systems with one scalar nonlinearity was the Andronov pointmapping
method, which is applicable only to piecewise linear systems of low
order. Van der Pol, Krylov and Bogolyubov suggested the harmonic balance
method, which is applicable to systems of arbitrary dimension with scalar nonlinearity
of a general form. However, this method is approximate and may incorrectly
predict the loss of stability and existence of oscillations.
In this dissertation, for systems with one scalar nonlinearity, the discussion
of the classical harmonic balance and the point-mapping methods has been carried
out. Advantages and disadvantages of the locus of a perturbed relay system
(LPRS) method, which is an extension of the harmonic balance method, were discussed
and new examples demonstrating difficulties of studying scenarios of the
loss of stability and onset of oscillations in relay systems were presented.
None of the above mentioned methods are applicable when oscillations
emerging in the system after the loss of stability demonstrate complex chaotic
behavior. Such phenomenon was first noticed by famous scientist Lorenz in the
study of turbulent convection of a fluid layer. One of the first explanations to the
birth of such oscillations was given via a homoclinic bifurcation, in which a homoclinic
oscillation appears in the phase space. In general, proving the existence
of a homoclinic oscillation and giving a full description of the loss of stability and
the onset of chaos via a homoclinic bifurcation remain open challenges.
In this dissertation, for a class of Lorenz-like systems, the conditions of the
existence of a homoclinic oscillation have been analytically obtained and a numerical
investigation of several new homoclinic bifurcation scenarios have been
carried out. For the Lorenz system, to visualize unstable periodic oscillations,
which may appear during homoclinic bifurcations and are embedded in chaotic
attractor, the Pyragas control algorithm has been implemented.
Keywords: global stability, periodic and homoclinic oscillations, chaos
...
Julkaisija
Jyväskylän yliopistoISBN
978-951-39-7989-8ISSN Hae Julkaisufoorumista
2489-9003Julkaisuun sisältyy osajulkaisuja
- Artikkeli I: G.A. Leonov, N.V. Kuznetsov, M.A. Kiseleva, R.N. Mokaev. (2017). Global Problems for Differential Inclusions. Kalman and Vyshnegradskii Problems and Chua Circuits. Differential Equations, Vol. 53 (13), 1671–1702. DOI: 10.1134/S0012266117130018
- Artikkeli II: E.D. Akimova, I.M. Boiko, N.V. Kuznetsov, R.N. Mokaev (2019). Analysis of oscillations in discontinuous Lurie systems via LPRS method. Vibroengineering PROCEDIA, Vol. 25, PP. 177–181. DOI: 10.21595/vp.2019.20817
- Artikkeli III: N.V. Kuznetsov, O.A. Kuznetsova, D.V. Koznov, R.N. Mokaev, B.R.Andrievsky (2018). Counterexamples to the Kalman Conjectures. IFAC-PapersOnLine 51,I.33, 138–143. DOI: 10.1016/j.ifacol.2018.12.107
- Artikkeli IV: N.V. Kuznetsov, O.A. Kuznetsova, T.N. Mokaev, R.N. Mokaev, M.V. Yul-dashev, R.V. Yuldashev (2019). Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture. Proceedings of the11thIFAC Symposium on Nonlinear Control Systems. Accepted to IFAC-PapersOnLine. DOI: 10.1016/j.ifacol.2019.11.747
- Artikkeli V:E.V. Kudryashova; E.V., Kuznetsov; N.V., Kuznetsova; O.A., Leonov; G.A.,Mokaev; R.N. (2019). Harmonic Balance Method and Stability of Discontinuous Systems. In Matveenko V., Krommer M., Belyaev A., Irschik H. (eds) Dynamicsand Control of Advanced Structures and Machines. Springer, Cham, 99–107. DOI: 10.1007/978-3-319-90884-7_11
- Artikkeli VI: N.V. Kuznetsov, T.N. Mokaev, E.V. Kudryashova, O.A. Kuznetsova, R.N.Mokaev, M.V. Yuldashev, R.V. Yuldashev (2020). Stability and Chaotic Attractors of Memristor-Based Circuit with a Line of Equilibria. Lecture Notes in Electrical Engineering, 639–644. DOI: 10.1007/978-3-030-14907-9_62
- Artikkeli VII: G.A. Leonov, R.N. Mokaev, N.V. Kuznetsov, T.N. Mokaev (2020). Homoclinic Bifurcations and Chaos in the Fishing Principle for the Lorenz-like Systems. International Journal of Bifurcation and Chaos, Vol. 30. DOI: 10.1142/S0218127420501242
- Artikkeli VIII: N.V. Kuznetsov, T.N. Mokaev, R.N. Mokaev, O.A. Kuznetsova, E.V. Kudryashova (2019). A lower-bound estimate of the Lyapunov dimension for the global attractor of the Lorenz system. Preprint. Arxiv:1910.08740
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