Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture
Kuznetsov, N. V., Kuznetsova, O. A., Mokaev, T. N., Mokaev, R. N., Yuldashev, M. V., & Yuldashev, R. V. (2019). Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture. In L. Jadachowski (Ed.), 11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019 Vienna, Austria, 4-6 September 2019 (52, pp. 7-12). IFAC; Elsevier. IFAC-PapersOnLine. https://doi.org/10.1016/j.ifacol.2019.11.747
Published in
IFAC-PapersOnLineAuthors
Editors
Date
2019Copyright
© 2019 IFAC
The Aizerman and Kalman conjectures played an important role in the theory of global stability for control systems and set two directions for its further development – the search and formulation of sufficient stability conditions, as well as the construction of counterexamples for these conjectures. From the computational perspective the latter problem is nontrivial, since the oscillations in counterexamples are hidden, i.e. their basin of attraction does not intersect with a small neighborhood of an equilibrium. Numerical calculation of initial data of such oscillations for their visualization is a challenging problem. Up to now all known counterexamples to the Kalman conjecture were constructed in such a way that one locally stable limit cycle (hidden oscillation) co-exists with a locally stable equilibrium. In this paper we demonstrate a multistable configuration of three co-existing hidden oscillations (limit cycles) and a locally stable equilibrium in the phase space of the fourth-order system, which provides a new class of counterexamples to the Kalman conjecture.
...
Publisher
IFAC; ElsevierConference
IFAC Symposium on Nonlinear Control SystemsIs part of publication
11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019 Vienna, Austria, 4-6 September 2019ISSN Search the Publication Forum
2405-8971Keywords
Publication in research information system
https://converis.jyu.fi/converis/portal/detail/Publication/33903502
Metadata
Show full item recordCollections
Additional information about funding
We acknowledge support form the Russian Scientific Foundation (project 19-41-02002, sections 2-4) and the Leading Scientific Schools of Russia (project NSh-2858.2018.1, section 1).License
Related items
Showing items with similar title or keywords.
-
Counterexamples to the Kalman Conjectures
Kuznetsov, Nikolay; Kuznetsova, O. A.; Koznov, D. V.; Mokaev, R. N.; Andrievsky, B. (IFAC; Elsevier Ltd., 2018)In the paper counterexamples to the Kalman conjecture with smooth nonlinearity basing on the Fitts system, that are periodic solution or hidden chaotic attractor are presented. It is shown, that despite the fact that ... -
The Lorenz system : hidden boundary of practical stability and the Lyapunov dimension
Kuznetsov, N. V.; Mokaev, T. N.; Kuznetsova, O. A.; Kudryashova, E. V. (Springer, 2020)On the example of the famous Lorenz system, the difficulties and opportunities of reliable numerical analysis of chaotic dynamical systems are discussed in this article. For the Lorenz system, the boundaries of global ... -
Hidden and self-excited attractors in radiophysical and biophysical models
Stankevich, Nataliya (University of Jyväskylä, 2017)One of the central tasks of investigation of dynamical systems is the problem of analysis of the steady (limiting) behavior of the system after the completion of transient processes, i.e., the problem of localization and ... -
The Egan problem on the pull-in range of type 2 PLLs
Kuznetsov, Nikolay V.; Lobachev, Mikhail Y.; Yuldashev, Marat V.; Yuldashev, Renat V. (Institute of Electrical and Electronics Engineers (IEEE), 2021)In 1981, famous engineer William F. Egan conjectured that a higher-order type 2 PLL with an infinite hold-in range also has an infinite pull-in range, and supported his conjecture with some third-order PLL implementations. ... -
Harmonic balance analysis of pull-in range and oscillatory behavior of third-order type 2 analog PLLs
Kuznetsov, N.V.; Lobachev, M.Y.; Yuldashev, M.V.; Yuldashev, R.V.; Kolumbán, G. (Elsevier, 2020)The most important design parameters of each phase-locked loop (PLL) are the local and global stability properties, and the pull-in range. To extend the pull-in range, engineers often use type 2 PLLs. However, the engineering ...