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Contemporary Mathematics. Fundamental Directions, 2012, Volume 45, Pages 105–121 (Mi cmfd216)  
This article is cited in 26 scientific papers (total in 26 papers)

Hidden oscillations in dynamical systems. 16 Hilbert's problem, Aizerman's and Kalman's conjectures, hidden attractors in Chua's circuits

G. A. Leonova, N. V. Kuznetsovab

a St. Petersburg State University, Faculty of Mathematics and Mechanics, St. Petersburg, Russia
b University of Jyväskylä, Department of Mathematical Information Technology, Jyväskylä, Finland
Full-text PDF (4985 kB) Citations (26)
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Abstract: The present survey is devoted to efficient methods for localization of hidden oscillations in dynamical systems. Their application to Hilbert's sixteenth problem for quadratic systems, Aizerman's problem, and Kalman's problem on absolute stability of control systems, and to the localization of chaotic hidden attractors (the basin of attraction of which does not contain neighborhoods of equilibria) is considered. The synthesis of the describing function method with the applied bifurcation theory and numerical methods for computing hidden oscillations is described.
English version:
Journal of Mathematical Sciences, 2014, Volume 201, Issue 5, Pages 645–662
DOI: https://doi.org/10.1007/s10958-014-2017-6
Bibliographic databases:
Document Type: Article
UDC: 531.36+534.1
Language: Russian
Citation: G. A. Leonov, N. V. Kuznetsov, “Hidden oscillations in dynamical systems. 16 Hilbert's problem, Aizerman's and Kalman's conjectures, hidden attractors in Chua's circuits”, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 1, CMFD, 45, PFUR, M., 2012, 105–121; Journal of Mathematical Sciences, 201:5 (2014), 645–662
Citation in format AMSBIB
\Bibitem{LeoKuz12}
\by G.~A.~Leonov, N.~V.~Kuznetsov
\paper Hidden oscillations in dynamical systems. 16 Hilbert's problem, Aizerman's and Kalman's conjectures, hidden attractors in Chua's circuits
\inbook Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14--21, 2011). Part~1
\serial CMFD
\yr 2012
\vol 45
\pages 105--121
\publ PFUR
\publaddr M.
\mathnet{http://mi.mathnet.ru/cmfd216}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3087054}
\transl
\jour Journal of Mathematical Sciences
\yr 2014
\vol 201
\issue 5
\pages 645--662
\crossref{https://doi.org/10.1007/s10958-014-2017-6}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84905895534}
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    • This publication is cited in the following 26 articles:
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    2. Shaojuan Ma, Meibo Wang, Xiaoyan Ma, Hufei Li, “Multi-stability analysis and energy control for 4D hyperchaotic system with hidden attractors”, Eur. Phys. J. Spec. Top., 2023  crossref
    3. Alexander Pesterev, Yury Morozov, Lecture Notes in Computer Science, 14395, Optimization and Applications, 2023, 129  crossref
    4. Prakash Chandra Gupta, Piyush Pratap Singh, “Chaos, multistability and coexisting behaviours in small-scale grid: Impact of electromagnetic power, random wind energy, periodic load and additive white Gaussian noise”, Pramana - J Phys, 97:1 (2022)  crossref
    5. Fahimeh Nazarimehr, Mohammad-Ali Jafari, Sajad Jafari, Viet-Thanh Pham, Xiong Wang, Guanrong Chen, Emergence, Complexity and Computation, 40, Chaotic Systems with Multistability and Hidden Attractors, 2021, 565  crossref
    6. L. A. Klimina, “Method for Constructing Periodic Solutions of a Controlled Dynamic System with a Cylindrical Phase Space”, J. Comput. Syst. Sci. Int., 59:2 (2020), 139  crossref
    7. J. Kengne, H. Abdolmohammadi, V. Folifack Signing, S. Jafari, G. H. Kom, “Chaos and Coexisting Bifurcations in a Novel 3D Autonomous System with a Non-Hyperbolic Fixed Point: Theoretical Analysis and Electronic Circuit Implementation”, Braz J Phys, 50:4 (2020), 442  crossref
    8. V. M. Kuntsevich, “Bounded perturbations of nonlinear discrete systems: estimation of impact and minimization”, Autom. Remote Control, 80:9 (2019), 1574–1590  mathnet  crossref  crossref  isi  elib
    9. Mahdi Nourian Zavareh, Fahimeh Nazarimehr, Karthikeyan Rajagopal, Sajad Jafari, “Hidden Attractor in a Passive Motion Model of Compass-Gait Robot”, Int. J. Bifurcation Chaos, 28:14 (2018), 1850171  crossref
    10. Sara Kamali, Shahriar Gharibzadeh, Sajad Jafari, “A new look to coma from the viewpoint of nonlinear dynamics”, Nonlinear Dyn, 92:4 (2018), 2119  crossref
    11. Fahimeh Nazarimehr, Sajad Jafari, Seyed Mohammad Reza Hashemi Golpayegani, J. C. Sprott, “Categorizing Chaotic Flows from the Viewpoint of Fixed Points and Perpetual Points”, Int. J. Bifurcation Chaos, 27:02 (2017), 1750023  crossref
    12. Fahimeh Nazarimehr, Batool Saedi, Sajad Jafari, J. C. Sprott, “Are Perpetual Points Sufficient for Locating Hidden Attractors?”, Int. J. Bifurcation Chaos, 27:03 (2017), 1750037  crossref
    13. Sifeu Takougang Kingni, Sajad Jafari, Viet-Thanh Pham, Paul Woafo, “Constructing and analyzing of a unique three-dimensional chaotic autonomous system exhibiting three families of hidden attractors”, Mathematics and Computers in Simulation, 132 (2017), 172  crossref
    14. Fahimeh Nazarimehr, Sajad Jafari, Guanrong Chen, Tomasz Kapitaniak, Nikolay V. Kuznetsov, Gennady A. Leonov, Chunbiao Li, Zhouchao Wei, “A Tribute to J. C. Sprott”, Int. J. Bifurcation Chaos, 27:14 (2017), 1750221  crossref
    15. Sajad Jafari, Viet-Thanh Pham, Tomasz Kapitaniak, “Multiscroll Chaotic Sea Obtained from a Simple 3D System Without Equilibrium”, Int. J. Bifurcation Chaos, 26:02 (2016), 1650031  crossref
    16. Dawid Dudkowski, Sajad Jafari, Tomasz Kapitaniak, Nikolay V. Kuznetsov, Gennady A. Leonov, Awadhesh Prasad, “Hidden attractors in dynamical systems”, Physics Reports, 637 (2016), 1  crossref
    17. Fuchen Zhang, Da Lin, Min Xiao, Huanrong Li, “Dynamical behaviors of the chaotic Brushless DC motors model”, Complexity, 21:4 (2016), 79  crossref
    18. Nikolay V. Kuznetsov, Lecture Notes in Electrical Engineering, 371, AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, 2016, 13  crossref
    19. Sifeu Takougang Kingni, Viet-Thanh Pham, Sajad Jafari, Guy Richard Kol, Paul Woafo, “Three-Dimensional Chaotic Autonomous System with a Circular Equilibrium: Analysis, Circuit Implementation and Its Fractional-Order Form”, Circuits Syst Signal Process, 35:6 (2016), 1933  crossref
    20. Sajad Jafari, Viet-Thanh Pham, S. Mohammad Reza Hashemi Golpayegani, Motahareh Moghtadaei, Sifeu Takougang Kingni, “The Relationship Between Chaotic Maps and Some Chaotic Systems with Hidden Attractors”, Int. J. Bifurcation Chaos, 26:13 (2016), 1650211  crossref
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