Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
Analytical-numerical methods for hidden attractors' localization : The 16th Hilbert problem, Aizerman and Kalman conjectures, and Chua circuits. / Leonov, Gennady A.; Kuznetsov, Nikolay V.
Numerical Methods for Differential Equations, Optimization, and Technological Problems. ed. / Sergey Repin; Timo Tiihonen; Tero Tuovinen. Springer Nature, 2013. p. 41-64 (Computational Methods in Applied Sciences; Vol. 27).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Research › peer-review
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TY - GEN
T1 - Analytical-numerical methods for hidden attractors' localization
T2 - ECCOMAS Thematic Conference Computational Analysis and Optimization, CAO 2011
AU - Leonov, Gennady A.
AU - Kuznetsov, Nikolay V.
N1 - Publisher Copyright: © 2013 Springer Science+Business Media Dordrecht.
PY - 2013
Y1 - 2013
N2 - This survey is devoted to analytical-numerical methods for hidden attractors' localization and their application to well-known problems and systems. From the computation point of view, in nonlinear dynamical systems the attractors can be regarded as self-exciting and hidden attractors. Self-exciting attractors can be localized numerically by the following standard computational procedure: after a transient process a trajectory, started from a point of an unstable manifold in a small neighborhood of unstable equilibrium, reaches an attractor and computes it. In contrast, a hidden attractor is an attractor whose basin of attraction does not contain neighborhoods of equilibria. In well-known Van der Pol, Belousov- Zhabotinsky, Lorenz, Chua, and many other dynamical systems classical attractors are self-exciting attractors and can be obtained numerically by the standard computational procedure. However, for localization of hidden attractors it is necessary to develop special analytical-numerical methods, in which at the first step the initial data are chosen in a basin of attraction and then the numerical localization (visualization) of the attractor is performed. The simplest examples of hidden attractors are internal nested limit cycles (hidden oscillations) in two-dimensional systems (see, e.g., the results concerning the second part of the 16th Hilbert's problem). Other examples of hidden oscillations are counterexamples to Aizerman's conjecture and Kalman's conjecture on absolute stability in the automatic control theory (a unique stable equilibrium coexists with a stable periodic solution in these counterexamples). In 2010, for the first time, a chaotic hidden attractor was computed first by the authors in a generalized Chua's circuit and then one chaotic hidden attractor was discovered in a classical Chua's circuit.
AB - This survey is devoted to analytical-numerical methods for hidden attractors' localization and their application to well-known problems and systems. From the computation point of view, in nonlinear dynamical systems the attractors can be regarded as self-exciting and hidden attractors. Self-exciting attractors can be localized numerically by the following standard computational procedure: after a transient process a trajectory, started from a point of an unstable manifold in a small neighborhood of unstable equilibrium, reaches an attractor and computes it. In contrast, a hidden attractor is an attractor whose basin of attraction does not contain neighborhoods of equilibria. In well-known Van der Pol, Belousov- Zhabotinsky, Lorenz, Chua, and many other dynamical systems classical attractors are self-exciting attractors and can be obtained numerically by the standard computational procedure. However, for localization of hidden attractors it is necessary to develop special analytical-numerical methods, in which at the first step the initial data are chosen in a basin of attraction and then the numerical localization (visualization) of the attractor is performed. The simplest examples of hidden attractors are internal nested limit cycles (hidden oscillations) in two-dimensional systems (see, e.g., the results concerning the second part of the 16th Hilbert's problem). Other examples of hidden oscillations are counterexamples to Aizerman's conjecture and Kalman's conjecture on absolute stability in the automatic control theory (a unique stable equilibrium coexists with a stable periodic solution in these counterexamples). In 2010, for the first time, a chaotic hidden attractor was computed first by the authors in a generalized Chua's circuit and then one chaotic hidden attractor was discovered in a classical Chua's circuit.
UR - http://www.scopus.com/inward/record.url?scp=84964265507&partnerID=8YFLogxK
U2 - 10.1007/978-94-007-5288-7_3
DO - 10.1007/978-94-007-5288-7_3
M3 - Conference contribution
AN - SCOPUS:84964265507
SN - 9789400752870
T3 - Computational Methods in Applied Sciences
SP - 41
EP - 64
BT - Numerical Methods for Differential Equations, Optimization, and Technological Problems
A2 - Repin, Sergey
A2 - Tiihonen, Timo
A2 - Tuovinen, Tero
PB - Springer Nature
Y2 - 9 June 2011 through 11 June 2011
ER -
ID: 95267833