TY - JOUR

T1 - Algorithms for finding hidden oscillations in nonlinear systems. the Aizerman and Kalman conjectures and Chua's circuits

AU - Bragin, V. O.

AU - Vagaitsev, V. I.

AU - Kuznetsov, N. V.

AU - Leonov, G. A.

N1 - Funding Information: ACKNOWLEDGMENTS This work was financially supported by the Ministry of Education and Science of the Russian Federation, St. Petersburg University (grant no. 6.37.98.2011), and Academy of Finland (grant no. 138488).

PY - 2011/8

Y1 - 2011/8

N2 - An algorithm for searching hidden oscillations in dynamic systems is developed to help solve the Aizerman's, Kalman's and Markus-Yamabe's conjectures well-known in control theory. The first step of the algorithm consists in applying modified harmonic linearization methods. A strict mathematical substantiation of these methods is given using special Poincare maps. Subsequent steps of the proposed algorithms rely on the modern applied theory of bifurcations and numerical methods of solving differential equations. These algorithms help find and localize hidden strange attractors (i.e., such that a basin of attraction of which does not contain neighborhoods of equilibria), as well as hidden periodic oscillations. One of these algorithms is used here to discover, for the first time, a hidden strange attractor in the dynamic system describing a nonlinear Chua's circuit, viz. an electronic circuit with nonlinear feedback.

AB - An algorithm for searching hidden oscillations in dynamic systems is developed to help solve the Aizerman's, Kalman's and Markus-Yamabe's conjectures well-known in control theory. The first step of the algorithm consists in applying modified harmonic linearization methods. A strict mathematical substantiation of these methods is given using special Poincare maps. Subsequent steps of the proposed algorithms rely on the modern applied theory of bifurcations and numerical methods of solving differential equations. These algorithms help find and localize hidden strange attractors (i.e., such that a basin of attraction of which does not contain neighborhoods of equilibria), as well as hidden periodic oscillations. One of these algorithms is used here to discover, for the first time, a hidden strange attractor in the dynamic system describing a nonlinear Chua's circuit, viz. an electronic circuit with nonlinear feedback.

UR - http://www.scopus.com/inward/record.url?scp=80052988208&partnerID=8YFLogxK

U2 - 10.1134/S106423071104006X

DO - 10.1134/S106423071104006X

M3 - Article

VL - 50

SP - 511

EP - 543

JO - Journal of Computer and Systems Sciences International

JF - Journal of Computer and Systems Sciences International

SN - 1064-2307

IS - 4

ER -