TY - JOUR

T1 - Hidden attractors in dynamical systems

T2 - From hidden oscillations in hilbert-kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in chua circuits

AU - Leonov, G. A.

AU - Kuznetsov, N. V.

N1 - Funding Information: This work was supported by the Academy of Finland, Russian Ministry of Education and Science Funding Information: (Federal target program), Russian Foundation for Basic Research and Saint-Petersburg State University.

PY - 2013/1

Y1 - 2013/1

N2 - From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors. At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes. Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit. This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods.

AB - From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect with small neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure. For localization of hidden attractors it is necessary to develop special procedures, since there are no similar transient processes leading to such attractors. At first, the problem of investigating hidden oscillations arose in the second part of Hilbert's 16th problem (1900). The first nontrivial results were obtained in Bautin's works, which were devoted to constructing nested limit cycles in quadratic systems, that showed the necessity of studying hidden oscillations for solving this problem. Later, the problem of analyzing hidden oscillations arose from engineering problems in automatic control. In the 50-60s of the last century, the investigations of widely known Markus-Yamabe's, Aizerman's, and Kalman's conjectures on absolute stability have led to the finding of hidden oscillations in automatic control systems with a unique stable stationary point. In 1961, Gubar revealed a gap in Kapranov's work on phase locked-loops (PLL) and showed the possibility of the existence of hidden oscillations in PLL. At the end of the last century, the difficulties in analyzing hidden oscillations arose in simulations of drilling systems and aircraft's control systems (anti-windup) which caused crashes. Further investigations on hidden oscillations were greatly encouraged by the present authors' discovery, in 2010 (for the first time), of chaotic hidden attractor in Chua's circuit. This survey is dedicated to efficient analytical-numerical methods for the study of hidden oscillations. Here, an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods.

KW - 16th Hilbert problem

KW - absolute stability

KW - Aizerman conjecture

KW - Chua circuits

KW - describing function method

KW - drilling system

KW - harmonic balance

KW - hidden attractor

KW - Hidden oscillation

KW - induction motor

KW - Kalman conjecture

KW - large (normal amplitude) and small limit cycle

KW - Lienard equation

KW - Lyapunov focus values (Lyapunov quantities Poincaré-Lyapunov constants Lyapunov coefficients)

KW - nonlinear control system

KW - phase-locked loop (PLL)

KW - quadratic system

UR - http://www.scopus.com/inward/record.url?scp=84874642094&partnerID=8YFLogxK

U2 - 10.1142/S0218127413300024

DO - 10.1142/S0218127413300024

M3 - Review article

VL - 23

JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering

SN - 0218-1274

IS - 1

M1 - 1330002

ER -