TY - GEN

T1 - Hidden attractors in dynamical systems

T2 - 19th IFAC World Congress on International Federation of Automatic Control, IFAC 2014

AU - Kuznetsov, N. V.

AU - Leonov, G. A.

N1 - Publisher Copyright: © IFAC.

PY - 2014

Y1 - 2014

N2 - From a computational point of view it is natural to suggest the classification of attractors, based on the simplicity of finding basin of attraction in the phase space: an attractor is called a hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called a self-excited attractor. Self-excited attractors can be localized numerically by the standard computational procedure, in which after a transient process a trajectory, started from a point of unstable manifold in a neighborhood of unstable equilibrium, is attracted to the state of oscillation and traces it. Thus one can determine unstable equilibria and check the existence of self-excited attractors. In contrast, for the numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures in which an initial point can be chosen from the basin of attraction analytically. For example, hidden attractors are attractors in the systems with no-equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors); hidden attractors arise in the study of well-known fundamental problems such as 16th Hilbert problem, Aizerman and Kalman conjectures, and in applied research of Chua circuits, phase-locked loop based circuits, aircraft control systems, and others.

AB - From a computational point of view it is natural to suggest the classification of attractors, based on the simplicity of finding basin of attraction in the phase space: an attractor is called a hidden attractor if its basin of attraction does not intersect with small neighborhoods of equilibria, otherwise it is called a self-excited attractor. Self-excited attractors can be localized numerically by the standard computational procedure, in which after a transient process a trajectory, started from a point of unstable manifold in a neighborhood of unstable equilibrium, is attracted to the state of oscillation and traces it. Thus one can determine unstable equilibria and check the existence of self-excited attractors. In contrast, for the numerical localization of hidden attractors it is necessary to develop special analytical-numerical procedures in which an initial point can be chosen from the basin of attraction analytically. For example, hidden attractors are attractors in the systems with no-equilibria or with the only stable equilibrium (a special case of multistability and coexistence of attractors); hidden attractors arise in the study of well-known fundamental problems such as 16th Hilbert problem, Aizerman and Kalman conjectures, and in applied research of Chua circuits, phase-locked loop based circuits, aircraft control systems, and others.

KW - 16th Hilbert problem

KW - Absolute stability

KW - Aircraft control systems

KW - Aizerman conjecture

KW - Chua circuits

KW - Coexistence of attractors

KW - Coexisting attractors

KW - Describing function method

KW - Drilling system

KW - Harmonic balance

KW - Hidden attractor

KW - Hidden oscillation

KW - Kalman conjecture

KW - Multistability

KW - Multistable systems

KW - Nested limit cycles

KW - Nonlinear control system

KW - Phase-locked loop (PLL)

KW - Systems with no equilibria

UR - http://www.scopus.com/inward/record.url?scp=84929633386&partnerID=8YFLogxK

U2 - 10.3182/20140824-6-za-1003.02501

DO - 10.3182/20140824-6-za-1003.02501

M3 - Conference contribution

T3 - IFAC Proceedings Volumes (IFAC-PapersOnline)

SP - 5445

EP - 5454

BT - 19th IFAC World Congress IFAC 2014, Proceedings

A2 - Boje, Edward

A2 - Xia, Xiaohua

PB - International Federation of Automatic Control

Y2 - 24 August 2014 through 29 August 2014

ER -