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L.A. Cherkas (1997)
The Dulac Function for Polynomial Autonomous Systems on a PlaneDiffer. Uravn., 33
F. Dumortier, Joan Ferragud, J. Llibre (2006)
Qualitative Theory of Planar Differential Systems
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A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier (1967)
Teoriya bifurkatsii dinamicheskikh sistem na ploskosti
R. Reißing, G. Sansone, Roberto Conti (1963)
Qualitative theorie Nichtlinearer Differentialgleichungen
L. Cherkas, A. Grin (2006)
Spline approximations in the problem of estimating the number of limit cycles of autonomous systems on the planeDifferential Equations, 42
A.A. Grin’, L.A. Cherkas (2005)
Extrema of the Andronov-Hopf Function of a Polynomial Lienard SystemDiffer. Uravn., 41
L. Cherkas, A. Grin, K. Schneider (2007)
On the approximation of the limit cycles functionElectronic Journal of Qualitative Theory of Differential Equations
L.A. Cherkas (1977)
Estimation Methods for the Number of Limit Cycles of Autonomous SystemsDiffer. Uravn., 13
A.A. Grin’ (2006)
Reduction to Transversality of Curves in the Problem, of the Construction of a Dulac FunctionDiffer. Uravn., 42
A. Grin, L. Cherkas (2005)
Extrema of the Andronov-Hopf function of a polynomial Lienard systemDifferential Equations, 41
A.A. Andronov, E.A. Leontovich, I.I. Gordon, A.G. Maier (1966)
Kachestvennaya teoriya dinamicheskikh sistem vtorogo poryadka
Yen-chʿien Yeh, Suihua Cai (2009)
Theory of Limit Cycles
A.A. Grin, K.R. Schneider (2007)
On Some Classes of Limit Cycles of Planar Dynamical SystemsDyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 14
L.A. Cherkas (2003)
An Estimate for the Number of Limit Cycles Using Critical Points of a Conditional ExtremumDiffer. Uravn., 39
S. Smale (1998)
Mathematical problems for the next centuryThe Mathematical Intelligencer, 20
L.A. Cherkas, A.A. Grin’ (2006)
Spline Approximations in the Problem of Estimating the Number of Limit Cycles of Autonomous Systems on a PlaneDiffer. Uravn., 42
A.A. Grin’, L.A. Cherkas (2000)
Dulac Function for Lienard SystemsTr. Inst. Mat. NAN Belarusi, 4
A. Andronov (1971)
Theory of bifurcations of dynamic systems on a plane = Teoriya bifurkatsii dinamicheskikh sistem na ploskosti
For autonomous systems on the real plane, we develop a regular method for localizing and estimating the number of limit cycles surrounding the unique singular point. The method is to divide the phase plane into annulus-shaped domains with transversal boundaries in each of which a Dulac function is constructed by solving an optimization problem, which permits one to use the Bendixson-Dulac criterion. We state the principle of reduction to global uniqueness and use it in the case of existence of an Andronov-Hopf function of limit cycles to obtain a sharp global estimate of the number of limit cycles for an individual system as well as for a one-parameter family of such systems in an unbounded domain.
Differential Equations – Springer Journals
Published: Mar 11, 2010
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