A higher dimensional generalization of Bendixon's criterion

A. Pogromsky; G. Leonov; H. Nijmeijer

doi:10.3182/20020721-6-es-1901.00289

A higher dimensional generalization of Bendixon's criterion

A. Pogromsky, G. Leonov, H. Nijmeijer

Dynamics and Control

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Abstract

Conditions are given that guarantee the nonexistence of periodic orbits lying entirely in a simply connected set. The conditions are formulated in terms of matrix inequalities involving the variational equation. For systems defined in R the conditions are equivalent to Bendixon's criterion. A connection with analytic estimates of the Hausdorff dimension of invariant compact sets is emphasized.

Original language	English
Pages (from-to)	235-240
Number of pages	6
Journal	IFAC Proceedings Volumes
Volume	35
Issue number	1
DOIs	https://doi.org/10.3182/20020721-6-es-1901.00289
Publication status	Published - 2002
Event	15th World Congress of the International Federation of Automatic Control (IFAC 2002 World Congress) - Barcelona, Spain Duration: 21 Jul 2002 → 26 Jul 2002 Conference number: 15

Keywords

Direct Lyapunov method
Linearization
Periodic solutions

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10.3182/20020721-6-es-1901.00289

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abstract = "Conditions are given that guarantee the nonexistence of periodic orbits lying entirely in a simply connected set. The conditions are formulated in terms of matrix inequalities involving the variational equation. For systems defined in R the conditions are equivalent to Bendixon's criterion. A connection with analytic estimates of the Hausdorff dimension of invariant compact sets is emphasized.",

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AU - Leonov, G.

AU - Nijmeijer, H.

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N2 - Conditions are given that guarantee the nonexistence of periodic orbits lying entirely in a simply connected set. The conditions are formulated in terms of matrix inequalities involving the variational equation. For systems defined in R the conditions are equivalent to Bendixon's criterion. A connection with analytic estimates of the Hausdorff dimension of invariant compact sets is emphasized.

AB - Conditions are given that guarantee the nonexistence of periodic orbits lying entirely in a simply connected set. The conditions are formulated in terms of matrix inequalities involving the variational equation. For systems defined in R the conditions are equivalent to Bendixon's criterion. A connection with analytic estimates of the Hausdorff dimension of invariant compact sets is emphasized.

KW - Direct Lyapunov method

KW - Linearization

KW - Periodic solutions

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DO - 10.3182/20020721-6-es-1901.00289

M3 - Conference article

SN - 1474-6670

VL - 35

SP - 235

EP - 240

JO - IFAC Proceedings Volumes

JF - IFAC Proceedings Volumes

IS - 1

T2 - 15th World Congress of the International Federation of Automatic Control (IFAC 2002 World Congress)

Y2 - 21 July 2002 through 26 July 2002

ER -

A higher dimensional generalization of Bendixon's criterion

Abstract

Keywords

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