The Hilbert number of a class of differential equations

Jaume Llibre; Ammar Makhlouf

doi:10.11948/2015012

The Hilbert number of a class of differential equations

Jaume Llibre, Ammar Makhlouf

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

Abstract

© 2015, Journal of Applied Analysis and Computation. All Rights Reserved. The notion of Hilbert number from polynomial differential systems in the plane of degree n can be extended to the differential equations of the form (formula presented) defined in the region of the cylinder (θ; r) ∈ θS<sup>1</sup>× ℝ where the denominator of (*) does not vanish. Here a; a<inf>0</inf>; a<inf>1</inf>;...; a<inf>n</inf> are analytic 2π–periodic functions, and the Hilbert number ℍ (n) is the supremum of the number of limit cycles that any differential equation (*) on the cylinder of degree n in the variable r can have. We prove that ℍ (n) = ∞ for all n ≥ 1.

Original language	English
Pages (from-to)	141-145
Journal	Journal of Applied Analysis and Computation
Volume	5
Issue number	1
DOIs	https://doi.org/10.11948/2015012
Publication status	Published - 1 Jan 2015

Keywords

Averaging theory
Hilbert number
Periodic orbit
Trigonometric polynomial

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abstract = "{\textcopyright} 2015, Journal of Applied Analysis and Computation. All Rights Reserved. The notion of Hilbert number from polynomial differential systems in the plane of degree n can be extended to the differential equations of the form (formula presented) defined in the region of the cylinder (θ; r) ∈ θS1× ℝ where the denominator of (*) does not vanish. Here a; a0; a1;...; an are analytic 2π–periodic functions, and the Hilbert number ℍ (n) is the supremum of the number of limit cycles that any differential equation (*) on the cylinder of degree n in the variable r can have. We prove that ℍ (n) = ∞ for all n ≥ 1.",

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author = "Jaume Llibre and Ammar Makhlouf",

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N2 - © 2015, Journal of Applied Analysis and Computation. All Rights Reserved. The notion of Hilbert number from polynomial differential systems in the plane of degree n can be extended to the differential equations of the form (formula presented) defined in the region of the cylinder (θ; r) ∈ θS1× ℝ where the denominator of (*) does not vanish. Here a; a0; a1;...; an are analytic 2π–periodic functions, and the Hilbert number ℍ (n) is the supremum of the number of limit cycles that any differential equation (*) on the cylinder of degree n in the variable r can have. We prove that ℍ (n) = ∞ for all n ≥ 1.

AB - © 2015, Journal of Applied Analysis and Computation. All Rights Reserved. The notion of Hilbert number from polynomial differential systems in the plane of degree n can be extended to the differential equations of the form (formula presented) defined in the region of the cylinder (θ; r) ∈ θS1× ℝ where the denominator of (*) does not vanish. Here a; a0; a1;...; an are analytic 2π–periodic functions, and the Hilbert number ℍ (n) is the supremum of the number of limit cycles that any differential equation (*) on the cylinder of degree n in the variable r can have. We prove that ℍ (n) = ∞ for all n ≥ 1.

KW - Averaging theory

KW - Hilbert number

KW - Periodic orbit

KW - Trigonometric polynomial

U2 - 10.11948/2015012

DO - 10.11948/2015012

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SP - 141

EP - 145

JO - Journal of Applied Analysis and Computation

JF - Journal of Applied Analysis and Computation

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