THE HILBERT NUMBER OF A CLASS OF DIFFERENTIAL EQUATIONS

Jaume Llibre; Ammar Makhlouf

doi:10.11948/2015012

2015 Volume 5 Issue 1

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Jaume Llibre, Ammar Makhlouf. THE HILBERT NUMBER OF A CLASS OF DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2015, 5(1): 141-145. doi: 10.11948/2015012

Citation:

Jaume Llibre, Ammar Makhlouf. THE HILBERT NUMBER OF A CLASS OF DIFFERENTIAL EQUATIONS[J]. Journal of Applied Analysis & Computation, 2015, 5(1): 141-145. doi: 10.11948/2015012

THE HILBERT NUMBER OF A CLASS OF DIFFERENTIAL EQUATIONS

Jaume Llibre¹,
Ammar Makhlouf²

1 Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain;
2 Department of mathematics, UBMA University Annaba, Elhadjar, BP12, Annaba, Algeria

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Abstract

Abstract

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 dr/dθ=(a(θ))/???20150112??? a_j(θ)r^j (∗) defined in the region of the cylinder (θ, r) ∈ S¹×R where the denominator of (∗) does not vanish. Here a, a₀, a₁,…, a_n 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.
- Periodic orbit /
- averaging theory /
- trigonometric polynomial /
- Hilbert number
MSC: 34C29;37C27

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References

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