Center conditions for nilpotent cubic systems using the Cherkas method

doi:10.1016/j.matcom.2016.04.002

Mathematics and Computers in Simulation

Volume 129, November 2016, Pages 1-9

https://doi.org/10.1016/j.matcom.2016.04.002 Get rights and content

Abstract

In this study, we consider the center problem of a cubic polynomial differential system with a nilpotent linear part. The analysis is based on the application of the Cherkas method to the Takens normal form. The analysis requires many computations, which are verified by employing one algebraic manipulator and extensive use of the computer algebra system called Singular.

Section snippets

Introduction and preliminary results

We consider the following differential system with a nilpotent linear part $\dot{x} = y + P (x, y), \dot{y} = Q (x, y),$ where $P$ and $Q$ are analytic in a neighborhood of the origin without constants as linear terms. We assume that the origin is an isolated singular point. The monodromy problem involves characterizing when a singular point is either a focus or a center. Andreev [4] solved this problem for nilpotent singular points.

Theorem 1.1 Andreev

Let $y = ϕ (x)$ be the solution of $y + P (x, y) = 0$ that passes through the origin. Consider the

Proof of Theorem 1.7

By applying Theorem 1.1 to system (1.10), we find that the solution passing through the origin has the form $ϕ (x) = - a_{2} x^{2} + (a_{2} b_{1} - a_{3}) x^{3} + O (x^{3})$ , and by computing the first terms of the functions $ψ (x)$ and $Δ (x)$ , we obtain $ψ (x) = c_{2} x^{2} + (c_{3} - a_{2} d_{1}) x^{3} + O (x^{4}),$ $Δ (x) = (2 a_{2} + d_{1}) x + (3 a_{3} - a_{2} b_{1} + d_{2} - 2 a_{2} e_{0}) x^{2} + O (x^{3}) .$ A first necessary condition to have a center or a focus for system (1.10) is that the function $ψ (x)$ has a power development that begins with terms of odd degree. Hence, we require that $ψ_{2} = c_{2}$ is equal to zero.

Proof of Theorem 1.8

In order to apply Theorem 1.5, we consider the analytic function (1.9), which we write as $F (ξ (u)) = \sum_{i = 1}^{\infty} f_{i} u^{i} .$ Theorem 1.5 states that all the centers of (1.8) are characterized by the condition that $f_{2 i - 1} = 0$ for $i \geq 1$ . Therefore, computing the necessary conditions requires the computation of the function $ξ (u)$ , which can be achieved by a formal substitution of $ξ (u)$ and solving the recursive linear system obtained from the equation $u = ξ (u) \sqrt[2 l]{2 l \frac{G (ξ (u))}{ξ {(u)}^{2 l}}} .$ Next, we substitute the expression of $ξ (u)$

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^☆: The author was partially supported by MINECO/FEDER grant number MTM2014-53703-P and AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204.

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Original articlesCenter conditions for nilpotent cubic systems using the Cherkas method☆

Abstract

Section snippets

Introduction and preliminary results

Proof of Theorem 1.7

Proof of Theorem 1.8

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Original articles
Center conditions for nilpotent cubic systems using the Cherkas method☆