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Solution of center-focus problem for a class of cubic systems
Chinese Annals of Mathematics, Series B, 2015The center-focus problem is solved for the system \[ \begin{aligned} \dot x &= y+ c_{2,0}x^2+ c_{3,0} x^3,\\ \dot y &=-x+ d_{2,0} x^2+ d_{2,1}xy+ d_{3,0}x^3+ d_{3,1} x^2y+ d_{3,2}xy^2.\end{aligned} \]
Sang, Bo, Niu, Chuanze
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Qualitative Theory of Dynamical Systems, 2023
Consider the planar polynomial system \[ \begin{array}{l} \frac{{dx}}{{dt}}= y(1+a_{11}x+a_{21}x^2+a_{31}x^3), \\ \frac{{dy}}{{dt}} = b_{20}x^2+b_{11}xy+b_{02}y^2+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3 \end{array}\tag{1} \] having the origin as nilpotent singular point.
Li, Feng +3 more
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Consider the planar polynomial system \[ \begin{array}{l} \frac{{dx}}{{dt}}= y(1+a_{11}x+a_{21}x^2+a_{31}x^3), \\ \frac{{dy}}{{dt}} = b_{20}x^2+b_{11}xy+b_{02}y^2+b_{30}x^3+b_{21}x^2y+b_{12}xy^2+b_{03}y^3 \end{array}\tag{1} \] having the origin as nilpotent singular point.
Li, Feng +3 more
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Solution of the center-focus problem for a nine-parameter cubic system
Differential Equations, 2011The center problem for the cubic system of differential equations \[ \dot x = y (1 + D x + P x^2), \quad \dot y = -x + A x^2 + 3 B x y + C y^2 + K x^3 + 3 L x^2 y + M x y^ 2 + N y^ 3 \tag{1} \] is studied. In [\textit{Y. L. Bondar} and \textit{A. P. Sadovskii}, Bul. Acad. Stiinte Repub. Mold., Mat. 2004, No.
Sadovskii, A. P., Shcheglova, T. V.
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On the center-focus problem for systems with a degenerate singular point
Differential Equations, 2008zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Sadovskii, A. P., Cherginets, D. N.
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Center-focus problem and limit cycles bifurcations for a class of cubic Kolmogorov model
Nonlinear Dynamics, 2012zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Du, Chaoxiong, Huang, Wentao
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Solution of the center-focus problem for a cubic system reducible to a lienard system
Differential Equations, 2006ẋ = y(1 +Dx), ẏ = −x ( 1 + a1x+ a2x + a3x + a4x + a5x ) + x ( b0 + b1x+ b2x ) y +Hy, (2) where ai, bj ,H ∈ C, i = 1, 2, 3, 4, 5, j = 0, 1, 2. System (2) with a4 = a5 = b2 = 0 was considered in [3, 4]. Note that the change of variables y(x) = [Y (x)− x(R+Qx)]/(1 +Dx) (we do not change the notation of y) reduces system (1) to system (2) with a1 = 2D −A ...
Yu. L. Bondar’, A. P. Sadovskii
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Center-focus problem for analytic systems with nonzero linear part
Differential Equations, 2017zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On the Center-Focus Problem for a Family of High-Order Polynomial Differential Systems
Journal of Dynamics and Differential Equations, 2019zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Differential Equations, 2003
The authors consider the center-focus problem for system \[ \dot x=y+X_{2n+1}(x,y),\quad \dot y=Y_{2n+1}(x,y).\tag{1} \] They solve the center-focus problem for system (1) for \(n=3\) using the formal series method and prove the existence of systems (1) for \(n=1,2,3\) with 2, 5 and 9 limit cycles, respectively, in a neighborhood of the origin.
Andreev, A. F. +2 more
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The authors consider the center-focus problem for system \[ \dot x=y+X_{2n+1}(x,y),\quad \dot y=Y_{2n+1}(x,y).\tag{1} \] They solve the center-focus problem for system (1) for \(n=3\) using the formal series method and prove the existence of systems (1) for \(n=1,2,3\) with 2, 5 and 9 limit cycles, respectively, in a neighborhood of the origin.
Andreev, A. F. +2 more
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Navigating financial toxicity in patients with cancer: A multidisciplinary management approach
Ca-A Cancer Journal for Clinicians, 2022Grace Li Smith +2 more
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