Limit Cycles of Differential Equations pp 143-158 | Cite as

# A Unified Proof of the Weak Hilbert’s 16th Problem for n=2

Chapter

## Abstract

As we explained in Subsection 1.2.1, any cubic generic Hamiltonian, with at least one period annulus contained in its level curves, can be transformed into the normal form where Figure 1 (in Subsection 1.2.1) shows all five possible phase portraits of The vector field The oval

$$
H(x,y) = \frac{1}
{2}(x^2 + y^2 - )\frac{1}
{3}x^3 + axy^2 + \frac{1}
{3}by^3 ,
$$

(4.1)

*a, b*are parameters lying in the open region$$
G = \left\{ {(a,b): - \frac{1}
{2} < a < 1,0 < b < (1 - a)\sqrt {1 + 2a} } \right\}.
$$

(4.2)

*X*_{H}in the generic cases. Here*X*_{H}is the Hamiltonian vector field corresponding to*H*, i.e.,$$
X_H = H_y \frac{\partial }
{{\partial x}} - H_x \frac{\partial }
{{\partial y}}.
$$

(4.3)

*X*_{H}has a center at the origin in the (*x, y*)-plane, and the continuous family of ovals, surrounding the center, is$$
\{ \gamma h\} \subset \{ (x,y):H(x,y) = h,0 < h < 1/6\} .
$$

(4.4)

*γ*_{h}shrinks to the center as*h*→ 0, and the oval*γ*_{h}terminates at the saddle loop of the saddle point (1, 0) when*h*→ 1/6.## Keywords

Riccati Equation Tangent Line Tangent Point Hamiltonian Vector Quadratic System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Birkhäuser Verlag 2007