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

Part of the Advanced Courses in Mathematics CRM Barcelona book series (ACMBIRK)


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
$$ H(x,y) = \frac{1} {2}(x^2 + y^2 - )\frac{1} {3}x^3 + axy^2 + \frac{1} {3}by^3 , $$
where a, b are parameters lying in the open region
$$ G = \left\{ {(a,b): - \frac{1} {2} < a < 1,0 < b < (1 - a)\sqrt {1 + 2a} } \right\}. $$
Figure 1 (in Subsection 1.2.1) shows all five possible phase portraits of XH in the generic cases. Here XH is the Hamiltonian vector field corresponding to H, i.e.,
$$ X_H = H_y \frac{\partial } {{\partial x}} - H_x \frac{\partial } {{\partial y}}. $$
The vector field XH has a center at the origin in the (x, y)-plane, and the continuous family of ovals, surrounding the center, is
$$ \{ \gamma h\} \subset \{ (x,y):H(x,y) = h,0 < h < 1/6\} . $$
The oval γh shrinks to the center as h → 0, and the oval γh terminates at the saddle loop of the saddle point (1, 0) when h → 1/6.


Riccati Equation Tangent Line Tangent Point Hamiltonian Vector Quadratic System 
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© Birkhäuser Verlag 2007

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