\`x^2+y_1+z_12^34\`
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Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles

  • * Corresponding author: Claudia Valls

    * Corresponding author: Claudia Valls
The first author is partially supported by the MINECO grant MTM2016-77278-P (FEDER). The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013
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  • We provide normal forms and the global phase portraits in the Poincaré disk for all Hamiltonian planar polynomial vector fields of degree 3 symmetric with respect to the $ x- $axis having a nilpotent saddle at the origin.

    Mathematics Subject Classification: Primary: 34C05, 34C07; Secondary: 34C08.

    Citation:

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  • Figure 1.  Global phase portraits of Hamiltonian planar polynomial vector field of degree three with a nilpotent saddle at the origin and $ \mathbb{Z}_2 $-symmetric with $ R(x, y) = (x, -y) $. The separatrices are in bold

    Figure 2.  Blow up of the origin of $ U_2 $

    Figure 3.  The curves where the number of finite singular points can change. The plot of the curves $ b = b_2 $ (continous thick line), $ b = b_3 $ (dashed line), and $ b = a^2 $ (continuous thin line) for $ b>0 $

    Figure 4.  The black lines correspond to the curves where the number of finite singular points can change (see Figure 3 for more details), and the gray lines correspond to the curves where the connection of saddles can occur. The connection with the saddle at the origin can occur on the upper gray line, and the connection between two saddles different from the origin can occur on the lower gray line

    Figure 5.  Global phase portraits of systems (VII) for $ a = 7 $ and $ b = 34.7 $ (1.17') and $ b = b_2 $ and $ a = 4.5 $ (1.21'). The separatrices are in bold

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