engleski

Unfolding of the Hamiltonian triangle vector field

The paper considers a family of planar vector fields $$\multline X_\epsilon(x, y) = (x(\beta - \beta x - (\beta + 1) y) + \epsilon_0 x^2 + \epsilon _1 y^2)\, \partial / \partial x \\+ (y(-\alpha + (\alpha+ 1) x + \alpha y) + \epsilon_0 y^2 + \epsilon_2 x^2)\, \partial/\partial y\endmultline$$ depending on the parameters $\epsilon = (\epsilon_0, \dots, \epsilon_4)$, where $\epsilon_3=\alpha - 1$ and $\epsilon_4 = \beta - 1$, that are close to $(0, \dots, 0)$. For $\epsilon_0 = \epsilon_1 = 0$the vector field has a first integral $H=x^\alpha y^\beta (1-x-y)$ and a center at $p_{; ; \alpha, \beta}; ; = \big(\alpha/(\alpha+\beta + 1), \beta/(1+(\alpha + \beta + 1)\big)$. For $\alpha=\beta =1$ the vector field is Hamiltonian, whence the title. This paper calculates the first terms in the development of the displacement function $\delta$ (the difference between the first return Poincaré map and the identity). The study is done `away' from the center $p_{; ; \alpha, \beta}; ; $ and from the separatrices. It is shown that $\delta$ is of the form $$\delta = \mu_1(J_1 + \cdots ) + \mu_2(J_2 + \cdots )+ \mu_3(J_3 + \cdots ) + \mu_4(J_4 + \cdots ), $$ where each $\mu_i$ is some explicit polynomial $\mu_i=O(\epsilon)$ of $\epsilon$. The displacement function $\delta$ is parametrized by the first integral. An explicit computation of $J_i$ is obtained. The authors compare their results with those of H. Żołądek [J. Differential Equations 109 (1994), no. 2, 223–273 ; J. Differential Equations 67 (1987), no. 1, 1–55], and explain the different value of $J_4$. It is conjectured that$(J_1, J_2, J_3, J_4)$ form a Chebyshev system ; if the conjecture is true then at most three limit cycles can be born away from the center and the separatrices.

Hamiltonian system; perturbation; Chebyshev system; limit cycle

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