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Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

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  • In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0 = \{0\}\times \mathbf{R}^n\subset \mathbf{R}^{2n}$ and $L_1=\mathbf{R}^n\times \{0\} \subset \mathbf{R}^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\mathbf{R}^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H''_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$.
    Mathematics Subject Classification: 58E05, 70H05, 34C25.


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