报告摘要：
| In this talk, we will briefly introduce the Maslov-type index for symplectic paths starting from identity and its iteration theory under periodic solution and brake orbit boundary conditions. As applications we study the minimal period problems for symmetric priodic solutions and brake orbits of nonlinear autonomous Hamiltonian systems and reversible Hamiltonian systems respectively. For first order nonlinear autonomous even Hamiltonian systems in $R^{2n}$, 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 $T$. For first
order nonlinear autonomous reversible Hamiltonian systems in
$R^{2n}$, 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 ${T,;frac{T}{3}}$.
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