Minimal period problems of periodic solutions and brake orbits in Hamiltonian systems
发布时间：2018-09-03 点击次数：
| 报告题目： | Minimal period problems of periodic solutions and brake orbits in Hamiltonian systems |
| 报 告 人： | 张端智 教授（南开大学） |
| 报告时间： | 2018年09月07日 10:00--11:00 |
| 报告地点： | 理学院东北楼四楼报告厅（404） |
| 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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