论文标题
部分可观测时空混沌系统的无模型预测
Monotonicity and critical points of the period function for potential system
论文作者
论文摘要
本文与潜在系统的周期函数$ \ ddot {x}+g（x）= 0 $。我们给出一些足够的标准以确定单调性和上限至关键时期的数量。该结论基于（riemann-liouville）订单$ \ frac {1} {2} $和罗尔定理的（riemann-liouville）的半组属性。在多项式电位设置中，可以将潜在中心的关键时期数量缩小为计算半代数系统的真实零。从中，我们证明，如果非线性势$ g $是奇怪的，那么潜在的中心最多具有$ \ frac {deg（g）-3} {2} {2} $关键时期。为了说明其适用性，以更有效的方式证明了一些已知结果，并且讨论了一些具有复杂临界点的高纤维性汉密尔顿系统的关键时期，证明该系统可以完全具有两个关键时期。
This paper is concerned with the analytic behaviors (monotonicity, isochronicity and the number of critical points) of period function for potential system $\ddot{x}+g(x)=0$.We give some sufficient criteria to determine the monotonicity and upper bound to the number of critical periods. The conclusion is based on the semi-group properties of (Riemann-Liouville) fractional integral operator of order $\frac{1}{2}$ and Rolle's Theorem. In polynomial potential settings, bounding the the number of critical periods of potential center can be reduced to counting the real zeros of a semi-algebraic system. From which we prove that if nonlinear potential $g$ is odd, the potential center has at most $\frac{deg(g)-3}{2}$ critical periods. To illustrate its applicability some known results are proved in more efficient way, and the critical periods of some hyper-elliptic Hamiltonian systems of degree five with complex critical points are discussed, it is proved the system can have exactly two critical periods.
