论文标题
动态系统理论的三个示例
Three Examples in the Dynamical Systems Theory
论文作者
论文摘要
我们在动态系统理论中提出了三个明确的奇怪简单示例。第一个是两个分析性差异$ r $，$ s $ s $的示例，其封闭的二维环具有具有相交属性的封闭二维环，但其成分$ rs $却没有（$ r $仅是$π/2 $的旋转）。第二个示例是cotangent捆绑包中的非拉格朗日$ n $ -torus $ l_0 $ $也不是$ t^\ ast {\ mathbb t}^n $的零部分。第三个示例是两个单参数的分析性可逆自主普通微分方程的家庭，形式的$ \ dot {x} = f（x，y）$，$ \ dot {y} =μg（x，y）$在封闭的上半$ \ \ \ e \ geq 0 \ geq 0 \ geq 0 \ for $ $ $ $ $ $ $ 0的$ 0中，以$ 0的$ 0 < $μ> 1 $在拓扑上是非等效的。前两个示例在符号拓扑的一般环境中被阐述。
We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms $R$, $S$ of a closed two-dimensional annulus that possess the intersection property but their composition $RS$ does not ($R$ being just the rotation by $π/2$). The second example is that of a non-Lagrangian $n$-torus $L_0$ in the cotangent bundle $T^\ast{\mathbb T}^n$ of ${\mathbb T}^n$ ($n\geq 2$) such that $L_0$ intersects neither its images under almost all the rotations of $T^\ast{\mathbb T}^n$ nor the zero section of $T^\ast{\mathbb T}^n$. The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form $\dot{x}=f(x,y)$, $\dot{y}=μg(x,y)$ in the closed upper half-plane $\{y\geq 0\}$ such that for each family, the corresponding phase portraits for $0<μ<1$ and for $μ>1$ are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.
