Three Examples in the Dynamical Systems Theory

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 $\pi/2$). 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 $\dot{x}=f(x,y)$, $\dot{y}=\mu g(x,y)$ in the closed upper half-plane $\{y\geq 0\}$ such that for each family, the corresponding phase portraits for $0<\mu<1$ and for $\mu>1$ are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.