Three Examples in the Dynamical Systems Theory

TL;DR

This paper presents three explicit, simple examples in dynamical systems theory illustrating complex behaviors, including intersection property failure, non-Lagrangian tori, and topologically distinct phase portraits.

Contribution

It introduces three novel explicit examples in dynamical systems, highlighting subtle phenomena and expanding understanding within symplectic topology and differential equations.

Findings

Intersection property does not imply composition property for certain diffeomorphisms.

Existence of non-Lagrangian tori with unique intersection properties.

Phase portraits of differential equations can be topologically non-equivalent for different parameters.

Abstract

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.