Absence of Periodic Orbits in Two-Dimensional Autonomous Systems
Considerations on Periodic Orbits
The question of whether periodic orbits exist in an autonomous system is generally quite complicated, but if we’re talking about a $1,2$-dimensional space, we can discuss their absence relatively simply. Let’s say we have the following vector field given by a differential equation for spaces $X = \mathbb{R}$ or $X = \mathbb{R}^{2}$ and function $f : X \to X$: $$ \dot{x} = f(x) $$
1 Dimension
In a $1$-dimensional autonomous system, periodic orbits do not exist. This might seem obvious and can be simply proven by reductio ad absurdum, but before proving it, it’s significantly meaningful to geometrically and intuitively understand what a $1$-dimensional vector field is.
Even in $1$ dimensions, a vector field is a vector field, and for every $x \in \mathbb{R}$, there must be exactly one direction and magnitude determined by $\dot{x} = f(x)$. However, if periodic orbits existed, the points in between could go both left and right. The existence of such points contradicts the system being defined by a vector field, ultimately meaning no $1$-dimensional autonomous system can have a periodic orbit.
2 Dimensions
In a $2$-dimensional autonomous system, one can demonstrate the absence of periodic orbits depending on the region and parameters.
(2-1): For example, consider the damped oscillator: $$ \begin{align*} \dot{x} =& f(x,y) = y \\ \dot{y} =& g(x,y) = x - x^{3} - \delta y \qquad , \delta \ge 0 \end{align*} $$
Bendixson’s Criterion: In a simply connected region $D \subset \mathbb{R}^{2}$, $$ {{ \partial f } \over { \partial x }} + {{ \partial g } \over { \partial y }} \ne 0 $$ if the sign does not change, then the given $2$-dimensional vector field does not have a closed orbit within $D$ interior.
Since for all $(x,y) \in \mathbb{R}^{2}$, $\displaystyle {{ \partial f } \over { \partial x }} = 0$ and $\displaystyle {{ \partial g } \over { \partial y }} = - \delta$, $$ {{ \partial f } \over { \partial x }} + {{ \partial g } \over { \partial y }} = - \delta $$ here if $\delta >0$, then in the whole space $\mathbb{R}^{2} \subset \mathbb{R}^{2}$, ${{ \partial f } \over { \partial x }} + {{ \partial g } \over { \partial y }} <0$, hence there are no periodic orbits. This can also be explained with physical intuition. $\delta>0$ means that energy continuously leaks and eventually stops, implying the absence of periodic orbits and if $\delta=0$, there would be no damping, and like Galileo’s thought experiment, energy would be conserved to repeat the same motion forever.
(2-2): In a $2$-dimensional autonomous system, periodic orbits can exist. Pointing this out is necessary because, despite the clear indication that periodic orbits exist when $\delta = 0$ in (2-1), Bendixson’s striking impression often leads to misreading that periodic orbits do not exist in $2$ dimensions. For example, consider the following simple autonomous system without separate parameters: $$ \dot{x} = -y \\ \dot{y} = x $$ The solution of this differential equation for time $t$ can be represented as $$ (x,y) = \left( \cos t , \sin t \right) $$ thus, if the initial value is $p_{0} = (1,0)$, the flow will take the form of circling around a unit circle with radius $1$. Hence, the orbit passing through $p_{0}$ can be represented as: $$ O(p_{0}) := \left\{ (x,y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1 \right\} $$
3 Dimensions
On the other hand, the reason for such discussions being limited to $2$ dimensions is that from the moment it becomes $3$ dimensions, there are infinitely many ways to circumnavigate a flow. In a dynamical system defined by a vector field, a 2-dimensional space can be accurately split into two parts by one curve. This means there can’t be any peculiar flow that behaves independently without meeting such a curve, and if such a curve contracts or expands, the flow within two factions created by the curve, like its interior and exterior, is also affected.
Dividing a $3$ dimensional space is possible with a surface, and to drive the interest of dynamics up to this point, the system would need to be defined by partial differential equations. Likewise, to consider higher dimensions or a homeomorphic manifold, the level of handling space must be elevated.