📂Dynamics

Autonomous Systems: Orbits and Limit Cycles

Definition

Let’s assume we have a vector field given by a differential equation with respect to space $X$ and function $f : X \to X$ as follows: $$ \dot{x} = f(x) $$ Consider the flow of the autonomous system at the initial time $t_{0}$ and the initial point $x_{0}$ to be represented as $x(t,t_{0},x_{0})$.

1. Then, an Orbit passing through $x_{0} \in X$ is represented as follows[^1]: $$ O(x_{0}) := \left\{ x \in X : x = x(t, t_{0} , x_{0}) \right\} $$ Of course, for all time points $T \in I$, $O\left( x (T , t_{0} , x_{0}) \right) = O (x_{0})$ holds true.
2. If an orbit satisfies the following for all $t \in \mathbb{R}$ and there exists a $T > 0$, it is said to be $T$-periodic, and that orbit is called a Periodic Orbit. $$ x(t,t_0) = x(t + T,t_0) $$
3. A periodic orbit that is not a singleton set containing only one fixed point is called a Cycle.
4. A cycle with no other cycle in its neighborhood is called a Limit Cycle[^3].

Reference 1

Example

Let’s consider the following simple autonomous system as an example: $$ \dot{x} = -y \\ \dot{y} = x $$ Since the solution to this differential equation can be represented as for time $t$ $$ (x,y) = \left( \cos t , \sin t \right) $$ assuming the initial value is $p_{0} = (1,0)$, the flow will form a path that circles around a unit circle with a radius of $1$. Therefore, the orbit passing through $p_{0}$ can be represented as follows. $$ O(p_{0}) := \left\{ (x,y) \in \mathbb{R}^{2} : x^{2} + y^{2} = 1 \right\} $$ Especially, this orbit is periodic as the flow passes through the same points, making it $2 \pi$-periodic.

Reference 2[^2]: Wiggins. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos Second Edition(2nd Edition): p71.