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Autonomous Systems: Orbits and Limit Cycles 📂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 XX and function f:X→Xf : X \to X as follows: x˙=f(x) \dot{x} = f(x) Consider the flow of the autonomous system at the initial time t0t_{0} and the initial point x0x_{0} to be represented as x(t,t0,x0)x(t,t_{0},x_{0}).

  1. Then, an Orbit passing through x0∈Xx_{0} \in X is represented as follows[^1]: O(x0):={x∈X:x=x(t,t0,x0)} O(x_{0}) := \left\{ x \in X : x = x(t, t_{0} , x_{0}) \right\} Of course, for all time points T∈IT \in I, O(x(T,t0,x0))=O(x0)O\left( x (T , t_{0} , x_{0}) \right) = O (x_{0}) holds true.
  2. If an orbit satisfies the following for all t∈Rt \in \mathbb{R} and there exists a T>0T > 0, it is said to be TT-periodic, and that orbit is called a Periodic Orbit. x(t,t0)=x(t+T,t0) 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: x˙=−yy˙=x \dot{x} = -y \\ \dot{y} = x Since the solution to this differential equation can be represented as for time tt (x,y)=(cos⁡t,sin⁡t) (x,y) = \left( \cos t , \sin t \right) assuming the initial value is p0=(1,0)p_{0} = (1,0), the flow will form a path that circles around a unit circle with a radius of 11. Therefore, the orbit passing through p0p_{0} can be represented as follows. O(p0):={(x,y)∈R2:x2+y2=1} 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 2π2 \pi-periodic.

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