📂Dynamics

Lyapunov Stability and Orbit Stability

Definition

Liapunov Stability ¹

Given a metric space $\left( X , \left\| \cdot \right\| \right)$ and a function $f : X \to X$, consider the following vector field presented as a differential equation: $$ \dot{x} = f(x) $$

1. Let $t_{0} \in \mathbb{R}$. If for every solution $\overline{x}(t)$ of the given differential equation, when $\varepsilon > 0$ is given, $$ \left\| \overline{x} \left( t_{0} \right) - y \left( t_{0} \right) \right\| < \delta \implies \left\| \overline{x}(t) - y(t) \right\| < \varepsilon \qquad , t > t_{0} $$ is satisfied for all other solutions $y(t)$ for which $\delta ( \varepsilon ) > 0$ exists, then $\overline{x}(t)$ is said to be Liapunov Stable.
2. If $\overline{x}(t)$ is Liapunov Stable and $$ \left\| \overline{x} \left( t_{0} \right) - y \left( t_{0} \right) \right\| < b \implies \lim_{t \to \infty} \left\| \overline{x}(t) - y(t) \right\| = 0 $$ is satisfied for all other solutions $y(t)$ for which a constant $b > 0$ exists, then $\overline{x}(t)$ is said to be Asymptotically Liapunov Stable.

Orbital Stability ²

3. Let $t_{0} \in \mathbb{R}$. If for every solution $\overline{x}(t)$ of the given differential equation, when $\varepsilon > 0$ is given, $$ \left\| \overline{x} \left( t_{0} \right) - y \left( t_{0} \right) \right\| < \delta \implies d \left( y(t) , O^{+} \left( x_{0} , t_{0} \right) \right) < \varepsilon \qquad , t > t_{0} $$ is satisfied for all other solutions $y(t)$ for which $\delta ( \varepsilon ) > 0$ exists, then $\overline{x}(t)$ is said to be orbitally Stable.
4. If $\overline{x}(t)$ is orbitally Stable and $$ \left\| \overline{x} \left( t_{0} \right) - y \left( t_{0} \right) \right\| < b \implies \lim_{t \to \infty} d \left( y(t) , O^{+} \left( x_{0} , t_{0} \right) \right) = 0 $$ is satisfied for all other solutions $y(t)$ for which a constant $b > 0$ exists, then $\overline{x}(t)$ is said to be Asymptotically orbitally Stable.

• $O^{+}(x_{0} , t_{0})$ is a notation representing the orbit after a fixed time $t_{0}$, defined as: $$ O^{+} \left( x_{0} , t_{0} \right) := \left\{ x \in X : x = \overline{x} (t) , t \ge t_{0} , \overline{x} \left( t_{0} \right) = x_{0} \right\} $$
• $d \left( p, S \right)$ is defined as the shortest distance between a point $p \in X$ and a subspace $S \subset X$: $$ d \left( p, S \right) := \inf_{x \in S} \left\| p - x \right\| $$

Explanation

Liapunov stability and orbital stability are concepts used to discuss whether the flow of an autonomous system is stable. A stable flow means that even if the initial point $\overline{x} \left( t_{0} \right)$ changes slightly, the flow still proceeds in a similar manner.

Stability, though described in relation to flow, can be directly transferred to fixed points, i.e., solutions that do not move, as $\overline{x} (t) = x_{0}$ is also a flow. In dynamics, interest usually lies in the stability of these fixed points.

• The difference between stability and asymptotic stability lies in whether the flow $y(t)$, after a slight change in initial value, actually converges to the flow $\overline{x}(t)$ or not. If the term ‘asymptotic’ is not included, $y(t)$ must simply remain as close as specified by $\varepsilon > 0$ to $\overline{x}(t)$, without necessarily converging over time. However, asymptotic stability means that convergence is required, and this convergence refers to convergence over time. To understand this without confusion, it’s essential to pay close attention to both the epsilon-delta argument and the limit expression $\lim_{t \to \infty}$ presented in the definition of stability.
• The difference between Liapunov stability and orbital stability can simply be thought of as the difference between a point and space. The definition of Liapunov stability focuses on the distance between two precise points $\overline{x}(t)$ and $y(t)$ at a precise time $t$. If, when observing the vector field, it essentially maintains a similar shape but the time required to reach the same point varies, it can’t be considered Liapunov stable. This is precisely the situation to think about orbital stability, which focuses on the distance between $O^{+} \left( x_{0} , t_{0} \right)$ and $y(t)$, meaning it considers all trajectories $\overline{x}(t)$ will eventually follow from the start. As $y(t)$ moves over time, and all future events that $\overline{x}(t)$ will experience are already encapsulated in the set $O^{+} \left( x_{0} , t_{0} \right)$, there’s no further concern about the passage of time. Hence, compared to Liapunov stability, orbital stability can be considered more lenient and generous.