Flows and invariant sets

Linearisation describes a system near one equilibrium. Dynamical systems asks about the long run: where do orbits end up, and how big is each basin?

1. Flows (Def. 1.1) and the group property (Prop. 1.2); invariant sets (Def. 1.3).
2. Omega-limit sets (Def. 1.4) and their properties (Prop. 1.5).
3. Lyapunov's theorem (Thm 1.6), LaSalle's invariance principle (Thm 1.7), with the damped pendulum (Example 1.8), and basins of attraction (Cor. 1.9).

For $\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x})$ with $\mathbf{f}$ locally Lipschitz, the flow $\varphi_t(\mathbf{x}_0)$ is the value at time $t$ of the unique solution with $\mathbf{x}(0) = \mathbf{x}_0$. The orbit of $\mathbf{x}_0$ is $\{\varphi_t(\mathbf{x}_0)\}$ and the forward orbit $\{\varphi_t(\mathbf{x}_0): t\ge0\}$. Distinct orbits never cross.

Wherever defined, $\varphi_0 = \mathrm{id}$ and $\varphi_{t + s} = \varphi_t\circ\varphi_s$. Hence each $\varphi_t$ is invertible with inverse $\varphi_{-t}$, and orbits are either equilibria, closed (periodic) orbits, or one-to-one images of an interval.

A set $A$ is invariant if $\varphi_t(A)\subseteq A$ for all $t$, and positively invariant if this holds for $t\ge0$. A compact positively invariant set is a trapping region; one is found by checking $\mathbf{f}$ points inward (or along) on its boundary, e.g. $\nabla g\cdot\mathbf{f}\le0$ on $\{g = c\}$ makes $\{g\le c\}$ positively invariant.

The omega-limit set of $\mathbf{x}$ is $\omega(\mathbf{x}) = \{\mathbf{y}: \varphi_{t_k}(\mathbf{x})\to\mathbf{y}\text{ for some }t_k\to\infty\}$; the alpha-limit set $\alpha(\mathbf{x})$ uses $t_k\to-\infty$. Equivalently $\omega(\mathbf{x}) = \bigcap_{T\ge0}\overline{\{\varphi_t(\mathbf{x}): t\ge T\}}$.

If the forward orbit of $\mathbf{x}$ lies in a compact set $K$, then $\omega(\mathbf{x})$ is non-empty, compact, connected and invariant, and $\operatorname{dist}(\varphi_t(\mathbf{x}), \omega(\mathbf{x}))\to0$ as $t\to\infty$.

Let $\mathbf{x}^*$ be an equilibrium and $V$ a $C^1$ function on a neighbourhood $U$ with $V(\mathbf{x}^*) = 0$, $V > 0$ on $U\setminus\{\mathbf{x}^*\}$. If $\dot V = \nabla V\cdot\mathbf{f}\le0$ on $U$, then $\mathbf{x}^*$ is stable; if $\dot V < 0$ on $U\setminus\{\mathbf{x}^*\}$, it is asymptotically stable.

Let $K$ be a trapping region and $V$ a $C^1$ function with $\dot V\le0$ on $K$. Let $M$ be the largest invariant set contained in $\{\mathbf{x}\in K: \dot V(\mathbf{x}) = 0\}$. Then every orbit starting in $K$ approaches $M$ as $t\to\infty$.

For $\ddot\theta + k\dot\theta + \sin\theta = 0$ ($k > 0$), write $\omega = \dot\theta$ and $V = \frac12\omega^2 + 1 - \cos\theta$. Then $\dot V = -k\omega^2\le0$. On $\{\omega = 0\}$ an invariant set needs $\dot\omega = -\sin\theta = 0$, so near the origin $M = \{(0, 0)\}$. By LaSalle the hanging position is asymptotically stable, with basin containing $\{V < 2\}\cap\{|\theta| < \pi\}$.

The basin of attraction of an asymptotically stable $\mathbf{x}^*$ is the set of $\mathbf{x}$ with $\varphi_t(\mathbf{x})\to\mathbf{x}^*$. If $\Omega_c = \{V\le c\}$ (the component containing $\mathbf{x}^*$) is bounded, $\dot V < 0$ on $\Omega_c\setminus\{\mathbf{x}^*\}$, then $\Omega_c$ lies in the basin.

• Try energy-like $V$: quadratic forms $ax^2 + by^2$ with $a, b$ chosen to cancel cross terms, or the physical energy.
• $\dot V < 0$ off the equilibrium: Lyapunov. Only $\dot V\le0$: find the largest invariant set in $\{\dot V = 0\}$ (LaSalle).
• Basin: largest bounded sublevel set of $V$ on which $\dot V < 0$.
• Trap: $V$ must be positive definite near $\mathbf{x}^*$; $\dot V\le0$ alone gives only stability.