3.3 Stability of Fixed Points

Lyapunov Stability

A fixed point $x^{*} \in R^{n}$ is Lyapunov Stable if

$\forall ε > 0, \exists δ > 0,$ such that $\forall x_{0} \in Ω$

| x_{0} - x^{*} | < δ ⟹ | ϕ^{t} (x_{0}) - x^{*} | < ε, \forall t \geq 0

or $\forall B_{ε} (x^{*}), \exists B_{δ} (x^{*}) so that ϕ^{t} (B_{δ} (x^{*})) \subseteq B_{ε} (x^{*})$

A fixed point $x^{*} \in R^{n}$ is called $ω -$ attracting if

$\exists δ > 0$ such that $\forall x_{0} \in Ω$

| x_{0} - x^{*} | < δ ⟹ lim_{t \to \infty} ϕ^{t} (x_{0}) = x^{*}

or $\exists B_{δ} (x^{*}) \subseteq W^{s} (x^{*})$ which is the stable set

A fixed point is asymptotically stable if it is bother Lyapunov Stable and $ω -$ attracting.

\begin{array}{ccc} Type & L-Stable & Asymptotic Stable \\ Nodal Source & \times & \times \\ Nodal Sink & ✓ & ✓ \\ Spiral Source & \times & \times \\ Spiral Sink & ✓ & ✓ \\ Center & ✓ & \times \\ Saddle & \times & \times \end{array}

A fixed point $x^{*}$ is called hyperbolic, provided that

Re (λ) \neq 0

for each eigenvalue $λ$ of the matrix $D F (x^{*})$
If $x^{*}$ is a hyperbolic point of the nonlinear system:

\dot{x} = F (x)

Where $F$ is smooth. Then the stability of $x^{*}$ is equivalent to the stability of the linearized:

\dot{u} = D F (x^{*}) u

at $0$ which is our first variation equation but in a higher dimension