Limit Sets

Omega Limit Point

$a\in {\mathbb{R}}^n$ is a $\omega -$limit point of ${\varphi }^t(x_0)$ if $\mathrm{\exists }$ a sequence of time $t_n\to \mathrm{\infty }$ such that

we denote this as $\omega (x_0)$
it gets arbitrarily close infinite times

Alpha Limit Point

$a\in {\mathbb{R}}^n$ is a $\alpha -$limit point of ${\varphi }^t(x_0)$ if $\mathrm{\exists }$ a sequence of time $t_n\to -\mathrm{\infty }$ such that

we denote this as $\alpha (x_0)$

Examples

Limit Points and Fixed points

if $x_0$ is a fixed point then ${\varphi }^t(x_0)=x_0,\mathrm{\forall }t$ so:

Limit Points and Periodic Orbits

Let ${\varphi }^t(x_0)$ be a periodic orbits for

$\mathrm{\forall }t\in \mathbb{R},{\varphi }^{T+t}(x_0)={\varphi }^t(x_0)$

Limit Points and Heteroclinic Orbits

Let ${\varphi }^t(x_0)$ be a heteroclinic orbit connecting two fixed points $x_1$ and $x_2$

Then one sees that $\alpha (x_0)=\{x_1\}$ and $\omega (x_0)=\{x_2\}$.

Limit Points and Homoclinic Orbits

Let ${\varphi }^t(x_0)$ be a homoclinic orbit connecting a fixed point $x^{\ast }$ to itself

Then one sees that $\alpha (x_0)=\omega (x_0)=\{x^{\ast }\}$.

Limit Sets and Homoclinic Orbits

Consider the IVP of the system

We have discussed whether the origin $0$ belongs to $\omega (x_0)$ or $\alpha (x_0)$ in the table below.

Bounded and Closed Sets

For $A\in {\mathbb{R}}^2$
Bounded: $\mathrm{\exists }r>0$ st $A\subseteq Br(0)$
Closed: $\mathrm{\forall }\{a_n{\}}_{n\in \mathbb{N}}\subseteq A,\underset{n\to \mathrm{\infty }}{lim}{z}_n\in A$
Compact: Closed and bounded

Invariant Sets:

Positively Invariant

if $x_0\in A⟹\mathrm{\forall }t\ge 0,{\varphi }^t(x_0)\in A$

Negatively Invariant

if $x_0\in A⟹\mathrm{\forall }t\le 0,{\varphi }^t(x_0)\in A$

Invariant

If it is both positively and negatively invariant

Stable/Unstable Sets

Stable Set

For a fixed point $x^{\ast }\in {\mathbb{R}}^n$ the stable set of $x^{\ast }:$

note $\omega (p)=\{x^{\ast }\}$

Unstable Set

For a fixed point $x^{\ast }\in {\mathbb{R}}^n$ the unstable set of $x^{\ast }:$

note $\alpha (p)=\{x^{\ast }\}$

Example

The coefficient matrix has eigenvalues ${\lambda }_1=1$, ${\lambda }_2=-1$, and eigenvectors ${\mathbf{v}}_1=(1,1{)}^t$ and ${\mathbf{v}}_2=(-1,1{)}^t$ respectively.

The general solution reads

Thus we see that