Phase Portraits for Linear Systems

For second order linear systems $\bold x'=A\bold x$ with $\det(\bold A-r\bold I)=0$ and $\det \bold A \ne 0$ trajectories follow particular stability patterns and types.

For higher order the cases are essentially combinations of these patterns

For Linear Systems we described the stability properties for the equilibrium solution $\bold x=\bold 0$

For the system $\bold x'=A\bold x$ in which $\det A\ne0$$~~~\bold x=\bold 0$ is the only critical point of the system by definition of invertibility (Linear Algebra Review)

<aside> ▫️ For the critical point $\bold x=0$ of $\bold x' =A\bold x$ where $A$ is $2\times 2$

  1. If the eigenvalues $r_1,r_2$ are real and negative or have negative real part, the point is asymptotically stable 2**.** If $r_1,r_2$ are purely imaginary the point is stable
  2. If $r_1,r_2$ are real and either is positive or if they have positive real part is unstable

</aside>

Description Eigenvalues Type of Critical Point Stability Notes
Distinct, real, positive $r_1>r_2>0$ Node Unstable (source)
Distinct, real, negative $r_1<r_2<0$ Node Asymptotically Stable (sink)
Oppositely Signed, real $r_2<0<r_1$ Saddle Point Unstable
Repeated, positive $r_1=r_2>0$ Proper or Improper Node Unstable
Repeated, negative $r_1=r_2<0$ Proper or Improper Node Asymptotically Stable
Complex conjugates $r_1,r_2=\lambda\pm i\mu\\~~~\lambda>0\\ 
Unstable
Asymptotically Stable |  |
| Imaginary | $r_1,r_2=\\pm i \\mu$ | Center | Stable | To determine direction plot $A\\begin{bmatrix}x_0\\\\y_o\\end{bmatrix}$ |

> Def: When a repeated eigenvalue only gives a single independent eigenvector, the ciritcal point is **improper**
> 

## Distinct Real:

![Untitled](<https://s3-us-west-2.amazonaws.com/secure.notion-static.com/3a446c8d-e58d-440f-9793-0e646b6ba82b/Untitled.png>)

## Saddle Point (Opposite Signs):

![Untitled](<https://s3-us-west-2.amazonaws.com/secure.notion-static.com/e0bd0fe0-4d03-42ce-9c76-642fc7f5a76b/Untitled.png>)

## Repeated Roots:

### Proper Nodes:

![Untitled](<https://s3-us-west-2.amazonaws.com/secure.notion-static.com/9b5ac3e8-588d-45e4-b6ba-aa6148b740f9/Untitled.png>)

### Improper Nodes:

![Untitled](<https://s3-us-west-2.amazonaws.com/secure.notion-static.com/0ca97341-7668-4cf4-857a-7ad48dfda11d/Untitled.png>)

![Untitled](<https://s3-us-west-2.amazonaws.com/secure.notion-static.com/e6af19e5-8504-4b7b-89ba-7286a1edfc11/Untitled.png>)

For improper nodes we find the independent eigenvalue solving $(A-\\lambda I)\\xi=0$ and the generalized vector $\\eta$ that satisfies $(A-\\lambda I)\\eta=\\xi$.

The general solution for improper nodes is:

$$
\\bold x=(c_1\\xi+c_2\\eta+c_2\\xi t)e^{rt}
$$

The direction is determined completely by $c_1\\xi+c_2\\eta+c_2\\xi t$  and $e^{rt}$ only effects the magnitude.