For some non-linear function, we can use its first Jacobian to create a linear system. Then, we can use that system to write the first order Taylor:

\begin{equation} y’ = \nabla F(crit)y \end{equation}

where \(crit\) are critical points.

## Phase Portrait stability

if

**all**\(Re[\lambda] < 0\) of \(\qty(\nabla F)(p)\) then \(p\) is considered**stable**—that is, points initially near \(p\) will exponentially approach \(p\)if

**at least one**\(Re[\lambda] > 0\) of \(\qty(\nabla F)(p)\) then \(p\) is considered**unstable**—that is, points initially near \(p\) will go somewhere elseif

**all**\(Re[\lambda] \leq 0\) and**at least one**\(\lambda\) is pure imaginary of \(\qty(\nabla F)(p)\), then there are no conclusions and \(p\) is considered**marginal**If there are

**no**purely imaginary values, then the solution paths of the ODE look like that of \(y’ = (\nambla F)(p) y\).

## Worked Example

Let’s Lotha-Volterra Prey-Predictor Equation again as an example

\begin{equation} \begin{cases} x_1’ = 2x_1-x_1x_2 \\ x_2’ = x_1x_2 - 3x_2 \end{cases} \end{equation}

we can stare at this (and factor \(x\) out) to understand that there are only two stationary points:

\begin{equation} (x_1,x_2) = (0,0), (3,2) \end{equation}

Let’s analyze this function for linearilzation.

Let’s write this expression in terms of the linear and non linear parts

\begin{equation} \begin{cases} x’ = \mqty(2 & 0 \\ 0 & -3) \mqty(x_1 \\ x_2) + \mqty(-x_1x_2 \\ x_1 x_2) \end{cases} \end{equation}

### Near \((0,0)\)

You will note that the right non-linear parts becomes very small near \((0,0)\), meaning we can analyze this in terms of a normal phase portrait.

### Near \((3,2)\)

We can translate this down:

Let:

\begin{equation} y = x - \mqty(3 \\2) \end{equation}

meaning:

\begin{equation} y’ = x’ = F\qty(y+\mqty(3 \\ 2)) \end{equation}

we can use a Taylor expansion to get:

\begin{equation} y’ = x’ = F\qty(y + \mqty(3\\2)) + \qty(\nabla F)y + \dots \end{equation}

Recall that \(F\) is given as:

\begin{equation} \mqty(2x_1 - x_1x_2 \\ x_1x_2-3x_2) \end{equation}

meaning:

\begin{equation} \nabla \mqty(2x_1 - x_1x_2 \\ x_1x_2-3x_2) = \mqty(2-x_2 & -x_1 \\ x_2 & x_1-3) \end{equation}

plugging in \((3, 2)\) obtains:

\begin{equation} y’ = \mqty(0 & -3 \\ 2 & 0) y \end{equation}

which we can analyze in the usual manners.