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Dane Sabo 2024-09-23 14:38:31 -04:00
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300s School/ME 2016 - Nonlinear Dynamical Systems 1/2024-09-23 Temporary Title.md
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@ -97,4 +97,9 @@ $$\dot y = y(b - d y - c x) = by - dy^2 -cxy$$
>[!note] Coupling Terms >[!note] Coupling Terms
>$\gamma x y$ and $c x y$ are coupling terms. >$\gamma x y$ and $c x y$ are coupling terms.
>These equations are coupled because of these. Without them x and y would just be doing their own thing. >These equations are coupled because of these. Without them x and y would just be doing their own thing.
## Computing our Jacobian
$$ {\bf J} =
\left[ \matrix{ \frac{\partial P}{\partial x} & \frac{\partial P}{\partial y} \\ \frac{\partial Q}{\partial x} & \frac{\partial Q}{\partial y}} \right] =
\left[ \matrix{ \beta -2\delta x - \gamma y & -\gamma x\\ - c y & b - 2dy - cx} \right]
$$
Now we can actually do stuff with this in SymPy