🗝️ Key Takeaways ...

• In uncoupled systems,

  • there is no interaction between equations,
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  • each variable evolves independently, and
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  These are the simplest systems — but they’re still systems!

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• In partially-coupled systems,

  • one equation is independent, and the other depends on it,
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  • You can solve these without any new ideas by solving the independent equation first, then substituting its solution into the dependent equation.
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  These systems model one-way influences, like a light level driving plant growth.

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• The most common systems are fully-coupled systems, where

  • both variables influence each other, and
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  • neither equation can be solved in isolation. That is, the system must be treated as a whole.
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• A system is linear if its unknowns appear without products, powers, or nonlinear functions. Such systems can be written in vector form:

  \begin{equation*} \dfrac{d\vec{X}}{dt} = A\vec{X} \end{equation*}

  where \(A\) is the constant coefficient matrix.

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• Solving such systems is an eigenvalue problem.
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• Euler’s Method updates each variable using its derivative at the current step. This update feeds into the coupled variables.
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• Phase plane diagrams reveal whether a system’s solutions decay, grow, or spiral.
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• You can often tell what will happen by the shape of the arrows — even without solving equations.
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