Phase portraits are powerful visual tools used to analyze systems of coupled differential equations. In the context of AHL 5.17, we focus on systems of the form:
\[\begin{cases} \frac{dx}{dt} = ax + by \\\frac{dy}{dt} = cx + dy \end{cases} \]
where $a$, $b$, $c$, and $d$ are constants. These equations describe how two variables, $x$ and $y$, change over time in relation to each other.
A phase portrait displays the trajectories of solutions in the $xy$-plane, allowing us to visualize the long-term behavior of the system without solving the equations explicitly.
For the system:
\[\begin{cases} \frac{dx}{dt} = 2x - y \\\frac{dy}{dt} = x + y \end{cases} \]
The equilibrium point is (0,0), as both equations equal zero when $x=y=0$.
Saddle points often occur when eigenvalues have different signs, creating a "saddle" shape in the phase portrait.
The nature of the eigenvalues of the system matrix determines the qualitative behavior of solutions.
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