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Section 17.2 Dynamical systems
Objectives
Interpret a vector field as defining a dynamical system.
Understand solutions as paths (trajectories) through the field.
Visualize and analyze streamlines of a vector field.
A dynamical system in two dimensions is given by:
\begin{equation*}
\qty(\dv{x}{t},\dv{y}{t})=(P(x,y),Q(x,y))
\end{equation*}
Solutions to this system describe how a point moves over time.
The resulting paths are called trajectories or streamlines.
Different initial points can lead to very different behavior, even within the same field.
These ideas connect directly to earlier work with differential equations and will later help us interpret line integrals and flow.
Predator-prey dynamics (Lotka-Volterra model):
\begin{align*}
\dv{x}{t} \amp = ax - bxy \\
\dv{y}{t} \amp = -cy + dxy
\end{align*}
Here,
\(x\) represents the prey population and
\(y\) the predator population. The vector field describes how both populations change in response to each other.
Disease spread (SIR model):
\begin{align*}
\dv{S}{t} \amp = -\dfrac{\beta}{n} SI \\
\dv{I}{t} \amp = \dfrac{\beta}{n} SI - \gamma I \\
\dv{R}{t} \amp = \gamma I
\end{align*}
This system tracks how individuals move between susceptible, infected, and recovered groups. The vector field determines how an outbreak evolves over time.
