Phase Portraits in Coupled Differential Equations
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.
Key Features of Phase Portraits
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.
Equilibrium Points
- Equilibrium points are locations where $\frac{dx}{dt} = \frac{dy}{dt} = 0$.
- These points represent states where the system doesn't change over time.
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 $x=y=0$.
Stable Populations
- Stable populations occur when trajectories converge to an equilibrium point.
- This indicates that regardless of initial conditions, the system will eventually settle into a steady state.
Saddle Points
- Saddle points are equilibrium points where trajectories approach from some directions but move away in others.
- They represent unstable equilibria.
Saddle points often occur when eigenvalues have different signs, creating a "saddle" shape in the phase portrait.
Eigenvalues and Their Impact on Solutions
The nature of the eigenvalues of the system matrix determines the qualitative behavior of solutions.
Positive Real Eigenvalues or Complex with Positive Real Part
- When eigenvalues are positive real numbers or complex numbers with positive real parts, all solutions move away from the origin.
- This results in unstable behavior.
Negative Real Eigenvalues or Complex with Negative Real Part
- Negative real eigenvalues or complex eigenvalues with negative real parts cause all solutions to move towards the origin, indicating stable behavior.
Complex Eigenvalues
- Complex eigenvalues lead to spiral trajectories.
- The real part determines whether the spiral moves inward (negative) or outward (positive).
For the system:
\[\begin{cases} \frac{dx}{dt} = -x + y \\ \frac{dy}{dt} = -2x - y \end{cases} \]
The eigenvalues are $-1 \pm i$. This results in an inward spiral, as the real part (-1) is negative.
Purely Imaginary Eigenvalues
- When eigenvalues are purely imaginary, solutions form closed orbits (circles or ellipses) around the origin.
Students often confuse purely imaginary eigenvalues with complex eigenvalues. Remember, purely imaginary eigenvalues have no real part, leading to closed orbits rather than spirals.
