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.
Real Eigenvalues with Different Signs
This scenario results in a saddle point at the origin. Some trajectories approach the origin along one eigenvector but then move away along the other.
Sketching Trajectories
To sketch trajectories in a phase portrait:
- Find the equilibrium points
- Calculate the eigenvalues and eigenvectors
- Determine the behavior near the equilibrium points based on the eigenvalues
- Sketch the eigenvectors (if real)
- Draw representative trajectories consistent with the determined behavior
Always start by analyzing the behavior near equilibrium points, as this gives crucial information about the overall structure of the phase portrait.
Exact Solutions for Real Distinct Eigenvalues
For systems with real distinct eigenvalues, we can find exact solutions. The general solution takes the form:
$$ \begin{pmatrix} x(t), \ y(t) \end{pmatrix} = c_1e^{\lambda_1t}\mathbf{v_1} + c_2e^{\lambda_2t}\mathbf{v_2} $$
where $\lambda_1$ and $\lambda_2$ are the eigenvalues, $\mathbf{v_1}$ and $\mathbf{v_2}$ are the corresponding eigenvectors, and $c_1$ and $c_2$ are constants determined by initial conditions.
ExampleFinding the Eigenvalues and Eigenvectors
We start with the system of differential equations written in matrix form:
\[\begin{bmatrix} \frac{dx}{dt} \\\frac{dy}{dt} \end{bmatrix} = \begin{bmatrix} 3 & -2 \\ 2 & -2 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} \]
Step 1: Find the Eigenvalues
Eigenvalues \(\lambda\) satisfy the characteristic equation:
\[\det\left( A - \lambda I \right) = 0 \] where \( A \) is the coefficient matrix: \[ A = \begin{bmatrix} 3 & -2 \\ 2 & -2 \end{bmatrix} \]
Subtract \(\lambda I \) from \( A \):
\[ A - \lambda I = \begin{bmatrix} 3 - \lambda& -2 \\ 2 & -2 - \lambda\end{bmatrix} \]
Compute the determinant: \[\begin{vmatrix} 3 - \lambda& -2 \\ 2 & -2 - \lambda\end{vmatrix} \]
Expanding: \[ (3 - \lambda)(-2 - \lambda) - (-2)(2) = (-6 - 3\lambda + 2\lambda + \lambda^2) + 4 \]\[\lambda^2 - \lambda - 2 = 0 \]
Factorizing: \[ (\lambda - 1)(\lambda + 2) = 0 \]
Thus, the eigenvalues are: \[\lambda_1 = 1, \quad\lambda_2 = 0 \]
Step 2: Find the Eigenvectors
For each eigenvalue \(\lambda\), solve: \[ (A - \lambda I) \mathbf{v} = 0 \]
Eigenvector for \(\lambda_1 = 1 \):
\[ A - I = \begin{bmatrix} 2 & -2 \\ 2 & -3 \end{bmatrix} \]
Solving: \[\begin{bmatrix} 2 & -2 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]\[ v_1 = v_2 \]\[\mathbf{v_1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \]
Eigenvector for \(\lambda_2 = 0 \):
\[ A = \begin{bmatrix} 3 & -2 \\ 2 & -2 \end{bmatrix} \]\[\mathbf{v_2} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \]
General Solution
\[\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = c_1 e^t \begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_2 \begin{bmatrix} 1 \\ 2 \end{bmatrix} \]
Connection to Eigenvalues and Eigenvectors
The eigenvalues and eigenvectors of the system matrix:
$$ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $$
play a crucial role in determining the behavior of solutions. The eigenvalues are found by solving the characteristic equation:
$$ \det(A - \lambda I) = 0 $$
and the corresponding eigenvectors are found by solving:
$$ (A - \lambda I)\mathbf{v} = \mathbf{0} $$
NoteIn AHL 5.17, we only consider systems with distinct, non-zero eigenvalues. This simplifies the analysis but still covers a wide range of interesting behaviors.
Applications and Extensions
While not explicitly required by the syllabus, understanding phase portraits has applications in various fields:
- Population dynamics: Modeling predator-prey relationships
- Economics: Analyzing supply and demand equilibria
- Control systems: Designing stable feedback systems
The Jacobian matrix, mentioned in the syllabus notes, extends these concepts to non-linear systems by providing a linear approximation near equilibrium points.
TipWhen analyzing real-world systems, start by identifying the variables and how they interact. This will help you set up the differential equations before moving on to the phase portrait analysis.
