Differential Equations Capstone and Phase Plane Analyzer — Theory
1. Autonomous systems
A two-dimensional autonomous system has the form:
\[
\begin{aligned}
x^{\prime}&=f(x,y)\\
y^{\prime}&=g(x,y)
\end{aligned}
\]
The word autonomous means that the right-hand sides do not explicitly depend on time.
The future motion depends only on the current point \((x,y)\).
2. Phase plane
In a phase plane, the horizontal axis is \(x\) and the vertical axis is \(y\).
A solution is drawn as a trajectory:
\[
t\mapsto (x(t),y(t))
\]
Instead of showing \(x(t)\) and \(y(t)\) separately as time graphs, the phase plane shows how the state moves
through the \(xy\)-plane.
3. Direction field
At each point \((x,y)\), the system gives a velocity vector:
\[
\mathbf{F}(x,y)=
\begin{pmatrix}
f(x,y)\\
g(x,y)
\end{pmatrix}
\]
A direction field draws small arrows showing the direction in which a solution would move if it passed through
that point.
4. Equilibrium points
An equilibrium point is a point where the system stops moving:
\[
f(x_0,y_0)=0,
\qquad
g(x_0,y_0)=0
\]
At an equilibrium point:
\[
x^{\prime}=0,
\qquad
y^{\prime}=0
\]
Therefore, a solution starting exactly at an equilibrium point stays there.
5. Nullclines
A nullcline is a curve where one derivative is zero.
\[
x^{\prime}=0
\]
is the \(x\)-nullcline, and:
\[
y^{\prime}=0
\]
is the \(y\)-nullcline. Equilibrium points occur where the two nullclines intersect.
6. Linearization
Near an equilibrium point, a nonlinear system can often be approximated by a linear system.
The matrix used for this local approximation is the Jacobian matrix:
\[
J(x,y)=
\begin{pmatrix}
f_x(x,y)&f_y(x,y)\\
g_x(x,y)&g_y(x,y)
\end{pmatrix}
\]
At an equilibrium point \((x_0,y_0)\), use:
\[
J(x_0,y_0)
\]
to study the local behavior.
7. Trace and determinant
For a \(2\times2\) Jacobian matrix:
\[
J=
\begin{pmatrix}
a&b\\
c&d
\end{pmatrix}
\]
the trace and determinant are:
\[
\begin{aligned}
\operatorname{tr}(J)&=a+d\\
\det(J)&=ad-bc
\end{aligned}
\]
The eigenvalues satisfy:
\[
\lambda^2-\operatorname{tr}(J)\lambda+\det(J)=0
\]
8. Stability classification
9. Hyperbolic and nonhyperbolic equilibria
An equilibrium point is called hyperbolic if no eigenvalue of the Jacobian has real part zero.
Hyperbolic equilibria are usually classified reliably by linearization.
If an eigenvalue has real part zero, the equilibrium is nonhyperbolic. In that case, linearization may not give
the full answer, and a deeper nonlinear analysis is needed.
10. Predator-prey model
A classic predator-prey model is:
\[
\begin{aligned}
x^{\prime}&=\alpha x-\beta xy\\
y^{\prime}&=\delta xy-\gamma y
\end{aligned}
\]
Here \(x\) can represent prey and \(y\) can represent predators. The interaction term \(xy\) couples the two
populations.
11. Damped pendulum preview
A damped pendulum can be written as a first-order system:
\[
\begin{aligned}
x^{\prime}&=y\\
y^{\prime}&=-\sin(x)-by
\end{aligned}
\]
Here \(x\) is angular position and \(y\) is angular velocity. The term \(-by\) models damping.
12. Van der Pol oscillator
The Van der Pol oscillator is a nonlinear oscillator often written as:
\[
\begin{aligned}
x^{\prime}&=y\\
y^{\prime}&=\mu(1-x^2)y-x
\end{aligned}
\]
It is famous because it can produce a stable closed curve called a limit cycle.
13. Numerical trajectory integration
Many nonlinear systems do not have simple closed-form solutions. Numerical methods approximate the trajectory.
This calculator uses fourth-order Runge-Kutta integration.
\[
\frac{d}{dt}
\begin{pmatrix}
x\\
y
\end{pmatrix}
=
\begin{pmatrix}
f(x,y)\\
g(x,y)
\end{pmatrix}
\]
The numerical curve shows the approximate solution through the selected initial point.
15. Common mistakes
- Confusing phase plots and time plots: a phase portrait plots \(y\) versus \(x\), not \(x\) versus \(t\).
- Ignoring equilibria: equilibrium points organize the entire phase portrait.
- Using only the vector field: trajectories show actual numerical solution paths through initial points.
- Forgetting nullclines: nullclines help locate equilibria and understand direction changes.
- Trusting linearization too much: nonhyperbolic equilibria may require deeper nonlinear analysis.
- Choosing a poor graph window: if no equilibrium appears, widen the phase-plane window.
- Ignoring units: axis numbers should be interpreted with the selected \(x\)- and \(y\)-units.
