Linear and Nonlinear Systems

2D Linear System

A linear system of ODEs can be written in two equivalent forms:

Equations

$\begin{cases}\dot{x} = ax + by \\ \dot{y} = cx + dy\end{cases}$

⇔

Matrix form

$\dot{\vec{x}} = A\vec{x},\quad A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}$

To find the general solution, we calculate the eigenvalues and eigenvectors of $A$. The form of the solution depends on whether the eigenvalues are real or complex.

Finding eigenvalues and eigenvectors

Solve the characteristic equation $\det(A - \lambda I) = 0$ to obtain eigenvalues $\lambda_1, \lambda_2$. For each $\lambda_i$, solve $(A - \lambda_i I)\vec{v_i} = 0$ to obtain the corresponding eigenvector $\vec{v_i}$.

$\det(A - \lambda I) = 0 \quad \Rightarrow \quad \lambda_1,\, \lambda_2,\, \vec{v_1},\, \vec{v_2}$

Case A

Real eigenvalues

When $\lambda_1, \lambda_2 \in \mathbb{R}$ and $\lambda_1 \neq \lambda_2$, the general solution is a direct linear combination.

Build the general solution as a linear combination of solutions associated with each eigenvalue/eigenvector

$\vec{x}(t) = c_1\, e^{\lambda_1 t}\, \vec{v_1} + c_2\, e^{\lambda_2 t}\, \vec{v_2}$

Apply initial conditions $\vec{x}(0) = \vec{x_0}$ to determine $c_1$ and $c_2$

Case B

Complex eigenvalues

When $\lambda_{1,2} = \alpha \pm i\omega$, eigenvalues and eigenvectors are complex conjugates. Two real linearly independent solutions must be extracted.

Form the complex solutions

$\vec{x}_+ = e^{\lambda_1 t}\, \vec{v_1} \quad\quad \vec{x}_- = e^{\lambda_2 t}\, \vec{v_2}$

Apply Euler's formula $e^{it} = \cos t + i\sin t$ to separate real and imaginary parts

Combine to obtain two real solutions linearly independent

$\vec{x}_1 = \dfrac{\vec{x}_+ + \vec{x}_-}{2} \quad\quad \vec{x}_2 = \dfrac{\vec{x}_+ - \vec{x}_-}{2i}$

General solution as a real linear combination

$\vec{x}(t) = c_1\, \vec{x}_1 + c_2\, \vec{x}_2$

Distinct real eigenvalues

Stable node

Both eigenvalues real negative

$\Delta > 0,\; \tau < 0,\; \tau^2 > 4\Delta$

Unstable node

Both eigenvalues real positive

$\Delta > 0,\; \tau > 0,\; \tau^2 > 4\Delta$

Saddle point

Real eigenvalues of opposite signs

$\Delta < 0$

Complex eigenvalues $\lambda = \alpha \pm i\omega$

Stable focus

Negative real part: $\alpha < 0$

$\Delta > 0,\; \tau < 0,\; \tau^2 < 4\Delta$

Unstable focus

Positive real part: $\alpha > 0$

$\Delta > 0,\; \tau > 0,\; \tau^2 < 4\Delta$

Centre

Zero real part: $\alpha = 0$

$\Delta > 0,\; \tau = 0$

Repeated eigenvalues $\lambda_1 = \lambda_2$

Star node

2 eigenvectors linearly independent

Diagonal matrix: $A = \lambda I$

stable if $\lambda < 0$ unstable if $\lambda > 0$

Degenerate node

1 eigenvector (non-diagonalisable matrix)

stable if $\lambda < 0$ unstable if $\lambda > 0$

Find fixed points

Values $(x^*, y^*)$ where the system does not evolve.

Set both equations to zero simultaneously:

$\dot{x} = f(x,y) = 0$
$\dot{y} = g(x,y) = 0$

Solve the system of equations for $(x^*, y^*)$. There may be multiple fixed points

Classify fixed points

Repeat for each fixed point $(x^*, y^*)$.

Calculate the Jacobian matrix: $$J(x,y) = \begin{pmatrix}\frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y}\end{pmatrix}$$

Substitute $(x^*, y^*)$ into $J$ and calculate the eigenvalues $\lambda_{1,2}$

Use the eigenvalues to classify the fixed point: the behaviour is identical to that of linear systems (see table in section 2)

Phase diagram

Combine the information from all fixed points.

Calculate the eigenvectors of each fixed point and draw the manifolds (stable and unstable)

Draw the nullclines: where $\dot{x}=0$ (purely vertical motion) and where $\dot{y}=0$ (purely horizontal motion)

Sketch the trajectories respecting the behaviour near each fixed point