Systems of ODEs

Many physical systems involve multiple variables that change simultaneously, leading to coupled systems of differential equations.

Linear Systems with Constant Coefficients

A first-order linear system has the form: $\mathbf{x}'(t) = \mathbf{A}\mathbf{x}(t)$ where $\mathbf{A}$ is a constant matrix. If $\mathbf{A}$ has eigenvalues $\lambda_i$ and eigenvectors $\mathbf{v}_i$, the general solution is: $\mathbf{x}(t) = \sum c_i e^{\lambda_i t} \mathbf{v}_i$

import numpy as np

# System: dx/dt = x + y, dy/dt = 4x + y
A = np.array([[1, 1], [4, 1]])

eigenvalues, eigenvectors = np.linalg.eig(A)

print("Eigenvalues:", eigenvalues)
print("Eigenvectors:\n", eigenvectors)

# Solution components involve exp(3t) and exp(-t)

Phase Plane Analysis

The behavior of the system near equilibrium points (where $\mathbf{x}' = 0$) can be classified by the eigenvalues:

• Sink (Stable node): All eigenvalues real and negative.
• Source (Unstable node): All eigenvalues real and positive.
• Saddle Point: Real eigenvalues of opposite signs.
• Center: Purely imaginary eigenvalues.
• Spiral: Complex eigenvalues with non-zero real part.

Numerical Integration (Runge-Kutta)

For non-linear systems or those without analytical solutions, we use numerical methods like the 4th-order Runge-Kutta (RK4).

def rk4_step(f, x, dt):
  k1 = f(x)
  k2 = f(x + 0.5 * dt * k1)
  k3 = f(x + 0.5 * dt * k2)
  k4 = f(x + dt * k3)
  return x + (dt / 6.0) * (k1 + 2*k2 + 2*k3 + k4)

# Predator-Prey (Lotka-Volterra)
def lotka_volterra(state):
  x, y = state
  a, b, c, d = 1.0, 0.1, 1.5, 0.75
  return np.array([a*x - b*x*y, -c*y + d*x*y])

state = np.array([10.0, 5.0]) # Initial populations
dt = 0.1
for _ in range(5):
  state = rk4_step(lotka_volterra, state, dt)
  print(f"Populations: {state}")