Systems of ODEs

What is it?

Formulas

$$\mathbf x' = A\mathbf x \implies \mathbf x(t) = e^{At}\mathbf x_0, \qquad e^{At} = \sum_{k\ge0}\frac{(At)^k}{k!}$$

$$\dot S = -\beta SI, \quad \dot I = \beta SI - \gamma I, \quad \dot R = \gamma I$$

SIR epidemic model

Where it shows up in computing

• Population and epidemic models★★★★★fundamentalPhysics and simulation

  SIR/SEIR epidemics and Lotka–Volterra predator–prey models are nonlinear ODE systems.

• N-body gravitational simulation★★★★★fundamentalPhysics and simulation

  $N$ gravitating bodies give $6N$ coupled first-order equations.

• Robot dynamics★★★★★frequentRobotics and control

  A robot's equations of motion $M(q)\ddot q + C(q,\dot q)\dot q + g(q) = \tau$ form a coupled system.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

What depends on it

• Dynamical systems