Circuits, population dynamics, epidemics, and mechanics
Epidemics, circuits, planets, and pendulums all run on the same equations. Adjust the parameters in each demo below and watch the theory play out.
Adjust the infection rate β and recovery rate γ. When R0 = β/γ rises above 1 the epidemic spikes; below 1 it dies out.
Every topic in this module -- direction fields, second-order oscillators, phase portraits, stability analysis, and chaos -- finds direct expression in real-world systems. This gallery showcases four applications that connect ODE theory to epidemiology, electrical engineering, celestial mechanics, and nonlinear dynamics.
The SIR model is a system of three coupled first-order ODEs describing the spread of an infectious disease through a population. The variables S (susceptible), I (infected), and R (recovered) satisfy dS/dt = -βSI, dI/dt = βSI - γI, dR/dt = γI. The basic reproduction number R0 = β/γ determines whether an epidemic occurs: when R0 > 1 the disease spreads exponentially before burning out.
Connection to theory: This is a nonlinear autonomous system whose phase portrait (shown in the S-I plane) reveals a single trajectory for each set of initial conditions. The nullcline S = γ/β marks the peak of the infection curve -- exactly the kind of qualitative analysis studied in the phase portraits module.
Adjust resistance, inductance, and capacitance. Resistance acts like damping: raise it and the ringing charge settles without oscillating.
An RLC circuit obeys the second-order ODE Lq″ + Rq′ + q/C = V(t), which is mathematically identical to the spring-mass system mx″ + bx′ + kx = F(t). Resistance R plays the role of damping, inductance L acts as inertia, and capacitance C corresponds to the spring compliance. The same three regimes appear: underdamped (oscillatory), critically damped, and overdamped.
Connection to theory: This is the same characteristic equation Lr2 + Rr + 1/C = 0 from the second-order oscillators module. The discriminant R2 - 4L/C determines the damping regime, exactly as b2 - 4mk does for the mechanical spring-mass system.
Change the initial velocity and watch the orbit go from circle to ellipse, then escape on a parabola or hyperbola once the energy reaches zero.
The two-body gravitational problem reduces to a second-order ODE for the radial coordinate: r″ = -GM/r2 + L2/(m2r3). The initial velocity determines the total energy and therefore the orbit type: circular, elliptical, parabolic, or hyperbolic. This is a conservative system where kinetic and potential energy trade off while total energy stays constant.
Connection to theory: Orbital mechanics illustrates stability analysis in a central force field. Circular orbits are equilibria of the effective radial potential. Small perturbations yield closed (elliptical) orbits -- an example of stable oscillation about an equilibrium, just as in the stability module.
Two pendulums start 0.001 radians apart. Set a small starting angle θ2and they swing together; set a large one and they tumble and soon go separate ways.
The double pendulum is governed by a system of two coupled, nonlinear second-order ODEs derived from the Lagrangian. For small angles the system is nearly integrable and periodic; for large angles it exhibits deterministic chaos. Two pendulums with initial angles differing by just 0.001 radians rapidly diverge -- the hallmark of sensitive dependence on initial conditions.
Connection to theory: This demonstration ties together nonlinear dynamics, bifurcations, and chaos. At small angles the linearized system predicts two normal modes; at large angles the full nonlinear equations produce the same kind of sensitive dependence seen in the Lorenz attractor module. The transition between regular and chaotic motion is itself a bifurcation.
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