Layer 22
Classical Mechanics, Phase Space & Chaos
From Newtonian motion to Hamiltonian phase space, nonlinear forcing, bifurcations and deterministic chaos.
interactivemodel assumptions visibleestablished science separated from analogy
Story step: Population and ecosystem dynamics return us to a deeper mathematical question: how do deterministic laws generate stable motion, bifurcations and chaos?
1 · Newtonian oscillator & phase space
A harmonic oscillator traces a closed ellipse in phase space because its total energy is conserved.
m x¨ = −kx H = ½mv² + ½kx²
—total energy
—angular frequency
—period
2 · Driven Duffing oscillator
A deterministic nonlinear oscillator can be regular or chaotic depending on damping, forcing and frequency.
x¨ + δx˙ − x + x³ = γ cos(ωt)
The two traces begin only 10⁻⁶ apart. Rapid separation is a signature of sensitive dependence, not randomness in the equations.
3 · Bifurcation & deterministic chaos in the logistic map
The logistic map is not a mechanical system, but it makes the route from a stable fixed point to period doubling and chaos unusually visible.
xₙ₊₁ = r xₙ(1 − xₙ)
—Lyapunov exponent
—qualitative behavior
—latest x
Scientific boundary: The harmonic oscillator is exactly integrable. The Duffing and logistic models illustrate nonlinear dynamics but are not universal descriptions of physical systems. Positive sensitivity depends on parameters and numerical resolution.