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Lorenz System Bifurcation
Strange attractor — sensitive dependence on initial conditions
The Lorenz system (1963): ẋ=σ(y−x), ẏ=x(ρ−z)−y, ż=xy−βz. Originally a simplified model of atmospheric convection. At σ=10, ρ=28, β=8/3 it exhibits a strange attractor — bounded, aperiodic, with fractal dimension ≈2.06. Two nearly-identical initial conditions diverge exponentially (Lyapunov exponent λ₁≈0.9), the mathematical essence of chaos.