Knowing every equation exactly is still not enough to predict tomorrow. The double pendulum here follows the Lagrangian equations integrated by RK4, without a grain of randomness in it, yet three copies set 0.001 radian apart cling together for a few seconds and then peel off into three different fates: the gap between nearby trajectories grows exponentially, a positive Lyapunov exponent. Chaos is not luck, it is a consequence of nonlinearity — and Lorenz’s three equations of 1963 trace a path winding around two butterfly wings forever without once repeating.
Knowing every equation exactly is still not enough to predict tomorrow. The double pendulum here follows the Lagrangian equations integrated by RK4, without a grain of randomness in it, yet three copies set 0.001 radian apart cling together for a few seconds and then peel off into three different fates: the gap between nearby trajectories grows exponentially, a positive Lyapunov exponent. Chaos is not luck, it is a consequence of nonlinearity — and Lorenz’s three equations of 1963 trace a path winding around two butterfly wings forever without once repeating.
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