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The Lorenz system is a classic example of the “butterfly effect,” where small changes in initial conditions lead to very different outcomes. In this visualization, we simulated the Lorenz attractor using MATLAB and Claude with 10 slightly varied starting points.

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Mathematics, physics. We live in a chaotic, unpredictable universe. This is a grid of 812 double pendulums, each with slightly different initial conditions for the two angles of the pendulum. By Jonathan Nafziger, Used with permission
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A double pendulum demonstrates chaos theory, tracing its unpredictable path The dependence on initial conditions, and sensitivity to variation is one of the fundamental properties of chaos This is commonly referred to as "the butterfly effect,"
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When the Lorenz Attractor Becomes a Cosmos Many of us have seen plots of the famous Lorenz system that famous butterfly-shaped chaos: ẋ = σ(y - x) ẏ = x(ρ - z) - y ż = xy - βz But here, we went one layer deeper. At every point (x, y, z), we computed the local Jacobian matrix, J(x, y, z) = [ -σ σ 0 ] [ ρ - z -1 - x ] [ y x - β ] Then we found its eigenvalues λ and plotted them in the complex plane. Re(λ) tells us whether nearby motion is locally growing or decaying. Im(λ) tells us whether that local motion is rotating. That gives us the spectral flow of the Lorenz system and not the attractor itself, but the hidden local linear geometry driving its instability. When I saw that flow, it looked strangely cosmic with two mirrored swarms curling like particles trapped in a mathematical whirlpool. So I mapped that spectral flow onto our living fabric, not as Einstein Spacetime this time, but as a creative geometric stage for chaos itself. #LorenzSystem #ChaosTheory #DynamicalSystems #Eigenvalues #SpectralFlow #ComplexPlane #NonlinearDynamics #Mathematics #Physics
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As someone who ships LLM systems in production, this double pendulum video is the closest thing to a "why one changed word breaks Claude and GPT-5" explainer I've ever seen released for free. Everyone thinks LLMs are deterministic machines. They're not. Every prompt to GPT-5 is a chaotic double pendulum - a tiny perturbation in the initial state (one comma, one whitespace) sends the entire response into a completely different attractor. Bookmark this 26-min video and watch tonight. Same butterfly-effect chaos that broke Poincaré's math, now the exact reason your API calls to Claude feel unreliable.
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There truly is an underlying order in every chaotic system In this scene, we have 2500 identical triple pendulums on a grid with almost the same initial conditions.
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