{"title":"The Lorenz system: the largest Lyapunov exponent across ρ","parts":[],"pages":[{"id":"lorenz","title":"The Lorenz system across ρ","part":null,"status":"trial","summary":null,"desc":"$\\dot x = \\sigma(y-x)$, $\\dot y = x(\\rho - z) - y$, $\\dot z = xy - \\beta z$ with\n$\\sigma = 10$ and $\\beta = 8/3$. The largest Lyapunov exponent $\\lambda_1$ comes from one\ntrajectory per $\\rho$ after a warm-up, through `DynamicalModels.lyapunov_exponent`.\nWhere $\\lambda_1 \u003e 0$ nearby trajectories separate: the system is chaotic. The textbook\nonset for these $\\sigma, \\beta$ is near $\\rho \\approx 24.74$.\n","figures":[],"tables":[],"codes":[],"content":[],"sections":[{"id":"sweep","title":"Largest Lyapunov exponent","desc":null,"figures":[{"id":"sweep_fig1","caption":"λ₁ against ρ; the dashed line is λ₁ = 0","code":"fig","params":null,"assets":["assets/figures/lorenz/sweep/sweep_fig1.svg"],"data":"assets/figures/lorenz/sweep/sweep_fig1.csv","table":{"header":["series","x","y"],"rows":[["y1",14.0,-0.39482806576746504],["y1",16.0,-0.30771039388554267],["y1",18.0,-0.22799185558640583],["y1",20.0,-0.15455204201711514],["y1",22.0,-0.08641389193838844],["y1",24.0,0.7628192501792441],["y1",24.74,0.8072655655087094],["y1",26.0,0.8568037408021609],["y1",28.0,0.9029252414256069],["y1",30.0,0.9509739887184903],["y1",35.0,1.044859479173399],["y1",40.0,1.1314122118153696],["y2",1.0,0.0],["y2",2.0,0.0],["y2",null,0.0]],"total":15},"comments":[]}],"tables":[{"id":"sweep_tbl1","caption":"one trajectory per ρ","code":"[[string(r), string(round(l; digits = 4)), if l \u003e 0\n        \"chaotic\"\n    else\n        \"not chaotic\"\n    end] for (r, l) = zip(rhos, lam)]","header":["ρ","λ₁","verdict"],"rows":[["14.0","-0.3948","not chaotic"],["16.0","-0.3077","not chaotic"],["18.0","-0.228","not chaotic"],["20.0","-0.1546","not chaotic"],["22.0","-0.0864","not chaotic"],["24.0","0.7628","chaotic"],["24.74","0.8073","chaotic"],["26.0","0.8568","chaotic"],["28.0","0.9029","chaotic"],["30.0","0.951","chaotic"],["35.0","1.0449","chaotic"],["40.0","1.1314","chaotic"]]}],"codes":[],"content":[{"kind":"figure","id":"sweep_fig1"},{"kind":"table","id":"sweep_tbl1"}]}]}]}