Simulate the Lorenz system in 3D. Adjust sigma, rho, beta parameters and watch butterfly-shaped trajectories evolve.
Unlike a static rendering of the familiar butterfly, the Lorenz Attractor Simulator turns sigma, rho, and beta into a three-dimensional state-space path for the coupled Lorenz equations. As the simulated state (x, y, z) advances, the plot reveals whether motion settles, circulates, or repeatedly switches between two lobes. Parameter changes can alter both the shape and the long-term regime. The main surprise is that the butterfly is not a fixed curve: in chaotic regimes, tiny differences in starting state or numerical progression can eventually produce visibly different paths even when the three displayed parameters match.
The system being traced is dx/dt = sigma*(y-x), dy/dt = x*(rho-z)-y, and dz/dt = x*y-beta*z. Each displayed point is one state of these three coupled variables, and the local derivatives give its direction of travel. Sigma multiplies the difference y-x, rho occurs in the y derivative, and beta contributes the -beta*z term. Changing any parameter therefore changes the vector field followed by the trajectory.
The visualization traces successive states through 3D rather than plotting one variable against time. With sigma=10, rho=28, and beta=8/3, suitable nonzero starting states typically approach the classic two-lobed attractor after transient motion. A path may wind around one lobe several times before switching to the other. Those revisits are nearby passages through state space, not evidence that the state has repeated exactly.
The classic butterfly is associated with particular parameter choices and a nonzero starting state. At x=y=z=0, all three derivatives are zero, so the system remains at the origin for any sigma, rho, and beta. Other settings can settle toward equilibria or show different transients, so the absence of two lobes is not automatically a malfunction.
No. The axes represent x, y, and z, while time determines progress along the curve. Moving farther right means that x is larger, not necessarily that the state is later in the run. Unless a separate time-series view is provided, use this plot to inspect state-space geometry rather than measure periods or event times.
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