Camera:
The source implements the deterministic Lorenz equations, not random forcing. We first inspect the derivative field at the initial point: x equals point one, y and z are zero. Hold rho at twenty eight and beta at eight thirds. Start with sigma ten. The x derivative is sigma times y minus x, which gives minus one. The y derivative is two point eight and the z derivative is zero. Yellow represents the x component and cyan the y component. Both component lengths use forty pixels per derivative unit. Increase only sigma from ten to twenty while keeping the inspected point fixed. The yellow endpoint moves farther left, because sigma multiplies the same negative difference, minus point one. The cyan component remains unchanged. This inspects the vector field at one fixed position; it does not integrate a trajectory while holding that position stationary. The source parameter slider also changes the equation and resets its numerical trajectory. Hold sigma at twenty. The x derivative is now minus two, so the yellow component is eighty pixels left of the origin. The y derivative remains two point eight, one hundred twelve pixels upward, and the z derivative remains zero. The component endpoints stop while these values are read. Increasing sigma changed a model parameter. It did not improve the accuracy of a numerical approximation, which is a different question. Restore sigma ten and beta eight thirds. Inspect the same initial point, point one, zero, zero. Start rho at twenty eight. The y derivative is x times rho minus z, minus y: here that is point one times twenty eight, or two point eight. The cyan component extends one hundred twelve pixels upward. The yellow x derivative remains minus one, forty pixels to the left. The fixed point and the shared derivative scale are the same as in the first scene. Increase only rho from twenty eight to forty eight. The cyan endpoint moves upward according to point one times the current rho. The yellow component stays fixed because rho does not appear in the x equation. This comparison isolates one parameter at the initial state. It does not claim that a complete trajectory scales proportionally, or that every rho value gives chaotic behavior. Hold rho at forty eight. The y derivative is four point eight, one hundred ninety two pixels upward. The x derivative is still minus one and the z derivative is still zero. Both components remain fixed while we read the result. Preset names by themselves are not proofs of periodicity or chaos. Those behaviors depend on the equations, parameters, initial conditions and the numerical interval examined. Now use the classic source parameters: sigma ten, rho twenty eight, beta eight thirds. Start at point one, zero, zero. The source advances with fourth order Runge Kutta, using a step of point zero one model time units. The cyan curve is precomputed from exactly one hundred of those steps. Its white marker begins at the initial state. This is an x y projection with eight pixels per coordinate unit. The upper marker tracks step index from zero to one hundred. Advance through the numerical states, one source Runge Kutta step at a time. Each step evaluates four derivative stages and combines them with weights one, two, two, one, divided by six. The marker visits the exact computed state at each integer step. Motion between adjacent states is smoothly interpolated for explanation; those intermediate pixels are not extra Runge Kutta steps or an exact solution of the continuous differential equation. The full curve is shown as a reference while the marker follows it. Hold at step one hundred, model time one. This calculation ends near x minus eight point one one, y minus nine point four zero, and z twenty five point six zero. The marker stops at that actual numerical endpoint. The z coordinate is omitted from this projection, not set to zero. A finite step trajectory should be checked for convergence when accuracy matters. This one unit segment does not establish a universal Lyapunov exponent or a physical forecast. The original pause, parameter sliders, camera views, reset and actual CSV and PNG exports remain available above.