I had a question about stochastic differential equations. I come from statistics, so I am more familiar with say random walks, which are of course, discrete. So if I had a process like where $x(t)$ traces the random walk evolution of a variable over time, then that might look like this recursively:
$$ x(t+1) = x(t) + \epsilon \sim Binomial(p,n) $$
where $p,n$ are parameters for the chance of success and number of trials, respectively. If I ran this simulation multiple and plotted time $t$ versus $x(t)$, I would get a fan or cone shaped plot with all trajectories starting at the same point, but then progressively spreading out with each time step.
My question is, when I look at plots of stochastic differential equations, I don't see this same kind of fanning behavior. When talking about an SDE of the form:
$$ du = f(u, p, t)dt + g(u, p, t)dW $$
In this case, $f$ is the deterministic part of the process and $g$ is the noise process, usually Brownian motion. This is of course a continuous time equation. Here is a plot of a stochastic version of the lorenz equation that conforms to the SDE form above--taken from the Julia Differential Equations documentation.
Again, my question is, why don't we see the trajectory fan out over time. It seems like the trajectory does wiggle, but it does not diffuse as in a random walk. I was just trying to understand what are the forces at play that prevent the noise process to overwhelm the deterministic part of the process over time?
The chaotic attractor of the Lorenz system is, as the name says, attracting. Thus any deviation introduced by the stochastic term is corrected towards the double spiral.
One still gets a trajectory fan if one starts at the same point and computes and plots several paths with different random walks. Again, any deviation from the plane or surface of the single spirals will be quickly corrected, the random movements spread and accumulate majorly inside that surface.
Above for the initial segment $t\in[0,2]$ after the start at $(x,y,z)=(5,5,5)$, below a little more evolution with $t\in[0,6]$.
As one can see, in the outer rim of the spirals, away from their intersection where the velocity of the Lorenz system is high the paths are almost smooth, while in the slower inner parts the noise terms have more influence. As was constructed, as the random motion increments are merely added to the velocity vector.
The bundle of trajectories visibly dissolves with the first change to the other spiral. The inane chaotic nature of the Lorenz system magnifies the deviations due to the accumulated random increments.