Not wishing to underestimate the power of the Dark Side, I've produced some renderings of the Aizawa attractor via MATLAB.
These were obtained using a very simple approach (Euler's method - the (x,y,z) coordinates of the current point are pushed through the differential equations, the resulting new values for the time-derivatives of (x,y,z) then multiplied by a time step, and then added to the current point, to give the next (x,y,z) point, and so on).
The following shows the xz-plane for a time step of 0.0001.
The published examples of the Aizawa attractor (for example, [1]) seem to use 0.01 as the time step - but it's apparent that the size of the time step has (perhaps not surprisingly) some effect on the solution.
The results below are for different time - step sizes (0.01, 0.001, 0.0001 seconds). In all cases the solutions show the first 600 seconds of the trajectory (i.e. for the 0.01 second time step, the equations were iterated 60,000 times, producing 60,000 points, and so on). The parameters are as in the previous posts (α = 0.95, β = 0.7, γ = 0.6, δ = 3.5, ε = 0.25 and ζ = 0.1), with initial conditions also as in previous post (x = 0.1, y = z = 0).
These tentative results tend to suggest that the 0.01 time step is a bit coarse: the 0.001 and 0.0001 cases look similar to each other, but different to the 0.01 case. Interestingly, the central tube-like structure becomes thicker (and more open) when the time step is smaller. Also shown is the attractor in the xy-plane.
The following shows the xy-plane for the 0.0001 case.
Evidently, the tube-like opening, although apparently circular, is not centred on the origin. This appears to agree with results obtained from the analog computer:
Reference
[1] http://jlswbs.blogspot.co.uk/2011/10/aizawa.html
These were obtained using a very simple approach (Euler's method - the (x,y,z) coordinates of the current point are pushed through the differential equations, the resulting new values for the time-derivatives of (x,y,z) then multiplied by a time step, and then added to the current point, to give the next (x,y,z) point, and so on).
The following shows the xz-plane for a time step of 0.0001.
The published examples of the Aizawa attractor (for example, [1]) seem to use 0.01 as the time step - but it's apparent that the size of the time step has (perhaps not surprisingly) some effect on the solution.
The results below are for different time - step sizes (0.01, 0.001, 0.0001 seconds). In all cases the solutions show the first 600 seconds of the trajectory (i.e. for the 0.01 second time step, the equations were iterated 60,000 times, producing 60,000 points, and so on). The parameters are as in the previous posts (α = 0.95, β = 0.7, γ = 0.6, δ = 3.5, ε = 0.25 and ζ = 0.1), with initial conditions also as in previous post (x = 0.1, y = z = 0).
These tentative results tend to suggest that the 0.01 time step is a bit coarse: the 0.001 and 0.0001 cases look similar to each other, but different to the 0.01 case. Interestingly, the central tube-like structure becomes thicker (and more open) when the time step is smaller. Also shown is the attractor in the xy-plane.
The following shows the xy-plane for the 0.0001 case.
 |
| xy-plane, 10 minutes' of trajectory |
Reference
[1] http://jlswbs.blogspot.co.uk/2011/10/aizawa.html
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