In this context jerk is the rate of change of acceleration...
Presumably this is all well known to most people, but I find it interesting that (certain?) chaotic systems can be written in jerk form...
dz/dt = J(x,y,z)
where J is (some?) function of x, dx/dt, and d2 x/dt2 i.e. with y = dx/dt and z = dy/dt = d2 x/dt2 .
For example (Sprott [1], equation (10)):
d3x/dt3 = -a d2 x/dt2 + (dx/dt)2 -x .
Which - as Sprott says - is chaotic for 'a small range of the single parameter a around 2.02'.
Writing out the three equations gives something to wire up:
dx/dt = y
dy/dt = z
dz/dt = -a z + y2 - x .
I tried patching these up directly - but the thing overloaded - so nothing for it but to have a go at scaling. After looking in several books (and being not much the wiser - I generally need to construct things for myself using more, smaller, steps than most books seem to offer), I decided to just write down:
u = x/2, v = y/2 and w = z/2 → x = 2u, y = 2v and z = 2w .
Hence
d(2u)/dt = 2v → du/dt = v;
d(2v)/dt = 2w → dv/dt = w;
d(2w)/dt = -2wa + (2v)2 → dw/dt = -aw + 2v2 - u.
I guess it's obvious (in hindsight) that it's the non-linear bit of the equations which ends up with the scaling. I patched these 'scaled' equations up:
One nice things is the elegance of the patching, in the sense that the integrators form a simple chain.
In FAST mode the computer produced this the following. This proved stable - but only for a small range of values for parameter a:
Slowing things down (i.e. 1 μF integrating capacitors / 1 M ohm input resistors) gave the following pen plot - but note it goes awry after half a dozen 'orbits'...
The final part of [1] alludes to jerky Lorenz-like systems, which involve - amongst other things - functions like tan and hyperbolic sine :) But these will have to wait awhile, whilst the final(?) rack of my computer gets some attention...this - it is planned - will furnish sine, cosine, tangent, and their hyperbolic cousins...
[1] Sprott, J. C., 'Simplifications of the Lorenz Attractor', Nonlinear Dynamics, Psychology, and Life Sciences, 2009, Vol. 13, No. 3, pp. 217 - 278.
Presumably this is all well known to most people, but I find it interesting that (certain?) chaotic systems can be written in jerk form...
dz/dt = J(x,y,z)
where J is (some?) function of x, dx/dt, and d2 x/dt2 i.e. with y = dx/dt and z = dy/dt = d2 x/dt2 .
For example (Sprott [1], equation (10)):
d3x/dt3 = -a d2 x/dt2 + (dx/dt)2 -x .
Which - as Sprott says - is chaotic for 'a small range of the single parameter a around 2.02'.
Writing out the three equations gives something to wire up:
dx/dt = y
dy/dt = z
dz/dt = -a z + y2 - x .
I tried patching these up directly - but the thing overloaded - so nothing for it but to have a go at scaling. After looking in several books (and being not much the wiser - I generally need to construct things for myself using more, smaller, steps than most books seem to offer), I decided to just write down:
u = x/2, v = y/2 and w = z/2 → x = 2u, y = 2v and z = 2w .
Hence
d(2u)/dt = 2v → du/dt = v;
d(2v)/dt = 2w → dv/dt = w;
d(2w)/dt = -2wa + (2v)2 → dw/dt = -aw + 2v2 - u.
I guess it's obvious (in hindsight) that it's the non-linear bit of the equations which ends up with the scaling. I patched these 'scaled' equations up:
One nice things is the elegance of the patching, in the sense that the integrators form a simple chain.
In FAST mode the computer produced this the following. This proved stable - but only for a small range of values for parameter a:
Slowing things down (i.e. 1 μF integrating capacitors / 1 M ohm input resistors) gave the following pen plot - but note it goes awry after half a dozen 'orbits'...
The final part of [1] alludes to jerky Lorenz-like systems, which involve - amongst other things - functions like tan and hyperbolic sine :) But these will have to wait awhile, whilst the final(?) rack of my computer gets some attention...this - it is planned - will furnish sine, cosine, tangent, and their hyperbolic cousins...
[1] Sprott, J. C., 'Simplifications of the Lorenz Attractor', Nonlinear Dynamics, Psychology, and Life Sciences, 2009, Vol. 13, No. 3, pp. 217 - 278.
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