By using a plotter to output a computer generated strange attractor solution to the Lorenz equation, that draws a line corresponding to the same fixed interval for every time step, it was found that the characteristic concentric circles of that attractor were approximated by a set of nested hexagons resembling a spider-web.

That is, these circular orbits had a period of 6 units independent of their radius. Of course, this period independence, of an orbit whose radius steadily increases is what one would expect from the motion of a charged a particle in a cyclotron.

The question is, does this observation go any deeper than pure coincidence?

  • $\begingroup$ Do you have a reference for the results you quote? The Lorenz attractor is chaotic, while the trajectory you describe is periodic, so there's something very off here. $\endgroup$ – stafusa Feb 6 at 20:04
  • $\begingroup$ I came to these conclusions from the derivation below given the observation above. The trajectories of a cyclotron are not periodic since they change their radius with every orbit-rather, the period of the orbits are radius independent until they are unpredictably kicked off the circular part of the attractor by the chaotic aspect of the motion $\endgroup$ – H. Cooper Feb 8 at 2:53

The answer is yes because there is a general relationship between 3-D strange attractors and the motion of a charged particle in an EM field.

To see this, write the equations for a 3-D system as v = dx/dt = A(r).

Form dv/dt = (v . grad)A and use familiar vector identities to obtain dv/dt = E - v x B,

E = -gradV

V = -A . A/2

B = curl(A).

The Lorenz equations thus give rise to that special case of a Lorentz eq. for a negatively charged particle moving in effective electric and magnetic fields where electric potential takes the form

V = - (A . A)/2

By taking the Laplacian of V, Maxwell's first equation returns the source density needed to implement this special case.

Although, it might seem that gauge invariance is violated by the form of V, since under these circumstances that transformation must be applied to V itself, there is no violation.

Finally, since the conserved energy of this system is E = (v.v - A . A)/2, the attractors, v = A, correspond to the set of zero energy initial conditions.

In general, a 3-D strange attractor corresponds to the trajectory of a charged particle trapped in an electro-magnetic bottle-one that can reproduce the action of a cyclotron using static E and B fields:

E = (charge to mass ratio)E

B = (charge to mass ratio)B.

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    $\begingroup$ please use mathjax for the formulas $\endgroup$ – hyportnex Feb 6 at 15:50
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    $\begingroup$ This does not really make any sense. The Lorenz equations are derived entirely separately from electromagnetism (from a model of order parameters of fluid convection). Are you really saying that you can imitate any continuous 3D dynamical system by having a sufficiently contrived electric and magnetic potential? $\endgroup$ – Anders Sandberg Feb 6 at 17:39
  • $\begingroup$ The answer to that question is yes. This conclusion follows from purely mathematical identities having nothing to do with physical context. The Lorenz and Rossler attractors are special cases that feature cyclotron like motion $\endgroup$ – H. Cooper Feb 8 at 2:42
  • $\begingroup$ Once the v = A(r) is specified in some physical context, the identity grad(A . A) = 2A x curl(A) + 2(A . grad)A derives the Lorentz equation. This result is thus independent of the physical context. $\endgroup$ – H. Cooper Feb 10 at 17:49
  • $\begingroup$ That is to say, once the chain rule is used to derive dv/dt = (v . grad)A, the identities do the rest. $\endgroup$ – H. Cooper Feb 10 at 18:02

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