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WOLFRAM NOTEBOOK
Chaotic Attractor for the Solar Cycle
Chaotic Attractor for the Solar Cycle
The solar activity model comprises two nearly independent nonlinear dynamical systems [1]. Model A represents the oscillatory mechanism underlying the solar cycle, and Model B represents the main convective dynamo of the Sun [3].
Model A is a two-dimensional nonlinear system, an oscillator localized in the tachocline, =βX-ωY-(+)X,=βY+ωX-(+)Y, where the parameters and are referred to as a growth rate of the oscillations and the characteristic oscillation frequency of the tachocline, respectively.
dX
dt
2
X
2
Y
dY
dt
2
X
2
Y
β
ω
Model B is a Moore–Spiegel dynamical system that represents the turbulent convection dynamo produced by the convection zone, =y,=Ax-+α-q-μy,=-εα+γx(-1).
dx
dt
dy
dt
3
x
2
X
dα
dt
2
x
Here, to reproduce the kind of on/off intermittency displayed by the solar cycle, the chaotic oscillator in Model B drives Model A via a bifurcation parameter of a nonlinear oscillator with a Hopf bifurcation. The driving is represented by , where controls the instability of Model A, and and are constants. In particular, is seen as the constant demand rate of magnetic flux down from the convection zone. To make spots, has to be at least as large as .
β=(x-χ)
Β
β
Β
χ
χ
x
χ
To reproduce perturbations in the solar cycle during periods of no activity (intermittency) feedback is introduced in the form of [2], for .
q
2
X
0<q⩽0.1
The theoretical sunspot number, , is defined in [3], inspired by the Zurich method of weighting groups of spots for .
N=
2
(X-px)
p=0.1