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Modelling the solar dynamo using the Lorenz equations

Some Properties of the Lorenz equations

There are three fixed points of the system

When x=y=0 the equations reduce to dz/dt=-αz, which has solution z(t)=Ae-αt where A is a constant of integration. Thus any solutions initially on the z-axis will remain on it (i.e. the z-axis is invariant under the flow), and tend toward the origin. Physically this means that we will need a seed magnetic field in order for dynamo action to occur, just as we would hope for a dynamo model.

Taking the divergence of the flow shows that the equations are dissipative:

dissipative property
This tells us the phase space volume contracts at an exponential rate of σ+α+1. The theory of dissipative systems now tells us there exists a global attractor that is a compact, connected invariant set.

We use a Lyapunov function to show that the system has a trapping region into which all solutions eventually enter but never leave.
Consider the function

an energy function
for which
an energy function

Since σ and α are positive, the first three terms on the right hand side of this expression are negative.
Defining λ=min{1,σ,α}>0 shows

an energy function
This can easily be transformed into spherical coordinates:
an energy function

The Lyapunov function increases as R→∞ and dE/dt<0 for R>D, so all solutions of the equations enter the sphere of radius D and cannot leave. Hence x, y, and z are bounded and the phase space is Cartesian with axes x, y, and z.

For all parameter choices the equations are invariant under the symmetry (x,y,z) → (-x,-y,z), which corresponds to reversal of the field

We use the Jacobian matrix at the fixed point 0 to determine its stability to small perturbations:

Jacobian at zero
The characteristic polynomial of this matrix is
(λ+α)(λ2 + (σ+1)λ + σ(1-b))=0
which has roots
roots
λ1 and λ3 are always negative. Examining the sign of λ2 shows the equilibrium will be stable for b < 1 and unstable for b > 1.

At b = 1 stability is transferred to the steady states C±. Subsequent behaviour depends on our choices of α, σ, and b, so determining their sizes becomes important.


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