Nonlinear quenching of current fluctuations in a self-exciting homopolar dynamo

Abstract. In the interpretation of geomagnetic polarity reversals with their highly variable frequency over geological time it is necessary, as with other irregularly fluctuating geophysical phenomena, to consider the relative importance of forced contributions associated with changing boundary conditions and of free contributions characteristic of the behaviour of nonlinear systems operating under fixed boundary conditions.  New evidence -albeit indirect- in favour of the likely predominance of forced contributions is provided by the discovery reported here of the possibility of complete quenching by nonlineax effects of current fluctuations in a self-exciting homopolar dynamo with its single Faraday disk driven into rotation with angular speed y(τ) (where τ denotes time) by a steady applied couple.  The armature of an electric motor connected in series with the coil of the dynamo is driven into rotation' with angular speed z(τ) by a torque xf (x) due to Lorentz forces associated with the electric current x(τ) in the system (just as certain parts of the spectrum of eddies within the liquid outer core are generated largely by Lorentz forces associated with currents generated by the self-exciting magnetohydrodynamic (MHD) geodynamo).   The discovery is based on bifurcation analysis supported by computational studies of the following (mathematically novel) autonomous set of nonlinear ordinary differential equations:

dx/dt = x(y - 1) - βzf(x),
dy/dt = α(1 - x²) - κy,
dz/dt = xf (x) -λz,          where f (x) = 1 - ε + εσx,

in cases when the dimensionless parameters (α, β, κ, λ, σ) are all positive and 0 ≤ ε ≤ 1. Within those regions of (α, β, κ, λ, σ) parameter space where the applied couple, as measured by α, is strong enough for persistent dynamo action (i.e. x ≠ 0) to occur at all, there are in general extensive regions where x(τ) exhibits large amplitude regular or irregular (chaotic) fluctuations.  But these fluctuating régimes shrink in size as increases from zero, and they disappear altogether when ε = 1, leaving only steady régimes of dynamo action.