## Abstract

We investigate the novel set of nonlinear ordinary differential equations $\dot{x}$ = x(y - 1) - $\beta $z, $\quad \dot{y}$ = $\alpha $(1-x$^{2}$)-$\kappa $y and $\dot{z}$ = x - $\lambda $z, where $\dot{x}\equiv $ dx/d$\tau $, etc. They govern the behaviour of two (mathematically equivalent) self-exciting homopolar dynamo systems, each comprising a Faraday disk and coil arrangement, the first with a capacitor and the second with a motor connected in series with the coil, where in each case the applied couple G that drives the disk (of moment of inertia A) into rotation with angular speed $\Omega $(t) is assumed steady. The independent variable $\tau $ denotes time t measured in units of L/Rs, where L is the self-inductance of the system and R the total series resistance. The dependent variable x($\tau $) is the electric current I(t) generated in the system measured in units of (G/M)$^{1/2}$ A, where 2$\pi $M is the mutual inductance between the disk and coil, and y($\tau $) corresponds to $\Omega $(t) measured in units R/M rad s$^{-1}$. In the case of the series capacitor (of capacitance C), the third independent variable z($\tau $) is the charge Q(t) on the capacitor measured in units (G/M)$^{1/2}$(L/R) C; in the case of the series motor, z($\tau $) is the angular speed $\omega (<I>t</I>)$ with which the armature of the motor is driven into rotation by the torque HI(t) due to the current I(t) passing through it, measured in units (L/R)(M/G)$^{1/2}$(H/B) rad s$^{-1}$, where B is the moment of inertia of the armature. Common to both systems are the dimensionless parameters $\alpha \equiv $ GLM/R$^{2}$A and $\kappa \equiv $ KL/RA, where K is the coefficient of (linear) mechanical friction in the disk. In the case of the capacitor, $\beta \equiv $ L/CR$^{2}$ and $\lambda \equiv $ L/RrC where r is the leakage resistance of the capacitor; in the case of the motor, $\beta \equiv $ H$^{2}$L/R$^{2}$B and $\lambda \equiv $ DL/RB where D is the coefficient of (linear) mechanical friction in the motor. The behaviour of the system, including its sensitivity to initial conditions, depends on the four parameters ($\alpha $, $\beta $, $\kappa $, $\lambda $), the least interesting case being when the system fails to function as a self-exciting dynamo capable of amplifying a small adventitious electric current because $\alpha $/$\kappa $ is not large enough for motional induction to overcome ohmic dissipation. Otherwise, i.e. where $\alpha $/$\kappa $ exceeds a critical value dependent on $\beta $ and $\lambda $, dynamo action occurs in which the detailed time dependence of the current x($\tau $) and of the other variables y($\tau $) and z($\tau $) depends critically on the exact values of ($\alpha $, $\beta $, $\kappa $, $\lambda $) and in some cases also on the initial conditions. In the simplest cases, x($\tau $) tends to solutions which (apart from the sign of x($\tau $), which depends of the sign of the initial disturbance) are either independent of $\tau $ or vary harmonically with $\tau $. At other values of ($\alpha $, $\beta $, $\kappa $, $\lambda $), multiple solutions are found, some of which are periodic (but non-harmonic), including square x($\tau $) and saw-tooth y($\tau $) and z($\tau $) waveforms, and others chaotic. A full elucidation of this behaviour will require extensive numerical studies over wide ranges of all these parameters, but bifurcation theory applied to the stability or otherwise of the equilibrium solutions (x$_{0}$, y$_{0}$, z$_{0}$) = (0, $\alpha $/$\kappa $, 0) and ($\pm $[1 - ($\kappa $/$\alpha $)(1 + $\beta $/$\lambda $)]$^{1/2}$, 1 + $\beta $/$\lambda $, x$_{0}$/$\lambda $) provides theoretical guidance. It shows in particular the usefulness of a regime diagram in parameter space of the first quadrant of the ($\beta $, $\alpha $/$\kappa $) plane where there is one line $\alpha $/$\kappa $ = 1 + $\beta $/$\lambda $ where symmetry breaking bifurcations occur and parts of two lines $\alpha $/$\kappa $ = 1 + $\lambda $ and $\alpha $/$\kappa $ = [(2$\beta $ - $\kappa \lambda $ - $\lambda ^{2}$)/2($\kappa $ - $\beta $/$\lambda $) + 3$\beta $/2$\lambda $ +] upon which Hopf bifurcations occur, all meeting at the point ($\beta $, $\alpha $/$\kappa $) = ($\lambda ^{2}$, 1 + $\lambda $) of the Takens-Bogdanov `double-zero eigenvalue' type, with reflectional symmetry. The equations governing self-exciting homopolar dynamos can be used as the basis of nonlinear low-dimensional analogues in the study of the temporal behaviour of certain phenomena of interest in geophysical fluid dynamics. These include the main geomagnetic field produced by self-exciting magnetohydrodynamic (MHD) dynamo action in the Earth's liquid metallic core and the `El Nino-Southern Oscillation' of the Earth's atmosphere-ocean system (considerations of which prompted the present study), which has certain characteristics resembling those found in nonlinear relaxation oscillators and is produced by complex global-scale interactions between the atmosphere and oceans. Geophysical implications of the findings of the present study of simple (but not over-simplified) physically realistic dynamos will be discussed elsewhere, in the context of the further computational investigations needed to elucidate more fully the rich and complex behaviour indicated by the results obtained to date.

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