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tokens: . For the bifurcation point μ = 0, we derive from ( 6 ) the condition for the cycle-frequency ν to be given by equationThe solution with the lowest value, ν *, defines the critical dynamo number, at which the zero solution becomes unstable to be given byAt this point, Equation ( 4 ) has the harmonic solution B ( t ) = A sin ν * t . However, the amplitude A is unstable. It decays to zero, for dynamo numbers below , and would grow unbounded for higher values, if it were not for B max , which stabilizes the solution again. Simulations show that, at , the zero solution bifurcates to the harmonic solution B ( t ) ∝ sin ν * t , with amplitude close to B max . Further increasing leads to periodic and chaotic modulations of both the amplitude and the period. (Figure 3 ). This period-doubling route to chaos is a robust feature of BL-type dynamos, which has been observed also in simpler iterative map models as well as more complicated spatially resolved models (Charbonneau et al. 2005 ). The amplitude-modulations are superimposed to an increasing trend, starting at around B max . The period, on the other hand, is modulated around the linear solution ν *. For the sake of simplicity, we refer to this entire mode of oscillation as the weak mode. Figure 3. Field amplitudes B 2 as a function of the dynamo number , for q = 0.82, exhibiting the transition between the weak and the strong mode around . The inset shows the corresponding periods against (same units). Gray vertical lines indicate the dynamo number used for the realization in Figure 4 . If we introduce B min , this generic picture persists. However, the zero solution remains stable for dynamo numbers beyond . Hence, there is a range of dynamo numbers, roughly between and , for which the dynamo admits at least two stable modes (the weak mode and the zero mode). For even larger values of the dynamo number, a fundamentally different mode of oscillation emerges (the branch in Figure 3 above dynamo number ). This is a mode where B ( t ) is above B max most of the time and both the amplitude and the period are constant. We call it the strong mode. It is the stable limit cycle that appears when we approximate in Equation ( 4 ). This approximation is valid for large , where the cycle period is much larger than q . For smaller values of , the approximation fails and the system can no longer be approximated by a second-order ODE, hence the emergence of the chaotically modulated weak mode. The cycles of the strong mode have a characteristic shape, similar to the large solar cycles that were observed in the sunspot records (Figure 2 ): a steep rising limb, followed by a falling limb with a kink. This shape has been interpreted as an effect of the B -field exceeding B max before by Stix ( 1972 ), albeit with a non-BL-type dynamo model. The fit in Figure 2 has been achieved through optimizing parameters q and . Interestingly, the optimized value turned out to be quite close to the critical point where the strong