A Model of Sunspot Number with Modified Logistic Function

Solar cycles are studied with the Version 2 monthly smoothed international sunspot number, the variations of which are found to be well represented by the modified logistic differential equation with four parameters: maximum cumulative sunspot number or total sunspot number $x_m$, initial cumulative sunspot number $x_0$, maximum emergence rate $r_0$, and asymmetry $\alpha$. A two-parameter function is obtained by taking $\alpha$ and $r_0$ as fixed value. In addition, it is found that $x_m$ and $x_0$ can be well determined at the start of a cycle. Therefore, a prediction model of sunspot number is established based on the two-parameter function. The prediction for cycles $4-23$ shows that the solar maximum can be predicted with average relative error being 8.8\% and maximum relative error being 22\% in cycle 15 at the start of solar cycles if solar minima are already known. The quasi-online method for determining solar minimum moment shows that we can obtain the solar minimum 14 months after the start of a cycle. Besides, our model can predict the cycle length with the average relative error being 9.5\% and maximum relative error being 22\% in cycle 4. Furthermore, we predict the sunspot number variations of cycle 24 with the relative errors of the solar maximum and ascent time being 1.4\% and 12\%, respectively, and the predicted cycle length is 11.0 (95\% confidence interval is 8.3$-$12.9) years. The comparison to the observation of cycle 24 shows that our prediction model has good effectiveness.