Solar magnetic cycles as a Van Der Pol-Duffing oscillator: new insights

Abstract

ABSTRACT In this paper, we use an approximative stable limit cycle solution of the hybrid Van der Pol-Duffing differential equation, obtained by homotopy and Poincaré–Lindstedt perturbation methods, to describe the toroidal component of the solar magnetic field B(t). This analytic approach allows us to recover an explicit relationship between the parameter μ, which is related to the meridional circulation, and the period of the Hale’s magnetic cycle with a correlation coefficient of r = −0.58. Furthermore, assuming that the sunspot number (SN) is proportional to the square of the toroidal magnetic field (SN∝B2), our solution accurately predict the presence of an harmonic oscillation in the SN data, occurring at a period of T/4 = 5.52 ± 0.44 yr. This prediction has been validated through Lomb–Scargle analysis, with a high statistical significance. Additionally, we find that the ratio of spectral powers between the T/4 harmonic and the main T/2 oscillation is almost equal to the value obtained from our solution using the mean values of the parameters. Interestingly, this study also reveals a correlation between the intermittent 5.52-yr cycle and μ, the parameter associated with the meridional circulation of the Sun. Both follow a similar pattern, suggesting that the origin of the five-year cycle lies within the meridional circulation. Finally, we will see how, using this model, we can overcome the limitations of direct observations and reconstruct the variation profile of the meridional circulation over two centuries using a single observation (from the last magnetic cycle).