The dynamics of the evolution of solar cycles is considered a result of the activity manifestation in the northern and southern hemispheres. A study was performed based on separate datasets for the northern and southern hemispheres that contain the monthly and daily averages of the areas of sunspot groups for the period covering activity cycles from 12 to 24 (1874-2013) and the daily values of the Wolf number in the northern and southern hemispheres during cycles 23-24 (1992-2013).To obtain a pattern of development of the "northern" and "southern" solar cycles in detail, a special technique for the extended application of the wavelet analysis has been developed. It allows different the wave processes forming a solar cycle to be distinguished, together with the time of their existence. The application of bandpass Fourier filtering to the obtained data shows that the length of "11-year" cycles by the index S (p) varies from 10.2 to 11.5 years in the northern hemisphere and from 9.7 to 13.2 years in the southern. The 19th "northern" and 18th "southern" cycles turned out to be maximal. The formation of each of the cycles by all activity indices is determined by the joint effect of long-period processes lasting from 3 to 7 years and short-period processes lasting less than 2 years. When moving from one cycle to another, the long-period processes demonstrate mergings, separations, modulation, and periodic decays. The abnormal activity that appears during the growth, maximum, or decay phase of a cycle is formed at the expense of the simultaneous strengthening of short-period processes, the lengths and "period spectra" of which noticeably differ in the northern and southern hemispheres.
