In the 1970s, Feigenbaum, and independently, Coullet and Tresser, discovered an interesting phenomenon in physics called period doubling that showed how a sequence of dynamical systems with stable dynamics can converge to one with chaotic dynamics. In this paper we study analogous phenomena for the tangent family $\{T_t(z)=i t\tan z\}_{\pi/2\leq t\leq \pi}$ restricted to the real and imaginary axes. Because tangent maps have no critical points but have an essential singularity at infinity and two asymptotic values, the phenomena are related but different. We find single instances of "period doubling", "period quadrupling" and "period splitting". Then we prove there is a general pattern of "period merging" where two attracting cycles of period $2^n$ "merge" into one attracting cycle of period $2^{n+1}$ and "cycle doubling" where an attracting cycle of period $2^{n+1}$ "becomes" two attracting cycles of the same period. We adapt the renormalization techniques used to study period doublin 查看全文>>