Measurements were made of the absorption spectrum of hydrogen atoms to levels near the ionization threshold in a strong magnetic field. Taking advantage of a classical scaling law, we varied the photon energy and the magnetic-field strength simultaneously, and measured absorption versus B−1/3 at fixed scaled energy, ɛ=E/(B/Bo)2/3. The absorption rate has sinusoidal fluctuations which are correlated with closed classical orbits of the electron. Fourier transformation of this signal gives peaks, which we interpret as ‘‘recurrence strength,’’ as a function of the classical action of the closed orbit. Closed-orbit theory gives formulas for these recurrence strengths. We find that the formulas are in good agreement with the measurements. As the scaled energy is increased, observed recurrences proliferate, consistent with a change from orderly to chaotic motion of the electron. Bifurcation theory provides organizing principles for understanding this proliferation and for interpreting the data. New ‘‘exotic’’ orbits suddenly appear through saddle-node bifurcations. The ‘‘main sequence’’ of orbits is produced from an orbit parallel to B through a sequence of pitchfork and period-doubling bifurcations. Other recurrences are created by period-tripling and higher-order bifurcations of existing orbits. These bifurcations can have ‘‘generic’’ structure, or sometimes the structures are modified by symmetries of the system. Focusing effects associated with these bifurcations cause some recurrences to be particularly strong.