While attractors of dissipative dynamical systems describe their long-time behavior, bifurcations are often studied experimentally by dynamically sweeping structural parameters. If the parameter sweep is slow enough, the system follows adiabatically the time-dependent attractor. However, the time scale for convergence to the attractor typically diverges near the bifurcation, so that the adiabatic approximation fails near the bifurcation, and parameter sweep cannot directly reconstruct the bifurcation diagram; in particular, the system follows distinct trajectories during an upward and downward sweep through the bifurcations.

In the context of period-doubling and pitchfork bifurcations, I will show that full bifurcation structure can nevertheless be recovered by reducing the system to a time-dependent normal form with universal families of upsweep and downsweep trajectories that depend on a small sweep rate parameter $s$.

The area of the hysteresis loop in phase space that serves as an experimental measure of the effective sweep rate tends to zero in the slow sweep limit as $s^{1/4}$ times a distinctive logarithmically divergent factor.