  Pattern-forming systems can exhibit a diverse array of complex behaviors as
external parameters are varied, enabling a variety of useful functions in
biological and engineered systems. First-principles derivations of the
underlying transitions can be characterized using bifurcation theory on model
systems whose governing equations are known. In contrast, data-driven methods
for more complicated and realistic systems whose governing evolution dynamics
are unknown have only recently been developed. Here we develop a data-driven
approach, the {\em sparse identification for nonlinear dynamics with control
parameters} (SINDyCP), to discover dynamics for systems with adjustable control
parameters, such as an external driving strength. We demonstrate the method on
systems of varying complexity, ranging from discrete maps to systems of partial
differential equations. To mitigate the impact of measurement noise, we also
develop a weak formulation of SINDyCP and assess its performance on noisy data.
We demonstrate applications including the discovery of universal
pattern-formation equations, and their bifurcation dependencies, directly from
data accessible from experiments and the extrapolation of predictions beyond
the weakly nonlinear regime near the onset of an instability.
