Symmetries shape the current in ratchets induced by a bi-harmonic force

Equations describing the evolution of particles, solitons, or localized structures, driven by a zero-average, periodic, external force, and invariant under time reversal and a half-period time shift, exhibit a ratchet current when the driving force breaks these symmetries. The bi-harmonic force $f(t)={ϵ}_1\cos (q\omega t+{\varphi }_1)+{ϵ}_2\cos (p\omega t+{\varphi }_2)$ does it for almost any choice of ${\varphi }_1$ and ${\varphi }_2$, provided $p$ and $q$ are two co-prime integers such that $p+q$ is odd. It has been widely observed, in experiments in Josephson-junctions, photonic crystals, etc., as well as in simulations, that the ratchet current induced by this force has the shape $v\propto {ϵ}_1^p{ϵ}_2^q\cos (p{\varphi }_1-q{\varphi }_2+{\theta }_0)$ for small amplitudes, where ${\theta }_0$ depends on the damping (${\theta }_0=\pi \mathrm{/}2$ if there is no damping, and ${\theta }_0=0$ for overdamped systems). We rigorously prove that this precise shape can be obtained solely from the broken symmetries of the system and is independent of the details of the equation describing the system.