Resonance condition

A resonance condition specifies a match between forcing parameters and a homogeneous mode. For the constant-coefficient system $x^{\prime }-Ax=e^{\lambda t}v$, an exponential trial $x=e^{\lambda t}u$ requiresIf $\lambda$ is not an eigenvalue, this is invertible. If it is an eigenvalue, solvability depends on the direction $v$, not merely on the parameter match. For a diagonalizable matrix, every left eigenvector $\ell$ with eigenvalue $\lambda$ requires $\ell v=0$ for a same-shape trial; a nonzero projection drives a polynomial-exponential response.

For $A=\mathrm{diag}(1,-1)$ and $\lambda =1$, forcing $e^t(1,0)$ gives a $t{e}^t$ term, whereas forcing $e^t(0,1)$ has the ordinary particular solution $e^t(0,1/2)$. Thus a spectral match is a potential resonance, and coupling to that mode determines whether the resonant response actually appears.