  In this paper, we introduce an angle notion, called the singular angle, for
stable nonlinear systems from an input-output perspective. The proposed system
singular angle, based on the angle between $\mathcal{L}_2$-signals, describes
an upper bound for the "rotating effect" from the system input to output
signals. It is, thus, different from the recently appeared nonlinear system
phase which adopts the complexification of real-valued signals using the
Hilbert transform. It can quantify the passivity and serve as an angular
counterpart to the system $\mathcal{L}_2$-gain. It also provides an alternative
to the nonlinear system phase. A nonlinear small angle theorem, which involves
a comparison of the loop system angle with $\pi$, is established for feedback
stability analysis. When dealing with multi-input multi-output linear
time-invariant (LTI) systems, we further come up with the frequency-wise and
$\mathcal{H}_\infty$ singular angle notions based on the matrix singular angle,
and develop corresponding LTI small angle theorems.
