  In the sciences and engineering, the concept of bandwidth is often subject to
interpretation depending upon context and the requirements of a specific
community. The focus of this work is to formulate this concept for a general
class of passive oscillatory dynamical systems, including but not limited to
mechanical, structural, acoustic, electrical, and optical. Typically, the
bandwidth of these systems is determined by the half-power (-3 dB) method, and
the result is often referred to as half-power bandwidth. The fundamental
assumption underlying this definition is that the system performance degrades
once its power decreases by 50%; moreover, there are restrictive conditions
that are rarely met. Here the concept of root mean square bandwidth is
considered, justified by the Fourier uncertainty principle, to generalize the
definition of bandwidth to encompass linear and nonlinear, single and
multi-mode, low and high loss and time-varying and invariant oscillating
systems. By tying the bandwidth of an oscillatory dynamical system directly to
its dissipative capacity, one can formulate a definition based solely on its
transient energy evolution, effectively circumventing the previous
restrictions. Further, applications are given that highlight the limitations of
the traditional half-power bandwidth; including a Duffing oscillator with
hardening nonlinearity, and a geometrically nonlinear oscillator with tunable
hardening or softening nonlinearity. The resulting energy-dependent bandwidth
computations are compatible with the nonlinear dynamics of these systems, since
at low energies they recover the half-power bandwidth, whereas at high energies
they accurately capture the nonlinear physics. Moreover, the bandwidth
computation is directly tied to nonlinear harmonic generation in the transient
dynamics.
