  The clustering of small heavy inertial particles subjected to Stokes drag in
turbulence is known to be minimal at small and large Stokes number and
substantial at $\rm St = \mathcal O(1)$. This non-monotonic trend, which has
been shown computationally and experimentally, is yet to be explained
analytically. In this study, we obtain an analytical expression for the
Lyapunov exponents that quantitatively predicts this trend. The sum of the
exponents, which is the normalized rate of change of the signed-volume of a
small cloud of particles, is correctly predicted to be negative and positive at
small and large Stokes numbers, respectively, asymptoting to $\tau Q$ as $\tau
\to 0$ and $\tau^{-1/2} |Q|^{1/4}$ as $\tau \to \infty$, where $\tau$ is the
particle relaxation time and $Q(\tau)$ is the difference between the norm of
the rotation- and strain-rate tensors computed along the particle trajectory.
Additionally, the trajectory crossing is predicted only in hyperbolic flows
where $Q<0$ for sufficiently inertial particles with a $\tau$ that scales with
$|Q|^{-1/2}$. Following the onset of crossovers, a transition from clustering
to dispersion is predicted correctly. We show these behaviors are not unique to
three-dimensional isotropic turbulence and can be reproduced closely by a
one-dimensional mono-harmonic flow, which appears as a fundamental canonical
problem in the study of particle clustering. Analysis of this one-dimensional
canonical flow shows that the rate of clustering, quantified as the product of
the Lyapunov exponent and particle relaxation time, is bounded by $-1/2$,
behaving with extreme nonlinearity in the hyperbolic flows and always remaining
positive in the elliptic flows. These findings, which are stemmed from our
analysis, are corroborated by the direct numerical simulations.
