  We investigate the dynamics of a point-like active particle suspended in
fluid flow through a straight channel. For this particle-fluid system, we
derive a constant of motion for a general unidirectional fluid flow, and apply
it to an approximation of Poiseuille flow through channels with rectangular
cross-sections. We obtain a $4$D nonlinear conservative dynamical system with
one constant of motion and a dimensionless parameter describing the ratio of
maximum flow speed to intrinsic active particle speed. Applied to square
channels, we observe a diverse set of active particle trajectories with
variations in system parameters and initial conditions which we classify into
different types of swinging, trapping, tumbling and wandering motion. Regular
(periodic/quasiperiodic) motion as well as chaotic active particle motion are
observed for these trajectories and quantified using largest Lyapunov
exponents. We explore the transition to chaotic motion using Poincar\'e maps
and show ``sticky" chaotic tumbling trajectories that have long transients near
a periodic state. We briefly illustrate how these results extend to rectangular
cross-sections with width/height ratio larger than one. Outcomes of this work
may have implications for dynamics of natural and artificial microswimmers in
experimental microfluidic channels that typically have rectangular
cross-sections.
