  The Rate Control Protocol (RCP) uses explicit feedback from routers to
control network congestion. RCP estimates it's fair rate from two forms of
feedback: rate mismatch and queue size. An important design question that
remains open in RCP is whether the presence of queue size feedback is helpful,
given the presence of feedback from rate mismatch. The feedback from routers to
end-systems is time delayed, and may introduce instabilities and complex
non-linear dynamics. Delay dynamical systems are often modeled using delay
differential equations to facilitate a mathematical analysis of their
performance and dynamics. The RCP models with and without queue size feedback
give rise to two distinct non-linear delay differential equations. Earlier work
on this design question was based on methods of linear systems theory. For
further progress, it is quite natural to employ nonlinear techniques. In this
study, we approach this design question using tools from control and
bifurcation theory. The analytical results reveal that the removal of queue
feedback could enhance both stability and convergence properties. Further,
using Poincar\'{e} normal forms and center manifold theory, we investigate two
nonlinear properties, namely, the type of Hopf bifurcation and the asymptotic
stability of the bifurcating limit cycles. We show that the presence of queue
feedback in the RCP can lead to a sub-critical Hopf bifurcation, which would
give rise either to the onset of large amplitude limit cycles or to unstable
limit cycles. Whereas, in the absence of queue feedback, the Hopf bifurcation
is always super-critical and the bifurcating limit cycles are stable. The
analysis is complemented with computations and some packet-level simulations as
well. In terms of design, our study suggests that the presence of both forms of
feedback may be detrimental to the performance of RCP.
