  We conduct a local stability and Hopf bifurcation analysis for Compound TCP,
with small Drop-tail buffers, in three topologies. The first topology consists
of two sets of TCP flows having different round trip times, and feeding into a
core router. The second topology corresponds to two queues in tandem, and
consists of two distinct sets of TCP flows, regulated by a single edge router
and feeding into a core router. The third topology comprises of two distinct
sets of TCP flows, regulated by two separate edge routers, and feeding into a
common core router. For each of these cases, we conduct a detailed local
stability analysis and obtain conditions on the network and protocol parameters
to ensure stability. If these conditions get marginally violated, our analysis
shows that the underlying systems would lose local stability via a Hopf
bifurcation. After exhibiting a Hopf, a key concern is to determine the
asymptotic orbital stability of the bifurcating limit cycles. We present a
detailed analytical framework to address the stability of the limit cycles, and
the type of the Hopf bifurcation by invoking Poincare normal forms and the
center manifold theory. We conduct packet-level simulations to highlight the
existence and stability of the limit cycles in the queue size dynamics.
