  We consider a heteroclinic network in the framework of winnerless competition
of species. It consists of two levels of heteroclinic cycles. On the lower
level, the heteroclinic cycle connects three saddles, each representing the
survival of a single species; on the higher level, the cycle connects three
such heteroclinic cycles, in which nine species are involved. We show how to
tune the predation rates in order to generate the long time scales on the
higher level from the shorter time scales on the lower level. Moreover, when we
tune a single bifurcation parameter, first the motion along the lower and next
along the higher-level heteroclinic cycles are replaced by a heteroclinic cycle
between 3-species coexistence-fixed points and by a 9-species coexistence-fixed
point, respectively. We also observe a similar impact of additive noise. Beyond
its usual role of preventing the slowing-down of heteroclinic trajectories at
small noise level, its increasing strength can replace the lower-level
heteroclinic cycle by 3-species coexistence fixed-points, connected by an
effective limit cycle, and for even stronger noise the trajectories converge to
the 9-species coexistence-fixed point. The model has applications to systems in
which slow oscillations modulate fast oscillations with sudden transitions
between the temporary winners.

Adaptation and Self-Organizing Systems;Dynamical Systems;Neurons and Cognition