Topology of optimal flows with collective dynamics on closed orientable surfaces

We consider flows on a closed surface with one or more heteroclinic cycles that divide the surface into two regions. One of the region has gradient dynamics, like Morse fields. The other region has Hamiltonian dynamics generated by the field of the skew gradient of the simple Morse function. We construct the complete topological invariant of the flow using the Reeb and Oshemkov-Shark graphs and study its properties. We describe all possible structures of optimal flows with collective dynamics on oriented surfaces of genus no more than 2, both for flows containing a center and for flows without it.

Dynamic Contest Games with Time Delays

Dynamic asymmetric contest games are examined under the assumption that the assessed value of the prize by each agent depends on the total effort of all agents, and each agent has only delayed information about the efforts of the competitors. Assuming gradient dynamics with continuous time scales, first the resulting one-delay model is investigated. Then, assuming additional delayed information about the agents’ own efforts, a two-delay model is constructed and analyzed. In both cases, first the characteristic equation is derived in the general case, and then two special cases are considered. First, symmetric agents are assumed and then general duopolies are examined. Conditions are derived for the local stability of the equilibrium including stability thresholds and stability switching curves.