Generic profit singularities in time averaged optimization for phase transitions in polydynamical systems

We consider the optimization problem that consists in maximizing the time averaged profit for a motion of a smooth polydynamical system on the circle in the presence of a smooth profit density. When the problem depends on a k-dimensional parameter the optimal averaged profit is a function of the parameter. It is known that an optimal motion can always be selected among stationary strategies and a special type of periodic cyclic motions called "level cycles". We present the classification of all generic singularities of the optimal averaged profit when k <= 2 for phase transitions between these two optimal strategies