Generic profit singularities in time-averaged optimization for cyclic processes in polydynamical systems

We consider the optimization problem of maximizing the time-averaged profit for the motion of a smooth polydynamical system on the circle in the presence of a smooth profit density. If the problem depends on a $k$-dimensional parameter, then the optimal averaged profit is a function of the parameter. It is known from [4] that an optimal motion can always be selected among stationary strategies and a special type of periodic motions called $level cycles$. We present a classification of all generic singularities of the optimal averaged profit if $k\le2$ and the maximum is provided by level cycles.