Bifurcation analysis of a non-cooperative differential game with one weak player

We study a bifurcation problem for a system of two differential equations in implicit form. For each value of the parameter θ, the solution yields a pair of Nash equilibrium strategies in feedback form, for a non-cooperative differential game. When θ = 0, the second player has no power to influence the dynamics of the system, and his optimal strategy is myopic. The game thus reduces to an optimal control problem for the first player. By studying the bifurcation in the solutions to the corresponding system of Hamilton-Jacobi equations, one can establish existence and multiplicity of solutions to the differential game, as θ becomes strictly positive.