Simplest nonlinear vector fields on the hyperbolic paraboloid and their bifurcations

Samriddha Das, Supriyo Jana, and Soumen Sarkar

Abstract. In this paper, we describe the structure of arbitrary polynomial vector fields defined on the doubly ruled surface hyperbolic paraboloid. We then emphasize on the simplest nonlinear vector fields on this surface and obtain several results concerning the existence and nonexistence of first integrals as well as phase portraits. We also examine invariant lines generated by the intersection of the surface with appropriate planes and give an upper bound for how many such invariant lines can occur. We show that the bound is sharp when corresponding planes are invariant. Finally, using second-order averaging theory, we show that certain nonlinear vector fields on the hyperbolic paraboloid can bifurcate at most four limit cycles. An example is given where exactly four limit cycles bifurcate from the vector field.