The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x = f (t,x) where f ∈ C3(R2) is T -periodic in t , fx3 (t, x) < 0 for every (t, x) ∈ R2, and f (t,x)→∓∞ as x→∞, uniformly on t . We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions.Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold. © 2007 Elsevier Inc. All rights reserved