Abstract
We study the structure induced by the number of periodic solutions on the set of differential equations x ′ = f ( t , x ) where f ∈ C 3 ( R 2 ) is T-periodic in t, f x 3 ( t , x ) < 0 for every ( t , x ) ∈ R 2 , 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.
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