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In this article, we study the existence of periodic solutions to a class of distributed delay differential equations. We transform the search for periodic solutions with the special symmetry of a delay differential equation to the problem of finding periodic solutions of an associated Hamiltonian system. Using the critical point theory and the pseudo-index theory, we obtain some sufficient conditions for the multiplicity of periodic solutions. This is the first time that critical point theory has been used to study the existence of periodic solutions to distributed delay differential equations.

In this paper, we study the existence of periodic solutions of the following differential delay equations z ′ ′ (t) = ∑ k = 1 M − 1 (− 1) k f (z (t − k)) , z ′ ′ (t) = ∑ k = 1 M − 1 (− 1) k f (z (t − k)) , where f ∈ C (R N , R N) f ∈ C (R N , R N) , M , N ∈ N M , N ∈ N and M M is odd. By making use of S 1 S 1 -geometrical index theory, we obtain an estimation about the number of periodic solutions in term of the difference between eigenvalues of asymptotically linear matrices at the origin and at infinity. [ABSTRACT FROM AUTHOR]

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder O (2) × Γ × ℤ 2 -equivariant degree theory. Here, O (2) is related to the reversal symmetry combined with the autonomous form of the system, Γ reflects extra spacial symmetries of the system, and ℤ 2 is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with Γ = D 6 — the dihedral group of order 12. [ABSTRACT FROM AUTHOR]