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Abstract: We are concerned with a subharmonic bifurcation from infinity for scalar higher order ordinary differential equations. The equations contain principal linear parts depending on a scalar parameter, 2π-periodic forcing terms, and continuous nonlinearities with saturation. We suggest sufficient conditions for the existence of subharmonics (i.e., periodic solutions of multiple periods ) with arbitrarily large amplitudes and periods. We prove that this type of the subharmonic bifurcation occurs whenever a pair of simple roots of the characteristic polynomial crosses the imaginary axis at the points with an irrational α. Under some further assumptions, we estimate asymptotically the parameter intervals, where large subharmonics of periods exist. These assumptions relate the quality of the Diophantine approximations of α, the rate of convergence of the nonlinearity to its limits at infinity, and the smoothness of the forcing term. [Copyright &y& Elsevier]

The paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained. [Copyright &y& Elsevier]

Abstract The goal of this paper is to develop the Equivariant Degree based method for studying relative periodic solutions in the setiings with lack of smoothness and/or genericity. In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting Γ × S 1 -spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted Γ × S 1 -degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group Γ = D 8 ; and, a system of hysterestic electro-mechanical oscillators coupled in the same symmetric fashion. [ABSTRACT FROM AUTHOR]