Homoclinic and Periodic Orbits for Scalar Delay Differential Equations

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We study scalar delay differential equations with smooth unimodal feedbacks converging to a discontinuous limiting nonlinearity. If the smooth unimodal feedbacks replace with the function f(ξ) = ξ^k*e^(−ξ) or f(ξ) = ξ^k / (1 + ξ^n) with n > k > 0, we get the well-knownequations introduced by Lasota and by Mackey and Glass in 1977 to model the feedback control of blood cells. Despite the large number of theoretical results, so far, rigorous mathematical results have explained very little about the high complexity of the dynamics obtained numerically on the above nonlinearities. The thesis motivate from previous research. The thesis discusses the subsequent results in detail, which are in the Thesis. First, we build on earlier results on the one-dimensional leading unstable sets of an unstable equilibrium; we show that an analogous separation mechanism occurs along suitable periodic orbits. More precisely, for each unstable oscillatory mode of the intermediate equilibrium, we construct a periodic orbit whose leading two-dimensional unstable set has two branches with different forward asymptotic behavior: one approaches the trivial equilibrium, whereas the other enters a nontrivial recurrent oscillatory region near the switching level. We also establish connecting orbits from the unstable equilibrium to these periodic orbits and prove that the resulting structure persists for all sufficiently large smooth approximations. Second, for a certain range of parameters, we prove the existence of a threshold value such that a equation with the discontinuous feedback has a homoclinic solution to its unique positive stationary point. The homoclinic solution crosses the discontinuity of the feedback twice, both times transversally. We then establish a homoclinic bifurcation for every parameter that is close enough to the threshold value from above such that the equation with the discontinuous nonlinearity has a periodic solution whose minimal period tends to infinity and whose orbit has small Hausdorff distance from the homoclinic orbit. This bifurcation result does not require the usual saddle-quantity condition involving the leading stable and unstable characteristic values. Moreover, the periodic orbit has a neighborhood from which every solution reaches the orbit in finite time. Finally, the equation with the discontinuous nonlinearity arises as a limiting equation for a class of smooth Mackey–Glass type equations, and the above results yield, as a corollary, orbitally asymptotically stable periodic orbits with arbitrarily large minimal periods for equations in this class.

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