On a Model of a Currency Exchange Rate ? Local Stability and Periodic Solutions

Bifurcation of periodic solutions with large periods for a delay differential equation

“…The proof of the desired winding behaviour and of the limit relation is based on the exclusion of small solutions without zeros on unbounded intervals in Propositions 5.3 and 5.5. Proposition 5.3 generalizes Corollary 5.2 of [1]; its proof by a short comparison argument is new.…”

“…1.1 with parameter a > 0. For proofs which are omitted, compare [1] and [4]. If (ii) If sign(ẋ(s)) = sign(x(s)) for some s > t thenẋ has a zero z > s.…”

“…For further references, see [8][9][10][11]. The identification of the evaluation map with coordinates on invariant manifolds in [1] is similar to an earlier construction in Chapter 6 of [9]. Section 3 deals with a centre manifold W * for the augmented systeṁ…”

“…The main results in [1] establish asymptotic stability of the equilibrium for a < 1, instability for a > 1, and existence of a slowly oscillating periodic solution with orbit in the boundary of a global centre-unstable manifold for a > 1. In [12] the asymptotic shape of the periodic solutions found in [1], for a → ∞, is determined. There are also other periodic orbits, see Sect.…”

“…2 with preliminary results on the semiflows F a , on linearization at the stationary point 0, and on the change of stability of the stationary point when the parameter a crosses the critical level 1. As in [1] we choose a certain complementary subspace N ⊂ C of the two-dimensional centre space L 1 at a = 1 and a basis of L 1 , to the effect that projection along N onto L 1 and taking coordinates coincides with the evaluation map…”