Global bifurcations of phase-locked oscillators

Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

“…Recall that if u n (t)= x(t&nc) is a spatially p-periodic traveling wave of (2.9), then x(t+pc)= x(t) and x : R Ä R satisfies the mixed functional differential equation (2.10). We now normalize the period of x by The Hopf bifurcation problem of (5.2) was discussed in Alexander and Auchmuty [2]. From now on, we will fix the positive integer p. Then, for given constants c and { and for a given 2?-periodic mapping y : R Ä R, we can define To apply the S 1 -bifurcation theory developed by Erbe, Geba, Krawcewicz and Wu [19] for parameterized mixed functional differential equations, we need to verify that D z F (z, {, c), the derivative of F with respect to the first argument, is an isomorphism at given (0, {, c).…”

Hopf Bifurcations of Functional Differential Equations with Dihedral Symmetries

“…[26]) and provides models for various situations in biology, chemistry, and electrical engineering. The local Hopf bifurcation of this Turing ring has been extensively studied in the literature, see [1,7,11,18,24,30] and references therein.…”

“…As for FDEs, an analytic (local) Hopf bifurcation theorem was obtained in [31] as an analogy of the Golubitsky Stewart theorem [9]. Moreover, a topological Hopf bifurcation theory was developed in [18] for FDEs in the case where the spatial symmetry group is the abelian group Z N or Z :=S 1 . While the problem of looking for local bifurcations of periodic solutions with prescribed symmetries can always be reduced to one where the spatial symmetry group is Z N or Z (see [4,9]), examining the global interaction of all bifurcated periodic solutions requires the consideration of the full symmetry group of the equation.…”

Global Analysis of Dynamical Systems

“…We give three examples next. For further details we refer to section 12 below, as well as to [AA86], [AF89], [Far84], [Lie97], [Lie00], [FLA00a], [FL00], [FLA00b].…”

“…Example 1.3 Our third example is based on an observation for coupled oscillators due to [AA86], see also [AF89], [Lie97], [FLA00b]. Let i ∈ {±1, .…”