Towards a classification of bifurcations in Vlasov equations

Discontinuous codimension-two bifurcation in a Vlasov equation

“…On the theoretical side, these results are a further step towards a classification of bifurcations in Vlasov systems [15]. Several questions remain open however.…”

“…The model dependence appears solely in the coefficient K, i.e., κ α in (15), so the linear theory is universal for all models satisfying (3). We see from this expression that 1 is holomorphic on the domains Re λ > 0 and Re λ < 0, but not on the whole complex plane.…”

“…We see from this expression that 1 is holomorphic on the domains Re λ > 0 and Re λ < 0, but not on the whole complex plane. On the stable side of the bifurcation (κ α < 0), there are no eigenvalues; there are however Landau poles, which are roots of the analytically continued spectral function (15) from the right half plane Reλ > 0 to the left half plane Reλ 0. The continuation is performed by continuously deforming the integration contour from R to a new contour L so as to avoid the singular point p = iλ, which is in the upper half of the complex p plane for Reλ > 0, goes down on the real axis for Reλ = 0, and moves to the lower half for Reλ < 0 (see Appendix A 1 for more details).…”

“…At variance with [12], which uses nonstationary water-bag initial states, we consider in the present work small perturbations of smooth stationary reference states. Beyond the case of homogeneous states, bifurcations of Vlasov equations have also been studied for families of nonhomogeneous (position depending) distributions, in the context of self-gravitating systems [13], and more recently in [14,15]; these studies are restricted however to codimension-one bifurcations.…”