Towards a classification of bifurcations in Vlasov equations
arXiv:1909.11344 · doi:10.1103/PhysRevE.102.052208
Abstract
We propose a classification of bifurcations of Vlasov equations, based on the strength of the resonance between the unstable mode and the continuous spectrum on the imaginary axis. We then identify and characterize a new type of generic bifurcation where this resonance is weak, but the unstable mode couples with the Casimirs, which are constants of motion, to form a size 3 Jordan block. We derive a three-dimensional reduced noncanonical Hamiltonian system describing this bifurcation: coupling with the Casimirs controls the phase space portrait. Comparison of the reduced dynamics with direct numerical simulations on a test case gives excellent agreement. We finally discuss the relevance of this bifurcation to specific physical situations.
arXiv comment6 pages, 2 figures. Supplemental material available at the URL https://www.idpoisson.fr/barre/publications-et-preprints/
References in corpus (16)
- GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
- Structure and structure-preserving algorithms for plasma physics
- A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
- Finite-dimensional collisionless kinetic theory
- Lie-Poisson methods for isospectral flows
- Amplitude Equations for Electrostatic Waves: universal singular behavior in the limit of weak instability
- Universal trapping scaling on the unstable manifold for a collisionless electrostatic mode
- Nonlinear saturation of electrostatic waves: mobile ions modify trapping scaling
- Solving the Vlasov equation for one-dimensional models with long range interactions on a GPU
- Pattern formation in Hamiltonian systems with continuous spectra; a normal-form single-wave model
- Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas
- Langmuir wave filamentation in the kinetic regime. I. Filamentation instability of Bernstein-Greene-Kruskal modes in multidimensional Vlasov simulations
- Instability conditions for some periodic BGK waves in the Vlasov-Poisson system
- First Principles Justification of a ``Single Wave Model'' for Electrostatic Instabilities
- Trapping scaling for bifurcations in Vlasov systems
- Gravitational Loss-Cone Instability in Stellar Systems with Retrograde Orbit Precession