62 research outputs found
Generic perturbations of linear integrable Hamiltonian systems
In this paper, we investigate perturbations of linear integrable Hamiltonian
systems, with the aim of establishing results in the spirit of the KAM theorem
(preservation of invariant tori), the Nekhoroshev theorem (stability of the
action variables for a finite but long interval of time) and Arnold diffusion
(instability of the action variables). Whether the frequency of the integrable
system is resonant or not, it is known that the KAM theorem does not hold true
for all perturbations; when the frequency is resonant, it is the Nekhoroshev
theorem which does not hold true for all perturbations. Our first result deals
with the resonant case: we prove a result of instability for a generic
perturbation, which implies that the KAM and the Nekhoroshev theorem do not
hold true even for a generic perturbation. The case where the frequency is
non-resonant is more subtle. Our second result shows that for a generic
perturbation, the KAM theorem holds true. Concerning the Nekhrosohev theorem,
it is known that one has stability over an exponentially long interval of time,
and that this cannot be improved for all perturbations. Our third result shows
that for a generic perturbation, one has stability for a doubly exponentially
long interval of time. The only question left unanswered is whether one has
instability for a generic perturbation (necessarily after this very long
interval of time)
Symmetries and reversing symmetries of toral automorphisms
Toral automorphisms, represented by unimodular integer matrices, are
investigated with respect to their symmetries and reversing symmetries. We
characterize the symmetry groups of GL(n,Z) matrices with simple spectrum
through their connection with unit groups in orders of algebraic number fields.
For the question of reversibility, we derive necessary conditions in terms of
the characteristic polynomial and the polynomial invariants. We also briefly
discuss extensions to (reversing) symmetries within affine transformations, to
PGL(n,Z) matrices, and to the more general setting of integer matrices beyond
the unimodular ones.Comment: 34 page
Hamiltonian Hopf bifurcations in the discrete nonlinear Schr\"odinger trimer: oscillatory instabilities, quasiperiodic solutions and a 'new' type of self-trapping transition
Oscillatory instabilities in Hamiltonian anharmonic lattices are known to
appear through Hamiltonian Hopf bifurcations of certain time-periodic solutions
of multibreather type. Here, we analyze the basic mechanisms for this scenario
by considering the simplest possible model system of this kind where they
appear: the three-site discrete nonlinear Schr\"odinger model with periodic
boundary conditions. The stationary solution having equal amplitude and
opposite phases on two sites and zero amplitude on the third is known to be
unstable for an interval of intermediate amplitudes. We numerically analyze the
nature of the two bifurcations leading to this instability and find them to be
of two different types. Close to the lower-amplitude threshold stable
two-frequency quasiperiodic solutions exist surrounding the unstable stationary
solution, and the dynamics remains trapped around the latter so that in
particular the amplitude of the originally unexcited site remains small. By
contrast, close to the higher-amplitude threshold all two-frequency
quasiperiodic solutions are detached from the unstable stationary solution, and
the resulting dynamics is of 'population-inversion' type involving also the
originally unexcited site.Comment: 25 pages, 11 figures, to be published in J. Phys. A: Math. Gen.
Revised and shortened version with few clarifying remarks adde
Stability of invariant curves in four-dimensional reversible mappings near 1:1 resonance
Recent work on reversible Neimark-Sacker bifurcation in 4D maps is summarized. Linear stability analysis of families of invariant curves appearing in this bifurcation is presented by (a) referring to the analogous stability problem in reversible Hopt bifurcation in vector fields and (b) perturbatively calculating a set of quantities, termed quasi-multipliers, for the invariant curves. In particular, the critical rotation numbers corresponding to transition from elliptic to hyperbolic invariant curves on the subthreshold side in the so-called inverted bifurcation are calculated. Results of numerical iterations corroborating the above analysis are presented. The question of exploring the structure of the phase space close to the invariant curves is briefly addressed in the conclusion
- …