20,301 research outputs found
Multi-site breathers in Klein-Gordon lattices: stability, resonances, and bifurcations
We prove the most general theorem about spectral stability of multi-site
breathers in the discrete Klein-Gordon equation with a small coupling constant.
In the anti-continuum limit, multi-site breathers represent excited
oscillations at different sites of the lattice separated by a number of "holes"
(sites at rest). The theorem describes how the stability or instability of a
multi-site breather depends on the phase difference and distance between the
excited oscillators. Previously, only multi-site breathers with adjacent
excited sites were considered within the first-order perturbation theory. We
show that the stability of multi-site breathers with one-site holes change for
large-amplitude oscillations in soft nonlinear potentials. We also discover and
study a symmetry-breaking (pitchfork) bifurcation of one-site and multi-site
breathers in soft quartic potentials near the points of 1:3 resonance.Comment: 34 pages, 12 figure
Localization and Coherence in Nonintegrable Systems
We study the irreversible dynamics of nonlinear, nonintegrable Hamiltonian
oscillator chains approaching their statistical asympotic states. In systems
constrained by more than one conserved quantity, the partitioning of the
conserved quantities leads naturally to localized and coherent structures. If
the phase space is compact, the final equilibrium state is governed by entropy
maximization and the final coherent structures are stable lumps. In systems
where the phase space is not compact, the coherent structures can be collapses
represented in phase space by a heteroclinic connection to infinity.Comment: 41 pages, 15 figure
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
On the detuned 2:4 resonance
We consider families of Hamiltonian systems in two degrees of freedom with an
equilibrium in 1:2 resonance. Under detuning, this "Fermi resonance" typically
leads to normal modes losing their stability through period-doubling
bifurcations. For cubic potentials this concerns the short axial orbits and in
galactic dynamics the resulting stable periodic orbits are called "banana"
orbits. Galactic potentials are symmetric with respect to the co-ordinate
planes whence the potential -- and the normal form -- both have no cubic terms.
This -symmetry turns the 1:2 resonance into a
higher order resonance and one therefore also speaks of the 2:4 resonance. In
this paper we study the 2:4 resonance in its own right, not restricted to
natural Hamiltonian systems where would consist of kinetic and
(positional) potential energy. The short axial orbit then turns out to be
dynamically stable everywhere except at a simultaneous bifurcation of banana
and "anti-banana" orbits, while it is now the long axial orbit that loses and
regains stability through two successive period-doubling bifurcations.Comment: 31 pages, 7 figures: On line first on Journal of Nonlinear Science
(2020
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