75,884 research outputs found
Positive periodic solutions of singular systems
The existence and multiplicity of positive periodic solutions for second
order non-autonomous singular dynamical systems are established with
superlinearity or sublinearity assumptions at infinity for an appropriately
chosen parameter. Our results provide a unified treatment for the problem and
significantly improve several results in the literature. The proof of our
results is based on the Krasnoselskii fixed point theorem in a cone.Comment: Journal of Differential Equations, 201
Non-trivial, non-negative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms
We study the existence of non-trivial, non-negative periodic solutions for
systems of singular-degenerate parabolic equations with nonlocal terms and
satisfying Dirichlet boundary conditions. The method employed in this paper is
based on the Leray-Schauder topological degree theory. However, verifying the
conditions under which such a theory applies is more involved due to the
presence of the singularity. The system can be regarded as a possible model of
the interactions of two biological species sharing the same isolated territory,
and our results give conditions that ensure the coexistence of the two species.Comment: 39 page
Anomalous Thermostat and Intraband Discrete Breathers
We investigate the dynamics of a macroscopic system which consists of an
anharmonic subsystem embedded in an arbitrary harmonic lattice, including
quenched disorder. Elimination of the harmonic degrees of freedom leads to a
nonlinear Langevin equation for the anharmonic coordinates. For zero
temperature, we prove that the support of the Fourier transform of the memory
kernel and of the time averaged velocity-velocity correlations functions of the
anharmonic system can not overlap. As a consequence, the asymptotic solutions
can be constant, periodic,quasiperiodic or almost periodic, and possibly weakly
chaotic. For a sinusoidal trajectory with frequency we find that the
energy transferred to the harmonic system up to time is proportional
to . If equals one of the phonon frequencies ,
it is . We prove that there is a full measure set such that for
in this set it is , i.e. there is no energy dissipation.
Under certain conditions there exists a zero measure set such that for (0 \leq
\alpha < 1)(1 <\alpha \leq 2)\Omega\Omega\mathcal{C}(k)\Omega\in\mathcal{C}(k)t$.Comment: Physica D in prin
Morse theory on spaces of braids and Lagrangian dynamics
In the first half of the paper we construct a Morse-type theory on certain
spaces of braid diagrams. We define a topological invariant of closed positive
braids which is correlated with the existence of invariant sets of parabolic
flows defined on discretized braid spaces. Parabolic flows, a type of
one-dimensional lattice dynamics, evolve singular braid diagrams in such a way
as to decrease their topological complexity; algebraic lengths decrease
monotonically. This topological invariant is derived from a Morse-Conley
homotopy index and provides a gloablization of `lap number' techniques used in
scalar parabolic PDEs.
In the second half of the paper we apply this technology to second order
Lagrangians via a discrete formulation of the variational problem. This
culminates in a very general forcing theorem for the existence of infinitely
many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification
of two proofs and one definition; 55 pages, 20 figure
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
On periodic solutions of 2-periodic Lyness difference equations
We study the existence of periodic solutions of the non--autonomous periodic
Lyness' recurrence u_{n+2}=(a_n+u_{n+1})/u_n, where {a_n} is a cycle with
positive values a,b and with positive initial conditions. It is known that for
a=b=1 all the sequences generated by this recurrence are 5-periodic. We prove
that for each pair (a,b) different from (1,1) there are infinitely many initial
conditions giving rise to periodic sequences, and that the family of
recurrences have almost all the even periods. If a is not equal to b, then any
odd period, except 1, appears.Comment: 27 pages; 1 figur
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