1,472 research outputs found
Traveling surface waves of moderate amplitude in shallow water
We study traveling wave solutions of an equation for surface waves of
moderate amplitude arising as a shallow water approximation of the Euler
equations for inviscid, incompressible and homogenous fluids. We obtain
solitary waves of elevation and depression, including a family of solitary
waves with compact support, where the amplitude may increase or decrease with
respect to the wave speed. Our approach is based on techniques from dynamical
systems and relies on a reformulation of the evolution equation as an
autonomous Hamiltonian system which facilitates an explicit expression for
bounded orbits in the phase plane to establish existence of the corresponding
periodic and solitary traveling wave solutions
On the number of limit cycles for perturbed pendulum equations
We consider perturbed pendulum-like equations on the cylinder of the form where
are trigonometric polynomials of degree , and study the number of
limit cycles that bifurcate from the periodic orbits of the unperturbed case
in terms of and . Our first result gives upper bounds on
the number of zeros of its associated first order Melnikov function, in both
the oscillatory and the rotary regions. These upper bounds are obtained
expressing the corresponding Abelian integrals in terms of polynomials and the
complete elliptic functions of first and second kind. Some further results give
sharp bounds on the number of zeros of these integrals by identifying
subfamilies which are shown to be Chebyshev systems
Non-integrability of measure preserving maps via Lie symmetries
We consider the problem of characterizing, for certain natural number ,
the local -non-integrability near elliptic fixed points of
smooth planar measure preserving maps. Our criterion relates this
non-integrability with the existence of some Lie Symmetries associated to the
maps, together with the study of the finiteness of its periodic points. One of
the steps in the proof uses the regularity of the period function on the whole
period annulus for non-degenerate centers, question that we believe that is
interesting by itself. The obtained criterion can be applied to prove the local
non-integrability of the Cohen map and of several rational maps coming from
second order difference equations.Comment: 25 page
Basin of attraction of triangular maps with applications
We consider some planar triangular maps. These maps preserve certain
fibration of the plane. We assume that there exists an invariant attracting
fiber and we study the limit dynamics of those points in the basin of
attraction of this invariant fiber, assuming that either it contains a global
attractor, or it is filled by fixed or 2-periodic points. Finally, we apply our
results to a variety of examples, from particular cases of triangular systems
to some planar quasi-homogeneous maps, and some multiplicative and additive
difference equations, as well.Comment: 1 figur
A theoretical basis for the Harmonic Balance Method
The Harmonic Balance method provides a heuristic approach for finding
truncated Fourier series as an approximation to the periodic solutions of
ordinary differential equations. Another natural way for obtaining these type
of approximations consists in applying numerical methods. In this paper we
recover the pioneering results of Stokes and Urabe that provide a theoretical
basis for proving that near these truncated series, whatever is the way they
have been obtained, there are actual periodic solutions of the equation. We
will restrict our attention to one-dimensional non-autonomous ordinary
differential equations and we apply the results obtained to a couple of
concrete examples coming from planar autonomous systems
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