273 research outputs found
Some results on homoclinic and heteroclinic connections in planar systems
Consider a family of planar systems depending on two parameters and
having at most one limit cycle. Assume that the limit cycle disappears at some
homoclinic (or heteroclinic) connection when We present a method
that allows to obtain a sequence of explicit algebraic lower and upper bounds
for the bifurcation set The method is applied to two quadratic
families, one of them is the well-known Bogdanov-Takens system. One of the
results that we obtain for this system is the bifurcation curve for small
values of , given by . We obtain
the new three terms from purely algebraic calculations, without evaluating
Melnikov functions
Complex dynamics in double-diffusive convection
The dynamics of a small Prandtl number binary mixture in a laterally heated
cavity is studied numerically. By combining temporal integration, steady state
solving and linear stability analysis of the full PDE equations, we have been
able to locate and characterize a codimension-three degenerate Takens-Bogdanov
point whose unfolding describes the dynamics of the system for a certain range
of Rayleigh numbers and separation ratios near S=-1.Comment: 8 pages, 5 figure
On the number of limit cycles which appear by perturbation of two-saddle cycles of planar vector fields
We prove that every heteroclinic saddle loop (a two-saddle cycle) occurring
in an analytic finite-parameter family of plane analytic vector fields, may
generate no more than a finite number of limit cycles within the family.Comment: 21 pages, 10 figures, a new section explaining the so called "Petrov
trick" in the context of the paper is added. The paper will appear in
"Functional Analysis and Its Applications" (2013
Transport in Transitory Dynamical Systems
We introduce the concept of a "transitory" dynamical system---one whose
time-dependence is confined to a compact interval---and show how to quantify
transport between two-dimensional Lagrangian coherent structures for the
Hamiltonian case. This requires knowing only the "action" of relevant
heteroclinic orbits at the intersection of invariant manifolds of "forward" and
"backward" hyperbolic orbits. These manifolds can be easily computed by
leveraging the autonomous nature of the vector fields on either side of the
time-dependent transition. As illustrative examples we consider a
two-dimensional fluid flow in a rotating double-gyre configuration and a simple
one-and-a-half degree of freedom model of a resonant particle accelerator. We
compare our results to those obtained using finite-time Lyapunov exponents and
to adiabatic theory, discussing the benefits and limitations of each method.Comment: Updated and corrected version. LaTeX, 29 pages, 21 figure
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