222 research outputs found
How does the Smaller Alignment Index (SALI) distinguish order from chaos?
The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from
ordered motion, has been demonstrated recently in several
publications.\cite{Sk01,GRACM} Basically it is observed that in chaotic regions
the SALI goes to zero very rapidly, while it fluctuates around a nonzero value
in ordered regions. In this paper, we make a first step forward explaining
these results by studying in detail the evolution of small deviations from
regular orbits lying on the invariant tori of an {\bf integrable} 2D
Hamiltonian system. We show that, in general, any two initial deviation vectors
will eventually fall on the ``tangent space'' of the torus, pointing in
different directions due to the different dynamics of the 2 integrals of
motion, which means that the SALI (or the smaller angle between these vectors)
will oscillate away from zero for all time.Comment: To appear in Progress of Theoretical Physics Supplemen
Particle Swarm Optimization: An efficient method for tracing periodic orbits in 3D galactic potentials
We propose the Particle Swarm Optimization (PSO) as an alternative method for
locating periodic orbits in a three--dimensional (3D) model of barred galaxies.
We develop an appropriate scheme that transforms the problem of finding
periodic orbits into the problem of detecting global minimizers of a function,
which is defined on the Poincar\'{e} Surface of Section (PSS) of the
Hamiltonian system. By combining the PSO method with deflection techniques, we
succeeded in tracing systematically several periodic orbits of the system. The
method succeeded in tracing the initial conditions of periodic orbits in cases
where Newton iterative techniques had difficulties. In particular, we found
families of 2D and 3D periodic orbits associated with the inner 8:1 to 12:1
resonances, between the radial 4:1 and corotation resonances of our 3D Ferrers
bar model. The main advantages of the proposed algorithm is its simplicity, its
ability to work using function values solely, as well as its ability to locate
many periodic orbits per run at a given Jacobian constant.Comment: 12 pages, 8 figures, accepted for publication in MNRA
A new ghost cell/level set method for moving boundary problems:application to tumor growth
In this paper, we present a ghost cell/level set method for the evolution of interfaces whose normal velocity depend upon the solutions of linear and nonlinear quasi-steady reaction-diffusion equations with curvature-dependent boundary conditions. Our technique includes a ghost cell method that accurately discretizes normal derivative jump boundary conditions without smearing jumps in the tangential derivative; a new iterative method for solving linear and nonlinear quasi-steady reaction-diffusion equations; an adaptive discretization to compute the curvature and normal vectors; and a new discrete approximation to the Heaviside function. We present numerical examples that demonstrate better than 1.5-order convergence for problems where traditional ghost cell methods either fail to converge or attain at best sub-linear accuracy. We apply our techniques to a model of tumor growth in complex, heterogeneous tissues that consists of a nonlinear nutrient equation and a pressure equation with geometry-dependent jump boundary conditions. We simulate the growth of glioblastoma (an aggressive brain tumor) into a large, 1 cm square of brain tissue that includes heterogeneous nutrient delivery and varied biomechanical characteristics (white matter, gray matter, cerebrospinal fluid, and bone), and we observe growth morphologies that are highly dependent upon the variations of the tissue characteristics—an effect observed in real tumor growth
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