4,036 research outputs found
One-Parameter Bifurcation Analysis of Dynamical Systems using Maple
This paper presents two algorithms for one-parameter local
bifurcations of equilibrium points of dynamical systems.
The algorithms are implemented in the computer algebra system Maple 13 © and designed as a package. Some examples are reported to demonstrate the packageâs facilities.* This paper is partially supported by the Bulgarian Science Fund under grant Nr. DO 02â359/2008
Accurately model the Kuramoto--Sivashinsky dynamics with holistic discretisation
We analyse the nonlinear Kuramoto--Sivashinsky equation to develop accurate
discretisations modeling its dynamics on coarse grids. The analysis is based
upon centre manifold theory so we are assured that the discretisation
accurately models the dynamics and may be constructed systematically. The
theory is applied after dividing the physical domain into small elements by
introducing isolating internal boundaries which are later removed.
Comprehensive numerical solutions and simulations show that the holistic
discretisations excellently reproduce the steady states and the dynamics of the
Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as
an example to show how holistic discretisation may be successfully applied to
fourth order, nonlinear, spatio-temporal dynamical systems. This novel centre
manifold approach is holistic in the sense that it treats the dynamical
equations as a whole, not just as the sum of separate terms.Comment: Without figures. See
http://www.sci.usq.edu.au/staff/aroberts/ksdoc.pdf to download a version with
the figure
A reversible bifurcation analysis of the inverted pendulum
The inverted pendulum with a periodic parametric forcing is considered as a bifurcation problem in the reversible setting. Parameters are given by the size of the forcing and the frequency ratio. Normal form theory provides an integrable approximation of the Poincaré map generated by a planar vector field. Genericity of the model is studied by a perturbation analysis, where the spatial symmetry is optional. Here equivariant singularity theory is used.
Subtropical Real Root Finding
We describe a new incomplete but terminating method for real root finding for
large multivariate polynomials. We take an abstract view of the polynomial as
the set of exponent vectors associated with sign information on the
coefficients. Then we employ linear programming to heuristically find roots.
There is a specialized variant for roots with exclusively positive coordinates,
which is of considerable interest for applications in chemistry and systems
biology. An implementation of our method combining the computer algebra system
Reduce with the linear programming solver Gurobi has been successfully applied
to input data originating from established mathematical models used in these
areas. We have solved several hundred problems with up to more than 800000
monomials in up to 10 variables with degrees up to 12. Our method has failed
due to its incompleteness in less than 8 percent of the cases
Resonances in a spring-pendulum: algorithms for equivariant singularity theory
A spring-pendulum in resonance is a time-independent Hamiltonian model system for formal reduction to one degree of freedom, where some symmetry (reversibility) is maintained. The reduction is handled by equivariant singularity theory with a distinguished parameter, yielding an integrable approximation of the Poincaré map. This makes a concise description of certain bifurcations possible. The computation of reparametrizations from normal form to the actual system is performed by Gröbner basis techniques.
Better Answers to Real Questions
We consider existential problems over the reals. Extended quantifier
elimination generalizes the concept of regular quantifier elimination by
providing in addition answers, which are descriptions of possible assignments
for the quantified variables. Implementations of extended quantifier
elimination via virtual substitution have been successfully applied to various
problems in science and engineering. So far, the answers produced by these
implementations included infinitesimal and infinite numbers, which are hard to
interpret in practice. We introduce here a post-processing procedure to
convert, for fixed parameters, all answers into standard real numbers. The
relevance of our procedure is demonstrated by application of our implementation
to various examples from the literature, where it significantly improves the
quality of the results
A robust numerical method to study oscillatory instability of gap solitary waves
The spectral problem associated with the linearization about solitary waves
of spinor systems or optical coupled mode equations supporting gap solitons is
formulated in terms of the Evans function, a complex analytic function whose
zeros correspond to eigenvalues. These problems may exhibit oscillatory
instabilities where eigenvalues detach from the edges of the continuous
spectrum, so called edge bifurcations. A numerical framework, based on a fast
robust shooting algorithm using exterior algebra is described. The complete
algorithm is robust in the sense that it does not produce spurious unstable
eigenvalues. The algorithm allows to locate exactly where the unstable discrete
eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows
for stable shooting along multi-dimensional stable and unstable manifolds. The
method is illustrated by computing the stability and instability of gap
solitary waves of a coupled mode model.Comment: key words: gap solitary wave, numerical Evans function, edge
bifurcation, exterior algebra, oscillatory instability, massive Thirring
model. accepted for publication in SIAD
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