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## The boundary of the Krein space tracial numerical range, an algebraic approach and a numerical algorithm

Fonte: Centro de Matemática da Universidade de Coimbra
Publicador: Centro de Matemática da Universidade de Coimbra

Tipo: Pré-impressão

Português

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In this article, tracial numerical ranges associated with matrices in an
inde nite inner product space are investigated. The boundary equations of these
sets are obtained and the case of the boundary being a polygon is studied. As
an application, a numerical algorithm for plotting the tracial numerical range of
an arbitrary complex matrix, is presented. Our approach uses the elementary idea
that the boundary may be traced by computing the supporting lines.

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## Integração das equações diferenciais do filtro digital de Butterworth mediante algoritmo de quadratura numérica de ordem elevada; Integration of the Butterworth digital filters differential equations using numerical algorithm of high order integrator

Fonte: Biblioteca Digitais de Teses e Dissertações da USP
Publicador: Biblioteca Digitais de Teses e Dissertações da USP

Tipo: Dissertação de Mestrado
Formato: application/pdf

Publicado em 27/03/2003
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#Butterworth#Butterworth#digital filter#filtro digital#hermitian#hermitiano#numerical solution of differential equations#processamento de sinais#signal processing#solução numérica de equações diferenciais

Neste trabalho se apresenta o desenvolvimento de algoritmos hermitianos de integração das equações diferenciais do filtro digital de Butterworth mediante operadores de integração numérica de ordem elevada com passo único. A teoria do filtro de Butterworth é apresentada mediante o emprego da transformada de Fourier. Exemplos de aplicação apresentados através destes algoritmos mostram que os resultados são, como esperado, mais precisos que os resultantes dos métodos usuais presentes na literatura especializada; In this work is presented the development of hermitian algorithm for integration of the Butterworth digital filters differential equations by means of high order numerical one step operators. The Butterworth filters theory is presented based on the Fourier transform. Numerical examples show that the results of the developed hermitian algorithm are more accurate than the usual methods present in the specialized literature, as expected

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## A New Numerical Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/01/2011
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We present a new algorithm for reconstructing an unknown source in
Thermoacoustic and Photoacoustic Tomography based on the recent advances in
understanding the theoretical nature of the problem. We work with variable
sound speeds that might be also discontinuous across some surface. The latter
problem arises in brain imaging. The new algorithm is based on an explicit
formula in the form of a Neumann series. We present numerical examples with
non-trapping, trapping and piecewise smooth speeds, as well as examples with
data on a part of the boundary. These numerical examples demonstrate the robust
performance of the new algorithm.

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## A Numerical Algorithm for Zero Counting. III: Randomization and Condition

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In a recent paper (Cucker, Krick, Malajovich and Wschebor, A Numerical
Algorithm for Zero Counting. I: Complexity and accuracy, J. Compl.,24:582-605,
2008) we analyzed a numerical algorithm for computing the number of real zeros
of a polynomial system. The analysis relied on a condition number kappa(f) for
the input system f. In this paper, we look at kappa(f) as a random variable
derived from imposing a probability measure on the space of polynomial systems
and give bounds for both the tail P{kappa(f) > a} and the expected value E(log
kappa(f)).

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## A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/11/2014
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We present a novel integral-equation algorithm for evaluation of Zaremba
eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the
Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course,
(slight modifications of) our algorithms are also applicable to the pure
Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of
an ansatz based on the single layer boundary operator, the Zaremba eigenproblem
is transformed into a nonlinear equation for the eigenvalue $\mu$. For smooth
domains the singular structure at Dirichlet-Neumann junctions is incorporated
as part of our corresponding numerical algorithm---which otherwise relies on
use of the cosine change of variables, trigonometric polynomials and, to avoid
the Gibbs phenomenon that would arise from the solution singularities, the
Fourier Continuation method (FC). The resulting numerical algorithm converges
with high order accuracy without recourse to use of meshes finer than those
resulting from the cosine transformation. For non-smooth (Lipschitz) domains,
in turn, an alternative algorithm is presented which achieves high-order
accuracy on the basis of graded meshes. In either case, smooth or Lipschitz
boundary, eigenvalues are evaluated by searching for zero minimal singular
values of a suitably stabilized discrete version of the single layer operator
mentioned above. (The stabilization technique is used to enable robust
non-local zero searches.) The resulting methods...

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## A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Numerical Analysis#High Energy Physics - Phenomenology#High Energy Physics - Theory#Mathematical Physics

We recently derived a very accurate and fast new algorithm for numerically
inverting the Laplace transforms needed to obtain gluon distributions from the
proton structure function $F_2^{\gamma p}(x,Q^2)$. We numerically inverted the
function $g(s)$, $s$ being the variable in Laplace space, to $G(v)$, where $v$
is the variable in ordinary space. We have since discovered that the algorithm
does not work if $g(s)\rightarrow 0$ less rapidly than $1/s$ as
$s\rightarrow\infty$, e.g., as $1/s^\beta$ for $0<\beta<1$. In this note, we
derive a new numerical algorithm for such cases, which holds for all positive
and non-integer negative values of $\beta$. The new algorithm is {\em exact} if
the original function $G(v)$ is given by the product of a power $v^{\beta-1}$
and a polynomial in $v$. We test the algorithm numerically for very small
positive $\beta$, $\beta=10^{-6}$ obtaining numerical results that imitate the
Dirac delta function $\delta(v)$. We also devolve the published MSTW2008LO
gluon distribution at virtuality $Q^2=5$ GeV$^2$ down to the lower virtuality
$Q^2=1.69$ GeV$^2$. For devolution, $ \beta$ is negative, giving rise to
inverse Laplace transforms that are distributions and not proper functions.
This requires us to introduce the concept of Hadamard Finite Part integrals...

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## A numerical algorithm for a class of BSDEs via branching process

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We generalize the algorithm for semi-linear parabolic PDEs in
Henry-Labord\`ere (2012) to the non-Markovian case for a class of Backward SDEs
(BSDEs). By simulating the branching process, the algorithm does not need any
backward regression. To prove that the numerical algorithm converges to the
solution of BSDEs, we use the notion of viscosity solution of path dependent
PDEs introduced by Ekren, Keller, Touzi and Zhang (2012) and extended in Ekren,
Touzi and Zhang (2013).; Comment: 31 pages

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## Harmonic Shears and Numerical Conformal Mappings

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Numerical Analysis#Mathematics - Complex Variables#Primary 30C99, Secondary 30C30, 31A05, 65E05

In this article we introduce a numerical algorithm for finding harmonic
mappings by using the shear construction introduced by Clunie and Sheil-Small
in 1984. The MATLAB implementation of the algorithm is based on the numerical
conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll.
Several numerical examples are given. In addition, we discuss briefly the
minimal surfaces associated with harmonic mappings and give a numerical example
of minimal surfaces.; Comment: 15 pages, 6 figures

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## A numerical algorithm for $L_2$ semi-discrete optimal transport in 3D

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/09/2014
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This paper introduces a numerical algorithm to compute the $L_2$ optimal
transport map between two measures $\mu$ and $\nu$, where $\mu$ derives from a
density $\rho$ defined as a piecewise linear function (supported by a
tetrahedral mesh), and where $\nu$ is a sum of Dirac masses.
I first give an elementary presentation of some known results on optimal
transport and then observe a relation with another problem (optimal sampling).
This relation gives simple arguments to study the objective functions that
characterize both problems.
I then propose a practical algorithm to compute the optimal transport map
between a piecewise linear density and a sum of Dirac masses in 3D. In this
semi-discrete setting, Aurenhammer et.al [\emph{8th Symposium on Computational
Geometry conf. proc.}, ACM (1992)] showed that the optimal transport map is
determined by the weights of a power diagram. The optimal weights are computed
by minimizing a convex objective function with a quasi-Newton method. To
evaluate the value and gradient of this objective function, I propose an
efficient and robust algorithm, that computes at each iteration the
intersection between a power diagram and the tetrahedral mesh that defines the
measure $\mu$.
The numerical algorithm is experimented and evaluated on several datasets...

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## On convergence of numerical algorithm of a class of the spatial segregation of reaction-diffusion system with two population densities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Recently, much interest has gained the numerical approximation of equations
of the Spatial Segregation of Reaction-diffusion systems with m population
densities. These problems are governed by a minimization problem subject to the
closed but non-convex set. In the present work we deal with the numerical
approximation of equations of stationary states for a certain class of the
Spatial Segregation of Reaction-diffusion system with two population densities
having disjoint support. We prove the convergence of the numerical algorithm
for two competing populations with non-negative internal dynamics $f_i(x)\geq
0.$ At the end of the paper we present computational tests.; Comment: 13 pages, 8 figures, Free boundary, Two-phase membrane problem,
Reaction-diffusion systems, Finite difference

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## A Stable Numerical Algorithm for the Brinkman Equations by Weak Galerkin Finite Element Methods

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/12/2013
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This paper presents a stable numerical algorithm for the Brinkman equations
by using weak Galerkin (WG) finite element methods. The Brinkman equations can
be viewed mathematically as a combination of the Stokes and Darcy equations
which model fluid flow in a multi-physics environment, such as flow in complex
porous media with a permeability coefficient highly varying in the simulation
domain. In such applications, the flow is dominated by Darcy in some regions
and by Stokes in others. It is well known that the usual Stokes stable elements
do not work well for Darcy flow and vise versa. The challenge of this study is
on the design of numerical schemes which are stable for both the Stokes and the
Darcy equations. This paper shows that the WG finite element method is capable
of meeting this challenge by providing a numerical scheme that is stable and
accurate for both Darcy and the Stokes dominated flows. Error estimates of
optimal order are established for the corresponding WG finite element
solutions. The paper also presents some numerical experiments that demonstrate
the robustness, reliability, flexibility and accuracy of the WG method for the
Brinkman equations.; Comment: 20 pages, 21 plots and figures

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## Particle velocity based universal algorithm for numerical simulation of hydraulic fractures

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In the paper, we propose a new effective mathematical formulation and
resulting universal numerical algorithm capable of tackling various HF models
in the framework of a unified approach. The presented numerical scheme is not
limited to any particular elasticity model or crack propagation regime. Its
basic assumptions are: i) proper choice of independent and dependent variables
(with the direct utilization of a new one - the reduced particle velocity), ii)
tracing the fracture front by use of the speed equation which can be integrated
in a closed form and sets an explicit relation between the crack propagation
speed and the coefficients in the asymptotic expansion of the crack opening,
iii) proper regularization techniques, iv) improved temporal approximation, v)
modular algorithm architecture. The application of the new dependent variable,
the reduced particle velocity, instead of the usual fluid flow rate,
facilitates the computation of the crack propagation speed from the local
relation based on the speed equation. As a result, the position of the crack
front is accurately determined from an explicit formula derived from the speed
equation. The underlying ideas employed in the algorithm are combined together
producing a robust and efficient numerical scheme. Its performance is
demonstrated using classical examples of 1D models for hydraulic fracturing:
PKN and KGD under various fracture propagation regimes. Solution accuracy is
verified against dedicated analytical benchmarks and other solutions available
in the literature. Most of the ideas developed here...

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## Revisiting the method of characteristics via a convex hull algorithm

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/09/2014
Português

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We revisit the method of characteristics for shock wave solutions to
nonlinear hyperbolic problems and we describe a novel numerical algorithm - the
convex hull algorithm (CHA) - in order to compute, both, entropy dissipative
solutions (satisfying all relevant entropy inequalities) and entropy
conservative (or multivalued) solutions to nonlinear hyperbolic conservation
laws. Our method also applies to Hamilton-Jacobi equations and other problems
endowed with a method of characteristics. From the multivalued solutions
determined by the method of characteristic, our algorithm "extracts" the
entropy dissipative solutions, even after the formation of shocks. It applies
to, both, convex or non-convex flux/Hamiltonians. We demonstrate the relevance
of the proposed approach with a variety of numerical tests including a problem
from fluid dynamics.; Comment: 14 pages

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## A conservation formulation and a numerical algorithm for the double-gyre nonlinear shallow-water model

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We present a conservation formulation and a numerical algorithm for the
reduced-gravity shallow-water equations on a beta plane, subjected to a
constant wind forcing that leads to the formation of double-gyre circulation in
a closed ocean basin. The novelty of the paper is that we reformulate the
governing equations into a nonlinear hyperbolic conservation law plus source
terms. A second-order fractional-step algorithm is used to solve the
reformulated equations. In the first step of the fractional-step algorithm, we
solve the homogeneous hyperbolic shallow-water equations by the
wave-propagation finite volume method. The resulting intermediate solution is
then used as the initial condition for the initial-boundary value problem in
the second step. As a result, the proposed method is not sensitive to the
choice of viscosity and gives high-resolution results for coarse grids, as long
as the Rossby deformation radius is resolved. We discuss the boundary
conditions in each step, when no-slip boundary conditions are imposed to the
problem. We validate the algorithm by a periodic flow on an f-plane with exact
solutions. The order-of-accuracy for the proposed algorithm is tested
numerically. We illustrate a quasi-steady-state solution of the double-gyre
model via the height anomaly and the contour of stream function for the
formation of double-gyre circulation in a closed basin. Our calculations are
highly consistent with the results reported in the literature. Finally...

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## A numerical algorithm for singular optimal LQ control systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/03/2012
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A numerical algorithm to obtain the consistent conditions satisfied by
singular arcs for singular linear-quadratic optimal control problems is
presented. The algorithm is based on the presymplectic constraint algorithm
(PCA) by Gotay-Nester \cite{Go78,Vo99} that allows to solve presymplectic
hamiltonian systems and that provides a geometrical framework to the
Dirac-Bergmann theory of constraints for singular Lagrangian systems
\cite{Di49}. The numerical implementation of the algorithm is based on the
singular value decomposition that, on each step allows to construct a
semi-explicit system. Several examples and experiments are discussed, among
them a family of arbitrary large singular LQ systems with index 3 and a family
of examples of arbitrary large index, all of them exhibiting stable behaviour.; Comment: An old paper (2009) posted for archival purposes

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## An efficient numerical algorithm for the L2 optimal transport problem with applications to image processing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We present a numerical method to solve the optimal transport problem with a
quadratic cost when the source and target measures are periodic probability
densities. This method is based on a numerical resolution of the corresponding
Monge-Amp\`ere equation. We extend the damped Newton algorithm of Loeper and
Rapetti \cite{LR} to the more general case of a non uniform density which is
relevant to the optimal transport problem, and we show that our algorithm
converges for sufficiently large damping coefficients. The main idea consists
of designing an iterative scheme where the fully nonlinear equation is
approximated by a non-constant coefficient linear elliptic PDE that we solve
numerically. We introduce several improvements and some new techniques for the
numerical resolution of the corresponding linear system. Namely, we use a Fast
Fourier Transform (FFT) method by Strain \cite{St}, which allows to increase
the efficiency of our algorithm against the standard finite difference method.
Moreover, we use a fourth order finite difference scheme to approximate the
partial derivatives involved in the nonlinear terms of the Newton algorithm,
which are evaluated once at each iteration; this leads to a significant
improvement of the accuracy of the method...

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## A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Computer Science - Computational Complexity#Computer Science - Numerical Analysis#Computer Science - Symbolic Computation#Mathematics - Numerical Analysis#F.2.1#G.1#I.1.2

We describe an algorithm to count the number of distinct real zeros of a
polynomial (square) system f. The algorithm performs O(n D kappa(f)) iterations
where n is the number of polynomials (as well as the dimension of the ambient
space), D is a bound on the polynomials' degree, and kappa(f) is a condition
number for the system. Each iteration uses an exponential number of operations.
The algorithm uses finite-precision arithmetic and a polynomial bound for the
precision required to ensure the returned output is correct is exhibited. This
bound is a major feature of our algorithm since it is in contrast with the
exponential precision required by the existing (symbolic) algorithms for
counting real zeros. The algorithm parallelizes well in the sense that each
iteration can be computed in parallel polynomial time with an exponential
number of processors.; Comment: We made minor but necessary improvements in the presentation

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## Physical Formulation and Numerical Algorithm for Simulating N Immiscible Incompressible Fluids Involving General Order Parameters

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/08/2014
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We present a physical formulation, and a numerical algorithm, based on a
class of general order parameters for simulating the motion of a mixture of $N$
($N\geqslant 2$) immiscible incompressible fluids with given densities, dynamic
viscosities, and pairwise surface tensions. The introduction of general order
parameters leads to a more strongly coupled system of phase field equations, in
contrast to that with certain special choice of the order parameters. However,
the general form enables one to compute the N-phase mixing energy density
coefficients in an explicit fashion in terms of the pairwise surface tensions.
From the simulation perspective, the increased complexity in the form of the
phase field equations with general order parameters in actuality does not cause
essential computational difficulties. Our numerical algorithm reformulates the
($N-1$) strongly-coupled phase field equations for general order parameters
into $2(N-1)$ Helmholtz-type equations that are completely de-coupled from one
another, leading to a computational complexity essentially the same as that of
the simpler phase field equations associated with special choice of order
parameters. We demonstrate the capabilities of the method developed herein
using several test problems involving multiple fluid phases and large contrasts
in densities and viscosities among the multitude of fluids. In particular...

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## Second-order hyperbolic Fuchsian systems. Gowdy spacetimes and the Fuchsian numerical algorithm

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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This is the second part of a series devoted to the singular initial value
problem for second-order hyperbolic Fuchsian systems. In the first part, we
defined and investigated this general class of systems, and we established a
well-posedness theory in weighted Sobolev spaces. This theory is applied here
to the vacuum Einstein equations for Gowdy spacetimes admitting, by definition,
two Killing fields satisfying certain geometric conditions. We recover, by more
direct and simpler arguments, the well-posedness results established earlier by
Rendall and collaborators. In addition, in this paper we introduce a natural
approximation scheme, which we refer to as the Fuchsian numerical algorithm and
is directly motivated by our general theory. This algorithm provides highly
accurate, numerical approximations of the solution to the singular initial
value problem. In particular, for the class of Gowdy spacetimes under
consideration, various numerical experiments are presented which show the
interest and efficiency of the proposed method. Finally, as an application, we
numerically construct Gowdy spacetimes containing a smooth, incomplete,
non-compact Cauchy horizon.; Comment: 22 pages. A shortened version is included in: F. Beyer and P.G.
LeFloch...

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## Numerical investigation into the existence of limit cycles in two-dimensional predator-prey systems

Fonte: South African Journal of Science
Publicador: South African Journal of Science

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 01/01/2013
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There has been a surge of interest in developing and analysing models of interacting species in ecosystems, with specific interest in investigating the existence of limit cycles in systems describing the dynamics of these species. The original Lotka-Volterra model does not possess any limit cycles. In recent years this model has been modified to take disturbances into consideration and allow populations to return to their original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional Lotka-Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with respect to the existence of limit cycles in non-linear systems. In this paper a class of predator-prey models of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to exist. Furthermore...

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