scholarly journals The granularity of parallel homotopy algorithms for polynomial systems of equations

1989 ◽  
Vol 29 (1) ◽  
pp. 21-37 ◽  
Author(s):  
D. C. S. Allison ◽  
S. Harimoto ◽  
L. T. Watson
Keyword(s):  
Polynomial Systems ◽  
Systems Of Equations ◽  
Homotopy Algorithms

scholarly journals Accelerated Solution of Multivariate Polynomial Systems of Equations

2003 ◽  
Vol 32 (2) ◽  
pp. 435-454 ◽  
Author(s):  
B. Mourrain ◽  
V. Y. Pan ◽  
O. Ruatta
Keyword(s):  
Polynomial Systems ◽  
Systems Of Equations ◽  
Multivariate Polynomial

Solving Polynomial Systems for the Kinematic Analysis and Synthesis of Mechanisms and Robot Manipulators

1995 ◽  
Vol 117 (B) ◽  
pp. 71-79 ◽  
Author(s):  
M. Raghavan ◽  
B. Roth
Keyword(s):  
Kinematic Analysis ◽  
State Of The Art ◽  
Polynomial Systems ◽  
The State ◽  
Polynomial Equations ◽  
Systems Of Equations ◽  
Synthesis Of Mechanisms ◽  
Analysis And Synthesis ◽  
Systems Of Polynomial Equations ◽  
And Robotics

Problems in mechanisms analysis and synthesis and robotics lead naturally to systems of polynomial equations. This paper reviews the state of the art in the solution of such systems of equations. Three well-known methods for solving systems of polynomial equations, viz., Dialytic Elimination, Polynomial Continuation, and Grobner bases are reviewed. The methods are illustrated by means of simple examples. We also review important kinematic analysis and synthesis problems and their solutions using these mathematical procedures.

Multiple bifurcation in a predator–prey system with nonmonotonic predator response

1992 ◽  
Vol 120 (3-4) ◽  
pp. 313-347 ◽  
Author(s):  
Franz Rothe ◽  
Douglas S. Shafer
Keyword(s):  
Stable Equilibrium ◽  
Polynomial Systems ◽  
Systems Of Equations ◽  
Predator Prey ◽  
Codimension Two ◽  
Prey System ◽  
Bifurcation Phenomena ◽  
Quartic Polynomial ◽  
Smooth Perturbation ◽  
Predator Response

SynopsisA model of a predator–prey system showing group defence on the part of the prey is formulated, and reduced to a three-parameter family of quartic polynomial systems of equations. Mathematically, this system contains the Volterra–Lotka system, and yields numerous kinds of bifurcation phenomena, including a codimension-two singularity of cusp type, in a neighbourhood of which the quartic system realises every phase portrait possible under small smooth perturbation. Biologically, the nonmonotonic behaviour of the predator response function allows existence of a second singularity in the first quadrant, so that the system exhibits an enrichment paradox, and, for certain choices of parameters, coexistence of stable oscillation and a stable equilibrium.

scholarly journals Generalized Linear Product Homotopy Algorithms and the Computation of Reachable Surfaces

2004 ◽  
Vol 4 (3) ◽  
pp. 226-234 ◽  
Author(s):  
Hai-Jun Su ◽  
J. Michael McCarthy ◽  
Layne T. Watson
Keyword(s):  
Polynomial Systems ◽  
The Body ◽  
Algebraic Surfaces ◽  
Degree Of Freedom ◽  
Homotopy Algorithm ◽  
Spatial Configurations ◽  
Spherical Wrist ◽  
Homotopy Algorithms ◽  
Three Degree Of Freedom ◽  
Serial Chains

In this paper, we apply a homotopy algorithm to the problem of finding points in a moving body that lie on specific algebraic surfaces for a given set of spatial configurations of the body. This problem is a generalization of Burmester’s determination of points in a body that lie on a circle for five planar positions. We focus on seven surfaces that we term “reachable” because they correspond to serial chains with two degree-of-freedom positioning structures combined with a three degree-of-freedom spherical wrist. A homotopy algorithm based on generalized linear products is used to provide a convenient estimate of the number of solutions of these polynomial systems. A parallelized version of this algorithm was then used to numerically determine all of the solutions.

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