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Computing critical points for invariant algebraic systems [article]

Jean-Charles Faugère
2020 arXiv   pre-print
We consider the problem of computing the points at which 𝐟 vanish and the Jacobian matrix associated to 𝐟, ϕ is rank deficient provided that this set is finite.  ...  We exploit the invariance properties of the input to split the solution space according to the orbits of 𝒮_n.  ...  Algorithms for computing critical points We can now turn to the main question in this article.  ... 
arXiv:2009.00847v1 fatcat:icnflywcivas5gxm3fqbktg5am

Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials [article]

Thi Xuan Vu
2022 arXiv   pre-print
., x_n] with s<n, we consider the problem of computing the set W(ϕ, 𝐟) of points at which 𝐟 vanishes and the Jacobian matrix of 𝐟, ϕ with respect to x_1, ..., x_n does not have full rank.  ...  The runtime of our algorithm is polynomial in the total number of points described by the output.  ...  critical points for algebraic systems.  ... 
arXiv:2203.16094v2 fatcat:cnbp5vlqxzdlpnv4u2j6g3unk4

Computing Critical Points for Algebraic Systems Defined by Hyperoctahedral Invariant Polynomials

Thi Xuan Vu
2022 Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation  
CCS CONCEPTS • Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms.  ...  . , x n ] with s < n, we consider the problem of computing the set W (ϕ, f ) of points at which f vanishes and the Jacobian matrix of f , ϕ with respect to x 1 , . . . , x n does not have full rank.  ...  in computing critical points for algebraic systems.  ... 
doi:10.1145/3476446.3536181 fatcat:lafepkefoffuvaa4g7gb7htjmi

Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems

J. Chavarriga, H. Giacomini, M. Grau
2005 Bulletin des Sciences Mathématiques  
This work deals with planar polynomial differential systems {ẋ} = P (x, y), {ẏ} = Q(x, y). We give a set of necessary conditions for a system to have an invariant algebraic curve.  ...  These conditions are determined from the value of the cofactor at the singular points of the system, once considered in a compact space.  ...  Critical points at infinity We first define a polynomial differential equation in CP(2) and the notion of invariant algebraic curve and critical point for the equation.  ... 
doi:10.1016/j.bulsci.2004.09.002 fatcat:hvl3g5ydcberfpax4pxmbxm56u

Synthesizing Switching Controllers for Hybrid Systems by Generating Invariants [chapter]

Hengjun Zhao, Naijun Zhan, Deepak Kapur
2013 Lecture Notes in Computer Science  
We extend a template-based approach for synthesizing switching controllers for semi-algebraic hybrid systems, in which all expressions are polynomials.  ...  This is achieved by combining a QE (quantifier elimination)-based method for generating invariants with a qualitative approach for predefining templates.  ...  Jiang Liu for his great contribution to our previous joint work on invariant generation. We also thank Dr. Matthias Horbach, Mr.  ... 
doi:10.1007/978-3-642-39698-4_22 fatcat:na6ge7pjandallod5rdqksmyd4

Homogeneous mechanical systems with symmetry, and geometry of the coadjoint action

Ernesto A. Lacomba
1976 Bulletin of the American Mathematical Society  
In this note we announce further results on the global behaviour of the so called transitive mechanical systems [4], i.e., mechanical systems with symmetry on homogeneous spaces with an invariant riemannian  ...  The germ of this idea appeared in Arnold [1, especially §6], linked to the Euler equations for Lie groups. Detailed proofs will appear in [5].  ...  If ad(G) C GL(G) is algebraic, then lm(J) and lm(I) are semialgebraic sets for any transitive mechanical system with G as its group of symmetries.  ... 
doi:10.1090/s0002-9904-1976-14002-5 fatcat:us6ofb6eanhk3iywitirpcelje

Conformal algebras of two-dimensional disordered systems

Victor Gurarie, Andreas W W Ludwig
2002 Journal of Physics A: Mathematical and General  
Remarkably, we also find this algebra, and thereby an ensuing hidden supersymmetry, realized at general replica-averaged critical points, for which we derive an explicit formula for b.  ...  We discuss the structure of 2D conformal field theories (CFT) at central charge c=0 describing critical disordered systems, polymers and percolation.  ...  Cardy, for useful discussions. One of us (V.G.) is supported by the NSF grant PHY-94-07194.  ... 
doi:10.1088/0305-4470/35/27/101 fatcat:6zkgxjcu4ncefm5ysq6wf7unr4

A cubic Kolmogorov system with six limit cycles

N.G. Lloyd, J.M. Pearson, E Saéz, I. Szántó
2002 Computers and Mathematics with Applications  
We show in particular that a maximum of six small amplitude limit cycles can bifurcate from a critical point in the first quadrant, and we discuss the number of invariant lines.  ...  we consider a class of cubic Kolmogorov systems.  ...  The derivation of necessary conditions often involves extensive computing. We use the computer algebra system Reduce predominantly, but occasionally also Maple.  ... 
doi:10.1016/s0898-1221(02)00161-x fatcat:qwsfpnmlunbbdfqi372yzxszay

Distinct small-distance scaling behavior of on-off intermittency in chaotic dynamical systems

Ying-Cheng Lai
1996 Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics  
Although the statistics of their laminar phase obeys the algebraic scaling, quantities such as the average transient time for trajectories to fall in a small neighborhood of the asymptotic off state exhibit  ...  On-off intermittency in chaotic dynamical systems refers to the situation where some dynamical variables exhibit two distinct states in their course of time evolution.  ...  These singularities come from the successive iterations of the critical point of the map x c ϭ0.5 ͓9͔.  ... 
doi:10.1103/physreve.54.321 pmid:9965075 fatcat:5dmx23babrgzjc3pcvjfz5k3da

Real Root Finding for Equivariant Semi-algebraic Systems

Cordian Riener, Mohab Safey el Din
2018 Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '18  
Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by s polynomials  ...  Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d − 1 distinct coordinates.  ...  With a more algebraic flavour, computer algebra has been developed to solve polynomial systems which are invariant under the action of some groups.  ... 
doi:10.1145/3208976.3209023 dblp:conf/issac/RienerD18 fatcat:2xpe5foybvgxfaw7mat3gays6i

Some criteria for the existence of limit cycles for quadratic vector fields

J. Chavarriga, I.A. García
2003 Journal of Mathematical Analysis and Applications  
In this paper we consider real quadratic systems. We present new criteria for the existence and uniqueness of limit cycles for such systems by using Darbouxian particular solutions.  ...  Some results are based on the study of such systems in CP 2 . We also generalize the well-know result of Bautin on the nonexistence of limit cycles for quadratic Lotka-Volterra systems. (J.  ...  It is well known (see, for instance, [16] ) that a limit cycle of a quadratic system surrounds an unique critical point (x 0 , y 0 ) of the system.  ... 
doi:10.1016/s0022-247x(03)00149-5 fatcat:3h7fciuyora7lhze6gvoh3rc4y

Non-nested configuration of algebraic limit cycles in quadratic systems

J. Chavarriga, I.A. García, J. Sorolla
2006 Journal of Differential Equations  
This work deals with algebraic limit cycles of planar polynomial differential systems of degree two.  ...  More concretely, we show among other facts that a quadratic vector field cannot possess two non-nested algebraic limit cycles contained in different irreducible invariant algebraic curves.  ...  He has also pointed out us Ref. [15] .  ... 
doi:10.1016/j.jde.2006.01.009 fatcat:viqe3xhzyzbnlf57as3adjakfa

Real root finding for equivariant semi-algebraic systems [article]

Cordian Riener
2018 arXiv   pre-print
Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by s polynomials  ...  Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most 2d-1 distinct coordinates.  ...  With a more algebraic flavour, computer algebra has been developed to solve polynomial systems which are invariant under the action of some groups.  ... 
arXiv:1806.08121v1 fatcat:3ba7dehbuffezkyndhud34lsme

Page 8530 of Mathematical Reviews Vol. , Issue 2002K [page]

2002 Mathematical Reviews  
The critical point at infinity occurs only in spatial problems. We compute the homology of the integral manifold for each regular value of the Jacobi constant.  ...  “There are five positive critical values of the Jacobi constant: one is due to a critical point at infinity, another is due to the Lagrangian critical points and three are due to the Eulerian critical  ... 

Noncommutative topological dynamics

C. Correia Ramos, Nuno Martins, Ricardo Severino, J. Sousa Ramos
2006 Chaos, Solitons & Fractals  
We study noncommutative dynamical systems associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-algebras and Cuntz-Krieger algebras.  ...  As an example we consider systems having equal topological entropy log(1 + /), where / is the golden number, but distinct chaotic behavior and we show how a new numerical invariant allows to distinguish  ...  The K 0 -group of F s;r is not a complete invariant for the bimodal dynamical system. Proof. Set s fixed. Let r 1 and r 2 be two distinct values in Z½s; s À1 .  ... 
doi:10.1016/j.chaos.2005.03.016 fatcat:4viewhmd3jd6xkcifi7kmkwjo4
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