A Hybrid Polynomial System Solving Method for Mixed Trigonometric Polynomial Systems

The problem of local linearizability of the planar linear center perturbed by cubic non- linearities in all generalities on the system parameters (14 parameters) is far from being solved. The synchronization problem (as noted in Pikovsky, A., Rosenblum, M., and Kurths, J., 2003, Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge Nonlinear Science Series, Cambridge University Press, UK, and Blekhman, I. I., 1988, Synchronisation in Science and Technology, ASME Press Translations, New York) consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a five parameters non isochronous Chouikha cubic system (Chouikha, A. R., 2007, “Isochronous Centers of Lienard Type Equations and Applications,” J. Math. Anal. Appl., 331, pp. 358–376) we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system. The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb (J. C. Faugère, “FGb Salsa Software,” http://fgbrs.lip6.fr) for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts are different since our object is the synchronisation rather than the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers, (in particular Bardet, M., and Boussaada, I., 2011, “Compexity Reduction of C-algorithm,” App. Math. Comp., in press; Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(1), pp. 89–112; Bardet, M., Boussaada, I., Chouikha, A. R., and Strelcyn, J.-M., 2011, “Isochronicity Conditions for some Planar Polynomial Systems,” Bull. Sci. Math, 135(2), pp. 230–249; and furthermore, it can be considered as an adaptation of a qualitative theory method to a synchronization problem.

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Mixed-volume computation by dynamic lifting applied to polynomial system solving

Discrete & Computational Geometry ◽

10.1007/bf02711134 ◽

1996 ◽

Vol 16 (1) ◽

pp. 69-112 ◽

Cited By ~ 30

Author(s):

J. Verschelde ◽

K. Gatermann ◽

R. Cools

Keyword(s):

Polynomial System ◽

Mixed Volume ◽

Polynomial System Solving ◽

Volume Computation

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Characterization by orthogonal polynomial systems of finite Markov chains

Journal of Applied Probability ◽

10.1239/jap/1085496590 ◽

2001 ◽

Vol 38 (A) ◽

pp. 42-52 ◽

Cited By ~ 2

Author(s):

E. Seneta

Keyword(s):

Probability Distribution ◽

Markov Chains ◽

Orthogonal Polynomial ◽

Polynomial System ◽

Polynomial Systems ◽

Stochastic Matrices ◽

Mutation Model ◽

Hahn Polynomials ◽

Finite Markov Chains

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

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Homotopies and polynomial system solving I

ACM SIGSAM Bulletin ◽

10.1145/504331.504334 ◽

2001 ◽

Vol 35 (1) ◽

pp. 19-32 ◽

Cited By ~ 5

Author(s):

Ilias S. Kotsireas

Keyword(s):

Polynomial System ◽

Polynomial System Solving

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State estimation for stochastic polynomial systems with switching in the state equation

Transactions of the Institute of Measurement and Control ◽

10.1177/0142331217737134 ◽

2017 ◽

Vol 40 (9) ◽

pp. 2732-2739

Author(s):

Miguel Hernandez-Gonzalez ◽

Michael V Basin

Keyword(s):

Covariance Matrix ◽

Polynomial System ◽

Random Variable ◽

Polynomial Systems ◽

State Equation ◽

The State ◽

Nonlinear Functions ◽

Degree Polynomial ◽

Error Covariance ◽

Finite Dimensional

The problem of designing a mean-square filter has been studied for stochastic polynomial systems, where the state equation switches between two different nonlinear functions, over linear observations. A switching signal depends on a random variable modelled as a Bernoulli distributed sequence that takes the quantities of zero and one. The differential equations for the state estimate and the error covariance matrix are obtained in a closed form by expressing the conditional expectation of polynomial terms as functions of the estimate and covariance matrix. Finite-dimensional filtering equations are obtained for a particular case of a third-degree polynomial system. Numerical simulations are carried out in two cases of switching between different linear and second degree polynomial functions.

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