scholarly journals Homogeneous-Like Generalized Cubic Systems

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
G. R. Nicklason
Keyword(s):  
Homogeneous System ◽  
Polynomial Systems ◽  
Abel Equation ◽  
Center Condition ◽  
Similar Idea ◽  
Cubic System ◽  
Center Conditions ◽  
Abel Equations ◽  
The One ◽  
Cubic Systems

We consider properties and center conditions for plane polynomial systems of the formsx˙=-y-p1(x,y)-p2(x,y),y˙=x+q1(x,y)+q2(x,y)wherep1,q1andp2,q2are polynomials of degreesnand2n-1, respectively, for integersn≥2. We restrict our attention to those systems for whichyp2(x,y)+xq2(x,y)=0. In this case the system can be transformed to a trigonometric Abel equation which is similar in form to the one obtained for homogeneous systems(p2=q2=0). From this we show that any center condition of a homogeneous system for a givenncan be transformed to a center condition of the corresponding generalized cubic system and we use a similar idea to obtain center conditions for several other related systems. As in the case of the homogeneous system, these systems can also be transformed to Abel equations having rational coefficients and we briefly discuss an application of this to a particular Abel equation.

Center conditions, compositions of polynomials and moments on algebraic curves

1999 ◽  
Vol 19 (5) ◽  
pp. 1201-1220 ◽  
Author(s):  
M. BRISKIN ◽  
J.-P. FRANCOISE ◽  
Y. YOMDIN
Keyword(s):  
Recurrence Relation ◽  
Vector Fields ◽  
Algebraic Curves ◽  
Abel Equation ◽  
Linear Recurrence ◽  
Polynomial Vector Fields ◽  
Taylor Coefficients ◽  
Center Condition ◽  
Center Conditions ◽  
Linear Recurrence Relation

We consider an Abel equation $(*)$ $y^{\prime}=p(x)y^2+q(x)y^3$ with $p(x)$, $q(x)$ polynomials in $x$. A center condition for ($*$) (closely related to the classical center condition for polynomial vector fields on the plane) is that $y_0=y(0)\equiv y(1)$ for any solution $y(x)$ of ($*$). This condition is given by the vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. A new basis for the ideals $I_k=\{v_2,\dots,v_k\}$ has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of $P=\int p$ and $Q=\int q$ in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments $\int P^kq$. On this base, explicit center equations are obtained for small degrees of $p$ and $q$.

scholarly journals Coupled Plasma-Catalytic System with Rang 19pr Catalyst for Conversion of Tar

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
Michał Młotek ◽  
Joanna Woroszył ◽  
Bogdan Ulejczyk ◽  
Krzysztof Krawczyk
Keyword(s):  
Catalytic System ◽  
Gas Flow ◽  
Model Compound ◽  
Calorific Value ◽  
Homogeneous System ◽  
Toluene Concentration ◽  
Gliding Discharge ◽  
Outlet Stream ◽  
The One ◽  
Pyrolysis Of Biomass

Abstract A coupled plasma-catalytic system (CPCS) for the conversion of toluene was investigated and compared to the homogeneous system of gliding discharge plasma. Toluene was used as a model compound, which is present in tars. The study was carried out at atmospheric pressure, in a gas composition similar to the one obtained during pyrolysis of biomass. The effect of the initial toluene concentration, energy supplied to gliding discharge (GD) and the presence of a catalyst on the conversion of toluene was studied. Both the composition of outlet gas and its calorific value were monitored. Based on the obtained results it can be concluded that the conversion of toluene increases with the increase of gliding discharge power. The highest toluene conversion (89%) was received in the coupled plasma-catalytic system (catalyst: RANG-19PR) under the following conditions: CO (0.13 mol. fr.), CO2 (0.12 mol. fr.), H2 (0.25 mol. fr.), N2 (0.50 mol. fr.) and 4400 ppm of toluene with a gas flow rate of 1000 Nl/h. The composition of the outlet gas in the homogeneous system and in the CPCS changed in the range of a few percents. Toluene levels were reduced tenfold. Benzene, C3 and C4 hydrocarbons, as well as acetylene, ethylene and ethane, were detected in the outlet stream in trace amounts. Carbon deposits were present in the reactor. The products of methanation of carbon oxides were detected in the both studied systems. A mechanism of toluene decomposition in the CPCS was proposed. The application of the catalyst brought about an increase in the calorific value of the outlet gas. It was above the minimal level demanded by engines and turbines.

scholarly journals Amplitude Independent Frequency Synchroniser for a Cubic Planar Polynomial System

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Islam Boussaada
Keyword(s):  
New York ◽  
Initial Conditions ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Algebraic Equations ◽  
Compact Region ◽  
Synchronization Problem ◽  
The Family ◽  
Cubic System ◽  
Isochronous Centers

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.

scholarly journals Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

2010 ◽  
Vol 2010 ◽  
pp. 1-17
Author(s):  
Xiangdong Xie ◽  
Fengde Chen ◽  
Qingyi Zhan
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Uniqueness Of Limit Cycles ◽  
Cubic System ◽  
Straight Lines ◽  
Cubic Systems

A class of cubic systems with two invariant straight linesdx/dt=y(1-x2),  dy/dt=-x+δy+nx2+mxy+ly2+bxy2.is studied. It is obtained that the focal quantities ofO(0,0)are,W0=δ; ifW0=0, thenW1=m(n+l); ifW0=W1=0, thenW2=−nm(b+1); ifW0=W1=W2=0, thenOis a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focalO(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009).

Bifurcations in a Cubic System with a Degenerate Saddle Point

2014 ◽  
Vol 24 (11) ◽  
pp. 1450144
Author(s):  
Desheng Shang ◽  
Yaoming Zhang
Keyword(s):  
Perturbation Theory ◽  
Qualitative Analysis ◽  
Saddle Point ◽  
Limit Cycles ◽  
Blow Up ◽  
Cubic System ◽  
Cubic Systems

Bifurcations in a cubic system with a degenerate saddle point are investigated using the technique of blow-up, the method of planar perturbation theory and qualitative analysis. It has been found that after appropriate perturbations, at least 12 limit cycles can bifurcate from a degenerate saddle point in a type of cubic systems.

scholarly journals Algebraic geometry of the center-focus problem for Abel differential equations

2014 ◽  
Vol 36 (3) ◽  
pp. 714-744 ◽  
Author(s):  
M. BRISKIN ◽  
F. PAKOVICH ◽  
Y. YOMDIN
Keyword(s):  
Algebraic Geometry ◽  
Differential Equations ◽  
Moment Problem ◽  
Explicit Solution ◽  
Order Approximation ◽  
Abel Equation ◽  
Composition Algebra ◽  
Center Conditions ◽  
Abel Differential Equations ◽  
Compositio Math

The Abel differential equation $y^{\prime }=p(x)y^{3}+q(x)y^{2}$ with polynomial coefficients $p,q$ is said to have a center on $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(a)=y(b)$. The problem of giving conditions on $(p,q,a,b)$ implying a center for the Abel equation is analogous to the classical Poincaré center-focus problem for plane vector fields. Center conditions are provided by an infinite system of ‘center equations’. During the last two decades, important new information on these equations has been obtained via a detailed analysis of two related structures: composition algebra and moment equations (first-order approximation of the center ones). Recently, one of the basic open questions in this direction—the ‘polynomial moments problem’—has been completely settled in Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728]. In this paper, we present a progress in the following two main directions: first, we translate the results of Pakovich and Muzychuk [Solution of the polynomial moment problem. Proc. Lond. Math. Soc. (3)99(3) (2009), 633–657] and Pakovich [Generalized ‘second Ritt theorem’ and explicit solution of the polynomial moment problem. Compositio Math.149 (2013), 705–728] into the language of algebraic geometry of the center equations. Applying these new tools, we show that the center conditions can be described in terms of composition algebra, up to a ‘small’ correction. In particular, we significantly extend the results of Briskin, Roytvarf and Yomdin [Center conditions at infinity for Abel differential equations. Ann. of Math. (2)172(1) (2010), 437–483]. Second, applying these tools in combination with explicit computations, we start in this paper the study of the ‘second Melnikov coefficients’ (second-order approximation of the center equations), showing that in many cases vanishing of the moments and of these coefficients is sufficient in order to completely characterize centers.

PERIODIC CONSTANTS AND TIME-ANGLE DIFFERENCE OF ISOCHRONOUS CENTERS FOR COMPLEX ANALYTIC SYSTEMS

2006 ◽  
Vol 16 (12) ◽  
pp. 3747-3757 ◽  
Author(s):  
YIRONG LIU ◽  
JIBIN LI
Keyword(s):  
Open Problem ◽  
Isochronous Center ◽  
Planar Complex ◽  
Angle Difference ◽  
Cubic System ◽  
Recursive Formulas ◽  
Center Conditions ◽  
Isochronous Centers

By introducing a concept of the time-angle difference for planar complex analytic systems, two recursive formulas of the function of the time-angle difference and period constants of complex isochronous centers can be given. Using these results to a class of real cubic system, the first five period constants and the isochronous center conditions of the origin are obtained. One open problem posed by [Cairo et al., 1999] is proved.

Growth Phenomena — Algebraic and Geometric Descriptions

2001 ◽  
Vol 08 (01) ◽  
pp. 63-71
Author(s):  
Andrzej Jamiołkowski
Keyword(s):  
Vector Fields ◽  
Nonlinear Dynamical Systems ◽  
Polynomial Systems ◽  
Polynomial Vector ◽  
Finsler Spaces ◽  
Polynomial Vector Fields ◽  
Associative Algebras ◽  
Nonlinear Dynamical ◽  
Volterra Systems ◽  
The One

An enormous variety of nonlinear dynamical systems can be — by suitable introduction of new coordinates — represented in the form of polynomial systems and then can be reduced to Volterra systems, where the nonlinearities are at most quadratic. In this paper, we discuss a link between systems of differential equations with homogeneous quadratic polynomial vector fields and non-associative algebras on the one hand and the question of representation of such systems as geodesics in some Finsler spaces on the other hand.

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