On General Properties of Quadratic Systems

doi:10.1080/00029890.1982.11995405

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On General Properties of Quadratic Systems

Journal Article•DOI•

On General Properties of Quadratic Systems

Carmen Chicone¹, Tian Jinghuang²•Institutions (2)

University of Missouri¹, Chinese Academy of Sciences²

01 Mar 1982-American Mathematical Monthly (Informa UK Limited)-Vol. 89, Iss: 3, pp 167-178

TL;DR: In this paper, the general properties of quadratic systems are discussed and a discussion of the properties of Quadratic Systems is presented. The American Mathematical Monthly: Vol 89, No. 3, pp. 167-178.

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Abstract: (1982). On General Properties of Quadratic Systems. The American Mathematical Monthly: Vol. 89, No. 3, pp. 167-178.

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Book•

Ordinary Differential Equations with Applications

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Carmen Chicone

30 Aug 2008

TL;DR: In this paper, the authors introduce the notion of Ordinary Differential Equations (ODE) for linear systems and the stability of nonlinear systems and apply it to Hyperbolic Theory.

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Abstract: Introduction to Ordinary Differential Equations * Linear Systems and Stability of Nonlinear Systems * Applications * Hyperbolic Theory * Continuation of Periodic Solutions * Homoclinic Orbits, Melnikov?s Method, and Chaos * Averaging * Local Bifurcation * References * Index.

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785 citations

Cites background from "On General Properties of Quadratic ..."

...The general question of the number of isolated periodic orbits for a polynomial system in the plane has been open since 1905; it is called Hilbert’s 16th problem (see [47], [97], [145], and [154])....
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Journal Article•DOI•

Hilbert's 16th problem and bifurcations of planar polynomial vector fields

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Jibin Li¹•Institutions (1)

Kunming University of Science and Technology¹

01 Jan 2003-International Journal of Bifurcation and Chaos

TL;DR: The progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 15th problem and bifurcations of planar vector fields is discussed.

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Abstract: The original Hilbert's 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert's 16th problem is presented, and the relationship between Hilbert's 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert's 16th problem? Section 2: The first part of Hilbert's 16th problem. Section 3: The second part of Hilbert's 16th problem: introduction. Section 4: Focal values, saddle values and finite cyclicity in a fine focus, closed orbit and homoclinic loop. Section 5: Finiteness problem. Section 6: The weakened Hilbert's 16th problem. Section 7: Global and local bifurcations of Zq–equivariant vector fields. Section 8: The rate of growth of Hilbert number H(n) with n.

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461 citations

Cites background from "On General Properties of Quadratic ..."

...Some valuable surveys and books: [Coppel, 1966; Qin, 1982; Chicone & Tian, 1982; Cai, 1989; Ye, 1996; etc.] for quadratic systems; [Coleman, 1983; Lloyd, 1988; Ilyashenko, 1991; Schlomiuk, 1993; Roussarie, 1998; Rousseau, 1993; Yang, 1991; Ye, 1995; etc.] for cubic systems and more general systems....
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...Section 8: The rate of growth of Hilbert number H(n) with n. Keywords: Real algebraic curve; scheme of ovals; limit cycle; bifurcation of vector fields; configuration of limit cycles; Hilbert number H(n)....
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Journal Article•DOI•

Non-autonomous equations related to polynomial two-dimensional systems

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M. A. M. Alwash¹, N. G. Lloyd¹•Institutions (1)

Aberystwyth University¹

01 Jan 1987-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences

TL;DR: In this article, the authors investigated the problem of estimating the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients, and they gave a number of sufficient conditions and investigated the implications for the corresponding two-dimensional systems.

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Abstract: Periodic solutions of certain one-dimensional non-autonomous differential equations are investigated (equation (1.4)); the independent variable is complex. The motivation, which is explained in the introductory section, is the connection with certain polynomial two-dimensional systems. Several classes of coefficients are considered; in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients. In particular, we need to know when there is a full neighbourhood of periodic solutions. We give a number of sufficient conditions and investigate the implications for the corresponding two-dimensional systems.

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139 citations

Journal Article•DOI•

The number of limit cycles of certain polynomial differential equations

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T. R. Blows¹, N. G. Lloyd¹•Institutions (1)

Aberystwyth University¹

01 Jan 1984-Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences

TL;DR: In this paper, the maximum possible number of limit cycles of two-dimensional differential systems in terms of the degree of P and Q is investigated. But the authors focus on polynomials and do not consider quadratic systems.

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Abstract: Two-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

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94 citations

Journal Article•

A new dynamical approach of Emden-Fowler equations and systems

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Marie-Françoise Bidaut-Véron, Hector Giacomini

23 Feb 2010-Advances in Differential Equations

TL;DR: In this article, the existence of ground states for a non-variational system of order 4 is studied and a conjecture of nonexistence of ground state for the system with p = q = 2$ and m = m+1$ and s = m>0.

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Abstract: We give a new approach on general systems of the form \[ (G)\left\{ \begin{array} [c]{c}% -\Delta_{p}u=\operatorname{div}(\left\vert abla u\right\vert ^{p-2} abla u)=\varepsilon_{1}\left\vert x\right\vert ^{a}u^{s}v^{\delta},\\ -\Delta_{q}v=\operatorname{div}(\left\vert abla v\right\vert ^{q-2} abla u)=\varepsilon_{2}\left\vert x\right\vert ^{b}u^{\mu}v^{m}, \end{array} \right. \] where $Q,p,q,\delta,\mu,s,m,$ $a,b$ are real parameters, $Q,p,q eq1,$ and $\varepsilon_{1}=\pm1,$ $\varepsilon_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $\varepsilon_{1}=\varepsilon_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is variational. We give an important result on existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0.$ In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$ and $\delta=m+1$ and $\mu=s+1.$

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86 citations

Cites background from "On General Properties of Quadratic ..."

...This change of unknown was mentioned in [11] in the case p = 2, ε = 1 and N = 3....
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References

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Book•

Differential Equations, Dynamical Systems, and Linear Algebra

[...]

Morris W. Hirsch, Steve Smale

12 May 1974

TL;DR: In this article, the structure theory of linear operators on finite-dimensional vector spaces has been studied and a self-contained treatment of that subject is given, along with a discussion of the relations between dynamical systems and certain fields outside pure mathematics.

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Abstract: This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.

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2,891 citations

"On General Properties of Quadratic ..." refers background in this paper

...In particular, this quadratic system contains the Volterra-Lotka equations [16] x = x(A - By) Y =y(Cx - D) which are used in mathematical ecology to model the populations of a predator-prey system and are used in chemistry to model the concentrations of two ideal chemical reactants....
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Book•

On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type

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N. N. Bautin

01 Jan 1954

397 citations

Journal Article•DOI•

Book Reviews : THEORY OF BIFURCATIONS OF DYNAMIC SYSTEMS ON A PLANE A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier J. Wiley & Sons, New York , New York (1973)

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J. Walker

01 Aug 1976-The Shock and Vibration Digest

248 citations

Journal Article•DOI•

A survey of quadratic systems

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W.A Coppel

01 Jul 1966-Journal of Differential Equations

TL;DR: In this paper, a general theory of functions of a complex variable is constructed corresponding to the theory of elliptic functions, which is a special case of the problem of obtaining more explicit information for special classes of such systems.

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238 citations

"On General Properties of Quadratic ..." refers background in this paper

...The equation (see [8] for detailed reference)...
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...In particular, [8] if a quadratic system has a focus and a center, they are oppositely oriented....
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...In addition to the references on this topic in [8], we also cite [12] as a typical recent example....
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...We refer the reader to [18], [25] for the proofs and to [8] for additional historical references....
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...But, before we begin, we wish to mention two indispensable sources: the survey of Coppel [8] which contains an exhaustive bibliography up to 1966 and the beautiful paper of Tung Chin-chu [24] which contains many of the results and ideas which we shall explore....
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Journal Article•DOI•

Sur les cycles limites

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H. Dulac

01 Jan 1923-Bulletin de la Société Mathématique de France

TL;DR: The Bulletin de la S. M. F. as discussed by the authors implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.html).

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Abstract: © Bulletin de la S. M. F., 1923, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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221 citations

"On General Properties of Quadratic ..." refers methods in this paper

...However, we do have the theorem of Dulac [10] which states that, given a particular choice of coefficients for a system of form (ln), then the particular differential equation has at most a finite number of limit cycles....
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