The Monotonicity of Period Function for Codimension Four Quadratic System Q4

Zhao, Yulin

doi:10.1006/jdeq.2002.4175

Cited by 43 publications

(21 citation statements)

References 43 publications

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“…For instance, the period function is monotone for Volterra-Lotka systems [15,18,23,24]. The same result was proved for quadratic Hamiltonian systems [11] and for codimension four quadratic system [27]. Additional results concerning period functions can be found in [6,12,25,26,19,21].…”

Section: Introductionsupporting

confidence: 51%

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

Bonorino

Brietzke

Lukaszczyk

et al. 2005

Journal of Differential Equations

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In this work we study the period function T of solutions to the conservative equationWe present conditions on f that imply the monotonicity and convexity of T. As a consequence we obtain the criterium established by C. Chicone and find conditions easier to apply. We also get a condition obtained by Cima, Gasull and Mañosas about monotonicity and, following some of their calculations, present results on the period function of Hamiltonian systems where H (x, y) = F (x) + n −1 |y| n . Using the monotonicity of T, we count the homogeneous solutions to the semilinear elliptic equation u = u −1 in two dimensions.

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Section: Introductionsupporting

confidence: 51%

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

Bonorino

Brietzke

Lukaszczyk

et al. 2005

Journal of Differential Equations

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“…Chicone has conjectured [2] that the reversible centers have at most two critical periods and that the centers of the three other families have a monotonic period function. The behaviour of the period function of the quadratic centers has been studied extensively, and there is much analytic evidence that the conjecture is true (see [4,7,9,11,13,15,16] and the references therein). It is clear therefore that in this setting the most interesting family of centers is the reversible one.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded

Mañosas¹,

Villadelprat

2008

Proc. Amer. Math. Soc.

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Abstract. This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namelyIn this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle K = (−7, 2)×(0, 4). More concretely, we prove that if (D, F ) / ∈ K, then the period function of the center is monotonically increasing.

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“…This case follows from the results of Zhao [16]. In that paper the author studies the period function of a subfamily of quadratic centers that intersects the one in (21) at F = 1 3 .…”

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confidence: 79%

On the return time function around monodromic polycycles

Marı́n

Villadelprat

2006

Journal of Differential Equations

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In this paper we study the period function of centers of planar polynomial differential systems. With a convenient compactification of the phase portrait, the boundary of the period annulus of the center has two connected components: the center itself and a polycycle. We are interested in the behaviour of the period function near the polycycle. The desingularization of its critical points gives rise to a new polycycle (monodromic as well) with hyperbolic saddles or saddle-nodes at the vertices. In this paper we compute the first terms in the asymptotic development of the time function around any orbitally linearizable saddle that may come from this desingularization process. In addition, we use these developments to study the bifurcation diagram of the period function of the dehomogenized Loud's centers. More generally, the tools developed here can be used to study the return time function around a monodromic polycycle. This work is a continuation of the results in [P. Mardešić, D. Marín, J. Villadelprat, On the time function of the Dulac map for families of meromorphic vector fields, Nonlinearity 16 (2003) 855-881; P. Mardešić, D. Marín, J. Villadelprat, The period function of reversible quadratic centers, J. Differential Equations 224 (2006) 120-171].

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The Monotonicity of Period Function for Codimension Four Quadratic System Q4

Cited by 43 publications

References 43 publications

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

Properties of the period function for some Hamiltonian systems and homogeneous solutions of a semilinear elliptic equation

The bifurcation set of the period function of the dehomogenized Loud's centers is bounded

On the return time function around monodromic polycycles

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