On the monotonicity of the period function of some second order equations

Chow, Shui-Nee; Wang, Duo

doi:10.21136/cpm.1986.118260

Cited by 47 publications

(37 citation statements)

References 3 publications

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“…Nonetheless, numerical simulations clearly indicate ∂ E T ≥ 0 and ∂ L T ≤ 0 for a wide range of steady states, including the ones satisfying the assumptions of Theorem 8.15, see Remark 8.16 for an explicit example. We again emphasize that whether period functions as the one above are monotonous is an involved question and has been widely studied [9,64], especially in the context of bifurcation theory for Hamiltonian ODEs [11,12,27]. The rigorous monotonicity properties of T will be treated in future work.…”

Section: B Properties Of the Radial Period Functionmentioning

confidence: 99%

On the Existence of Linearly Oscillating Galaxies

Hadžić

Rein

Straub

2021

Arch Rational Mech Anal

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We consider two classes of steady states of the three-dimensional, gravitational Vlasov-Poisson system: the spherically symmetric Antonov-stable steady states (including the polytropes and the King model) and their plane symmetric analogues. We completely describe the essential spectrum of the self-adjoint operator governing the linearized dynamics in the neighborhood of these steady states. We also show that for the steady states under consideration, there exists a gap in the spectrum. We then use a version of the Birman-Schwinger principle first used by Mathur to derive a general criterion for the existence of an eigenvalue inside the first gap of the essential spectrum, which corresponds to linear oscillations about the steady state. It follows in particular that no linear Landau damping can occur in the neighborhood of steady states satisfying our criterion. Verification of this criterion requires a good understanding of the so-called period function associated with each steady state. In the plane symmetric case we verify the criterion rigorously, while in the spherically symmetric case we do so under a natural monotonicity assumption for the associated period function. Our results explain the pulsating behavior triggered by perturbing such steady states, which has been observed numerically.

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Section: B Properties Of the Radial Period Functionmentioning

confidence: 99%

On the Existence of Linearly Oscillating Galaxies

Hadžić

Rein

Straub

2021

Arch Rational Mech Anal

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“…It is very easy to see that Y(h, A, z) = O{\hz\ + |Az| + \z\ 3 Equation (2 [5,6]). There is a sector in the (h, A) plane which contains a unique periodic orbit along any line segment h = <5A contained in this sector.…”

Section: The Perturbed Hamiltonianmentioning

confidence: 99%

From sine waves to square waves in delay equations

Chow

Hale

Huang

1992

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…There are now a number of authors who deal with various questions related to the period function. Most of this work has been motivated by a desire to find sufficient conditions for the period function to be monotone [11,15,16,37,41], since monotonicity is a nondegeneracy condition for the bifurcation of subharmonic solutions of periodically forced Hamiltonian systems [14,Chapter 11], and since monotonicity implies existence and uniqueness for certain boundary value problems [6,12,43].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation of critical periods for plane vector fields

Chicone¹,

Jacobs²

1989

Trans. Amer. Math. Soc.

185

179

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Abstract.A bifurcation problem in families of plane analytic vector fields which have a nondegenerate center at the origin for all values of a parameter lí R* is studied. In particular, for such a family, the period function (¿;,A) i-> P(Ç,X) is defined; it assigns the minimum period to each member of the continuous band of periodic orbits (parametrized by £, e R ) surrounding the origin. The bifurcation problem is to determine the critical points of this function near the center with X as bifurcation parameter.Generally, if the function p , given by í >-+ P(£,A.) -P{0,Xt), vanishes to order 2k at the origin, then it is shown that the period function, after a perturbation of X, , has at most k critical points near the origin. If p vanishes to infinite order, i.e., the center is isochronous, it is shown that the number of critical points of P for perturbations of A« depends on the number of generators of the ideal of all Taylor coefficients of p(£, X), where the coefficients are considered elements of the ring of convergent power series in X . Specifically, if the ideal is generated by the first 2k Taylor coefficients, then a perturbation of X, produces at most k critical points of P near the origin. These theorems are applied to the quadratic systems with Bautin centers and to one degree of freedom "kinetic+potential" Hamiltonian systems with polynomial potentials. For the quadratic systems a complete solution of the bifurcation problem is obtained. For the Hamiltonian systems a number of results are proved independent of the degree of the potential and a complete solution is obtained for potentials of degree less than seven.Aside from their intrinsic interest, monotonicity properties of the period function are important in the question of existence and uniqueness of autonomous boundary value problems, in the study of subharmonic bifurcation of periodic oscillations, and in the analysis of the problem of linearization. In this regard it is shown that a Hamiltonian system with a polynomial potential of degree larger than two cannot be linearized. However, while these topics are the subject of a large literature, the spirit of this paper is more akin to that of N. Bautin's treatment of the multiple Hopf bifurcation for quadratic systems and the work on various forms of the weakened Hubert's 16th problem first posed

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On the monotonicity of the period function of some second order equations

Cited by 47 publications

References 3 publications

On the Existence of Linearly Oscillating Galaxies

On the Existence of Linearly Oscillating Galaxies

From sine waves to square waves in delay equations

Bifurcation of critical periods for plane vector fields

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