On the wave length of smooth periodic traveling waves of the Camassa-Holm equation

This paper is concerned with the wave length $\lambda$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$ (or "peak-to-peak amplitude"). Our main result establishes monotonicity properties of the map $a\longmapsto \lambda(a)$, i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated to the equation, which distinguish between the two possible qualitative behaviours of $\lambda(a)$, namely monotonicity and unimodality. The key point is to relate $\lambda(a)$ to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.Comment: 14 pages, 5 figure

On the period function in a class of generalized Lotka-Volterra systems

In this note, motivated by the recent results of Wang et al. [Wang et al., Local bifurcations of critical periods in a generalized 2D LV system, Appl. Math. Comput. 214 (2009) 17-25], we study the behaviour of the period function of the center at the point (1,1) of the planar differential system {u' = up(1−vq),v'= μvq(up−1), where p, q, μ ∈ R with pq > 0 and μ > 0. Our aim is twofold. Firstly, we determine regions in the parameter space for which the corresponding system has a center with a monotonic period function. Secondly, by taking advantage of the results of Wang et al., we show some properties of the bifurcation diagram of the period function and we make some comments for further research. The differential system under consideration is a generalization proposed by Farkas and Noszticzius of the Lotka-Volterra model

Unfoldings of saddle-nodes and their Dulac time

In this paper we study unfoldings of saddle-nodes and their Dulac time. By unfolding a saddle-node, saddles and nodes appear. In the first result (Theorem A) we prove uniform regularity by which orbits and their derivatives arrive at a node. Uniformity is with respect to all parameters including the unfolding parameter bringing the node to a saddle-node and a parameter belonging to a space of functions. In the second part, we apply this first result for proving a regularity result (Theorem B) on the Dulac time (time of Dulac map) of an unfolding of a saddle-node. This result is a building block in the study of bifurcations of critical periods in a neighbourhood of a polycycle. Finally, we apply Theorems A and B to the study of critical periods of the Loud family of quadratic centers and we prove that no bifurcation occurs for certain values of the parameters (Theorem C)

On the cyclicity of Kolmogorov polycycles

In this paper we study planar polynomial Kolmogorov’s differential systems Xµ x˙ = x f(x, y; µ), y˙ = yg(x, y; µ), with the parameter µ varying in an open subset Λ ⊂ RN. Compactifying Xµ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all µ ∈ Λ. We are interested in the cyclicity of Γ inside the family {Xµ}µ∈Λ, i.e., the number of limit cycles that bifurcate from Γ as we perturb µ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N = 3 and N = 5, and in both cases we are able to determine the cyclicity of the polycycle for all µ ∈ Λ, including those parameters for which the return map along Γ is the identity

The period function of Hamiltonian systems with separable variables

In this paper we study the period function of those planar Hamiltonian differential systems for which the Hamiltonian function H(x, y) has separable variables, i.e., it can be written as H(x, y) = F1(x) + F2(y). More concretely we are concerned with the search of sufficient conditions implying the monotonicity of the period function, i.e., the absence of critical periodic orbits. We are also interested in the uniqueness problem and in this respect we seek conditions implying that there exists at most one critical periodic orbit. We obtain in a unified way several sufficient conditions that already appear in the literature, together with some other results that to the best of our knowledge are new. Finally we also investigate the limit of the period function as the periodic orbits tend to the boundary of the period annulus of the center

Bifurcation of critical periods from Pleshkan's isochrones

Pleshkan proved in 1969 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in the family of cubic centers with homogeneous nonlinearities ℓ3. In this paper we prove that if we perturb any of these isochrones inside ℓ3, then at most two critical periods bifurcate from its period annulus. Moreover, we show that, for each k=0, 1, 2, there are perturbations giving rise to exactly k critical periods. As a byproduct, we obtain a partial result for the analogous problem in the family of quadratic centers ℓ2. Loud proved in 1964 that, up to a linear transformation and a constant rescaling of time, there are four isochrones in ℓ2. We prove that if we perturb three of them inside ℓ2, then at most one critical period bifurcates from its period annulus. In addition, for each k=0, 1, we show that there are perturbations giving rise to exactly k critical periods. The quadratic isochronous center that we do not consider displays some peculiarities that are discussed at the end of the paper

Bifurcation of local critical periods in the generalized Loud's system

We study the bifurcation of local critical periods in the differential system (x˙ = −y + Bxn−1y,y˙ = x + Dxn + F xn−2y2, where B, D, F ∈ R and n > 3 is a fixed natural number. Here by "local" we mean in a neighbourhood of the center at the origin. For n even we show that at most two local critical periods bifurcate from a weak center of finite order or from the linear isochrone, and at most one local critical period from a nonlinear isochrone. For n odd we prove that at most one local critical period bifurcates from the weak centers of finite or infinite order. In addition, we show that the upper bound is sharp in all the cases. For n = 2 this was proved by Chicone and Jacobs in [Bifurcation of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989) 433-486] and our proof strongly relies on their general results about the issue

The criticality of centers of potential systems at the outer boundary

The number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied. We call this number the criticality at the outer boundary. Our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1. We apply them to study the bifurcation diagram of the period function of X = −y∂ x ((x 1) p − (x 1) q )∂ y with q < p. This family was previously studied for q = 1 by Y. Miyamoto and K. Yagasaki