Quadratic Perturbations Research Articles

Perturbations of quadratic centers

We study the bifurcation of limit cycles in general quadratic perturbations of plane quadratic vector fields having a center at the origin. For any of the cases, we determine the essential perturbation and compute the corresponding bifurcation function. As an application, we find the precise location of the subset of centers in Q 3 R surrounded by period annuli of cyclicity at least three. Two specific cases are considered in more detail: the isochronous center S 1 and one of the intersection points ( Q 4 +) of Q 4 and Q 3 R . We prove that the period annuli around S 1 and Q 4 + have cyclicity two and three respectively. The proof is based on the possibility to derive appropriate Picard-Fuchs equations satisfied by the independent integrals included in the related bifurcation function.

Inclusions in Fluctuating Membranes: Exact Results

Point-like inclusions in fluid, fluctuating membranes are considered. Here the term inclusion is used in a general sense and describes a number of seemingly disparate situations: particles in membranes or other external and localized forces (such as a laser tweezer) which i) make the membrane locally stiffer, ii) induce a local spontaneous curvature, iii) change the local membrane thickness, or iv) the local separation between neighboring membranes. All these situations can be described by linear or quadratic local perturbations, for which the partition function is calculated exactly using a Gaussian membrane model. The deformed shape of a membrane in response to the presence of one inclusion and the membrane-mediated interactions between inclusions are thus obtained without further approximations. The interaction between two inclusions described by linear perturbations is temperature independent and therefore not affected by membrane fluctuations. The interaction between two inclusions described by quadratic perturbations is solely due to membrane shape fluctuations and vanishes at zero temperatures; in the strong coupling limit it shows a universal logarithmic divergence at short length scales. Formulas for the interaction of n inclusions are derived, which show non-trivial multibody contributions for the case of quadratic inclusions. All these results are valid for all temperatures and for all coupling strengths and thus bridge previously obtained results obtained at zero temperatures (neglecting membrane shape fluctuations) or using perturbation theory (for small strengths of the coupling between the inclusions and the membrane). These exact results are obtained with general Gaussian Hamiltonians and are thus applicable to all systems described by Gaussians forms.

Inhomogeneous Fuchs equations and the limit cycles in a class of near-integrable quadratic systems

We study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension-five components in the intersection of two strata in the centre manifold, and Q4. The study of limit cycles for this degenerate case requires us to investigate not the first but the second variation M2 of the displacement function. We prove that up to three limit cycles can emerge from the period annulus surrounding the centre. This implies that the cyclicity of period annuli of nearby systems in and Q4 is at most three as well. Our approach relies upon the possibility of deriving appropriate Picard–Fuchs equations satisfied by the four independent integrals included in M2.

The Cyclicity of the Period Annulus of the Quadratic Hamiltonian Triangle

The paper studies quadratic Hamiltonian centers surrounded by a separatrix contour having a form of a triangle. It is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to three.

Perturbations of quadratic Hamiltonian systems with symmetry

It is proved that arbitrary quadratic perturbations of quadratic Hamiltonian systems in the plane possessing central symmetry can produce at most two limit cycles.

Higher-order Melnikov functions for degenerate cubic Hamiltonians

It is shown that, in general, the first four Melnikov functions have to be taken into account in order to obtain definitive results concerning the limit cycles in quadratic perturbations of Hamiltonian systems in the plane with degenerate cubic Hamiltonians. An application is done in completing the proof that no more than two limit cycles can bifurcate out of homoclinic loops of quadratic Hamiltonian systems.

Abelian integrals of quadratic hamiltonian vector fields with an invariant straight line

We prove that the lowest upper bound for the number of isolated zeros of the Abelian integrals associated to quadratic Hamiltonian vector fields having a center and an invariant straight line after quadratic perturbations is one.

Bifurcation of limit cycles in a particular class of quadratic systems

Within the class of quadratic perturbations we show analytically or numerically how many limit cycles can be bifurcated at first order out of the periodic orbits surrounding the center of the quadratic system $$ \dot x=x(1-x-ay)\quad\text{and}\quad \dot y=y(-1+ax+y),\quad\text{where }1<a<\infty. $$

Bifurcation of Limit Cycles in a Particular Class of Quadratic Systems with Two Centers

Within the class of quadratic perturbations we show analytically or numerically how many limit cycles can be bifurcated at first order out of the periodic orbits nested around the centre point in (0, 0) or nested around the centre point in (0, 1/ n) of the quadratic system ẋ = − y + ny 2, ẏ = x − xy with 0 < n < 1.

Perturbations of a Hamiltonian family of cubic vector fields

This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

Cyclicity of homoclinic loops and degenerate cubic Hamiltonians

New conditions for a planar homoclinic loop to have cyclicity two under multiple parameter perturbations have been obtained. As an application it is proved that a homoclinic loop of a nongeneric cubic Hamiltonian has cyclicity two under arbitrary quadratic perturbations.

Effects of symmetry-breaking perturbations on the three-state Potts model

The effects of linear and quadratic symmetry-breaking perturbations on the continuous version of the three-state Potts model are analysed, using both Landau's theory of phase transitions and renormalisation-group techniques. Variation of the strength of the perturbations produces many different types of phase diagrams, featuring lambda lines, first-order lines, critical, tricritical and triple points and other complexities. Universal amplitude ratios characterising the multicritical points are calculated to first order in epsilon =4-d.