Localizing limit cycles : from numeric to analytical results

Presentation given by participants of the joint international multidisciplinary workshop MURPHYS-HSFS-2016 (MUltiRate Processes and HYSteresis; Hysteresis and Slow-Fast Systems), which was dedicated to the mathematical theory and applications of multiple scale systems and systems with hysteresis, and held at the Centre de Recerca Matemàtica (CRM) in Barcelona from June 13th to 17th, 2016This note presents the results of [4]. It deals with the problem of location and existence of limit cycles for real planar polynomial differential systems. We provide a method to construct Poincaré-Bendixson regions by using transversal curves, that enables us to prove the existence of a limit cycle that has been numerically detected. We apply our results to several known systems, like the Brusselator one or some Liénard systems, to prove the existence of the limit cycles and to locate them very precisely in the phase space. Our method, combined with some other classical tools can be applied to obtain sharp bounds for the bifurcation values of a saddle-node bifurcation of limit cycles, as we do for the Rychkov syste

Variational approach to a class of nonlinear oscillators with several limit cycles

We study limit cycles of nonlinear oscillators described by the equation $\ddot x + \nu F(\dot x) + x =0$. Depending on the nonlinearity this equation may exhibit different number of limit cycles. We show that limit cycles correspond to relative extrema of a certain functional. Analytical results in the limits $\nu ->0$ and $\nu -> \infty$ are in agreement with previously known criteria. For intermediate $\nu$ numerical determination of the limit cycles can be obtained.Comment: 12 pages, 3 figure

Time delay for one-dimensional quantum systems with steplike potentials

This paper concerns time-dependent scattering theory and in particular the concept of time delay for a class of one-dimensional anisotropic quantum systems. These systems are described by a Schr\"{o}dinger Hamiltonian $H = -\Delta + V$ with a potential $V(x)$ converging to different limits $V_{\ell}$ and $V_{r}$ as $x \to -\infty$ and $x \to +\infty$ respectively. Due to the anisotropy they exhibit a two-channel structure. We first establish the existence and properties of the channel wave and scattering operators by using the modern Mourre approach. We then use scattering theory to show the identity of two apparently different representations of time delay. The first one is defined in terms of sojourn times while the second one is given by the Eisenbud-Wigner operator. The identity of these representations is well known for systems where $V(x)$ vanishes as $|x| \to \infty$ ($V_\ell = V_r$). We show that it remains true in the anisotropic case $V_\ell \not = V_r$, i.e. we prove the existence of the time-dependent representation of time delay and its equality with the time-independent Eisenbud-Wigner representation. Finally we use this identity to give a time-dependent interpretation of the Eisenbud-Wigner expression which is commonly used for time delay in the literature.Comment: 48 pages, 1 figur

On the number of limit cycles of the Lienard equation

In this paper, we study a Lienard system of the form dot{x}=y-F(x), dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system. This sequence seems to converge to the exact equation of each limit cycle. We obtain also a sequence of polynomials R_n(x) whose roots of odd multiplicity are related to the number and location of the limit cycles of the system.Comment: 10 pages, 5 figures. Submitted to Physical Review

T-matrix computations of light scattering by red blood cells

The electromagnetic far field, as well as the near field, originating from light interaction with a red blood cell ~RBC! volume-equivalent spheroid, was analyzed by utilizing the T-matrix theory. This method is a powerful tool that makes it possible to study the influence of cell shape on the angular distribution ofscattered light. General observations were that the three-dimensional shape, as well as the optical thickness apparent to the incident field, affects the forward scattering. The backscattering was influenced by the shape of the surface facing the incident beam. Furthermore sphering as well as elongation of an oblate RBC into a volume-equivalent sphere or a prolate spheroid, respectively, was theoretically modeled to imitate physiological phenomena caused, e.g., by heat or the increased shear stress of flowing blood. Both sphering and elongation were shown to decrease the intensity of the forward-directed scattering, thus yielding lower g factors. The sphering made the scattering pattern independent ofazimuthal scattering angle fs, whereas the elongation induced more apparent fs-dependent patterns. The light scattering by a RBC volume-equivalent spheroid was thus found to be highly influenced by the shape of the scattering object. A near-field radius rnf was evaluated as the distance to which the maximum intensity of the total near field had decreased to 2.5 times that of the incident field. It was estimated to 2–24.5 times the maximum radius of the scattering spheroid, corresponding to 12–69 mm. Because the near-field radius was shown to be larger than a simple estimation of the distance between the RBC’s in whole blood, the assumption of independent scattering, frequently employed in opticalmeasurements on whole blood, seems inappropriate. This also indicates that one cannot extrapolate the results obtained from diluted blood to whole blood by multiplying with a simple concentration factor