Some Applications of the Extended Bendixson-Dulac Theorem

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.Comment: 19 pages, 3 figure

Dulac-Cherkas functions for generalized Liénard systems

Dulac-Cherkas functions can be used to derive an upper bound for the number of limit cycles of planar autonomous differential systems including criteria for the non-existence of limit cycles, at the same time they provide information about their stability and hyperbolicity. In this paper, we present a method to construct  a special class of Dulac-Cherkas functions for  generalized Liénard systems of the type $\frac{dx}{dt} = y, \quad  \frac{dy}{dt} = \sum_{j=0}^l h_j(x) y^j$ with $l \ge 1$. In case $1 \le l \le 3$,  linear differential equations play a key role in this process, for $l \ge 4$, we have to solve a system of linear differential and algebraic equations, where the number of equations is larger than the number of unknowns. Finally,  we show that Dulac-Cherkas functions can be used to construct generalized Liénard systems with any $l$ possessing limit cycles

On the approximation of the limit cycles function

We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function $l$. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincar'e function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Li'enard system. The obtained result supports a conjecture by Lins, de Melo and Pugh

New advances on the Lyapunov constants of some families of planar differential systems

This note presents some advances regarding the Lyapunov constants of some families of planar polynomial differential systems, as a first step toward the resolution of the center and cyclicity problems. First, a parallelization approach is computationally implemented to achieve the 14th Lyapunov constant of the complete cubic family. Second, a technique based on interpolating some specific quantities so as to reconstruct the structure of the Lyapunov constants is used to study a Kukles system, some fifth-degree homogeneous systems, and a quartic system with two invariant lines

From real to virtual

Each of us is a part of the Progress. Every our dream or our desire is a fuel of the big machine named – technology. Originally, humans are lazy, and therefore we not only survived but we become cleverer. What if an ancient human hadn‘t been so lazy and had not invented a wheel or spear? I think, It would be much harder to survive. However, let‘s continue

The correlations of glycated hemoglobin and carbohydrate metabolism parameters with heart rate variability in apparently healthy sedentary young male subjects

Introduction Sedentary lifestyle is a major risk factor for diabetes, cardiovascular and many other age-related diseases. Heart rate variability (HRV) reflects the function of regulatory systems of internal organs and may sensitively indicate early metabolic disturbances. We hypothesize that quantitative and qualitative changes of HRV in young subjects may reflect early metabolic derangements responsible for further development of clinically significant disease. Aim The aim of our study was to determine whether the parameters of carbohydrate metabolism (fasting blood glucose, HBA1c and surrogate insulin sensitivity/resistance indices) correlate with anthropometric data and HRV. Methods The study group consisted of 30 healthy sedentary male subjects aged 20–40, nonsmokers, mainly office and research employees, medical staff and students. Athletes, actively training more than one hour per week, severely obese and men of physical work were excluded from the study. HRV parameters were derived from short term ECG records (five minutes intervals) in supine position and during orthostatic test. Anthropometric data included height, weight, body mass index (BMI), age and body composition (estimation by bioelectric impedance method). The fasting blood glucose, insulin and C-peptide, homeostatic model assessment (HOMA-IR) index and glycated hemoglobin (HbA1c) were evaluated. Linear correlation coefficient (r) was calculated using Statistica 10.0 software. Results and discussion HOMA-IR index correlated positively with body weight, visceral fat and BMI (p=0.047, 0.027 and 0.017 respectively). In supine position pNN50 positively correlated with glucose/insulin ratio (p=0.011) and heart rate with HOMA-IR (p=0.006). In orthostatic test negative correlations of HBA1c with standard deviation, total and low frequency power were determined (p=0.034, 0.400 and 0.403 respectively), which indicates a gradual worsening of functional capacity of cardiovascular system with low-grade increase (under the conventional threshold) of HBA1c. Conclusions In apparently healthy sedentary subjects HRV reduction correlates with the age advancement, subclinical deteriorations of carbohydrate metabolism and excessive fat accumulation

Renormalization group and isochronous oscillations

We show how the condition of isochronicity can be studied for two dimensional systems in the renormalization group (RG) context. We find a necessary condition for the isochronicity of the Cherkas and another class of cubic systems. Our conditions are satisfied by all the cases studied recently by Bardet et al \cite{bard} and Ghose Choudhury and Guh

Laser in the axial electric field as a tool to search for P-, T- invariance violation

We consider rotation of polarization plane of the laser light when a gas laser is placed in a longitudinal electric field (10~kV/cm). It is shown that residual anisotropy of the laser cavity 10^{-6} and the sensitivity to the angle of polarization plane rotation about 10^{-11} -10^{-12} rad allows one to measure an electron EDM with the sensitivity about 10^{-30} e cm.Comment: 12 page