Synchronization problems for unidirectional feedback coupled nonlinear systems

In this paper we consider three different synchronization problems consisting in designing a nonlinear feedback unidirectional coupling term for two (possibly chaotic) dynamical systems in order to drive the trajectories of one of them, the slave system, to a reference trajectory or to a prescribed neighborhood of the reference trajectory of the second dynamical system: the master system. If the slave system is chaotic then synchronization can be viewed as the control of chaos; namely the coupling term allows to suppress the chaotic motion by driving the chaotic system to a prescribed reference trajectory. Assuming that the entire vector field representing the velocity of the state can be modified, three different methods to define the nonlinear feedback synchronizing controller are proposed: one for each of the treated problems. These methods are based on results from the small parameter perturbation theory of autonomous systems having a limit cycle, from nonsmooth analysis and from the singular perturbation theory respectively. Simulations to illustrate the effectiveness of the obtained results are also presented.Comment: To appear in Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Ana

On the response of autonomous sweeping processes to periodic perturbations

If $x_0$ is an equilibrium of an autonomous differential equation $\dot x=f(x)$ and $\det \|f'(x_0)\|\not=0$, then $x_0$ persists under autonomous perturbations and $x_0$ transforms into a $T$-periodic solution under non-autonomous $T$-periodic perturbations. In this paper we discover a similar structural stability for Moreau sweeping processes of the form $-\dot u\in N_B(u)+f_0(u),$ $u\in\mathbb{R}^2,$ i. e. we consider the simplest case where the derivative is taken with respect to the Lebesgue measure and where the convex set $B$ of the reduced system is a non-moving unit ball of $\mathbb{R}^2.$ We show that an equilibrium $\|u_0\|=1$ persists under periodic perturbations, if the projection $\overline{f}:\partial B\to\mathbb{R}^2$ of $f_0$ on the tangent to the boundary $\partial B$ is nonsingular at $u_0$

Historical and current aspects of angiovisualization methods in CАD (a literature review)

The aim of the work. To summarize and expand knowledge about current methods used for visualization of coronary arteries, their evolution, capabilities, effectiveness, indications for use, safety for patients, guided by the principles of evidence-based medicine. Coronary artery disease (CАD) is an extremely common clinical cardiovascular disease, which is caused by atherosclerosis of the subepicardial coronary arteries (CAs) and can have both acute and chronic course. The incidence of CАD is increasing every year and getting younger. CАD has not only a high morbidity rate, but also a high mortality rate. In Ukraine, mortality from CАD is the main cause of population mortality. Without timely diagnosis and effective treatment, myocardial infarction or sudden cardiac death may develop. It is possible to ascertain the etiological cause of myocardial ischemia only after visualization of the СAs. A practicing physician is able to visualize the СAs using X-ray contrast coronary angiography or contrast-enhanced computed tomography of the chest. СA visualization methods have been used in clinical practice for more than half a century. The hardware and software are constantly upgraded, the diagnostic options of these methods are improved and expanded, and recommendations regarding their use in general clinical practice are updated. The main task for clinicians is to confirm or rule out the presence of a СA atherosclerotic lesion, as well as to determine its localization, extent, degree of stenosis and its significance for coronary blood flow, the presence of СA calcification, collateral pathways, plaque composition and its internal structure. Only after identifying the anatomical and physiological aspects of the atherosclerotic process in СA, it is possible to choose the right strategy for the treatment of patients by a multidisciplinary heart team including pharmacological therapy, a method of cardiac revascularization, and measures for primary or secondary prevention. Conclusions. CАD is a common disease worldwide. Today, two methods of the CА visualization are available – invasive coronary angiography and non-invasive coronary CT angiography. Further research is needed on the efficacy and safety of different СA imaging methods in CAD. Better results of the diagnostic search depend on both the capabilities of the clinic hardware component and on the optimal sequence for diagnostic processes rationally constructed by physicians

Analysis with respect to instrumental variables for the exploration of microarray data structures

BACKGROUND: Evaluating the importance of the different sources of variations is essential in microarray data experiments. Complex experimental designs generally include various factors structuring the data which should be taken into account. The objective of these experiments is the exploration of some given factors while controlling other factors. RESULTS: We present here a family of methods, the analyses with respect to instrumental variables, which can be easily applied to the particular case of microarray data. An illustrative example of analysis with instrumental variables is given in the case of microarray data investigating the effect of beverage intake on peripheral blood gene expression. This approach is compared to an ANOVA-based gene-by-gene statistical method. CONCLUSION: Instrumental variables analyses provide a simple way to control several sources of variation in a multivariate analysis of microarray data. Due to their flexibility, these methods can be associated with a large range of ordination techniques combined with one or several qualitative and/or quantitative descriptive variables

Limit cycles in uniform isochronous centers of discontinuous differential systems with four zones

We apply the averaging theory of first order for discontinuous differential systems to study the bifurcation of limit cycles from the periodic orbits of the uniform isochronous center of the differential systems ẋ = -y+x, y = x + xy, and ẋ = -y + xy, y = x + xy, when they are perturbed inside the class of all discontinuous quadratic and cubic polynomials differential systems with four zones separately by the axes of coordinates, respectively. Using averaging theory of first order the maximum number of limit cycles that we can obtain is twice the maximum number of limit cycles obtained in a previous work for discontinuous quadratic differential systems perturbing the same uniform isochronous quadratic center at origin perturbed with two zones separately by a straight line, and 5 more limit cycles than those achieved in a prior result for discontinuous cubic differential systems with the same uniform isochronous cubic center at the origin perturbed with two zones separately by a straight line. Comparing our results with those obtained perturbing the mentioned centers by the continuous quadratic and cubic differential systems we obtain 8 and 9 more limit cycles respectively

The regularized visible fold revisited

The planar visible fold is a simple singularity in piecewise smooth systems. In this paper, we consider singularly perturbed systems that limit to this piecewise smooth bifurcation as the singular perturbation parameter $\epsilon\rightarrow 0$. Alternatively, these singularly perturbed systems can be thought of as regularizations of their piecewise counterparts. The main contribution of the paper is to demonstrate the use of consecutive blowup transformations in this setting, allowing us to obtain detailed information about a transition map near the fold under very general assumptions. We apply this information to prove, for the first time, the existence of a locally unique saddle-node bifurcation in the case where a limit cycle, in the singular limit $\epsilon\rightarrow 0$, grazes the discontinuity set. We apply this result to a mass-spring system on a moving belt described by a Stribeck-type friction law