Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.Comment: submitted to EP

Cone-like Invariant Manifolds for Nonsmooth Systems

This thesis deals with rigorous mathematical techniques for higher-dimensional nonsmooth systems and their applications. The dynamical behaviour of these systems is a nonlocal problem due to the lack of smoothness. Motivated by various examples of nonsmooth systems in applications, we propose to explore the concept of invariant surfaces in the phase space which is separated by a discontinuity hypersurface. For such systems the corresponding Poincaré map can be determined; it turns out that under suitable conditions an invariant cone occurs which is characterized by a fixed point of the Poincaré map. The invariant cone seems to serve in a similar way as a generalisation of the classical center manifold for smooth differential systems. Hence, the stability of the whole system can be reduced to investigate the stability on the two-dimensional surface of the cone. Motivated to study the generation of invariant cones out of smooth systems, a numerical procedure to establish invariant cones and their stability is presented. It has been found that the flat degenerate cone in a smooth system develops under nonsmooth perturbations into a cone-like configuration. Also a simple example is used to explain a paradoxical situation concerning stability. Theoretical results concerning the existence of invariant cones and possible mechanisms responsible for the observed behavior for general three dimensional nonsmooth systems are discussed. These investigations reveal that the system possesses a rich dynamic behavior and new phenomena such as, for instance, the existence of multiple invariant cones for such system. Our approach is developed to include the case when sliding motion takes place on the manifold. Sliding dynamical equations are formulated by using Filippov's method. Existence of invariant cones containing a segment of sliding orbits are given as well as stability on these cones. Different sliding bifurcation scenarios are treated by theoretical analysis and simulation. As an application we have investigated the dynamics of an automotive brake system model under the excitation of dry friction force which has served as a motivating example to develop our concepts. This model belongs to the class of nonsmooth systems of Filippov type which is investigated from direct crossing and a sliding motion point of view. Existence of invariant cones and different types of bifurcation phenomena such as sliding periodic doubling and multiple periodic orbits are observed. Finally, extensions to nonlinear perturbations of nonsmooth linear systems have been obtained by using the nonsmooth linear system as basic system. If the basic system possesses an attractive invariant cone without sliding motion, we have shown that locally the Poincaré map contains the necessary information with regard to attractivity of the invariant cone. The existence of a generalized center manifold reduction of nonlinear system has been proven by using Hadamard graph transformation approach. A class of nonlinear systems having a cone-like invariant "manifold" is presented to illustrate the center manifold reduction and associated bifurcation. The scientific contributions of parts of this thesis are presented in [32,39,66]

Complexity, Emergent Systems and Complex Biological Systems:\ud Complex Systems Theory and Biodynamics. [Edited book by I.C. Baianu, with listed contributors (2011)]

An overview is presented of System dynamics, the study of the behaviour of complex systems, Dynamical system in mathematics Dynamic programming in computer science and control theory, Complex systems biology, Neurodynamics and Psychodynamics.\u

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Assessments of functional capacity in cardiac rehabilitation

Functional capacity is an important predictor of mortality in patients with cardiovascular disease. Exercise walk tests, such as the modified shuttle walking test (MSWT) and the six-minute walking test (6-MWT), are the recommended protocols in the United Kingdom for evaluating functional capacity and the effects of therapeutic interventions on patients enrolled in cardiac rehabilitation (CR) programmes. Due to a lack of research into factors associated with walking test performance in cardiac patients, the main aim of this study is to identify whether routinely taken measures (clinical and non-clinical) and biomechanical parameters can predict test performance. A further aim is to establish reference equations or normative values to predict performance in this population. Finally, this research aims to improve the safety of exercise testing and training in community-based CR settings. After determining the long-term reliability of the MSWT, this study investigated the claim that only sex, age and anthropometric parameters (stature and weight), and not biomechanical or simple clinical parameters, can predict functional capacity assessed by the MSWT, in patients with less severe cardiac diseases or by the 6-MWT, in heart failure patients. Overweight heart failure patients showed a 13-fold increase, and older patients (>75 years) had a 5-fold increase, in the risk of poor prognosis via 6-MWT assessment. Furthermore, it is shown here that the effect of stature on the magnitude of change in MSWT results is powerful, and can be used to estimate functional capacity improvement in the cardiac population. Finally, poor functional capacity was shown to have no association with risk of cardiovascular events during exercise, and that exercise testing and training can be carried out safely in supervised community-based CR settings. These findings have implications in clinical practice and CR programme improvement, as they can help clinicians to make better-informed decisions about cardiac patients entering CR