Localization and dimension estimation of attractors in the Glukhovsky-Dolzhansky system

Hidden attractors in Chua circuit: mathematical theory meets physical experiments

AbstractAfter the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real exi…

Numerical analysis of dynamical systems : unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

In this article, on the example of the known low-order dynamical models, namely Lorenz, Rössler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rössler system. Using the example of the Vallis system describing the El…

Harmonic Balance Method and Stability of Discontinuous Systems

The development of the theory of discontinuous dynamical systems and differential inclusions was not only due to research in the field of abstract mathematics but also a result of studies of particular problems in mechanics. One of first methods, used for the analysis of dynamics in discontinuous mechanical systems, was the harmonic balance method developed in the thirties of the 20th century. In our work the results of analysis obtained by the method of harmonic balance, which is an approximate method, are compared with the results obtained by rigorous mathematical methods and numerical simulation. peerReviewed

Charge pump phase-locked loop with phase-frequency detector: closed form mathematical model

Charge pump phase-locked loop with phase-frequency detector (CP-PLL) is an electrical circuit, widely used in digital systems for frequency synthesis and synchronization of the clock signals. In this paper a non-linear second-order model of CP-PLL is rigorously derived. The obtained model obviates the shortcomings of previously known second-order models of CP-PLL. Pull-in time is estimated for the obtained second-order CP-PLL.