Zhanybai T. Zhusubaliyev

Border collision route to quasiperiodicity: Numerical investigation and experimental confirmation

Numerical studies of higher-dimensional piecewise-smooth systems have recently shown how a torus can arise from a periodic cycle through a special type of border-collision bifurcation. The present article investigates this new route to quasiperiodicity in the two-dimensional piecewise-linear normal form map. We have obtained the chart of the dynamical modes for this map and showed that border-collision bifurcations can lead to the birth of a stable closed invariant curve associated with quasiperiodic or periodic dynamics. In the parameter regions leading to the existence of an invariant closed curve, there may be transitions between an ergodic torus and a resonance torus, but the mechanism of creation for the resonance tongues is distinctly different from that observed in smooth maps. The transition from a stable focus point to a resonance torus may lead directly to a new focus of higher periodicity, e.g., a period-5 focus. This article also contains a discussion of torus destruction via a homoclinic bifurcation in the piecewise-linear normal map. Using a dc-dc converter with two-level control as an example, we report the first experimental verification of the direct transition to quasiperiodicity through a border-collision bifurcation.

Torus birth bifurcations in a DC/DC converter

Considering a pulsewidth modulated dc/dc converter as an example, this paper describes a border-collision bifurcation that can lead to the appearance of quasi-periodicity in piecewise-smooth dynamical systems. We demonstrate how a two-dimensional torus can arise from a periodic orbit through a bifurcation in which two complex-conjugate Poincare characteristic multipliers jump abruptly from the inside to the outside of the unit circle. The torus may be ergodic or resonant. However, in both cases the diameter of the torus develops approximately linearly with the distance to the bifurcation point as opposed to the characteristic parabolic form of the well-known Neimark-Sacker bifurcation. The paper also considers the birth of a torus via a subcritical Neimark-Sacker bifurcation in the piecewise-smooth system. Particular emphasis is given to the development of resonance zones via border-collision bifurcations

BORDER-COLLISION BIFURCATIONS AND CHAOTIC OSCILLATIONS IN A PIECEWISE-SMOOTH DYNAMICAL SYSTEM

Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling. We show that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations. The characteristic peculiarities of border-collision bifurcational transitions in piecewise-smooth systems are described and we provide a comparison with some recent results.

Quasiperiodicity and torus breakdown in a power electronic dc/dc converter

This paper discusses the mechanisms of torus formation and torus destruction in a dc/dc converter with relay control and hysteresis. We establish a chart of the dynamical modes in the input voltage versus load resistance parameter plane. This chart displays several different torus bifurcations along with their associated resonance tongues where periodic dynamics is observed. We show how a quintruple-turn torus is transformed into a single-turn torus in a homoclinic bifurcation and examine different mechanisms of torus destruction (via horseshoe formation and through period doubling). Particular emphasis is paid to following the changes of the stable and unstable manifolds in detail.

Bifurcation phenomena in an impulsive model of non-basal testosterone regulation

Complex nonlinear dynamics in a recent mathematical model of non-basal testosterone regulation are investigated. In agreement with biological evidence, the pulsatile (non-basal) secretion of testosterone is modeled by frequency and amplitude modulated feedback. It is shown that, in addition to already known periodic motions with one and two pulses in the least period of a closed-loop system solution, cycles of higher periodicity and chaos are present in the model in hand. The broad range of exhibited dynamic behaviors makes the model highly promising in model-based signal processing of hormone data.

Torus-Bifurcation Mechanisms in a DC/DC Converter With Pulsewidth-Modulated Control

Pulse-modulated converter systems play an important role in modern power electronics. However, by virtue of the complex interplay between ordinary (smooth) and so-called border-collision bifurcations generated by the switching dynamics, the changes in behavior that can occur in multilevel converter systems under varying operational conditions still remain to be explored in full. Considering the dynamics of a three-level dc/dc-converter, we demonstrate a number of new scenarios for the birth or destruction of resonant and ergodic tori. One scenario involves the formation of a doubled-layered torus structure around a stable focus point through three subsequent border-collision fold bifurcations. Another scenario replaces one of the fold bifurcations by a global bifurcation. In both of these scenarios, the basic mode of the converter remains stable while other modes grow up and bifurcate around it. We also illustrate the subcritical birth of both an ergodic and a resonance torus from the basic operational mode.

BIFURCATIONS AND CHAOTIC OSCILLATIONS IN AN AUTOMATIC CONTROL RELAY SYSTEM WITH HYSTERESIS

The dynamics of relay system with hysteresis is investigated. Systems of this type have a broad range of applications for power control and temperature regulation or in order to obtain highly stabilized electric or magnetic fields. Examples are power supplies for radio-electronics, computer equipment or spacecrafts, test stands for investigations of high- or low-temperature superconductivity, electron microscopes, and nuclear magnetic resonance tomographs. We first describe a general approach to determine stable and unstable periodic orbits for systems with hysteresis. Considering a concrete example of a four-dimensional relay system with hysteresis we hereafter determine the regions of periodic and chaotic oscillations in parameter space. The regularities in the occurrence of periodic motions are studied, and the associated bifurcations are described. The causes of nondeterminate (i.e. chaotic) dynamics are discussed and the influence of an external noise signal is analyzed.

Onset of chaos in a single-phase power electronic inverter

Supported by experiments on a power electronic DC/AC converter, this paper considers an unusual transition from the domain of stable periodic dynamics (corresponding to the desired mode of operation) to chaotic dynamics. The behavior of the converter is studied by means of a 1D stroboscopic map derived from a non-autonomous ordinary differential equation with discontinuous right-hand side. By construction, this stroboscopic map has a high number of border points. It is shown that the onset of chaos occurs stepwise, via irregular cascades of different border collisions, some of which lead to bifurcations while others do not.

Formation and destruction of multilayered tori in coupled map systems

The paper first illustrates how multilayered tori can arise through one or more pitchfork bifurcations of the saddle cycle on an ordinary resonance torus. The paper hereafter describes three different scenarios by which a multilayered torus can be destructed. One scenario involves a saddle-node bifurcation in which the middle layer of a three-layered torus disappears in an abrupt transition to chaos while the outer-layer manifolds and their associated saddle and unstable-focus cycles continue to exist and to control the transient dynamics. In a second scenario, the unstable focus cycles of the intermediate layers in a five-layered torus turn into unstable nodes, and closed loop connections are established between the unstable nodes and the points of the stable resonance node on the torus. Finally, a third scenario describes a transition in which homoclinic bifurcations destroy first the outer layers and thereafter also the inner layer. The paper also illustrates how the formation and destruction of multilayered tori can occur in the cluster dynamics of an ensemble of globally coupled maps. This leads to three additional scenarios for the destruction of multilayered tori.

Multistability and Torus Reconstruction in a DC–DC Converter With Multilevel Control

By virtue of their limited size and relatively low costs, multilevel dc-dc converters have come to play an important role in modern industrial power supply systems. When operating in a regime of high corrector gain, such converters can display a variety of new dynamic phenomena associated with the appearance of additional oscillatory modes. Starting in a state where four pairs of stable and unstable period-6 cycles coexist with the basic period-1 cycle, the paper describes a sequence of smooth and nonsmooth bifurcations through which the cycles and their basins of attraction transform as the output voltage is increased. The paper also describes the birth of a multilayered resonance torus through a transverse pitchfork bifurcation of the saddle cycle on an ordinary resonance torus.