scholarly journals Hidden Degeneracies in Piecewise Smooth Dynamical Systems

2016 ◽  
Vol 26 (05) ◽  
pp. 1650087 ◽  
Author(s):  
Mike R. Jeffrey
Keyword(s):  
Convex Combination ◽  
Layer System ◽  
Switching Surface ◽  
Fast Switching ◽  
Switching Parameter ◽  
Equivalent Control ◽  
Stable Forms ◽  
Standard Regularization ◽  
Structurally Stable ◽  
Switching Layer

When a flow suffers a discontinuity in its vector field at some switching surface, the flow can cross through or slide along the surface. Sliding along the switching surface can be understood as the flow along an invariant manifold inside a switching layer. It turns out that the usual method for finding sliding modes — the Filippov convex combination or Utkin equivalent control — results in a degeneracy in the switching layer whenever the flow is tangent to the switching surface from both sides. We derive the general result and analyze the simplest case here, where the flow curves parabolically on either side of the switching surface (the so-called fold–fold or two-fold singularities). The result is a set of zeros of the fast switching flow inside the layer, which is structurally unstable to perturbation by terms nonlinear in the switching parameter, terms such as [Formula: see text] [where the superscript does mean “squared”]. We provide structurally stable forms, and show that in this form the layer system is equivalent to a generic singularity of a two timescale system. Finally we show that the same degeneracy arises when a discontinuity is smoothed using standard regularization methods.

Instabilities of Elastic and Spinning Systems: Concepts and Phenomena

2017 ◽  
Vol 27 (09) ◽  
pp. 1730029 ◽  
Author(s):  
J. Michael T. Thompson
Keyword(s):  
Magnetic Tweezers ◽  
Imperfection Sensitivity ◽  
Experimental Physics ◽  
Dna Molecule ◽  
Gyroscopic Stabilization ◽  
Stable Forms ◽  
Guyed Mast ◽  
Interactive Buckling ◽  
Secondary Bifurcations ◽  
Structurally Stable

Bifurcational instabilities of increasing complexity are described, with instructive and intriguing applications. Multiple folds in the gravitational collapse of a star triggers a discussion about possible sequencing of folds. Columns, rings and a quirky floating log illustrate safe and dangerous pitchforks, with self-weight models as lecture demonstrations. Localized buckling is employed by biologists using magnetic tweezers to twist a DNA molecule and evaluate a cutting and de-knotting enzyme. Unfolding of complex bifurcations into “structurally-stable” forms by a number (codimension) of perturbations illuminates the imperfection-sensitivity of shells, and the interactive buckling of a guyed mast at a hyperbolic-umbilic catastrophe. Spinning of primitive planets introduces secondary bifurcations, while Kirchhoff’s analogy is displayed for rods and spinning tops. Examples of gyroscopic stabilization and levitation mimic the “traps” that are used in modern experimental physics to hold in place individual electrons and neutrons.

scholarly journals Robust Tracking Control for Switched Fuzzy Systems with Fast Switching Controller

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Hong Yang ◽  
Le Zhang ◽  
Xiaodong Liu
Keyword(s):  
Hybrid Control ◽  
Convex Combination ◽  
Good Control ◽  
Smooth Transition ◽  
Robust Tracking ◽  
Robust Tracking Control ◽  
Switching Law ◽  
Fast Switching ◽  
Control Schemes ◽  
Switching Controller

This paper addresses the problem of designing robust tracking controls for a class of switched fuzzy (SF) systems with time delay. A switched fuzzy system, which differs from existing ones, is firstly employed to describe a nonlinear system. Next, a fast switching controller consisting of a number of simple subcontrollers is proposed. The smooth transition is governed by using the fast switching controller. Tracking hybrid control schemes which are based upon a combination of theH∞tracking theory, fast switching control algorithm, and switching law design are developed such that theH∞model referent tracking performance is guaranteed. Since convex combination techniques are used to derive the delay independent criteria, some subsystems are allowed to be unstable. Finally, various comparisons of the elaborated examples are conducted to demonstrate the effectiveness of the proposed control design approach. All results illustrate good control performances as desired.

scholarly journals Hidden Bifurcations and Attractors in Nonsmooth Dynamical Systems

2016 ◽  
Vol 26 (04) ◽  
pp. 1650068 ◽  
Author(s):  
Mike R. Jeffrey
Keyword(s):  
Vector Fields ◽  
Discontinuous Systems ◽  
Smooth Vector ◽  
Switching Surface ◽  
Blowing Up ◽  
Piecewise Smooth Vector Fields ◽  
Nonsmooth Dynamical Systems ◽  
Smooth System ◽  
Switching Layer

We investigate the role of hidden terms at switching surfaces in piecewise smooth vector fields. Hidden terms are zero everywhere except at the switching surfaces, but appear when blowing up the switching surface into a switching layer. When discontinuous systems do surprising things, we can often make sense of them by extending our intuition for smooth system to the switching layer. We illustrate the principle here with a few attractors that are hidden inside the switching layer, being evident in the flow, despite not being directly evident in the vector field outside the switching surface. These can occur either at a single switch (where we will introduce hidden terms somewhat artificially to demonstrate the principle), or at the intersection of multiple switches (where hidden terms arise inescapably). A more subtle role of hidden terms is in bifurcations, and we revisit some simple cases from previous literature here, showing that they exhibit degeneracies inside the switching layer, and that the degeneracies can be broken using hidden terms. We illustrate the principle in systems with one or two switches.

Organizational Culture in the Focus of Improving Risk Management in the Enterprise

2020 ◽  
Vol 17 (4) ◽  
pp. 86-95
Author(s):  
R. M. Kachalov ◽  
Yu. A. Sleptsova
Keyword(s):  
Risk Management ◽  
Organizational Culture ◽  
Point Of View ◽  
Economic Risk ◽  
Management Activity ◽  
Economic Agents ◽  
Cultural Aspects ◽  
Industrial Enterprises ◽  
Operational Theory ◽  
Stable Forms

The article considers the manifestations of the phenomenon of economic risk in the socioeconomic ecosystems of industrial enterprises, examines and differentiates the pragmatic and cultural aspects of the concept of "economic risk management". In terms of methodology, the study is based on the operational theory of risk management, and also uses tools to describe the organizational culture of risk management. Pragmatic and cultural differences in the characteristics of economic risk are identified at the level of stable forms of management activity with the involvement of the main provisions of the operational theory of risk management. The phenomenon of risk is considered in the ontological space as an artificial category of activity of industrial enterprises and other economic agents that form a socio-economic ecosystem. This phenomenon is studied as a specific form of social communication associated with the desire to assess the uncertain future in the present time, mainly from the point of view of analysis and management of the level of economic risk in the enterprise.

Discrete element method to model cracking for two layer systems

TAPPI Journal ◽  
2019 ◽  
Vol 18 (2) ◽  
pp. 101-108
Author(s):  
Daniel Varney ◽  
Douglas Bousfield
Keyword(s):  
Discrete Element Method ◽  
Flexural Modulus ◽  
Single Layer ◽  
Discrete Element ◽  
Flexural Stress ◽  
Layer System ◽  
Three Point Bending ◽  
Moving Force ◽  
Coating Layers ◽  
Element Method

Cracking at the fold is a serious issue for many grades of coated paper and coated board. Some recent work has suggested methods to minimize this problem by using two or more coating layers of different properties. A discrete element method (DEM) has been used to model deformation events for single layer coating systems such as in-plain and out-of-plain tension, three-point bending, and a novel moving force picking simulation, but nothing has been reported related to multiple coating layers. In this paper, a DEM model has been expanded to predict the three-point bending response of a two-layer system. The main factors evaluated include the use of different binder systems in each layer and the ratio of the bottom and top layer weights. As in the past, the properties of the binder and the binder concentration are input parameters. The model can predict crack formation that is a function of these two sets of factors. In addition, the model can predict the flexural modulus, the maximum flexural stress, and the strain-at-failure. The predictions are qualitatively compared with experimental results reported in the literature.

Fertilization System as a Factor of Transforming the Humus State of the Soil

2015 ◽  
Vol 2 (2) ◽  
pp. 39-44
Author(s):  
V. Lopushniak
Keyword(s):  
Crop Rotation ◽  
Podzolic Soil ◽  
Humus Content ◽  
Organic Fertilizers ◽  
Mineral Fertilizers ◽  
Forest Steppe ◽  
Content Increase ◽  
Energy Reserves ◽  
Gross Energy ◽  
Stable Forms

Aim. To establish the effect of different fertilization systems in short fi eld crop rotation on the change in the state of humus in the dark gray podzolic soils in Western Forest-Steppe of Ukraine. Methods. Field studies were carried out in a stationary experiment of the Department of Soil Science and Agricultural Chemistry of the L’viv National Agrarian University; determination of humus content – according to DSTU 4289:2004, and that of its labile forms – in accordance with DSTU 4732:2007, fraction-group composition – by Ponomare- va-Plotnikova’s method, according to the measurement procedure 31-497058-008-2002. Results. The use of organo-mineral fertilizer system in short fi eld crop rotation with the saturation of organic fertilizers of 15 t/ha of crop rotation contributes to the humus content increase by 0.7 % after the third rotation in 0–40 cm layer of dark gray podzolic soil, the increase in the share of stable forms of humic compounds up to 57–59 % and the increase in the ratio of R HA :R FA to 1.3–1.4. The study demonstrated very high dependence of amount of gross energy reserves in the humus on the R HA :R FA ratio. Conclusions. The combined application of organic fertili- zers in the form of manure, non-market of the crop (straw) and siderate, along with mineral fertilizers is re- commended in short fi eld crop rotations of Western Forest-Steppe of Ukraine toensure expanded reproduction of fertility of dark gray podzolic soil, improvement of its humus status, increase in gross energy reserves and the share of the stable forms and humic acids in the humus.

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