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1

Controlling bifurcations in high-speed rotors utilizing active gas foil bearings

Papadopoulos Anastasios, Gavalas Ioannis, Chasalevris Athanasios

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2023

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Vol. 71, nr 6

art. no. e146796

EN

High-speed rotors on gas foil bearings (GFBs) are applications of increasing interest due to their potential to increase the power-toweight ratio in machines and also formulate oil-free design solutions. The gas lubrication principles render lower (compared to oil) power loss and increase the threshold speed of instability in rotating systems. However, self-excited oscillations may still occur at circumferential speeds similar to those in oil-lubricated journal bearings. These oscillations are usually triggered through Hopf bifurcation of a fixed-point equilibrium (balanced rotor) or secondary Hopf bifurcation of periodic limit cycles (unbalanced rotor). In this work, an active gas foil bearing (AGFB) is presented as a novel configuration including several piezoelectric actuators that shape the foil through feedback control. A finite element model for the thin foil mounted in some piezoelectric actuators (PZTs), is developed. Second, the gas-structure interaction is modelled through the Reynolds equation for compressible flow. A simple physical model of a rotating system consisting of a rigid rotor and two identical gas foil bearings is then defined, and the dynamic system is composed with its unique source of nonlinearity to be the impedance forces from the gas to the rotor and the foil. The third milestone includes a linear feedback control scheme to stabilize (pole placement) the dynamic system, linearized around a speed-dependent equilibrium (balanced rotor). Further to that, linear feedback control is applied in the dynamic system utilizing polynomial feedback functions in order to overcome the problem of instability.

2

Determination of Hopf bifurcation sets of a chemicalreactors by two-parameter continuation method

Berezowski Marek

Chemical and Process Engineering

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2020

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Vol. 41, nr 3

221-–227

EN

A two-parameter continuation method was developed and shown in the form of an example, allowingdetermination of Hopf bifurcation sets in a chemical reactor model. Exemplary calculations weremade for the continuous stirred tank reactor model (CSTR). The set of HB points limiting the rangeof oscillation in the reactor was determined. The results were confirmed on the bifurcation diagramof steady states and on time charts. The method is universal and can be used for various models ofchemical reactors.

3

Global Stability Analysis of Logistically Grown SIR Model with Loss of Immunity, Inhibitory Effect, Crowding Effect and its Protection Measure

Ghosh U., Sarkar S.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 2

125--141

EN

In this paper we have considered an SIR model with logistically grown susceptible in which the rate of incidence is directly affected by the inhibitory factors of both susceptible and infected populations and the protection measure for the infected class. Permanence of the solutions, global stability and bifurcation analysis in the neighborhood of equilibrium points has been investigated here. The Center manifold theory is used to find the direction of bifurcations. Finally numerical simulation is carried out to justify the theoretical findings.

4

Impact of roguing and insecticide spraying on mosaic disease in Jatropha curcas

Al-Basir F., Roy P. K., Ray S.

Control and Cybernetics

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2017

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Vol. 46, no. 4

325--344

EN

Jatropha curcas plant is greatly impaired by mosaic disease, caused by the viruses (Begomovirus), transmitted by whiteﬂies, which act as the vector. Roguing (i.e. removal of infected plant) and spraying of insecticides are common methods, employed in order to get rid of the disease. In this article, a mathematical model has been developed to study the mosaic disease dynamics while considering preventive measures of roguing and insecticide spraying. Suﬃcient conditions for the stability of equilibrium points of the system are among the results obtained through qualitative analysis. We obtain the basic reproduction number R0 and show that the disease free system is stable for R0 < 1 and unstable for R0 > 1. The region of stability of equilibrium points in diﬀerent parameter spaces have also been analysed. Hopf bifurcation at the endemic steady state has been studied subsequently, as well. Finally, by formulating an optimal control problem, optimal application of roguing and spraying techniques has been determined, keeping in mind the cost eﬀective control of the mosaic disease. Pontryagin minimum principle has been utilized to solve the optimal control problem. Numerical simulations illustrate the validity of the analytical outcomes.

5

Bifurcation and control for a discrete-time prey–predator model with Holling-IV functional response

Chen Q., Teng Z., Hu Z.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 2

247--261

EN

The dynamics of a discrete-time predator–prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

6

On Hopf bifurcation of Liu chaotic system

Yassen M. T., El-Dessoky M. M., Saleh E., Aly E. S.

Demonstratio Mathematica

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2013

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Vol. 46, nr 1

111--122

EN

In this paper, we analyze the dynamical behaviors of Liu system using the complementary-cluster energy-barrier criterion (CCEBC). Moreover, the Hopf bifurcation of this system is investigated using the first Lyapunov coefficient. Also, it is proved that this system has two Hopf bifurcation points, at which these Hopf bifurcations are nondegenerate and subcritical.

7

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Xu C., Liao M., He X.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no 1

97-107

EN

In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

8

O równaniach różniczkowych z opóźnieniem - teoria i zastosowania

Bodnar M., Piotrowska M.J.

Matematyka Stosowana : matematyka dla społeczeństwa

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2010

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T. 11/52

17-56

PL

Równania różniczkowe z opóźnionym argumentem pojawiają się w modelach matematycznych dotyczących zagadnień biologicznych, biochemicznych czy medycznych. Chociaż sama struktura równań jest podobna do równań różniczkowych zwyczajnych, to jednak istnieje zasadnicza różnica: równanie czy układ równań z opóźnieniem jest problemem nieskończeniewymiarowym z odpowiadającą mu przestrzenią fazową będącą przestrzenią funkcyjną-zwykle rozważamy przestrzeń funkcji ciągłych.Wtej pracy przestawiamy podstawową teorię dotyczącą tej klasy równań, jak również kilka przykładów zastosowań równań z opóźnieniem do opisu zagadnień biologicznych, medycznych i biochemicznych.

EN

Delay differential equations are used in mathematical models of biological, biochemical or medical phenomenons. Although the structure of these equations is similar to ordinary differential equations, the crucial difference is that a delay differential equation (or a system of equations) is an infinite dimensional problem and the corresponding phase space is a functional space - usually the space of continuous functions is considered. In this paper we present the basic theory of delay differential equations as well as some example of applications to models of biological, medical and biochemical systems.

9

Bifurcation Behavior and Attractors in Vehicle Dynamics

Hochlenert D., Von Wagner U., Hornig S.

Machine Dynamics Problems

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2009

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Vol. 33, No. 2

57-73

EN

Nonlinear self-excited systems in vehicle dynamics are discussed using the examples of squealing automotive disk brakes and the stability behavior of a railway wheelset. Both systems show self-excited vibrations for specific operation states. The self-excited vibrations are due to friction forces between pad and disk in the case of the automotive disk and due to contact forces in the case of the railway wheelset respectively. The analysis of the nonlinear equations of motion shows that the trivial solution looses stability either through a sub- or through a supercritical Hopf bifurcation depending on the system's parameters. In the case of a subcritical Hopf bifurcation two stable solutions coexist and the initial conditions determine which solution emerges. The properties of the nonlinear systems such as critical velocities, limit cycle amplitudes and attractors of coexisting solutions are calculated using center manifold reduction and normal form theory.

10

Nonlinear analysis of the stability of hydrodynamic bearings

Amamou A., Chouchane M., Naimi S.

Diagnostyka

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2009

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nr 4(52)

3-10

EN

The linear analysis of the stability of a hydrodynamic bearing is used to determine the stability boundaries and to predict if the steady state is stable or not. A nonlinear or weakly nonlinear model is used to determine the behaviour of the system near the critical stability boundaries. By applying the Hopf bifurcation theory, the existence of stable or unstable limit cycles in the neighbourhood of the stability boundaries can be predicted depending on the characteristics of the bearing.A numerical integration of the nonlinear equations of motion is then carried out in order to verify the results obtained analytically.

11

Coloured-noise-induced transitions in nonlinear structures

Mankin R., Laas T., Soika E., Sauga A., Rekker A., Ainsaar A., Ugaste U.

Nukleonika

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2008

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Vol. 53, No. 3

127-134

EN

In a stochastic framework, macroscopic approaches are sought to describe microscopic interaction between different species. Coloured-noise-induced transitions in stochastic N-species Lotka-Volterra systems are considered analytically as an appropriate model extendable to many natural and nano-technological processes. All the results discussed are computed by means of a dynamical mean-field approximation. It is demonstrated that interplay of coloured noise and interaction intensities of species can generate a variety of cooperation effects, such as discontinuous transitions of the mean population density, noise-induced Hopf bifurcations and relaxation oscillation. The necessary conditions for the cooperation effects are also discussed. Particularly, it is established that, in the case of the Beddington functional response, in certain parameter regions of the model an increase in noise correlation time can cause multiple transitions (more than two) between relaxation oscillatory regimes and equilibrium states.

12

Logistic Equations in Tumour Growth Modelling

Foryś U., Marciniak-Czochra A.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 3

317-325

EN

The aim of this paper is to present some approaches to tumour growth modelling using the logistic equation. As the first approach the well-known ordinary differential equation is used to model the EAT in mice. For the same kind of tumour, a logistic equation with time delay is also used. As the second approach, a logistic equation with diffusion is proposed. In this case a delay argument in the reaction term is also considered. Some mathematical properties of the presented models are studied in the paper. The results are illustrated using computer simulations.

13

Asymptotic behaviour and existence of a limit cycle of cubic autonomous systems

Barakova L.

Demonstratio Mathematica

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2001

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Vol. 34, nr 3

559-576

EN

In this paper a 2-dimensional real autonomous system with polynomial right-hand sides of a concrete type is studied. Hopf bifurcation is analysed and existence of a limit cycle is proved. A new formula to determine stability or unstability of this limit cycle is introduced. A positively invariant set, which is globally attractive, is found. Consequently, existence of a stable limit cycle around an unstable critical point is proved and also a sufficient condition for non-existence of a closed trajectory in the phase space is given. Global characteristics of the system are studied. An application in economics to the dynamic version of the neo-keynesian macroeconomic IS-LM model is presented.

14

Analytical parametric studies on BWR nonlinear dynamics

van der Hagen T. H. J. J., van Bragt D. D. B., Zboray R., Uddin R., de Kruijf W. J. M.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Mechanika

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1999

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Z. 67

61--70

EN

The main results are presented of parametric studies on the character and magnitude of nonlinear effects on BWR dynamics. Bifurcation analyses are performed on heated coolant channels and on nuclear reactors. Parametric studies are carried out with respect to reactivity feedback strength, cross-sectional void distribution and the axial power profile. Period doublings are encountered at higher-order bifurcations.