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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

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.

3

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.

4

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.

5

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.

6

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.

7

Dynamics of a 2L-baroclinic model of the troposphere. Part 2: impact of the zonal drift on bifurcations in the model

Brojewski R., Jakubiak B., Jasiński J.

Journal of Technical Physics

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2005

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Vol. 46, no 4

203-218

EN

The paper presents results of numerical analyses related with assessment of the impact of mean zonal drift in the troposphere on the properties of the dynamics of a simple two-layer model of the baroclinic troposphere. It was proved that the model dynamics is mainly influenced by zonally differentiated surface temperature. It causes numerous bifurcations. They are often Hopf bifurcations of the second kind during which tropospheric states quite distant from the states before the bifurcations are generated. It significantly influences the model predictability. The paper contributes to the assessment of applicability of hydrodynamic models to weather forecasting for periods longer than two weeks.