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1
FERRARI, FRANCO. "FIELD THEORIES ON THE POINCARÉ DISK." International Journal of Modern Physics A 11, no. 30 (1996): 5389–404. http://dx.doi.org/10.1142/s0217751x96002467.
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The massive scalar field theory and the chiral Schwinger model are quantized on a Poincaré disk of radius ρ. The amplitudes are derived in terms of Legendre functions. The behavior at long distances and near the boundary of some of the relevant correlation functions is studied. The exact computation of the chiral determinant appearing in the Schwinger model is obtained exploiting perturbation theory. This calculation poses interesting mathematical problems, as the Poincaré disk is a noncompact manifold with a metric tensor which diverges when it approaches the boundary. The results presented in this paper are very useful in view of possible extensions to general Riemann surfaces. Moreover, they could also shed some light in the quantization of field theories on manifolds with constant curvature scalars in higher dimensions.
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Nielsen, Frank. "The Siegel–Klein Disk: Hilbert Geometry of the Siegel Disk Domain." Entropy 22, no. 9 (2020): 1019. http://dx.doi.org/10.3390/e22091019.
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We study the Hilbert geometry induced by the Siegel disk domain, an open-bounded convex set of complex square matrices of operator norm strictly less than one. This Hilbert geometry yields a generalization of the Klein disk model of hyperbolic geometry, henceforth called the Siegel–Klein disk model to differentiate it from the classical Siegel upper plane and disk domains. In the Siegel–Klein disk, geodesics are by construction always unique and Euclidean straight, allowing one to design efficient geometric algorithms and data structures from computational geometry. For example, we show how to approximate the smallest enclosing ball of a set of complex square matrices in the Siegel disk domains: We compare two generalizations of the iterative core-set algorithm of Badoiu and Clarkson (BC) in the Siegel–Poincaré disk and in the Siegel–Klein disk: We demonstrate that geometric computing in the Siegel–Klein disk allows one (i) to bypass the time-costly recentering operations to the disk origin required at each iteration of the BC algorithm in the Siegel–Poincaré disk model, and (ii) to approximate fast and numerically the Siegel–Klein distance with guaranteed lower and upper bounds derived from nested Hilbert geometries.
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Anguelova, Lilia, John Dumancic, Richard Gass, and L. C. R. Wijewardhana. "Dark energy from inspiraling in field space." Journal of Cosmology and Astroparticle Physics 2022, no. 03 (2022): 018. http://dx.doi.org/10.1088/1475-7516/2022/03/018.
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Abstract We find an exact solution of the equations of motion of a two-field cosmological model, which realizes multi-field dark energy. The latter is characterized by field-space trajectories with turning rates that are always large. We study a class of two-field models and show that it is possible to have such trajectories, giving accelerated space-time expansion, even when the scalar potential preserves the rotational invariance of the field-space metric. For the case of Poincaré-disk field space, we derive the form of the scalar potential compatible with such background solutions and, furthermore, we find the exact solutions analytically. Their field-space trajectories are spirals inward, toward the center of the Poincaré disk. Interestingly, the functional form of the relevant scalar potential is compatible with a certain hidden symmetry, although the latter is broken by the presence of a constant term.
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Wei, Yuan, Zhaobo Chen, and Earl H. Dowell. "Nonlinear Characteristics Analysis of a Rotor-Bearing-Brush Seal System." International Journal of Structural Stability and Dynamics 18, no. 05 (2018): 1850063. http://dx.doi.org/10.1142/s0219455418500633.
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The vibration response and nonlinear dynamic behavior of a rotor-bearing-brush seal system were investigated with a new seal force model of the brush seal. The nonlinear oil–film force model was adopted based on a short bearing assumption. The dimensionless equation of motion was solved using the fourth order Runge–Kutta method. The effects of key parameters including rotor speed, installation spacing of the brush seal, disk eccentricity, disk mass, and journal mass on the nonlinear dynamic characteristics of rotor-bearing-brush seal system were determined and compared under different operating conditions with a bifurcation diagram, time history, axis orbit, poincaré map, frequency spectrum, and spectrum cascade. The results showed that the system response contained various nonlinear phenomena, such as periodic motion, multi-periodic motion, and quasi-periodic motion. The interaction of the rotor speed, installation spacing of the brush seal, disk eccentricity, disk mass, and journal mass could seriously affect the stability and working condition of the system. This study provides a theoretical support for the selection of key design parameters and further understanding of the nonlinear characteristics of rotor-bearing-seal systems with a brush seal.
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Han, W. B. "Dynamics of particles in slowly rotating black holes with dipolar halos." Proceedings of the International Astronomical Union 3, S248 (2007): 498–99. http://dx.doi.org/10.1017/s1743921308019935.
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AbstractIn general, the model of galaxy assumes a central huge black hole surrounded by a massive halo, disk or ring. In this paper, we investigate the gravitational field structure of a slowly rotating black hole with a dipolar halo, and the dynamics and chaos of test particles moving in it. Using Poincaré sections and fast Lyapunov indicator (FLI) in general relativity, we investigate chaos under different dynamical parameters, and find that the FLI is suitable for detecting chaos and even resonant orbits.
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ZHANG, KUNPENG, and QIAN DING. "LATERAL AND TORSIONAL VIBRATIONS OF A TWO-DISK ROTOR-STATOR SYSTEM WITH AXIAL CONTACT/RUBS." International Journal of Applied Mechanics 01, no. 02 (2009): 305–26. http://dx.doi.org/10.1142/s1758825109000137.
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The dynamics of a rotor system with axial contact/rub events between the disks and stator are investigated by numerical simulations. The formula for determining the contact/rub points, axial contact forces and dry friction forces are deduced. To account for their influence, the axial contact forces are substituted by equivalent forces acting at the disk centers, based on the equivalent moment rule. One-parametric model is used to estimate the contact-induced dry friction forces. The coupled equations of lateral and torsional motions of rotor and the lateral motion of disk are then established. Numerical simulations are carried out to reveal the lateral and torsional vibrations for both two-disk contact/rubs with different axial clearances, and one disk contact/rubs. Bifurcation diagrams, orbits, phase portraits, amplitude-frequency spectra and Poincaré maps are adopted to demonstrate the dynamical behaviors of the system. The results show that though both the lateral and torsional vibrations can reflect the influences of contact/rubs on rotor dynamics, the spectrum analyses of the torsional vibrations are more suitable to determine straight the extent of their effect.
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Lócsi, Levente. "A hyperbolic variant of the Nelder–Mead simplex method in low dimensions." Acta Universitatis Sapientiae, Mathematica 5, no. 2 (2013): 169–83. http://dx.doi.org/10.2478/ausm-2014-0012.
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Abstract The Nelder-Mead simplex method is a widespread applied numerical optimization method with a vast number of practical applications, but very few mathematically proven convergence properties. The original formulation of the algorithm is stated in Rn using terms of Euclidean geometry. In this paper we introduce the idea of a hyperbolic variant of this algorithm using the Poincaré disk model of the Bolyai- Lobachevsky geometry. We present a few basic properties of this method and we also give a Matlab implementation in 2 and 3 dimensions
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NAZARENKO, A. V. "DIRECTED RANDOM WALK ON THE LATTICES OF GENUS TWO." International Journal of Modern Physics B 25, no. 26 (2011): 3415–33. http://dx.doi.org/10.1142/s0217979211101831.
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The object of the present investigation is an ensemble of self-avoiding and directed graphs belonging to eight-branching Cayley tree (Bethe lattice) generated by the Fuchsian group of a Riemann surface of genus two and embedded in the Poincaré unit disk. We consider two-parametric lattices and calculate the multifractal scaling exponents for the moments of the graph lengths distribution as functions of these parameters. We show the results of numerical and statistical computations, where the latter are based on a random walk model.
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Petoukhov, S. V. "GENETIC CODING SYSTEM ANDALGEBRAIC HOLOGRAPHY." Metaphysics, no. 2 (August 25, 2022): 113–27. http://dx.doi.org/10.22363/2224-7580-2022-2-113-127.
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The article is devoted to the structural features of the molecular genetic coding system. These features in their matrix representation turn out to be conjugate with the matrix structures of algebraic holography, which have long been used in digital informatics. The relationship between ensembles of genetic structures and bit-reversing holography, split-quaternions, and the Poincaré disk model of hyperbolic motions is described. This connection leads to well-known works on quantum holographic noise-immune codes and makes it possible to comprehend the facts of the realization of hyperbolic geometry in genetically inherited macrophysiological phenomena.
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BELYKH, VLADIMIR, IGOR BELYKH, and ERIK MOSEKILDE. "HYPERBOLIC PLYKIN ATTRACTOR CAN EXIST IN NEURON MODELS." International Journal of Bifurcation and Chaos 15, no. 11 (2005): 3567–78. http://dx.doi.org/10.1142/s0218127405014222.
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Strange hyperbolic attractors are hard to find in real physical systems. This paper provides the first example of a realistic system, a canonical three-dimensional (3D) model of bursting neurons, that is likely to have a strange hyperbolic attractor. Using a geometrical approach to the study of the neuron model, we derive a flow-defined Poincaré map giving an accurate account of the system's dynamics. In a parameter region where the neuron system undergoes bifurcations causing transitions between tonic spiking and bursting, this two-dimensional map becomes a map of a disk with several periodic holes. A particular case is the map of a disk with three holes, matching the Plykin example of a planar hyperbolic attractor. The corresponding attractor of the 3D neuron model appears to be hyperbolic (this property is not verified in the present paper) and arises as a result of a two-loop (secondary) homoclinic bifurcation of a saddle. This type of bifurcation, and the complex behavior it can produce, have not been previously examined.
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Liu, Yi, Heng Liu, and BoWen Fan. "Nonlinear dynamic properties of disk-bolt rotor with interfacial cutting faults on assembly surfaces." Journal of Vibration and Control 24, no. 19 (2017): 4369–82. http://dx.doi.org/10.1177/1077546317724602.
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Interfacial cutting faults on assembly surfaces are considered in a three-dimensional (3D) disk-bolt rotor system. The traditional finite element method is used to establish the 3D model of faulted disk-bolt rotor. A contact algorithm is applied to calculate the static features of this combined rotor. It is revealed that interfacial cutting faults produce rotor bending which is gradually strengthened as rotational speed increases besides disk’s mass eccentricity. The 3D dynamic equations of a faulted disk-bolt rotor system include these cutting faults’ static influences. The nonlinear dynamic properties are investigated by Poincaré mapping, Newton iteration and a prediction-correction algorithm. As a result, the rotor bending due to cutting faults reduces the global stability of the complicated rotor and enlarges the vibration amplitude obviously. This speed-variant bending also decides the feature that rotor vibration increases again after critical speed no matter whether dynamic balance is carried out. The maximum allowable fault depth is obtained and it gives an explanation as to why the machining precision of assembly surfaces should be strictly controlled in the disk-bolt rotor. Generally, this paper originally tries to provide a feasible approach to consider a 3D interfacial cutting fault with specific shape and to analyze the static–dynamic coupling characteristics for a disk-bolt rotor.
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Ferrarello, Daniela, Maria Flavia Mammana, and Eugenia Taranto. "Non-Euclidean Geometry with Art by Means of GeoGebra." International Journal for Technology in Mathematics Education 26, no. 3 (2019): 113–19. http://dx.doi.org/10.1564/tme_v26.3.02.
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In this paper, a teaching path on non-Euclidean geometry focused on the Poincaré-disk model, and its connection with art is shown. We consider as artistic contents the Escher’s hyperbolic art and some paradoxes on Magritte’s work. The mathematical aspects are illustrated thanks to a Dynamic Geometry System (GeoGebra) implemented with a laboratory methodology in a Vygotskijan perspective. The use of the artefact (the DGS) is crucial to build mathematical concepts starting by artistic objects. The path was experimented with inmates in a high security prison and with University students. Some comments on the first experimentation are given.
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Diz-Pita, Érika. "Global dynamics of a predator-prey system with immigration in both species." Electronic Research Archive 32, no. 2 (2024): 762–78. http://dx.doi.org/10.3934/era.2024036.
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<abstract><p>In nature, the vast majority of species live in ecosystems that are not isolated, and the same is true for predator-prey ecological systems. With this work, we extend a predator-prey model by considering the inclusion of an immigration term in both species. From a biological point of view, that allows us to achieve a more realistic model. We consider a system with a Holling type Ⅰ functional response and study its global dynamics, which allows to not only determine the behavior in a region of the plane $ \mathbb{R}^2 $, but also to control the orbits that either go or come to infinity. First, we study the local dynamics of the system, by analyzing the singular points and their stability, as well as the possible behavior of the limit cycles when they exist. By using the Poincaré compactification, we determine the global dynamics by studying the global phase portraits in the positive quadrant of the Poincaré disk, which is the region where the system is of interest from a biological point of view.</p></abstract>
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Hei, Di, Yong Fang Zhang, Mei Ru Zheng, Liang Jia, and Yan Jun Lu. "Stability and Bifurcation of Nonlinear Bearing-Flexible Rotor System with a Single Disk." Advanced Materials Research 148-149 (October 2010): 141–46. http://dx.doi.org/10.4028/www.scientific.net/amr.148-149.141.
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Dynamic model and equation of a nonlinear flexible rotor-bearing system are established based on rotor dynamics. A local iteration method consisting of improved Wilson-θ method, predictor-corrector mechanism and Newton-Raphson method is proposed to calculate nonlinear dynamic responses. By the proposed method, the iterations are only executed on nonlinear degrees of freedom. The proposed method has higher efficiency than Runge-Kutta method, so the proposed method improves calculation efficiency and saves computing cost greatly. Taking the system parameter ‘s’ of flexible rotor as the control parameter, nonlinear dynamic responses of rotor system are obtained by the proposed method. The stability and bifurcation type of periodic responses are determined by Floquet theory and a Poincaré map. The numerical results reveal periodic, quasi-periodic, period-5, jump solutions of rich and complex nonlinear behaviors of the system.
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Hollebrands, Karen F., AnnaMarie Conner, and Ryan C. Smith. "The Nature of Arguments Provided by College Geometry Students With Access to Technology While Solving Problems." Journal for Research in Mathematics Education 41, no. 4 (2010): 324–50. http://dx.doi.org/10.5951/jresematheduc.41.4.0324.
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Prior research on students' uses of technology in the context of Euclidean geometry has suggested it can be used to support students' development of formal justifications and proofs. This study examined the ways in which students used a dynamic geometry tool, NonEuclid, as they constructed arguments about geometric objects and relationships in hyperbolic geometry. Eight students enrolled in a college geometry course participated in a task-based interview that was focused on examining properties of quadrilaterals in the Poincaré disk model. Toulmin's argumentation model was used to analyze the nature of the arguments students provided when they had access to technology while solving the problems. Three themes related to the structure of students' arguments were identified. These involved the explicitness of warrants provided, uses of technology, and types of tasks.
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Schneider, Klaus R. "THE POINT CHARGE OSCILLATOR: QUALITATIVE AND ANALYTICAL INVESTIGATIONS." Mathematical Modelling and Analysis 24, no. 3 (2019): 372–84. http://dx.doi.org/10.3846/mma.2019.023.
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We study the mathematical model of the point charge oscillator which has been derived by A. Beléndez et al. [2]. First we determine the global phase portrait of this model in the Poincaré disk. It consists of a family of closed orbits surrounding the unique finite equilibrium point and of a continuum of homoclinic orbits to the unique equilibrium point at infinity. Next we derive analytic expressions for the relationship between period (frequency) and amplitude. Further, we prove that the period increases monotone with the amplitude and derive an expression for its growth rate as the amplitude tends to infinity. Finally, we determine a relation between period and amplitude by means of the complete elliptic integral of the first kind K(k) and of the Jacobi elliptic function cn.
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Zhu, Youfeng, Zibo Wang, Qiang Wang, Xinhua Liu, Hongyu Zang, and Liang Wang. "Nonlinear Dynamic Analysis of Rotor Rub-Impact System." Shock and Vibration 2019 (November 29, 2019): 1–20. http://dx.doi.org/10.1155/2019/4867364.
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A dynamic model of a double-disk rub-impact rotor-bearing system with rubbing fault is established. The dynamic differential equation of the system is solved by combining the numerical integration method with MATLAB. And the influence of rotor speed, disc eccentricity, and stator stiffness on the response of the rotor-bearing system is analyzed. In the rotor system, the time history diagram, the axis locus diagram, the phase diagram, and the Poincaré section diagram in different rotational speeds are drawn. The characteristics of the periodic motion, quasiperiodic motion, and chaotic motion of the system in a given speed range are described in detail. The ways of the system entering and leaving chaos are revealed. The transformation and evolution process of the periodic motion, quasiperiodic motion, and chaotic motion are also analyzed. It shows that the rotor system enters chaos by the way of the period-doubling bifurcation. With the increase of the eccentricity, the quasi-periodicity evolution is chaotic. The quasiperiodic motion evolves into the periodic three motion phenomenon. And the increase of the stator stiffness will reduce the chaotic motion period.
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Telesphore, Tiendrebeogo, Yacouba Rachid Coulibaly Cheick, and Badolo Maliki. "Robust Formal Watermarking Model Based on the Hyperbolic Geometry for Image Security." International Journal of Recent Technology and Engineering (IJRTE) 9, no. 3 (2020): 617–29. https://doi.org/10.35940/ijrte.C4651.099320.
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The digital revolution has led to an increase in the production and exchange of valuable digitized documents across institutions, companies and the general public alike. Ensuring the authenticity, integrity and ownership of these official or high-value documents is essential if they are to be considered useful. Digital watermarking is a possible solution to this challenge as it has already been used for copyright protection, source tracking, and video authentication to name just a few applications of its use. It also enables integrity protection, which is of value for numerous documents types (e.g., official documents, medical images). In this paper, we propose a new watermarking solution that is applicable to image watermarking and is based on hyperbolic geometry. Our new solution builds upon existing work in geometrical watermarking.
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Ebrahimi, Reza, Mostafa Ghayour, and Heshmatallah Mohammad Khanlo. "Effects of some design parameters on bifurcation behavior of a magnetically supported coaxial rotor in auxiliary bearings." Engineering Computations 34, no. 7 (2017): 2379–95. http://dx.doi.org/10.1108/ec-04-2017-0141.
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Purpose This paper aims to present bifurcation analysis of a magnetically supported coaxial rotor model in auxiliary bearings, which includes gyroscopic moments of disks and geometric coupling of the magnetic actuators. Design/methodology/approach Ten nonlinear equations of motion were solved using the Runge–Kutta method. The vibration responses were analyzed using dynamic trajectories, power spectra, Poincaré maps, bifurcation diagrams and the maximum Lyapunov exponent. The analysis was carried out for different system parameters, namely, the inner shaft stiffness, inter-rotor bearing stiffness, auxiliary bearing stiffness and disk position. Findings It was shown that dynamics of the system could be significantly affected by varying these parameters, so that the system responses displayed a rich variety of nonlinear dynamical phenomena, including quasi-periodicity, chaos and jump. Next, some threshold values were provided with regard to the design of appropriate parameters for this system. Therefore, the proposed work can provide an effective means of gaining insights into the nonlinear dynamics of coaxial rotor–active magnetic bearing systems with auxiliary bearings in the future. Originality/value This paper considered the influences of the inner shaft stiffness, inter-rotor bearing stiffness, auxiliary bearing stiffness and disk position on the bifurcation behavior of a magnetically supported coaxial rotor system in auxiliary bearings.
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Yan, Hongbo, Enzuo Liu, Pengbo Zhao, Pei Liu, and Rui Cao. "Study on Chaotic Peculiarities of Magnetic-Mechanical Coupled System of Giant Magnetostrictive Actuator." Mathematical Problems in Engineering 2020 (November 25, 2020): 1–15. http://dx.doi.org/10.1155/2020/6864795.
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We studied the chaotic peculiarities of magnetic-mechanical coupled system of GMA. Based on the working principle of GMA and according to Newton’s second law of motion, first piezomagnetic equation, disk spring design theory, and structural dynamics principle of GMA, the present study established a GMA magnetic-mechanical coupled system model. By carrying out data modeling of this coupled system model, the bifurcation chart of the system with the variation of damping factor, excitation force, and exciting frequency parameters as well as the homologous offset oscillogram, phase plane trace chart, and Poincaré diagram was obtained, and the chaotic peculiarities of the system were analyzed. The influence of parametric errors on the coupled system was studied. The analytical results showed that the oscillation equation of the GMA magnetic-mechanical coupled system had nonlinearity and the movement morphology was complicated and diversified. By adjusting the damping factor, exciting frequency, and excitation force parameters of the system, the system could work under the stable interval, which provided theoretical support for the stability design of GMA.
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Yousefi, Masoud, R. D. Firouz-Abadi, and Hassan Haddadpou. "Nonlinear support effects on the aeroelastic stability of multi-stage turbine rotors." Engineering Solid Mechanics 13, no. 3 (2025): 2434–258. https://doi.org/10.5267/j.esm.2025.5.002.
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This study presents a comprehensive analysis of aeroelastic stability in multi-stage turbine rotors mounted on nonlinear supports. A high-fidelity dynamic model is developed by coupling the structural behavior of a rotating shaft–disk–blade assembly with quasi-steady aerodynamic forces. The system incorporates nonlinear stiffness and damping in the bearing supports, and the governing equations of motion are derived using the Lagrangian method. Aerodynamic forces are modeled using cascade theory for incompressible subsonic flow and integrated with structural dynamics through coordinate transformation. The resulting nonlinear system is solved using the Runge-Kutta method, and its stability characteristics are investigated via bifurcation diagrams and Poincaré maps. A detailed parametric study is conducted to examine the influence of aerodynamic parameters, structural parameters and support characteristics on rotor response. Results show that nonlinear supports significantly alter stability boundaries, reduce critical flutter speeds, and introduce multi-periodic dynamic behavior. These findings provide valuable insights into the design and tuning of support systems to enhance the dynamic robustness of turbomachinery.
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Li, Zi Gang, and Ming Li. "Non-Linear Dynamics of a Flexible Multi-Rotor Bearing System with a Fault of Parallel Misalignment." Applied Mechanics and Materials 138-139 (November 2011): 104–10. http://dx.doi.org/10.4028/www.scientific.net/amm.138-139.104.
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The dynamic behaviors of a flexible multi-rotor system with a fault of parallel misalignment are investigated on the basis of assumptions, such as the long journal bearings, small rotor misalignment and mass disk unbalance. Firstly, based on the Lagrange equations with undetermined multiplier, the dynamic model of a rotor system under the action of the nonlinear oil film forces is developed after taking into account the holonomic constraint, which describes the misalignment relation between two rotors, and the theoretical analysis reveals that the system with eleven DOF is of strong nonlinear properties. Then the nonlinear dynamic characteristics on numerical technique, such as steady state response, rotor orbit, Poincaré section and the largest Lyapunov exponent, are paid more attention in this study. The results show that at low speed the components of the steady-state responses in lateral direction is of the synchronous frequency with rotating speed as well as its integer multiples frequencies. As the speed increases the dynamic characteristics become complicated, and the nT-period, quasi-period and chaotic oscillations occur.
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Prishlyak, Oleksandr. "Regular Octagons in Hyperbolic Geometry." In the world of mathematics, no. 1 (2) (2024): 88–103. https://doi.org/10.17721/1029-4171.2024/2.10.
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When constructing hyperbolic structures on closed surfaces, one can use hyperbolic geometry (Lobachevsky geometry) on the plane. To do this, the surface must be represented as a 2n-gon on the hyperbolic plane, and a discrete group action, which is a subgroup of the movements of the hyperbolic plane, must be defined, for which the 2n-gon serves as a fundamental domain. If such a surface is a double torus (an oriented surface of genus 2), it can be obtained by gluing opposite sides of an octagon. In fact, the Lobachevsky plane is divided into octagons. The presence of symmetries simplifies calculations. Therefore, a natural problem arises regarding the partitioning into regular octagons. Additionally, it is important to provide examples of such octagons by specifying the coordinates of their vertices in one of the models of hyperbolic geometry. The models of the upper half-plane and the Poincaré model on the unit disk are used, for which the Riemannian metric is defined (the formula for finding the lengths of arcs of curves). We describe the main properties of hyperbolic lines and the group of movements (the group of isometric transformations) of hyperbolic geometry on the plane using fractional-linear transformations of the complex plane with real coefficients.
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Illés, Eduárd, Dániel Jánosi, and Tamás Kovács. "Orbital dynamics in galactic potentials under mass transfer." Astronomy & Astrophysics 692 (December 2024): A240. https://doi.org/10.1051/0004-6361/202348274.
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Context. Time-dependent potentials are common in galactic systems that undergo significant evolution, interactions, or encounters with other galaxies, or when there are dynamic processes such as star formation and merging events. Recent studies show that an ensemble approach along with the so-called snapshot framework in the theory of dynamical systems provide a powerful tool to analyze the time-dependent dynamics. Aims. In this work, we aim to explore and quantify the phase space structure and dynamical complexity in time-dependent galactic potentials consisting of multiple components. Methods. We applied the classical method of Poincaré surface of sections to analyze the phase space structure in a chaotic Hamiltonian system subjected to parameter drift. This, however, makes sense only when the evolution of a large ensemble of initial conditions is followed. Numerical simulations explore the phase space structure of such ensembles while the system undergoes a continuous parameter change. The pair-wise average distance of ensemble members allowed us to define a generalized Lyapunov exponent, which might also be time-dependent, to describe the system stability. Results. We provide a comprehensive dynamical analysis of the system under circumstances where linear mass transfer occurs between the disk and bulge components of the model.
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Sarkar, Arunoday, Chitrak Sarkar та Buddhadeb Ghosh. "A novel way of constraining the α-attractor chaotic inflation through Planck data". Journal of Cosmology and Astroparticle Physics 2021, № 11 (2021): 029. http://dx.doi.org/10.1088/1475-7516/2021/11/029.
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Abstract Defining a scale of k-modes of the quantum fluctuations during inflation through the dynamical horizon crossing condition k = aH we go from the physical t variable to k variable and solve the equations of cosmological first-order perturbations self consistently, with the chaotic α-attractor type potentials. This enables us to study the behaviour of ns , r, nt and N in the k-space. Comparison of our results in the low-k regime with the Planck data puts constraints on the values of the α parameter through microscopic calculations. Recent studies had already put model-dependent constraints on the values of α through the hyperbolic geometry of a Poincaré disk: consistent with both the maximal supergravity model 𝒩 = 8 and the minimal supergravity model 𝒩 = 1, the constraints on the values of α are 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3. The minimal 𝒩 = 1 supersymmetric cosmological models with B-mode targets, derived from these supergravity models, predicted the values of r between 10-2 and 10-3. Both in the E-model and the T-model potentials, we have obtained, in our calculations, the values of r in this range for all the constrained values of α stated above, within 68% CL. Moreover, we have calculated r for some other possible values of α both in low-α limit, using the formula r = 12α/N 2, and in the high-α limit, using the formula r = 4n/N, for n = 2 and 4. With all such values of α, our calculated results match with the Planck-2018 data with 68% or near 95% CL.
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Barbaresco, Frédéric. "Lie Group Statistics and Lie Group Machine Learning Based on Souriau Lie Groups Thermodynamics & Koszul-Souriau-Fisher Metric: New Entropy Definition as Generalized Casimir Invariant Function in Coadjoint Representation." Entropy 22, no. 6 (2020): 642. http://dx.doi.org/10.3390/e22060642.
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In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in Statistical Mechanics in the framework of Geometric Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum entropy) that is covariant under the action of dynamic groups of physics (e.g., Galileo’s group in classical physics). Souriau Lie Group Thermodynamics was also addressed 30 years after Souriau by R.F. Streater in the framework of Quantum Physics by Information Geometry for some Lie algebras, but only in the case of null cohomology. Souriau method could then be applied on Lie groups to define a covariant maximum entropy density by Kirillov representation theory. We will illustrate this method for homogeneous Siegel domains and more especially for Poincaré unit disk by considering SU(1,1) group coadjoint orbit and by using its Souriau’s moment map. For this case, the coadjoint action on moment map is equivariant. For non-null cohomology, we give the case of Lie group SE(2). Finally, we will propose a new geometric definition of Entropy that could be built as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping.
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Pratama, Febriyana Putra, and Julan Hernadi. "KONSISTENSI AKSIOMA-AKSIOMA TERHADAP ISTILAH-ISTILAH TAKTERDEFINISI GEOMETRI HIPERBOLIK PADA MODEL PIRINGAN POINCARE." EDUPEDIA 2, no. 2 (2018): 161. http://dx.doi.org/10.24269/ed.v2i2.148.
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This research aims to know the interpretation the undefined terms on Hyperbolic geometry and it’s consistence with respect to own axioms of Poincare disk model. This research is a literature study that discusses about Hyperbolic geometry. This study refers to books of Foundation of Geometry second edition by Gerard A. Venema (2012), Euclidean and Non Euclidean Geometry (Development and History) by Greenberg (1994), Geometry : Euclid and Beyond by Hartshorne (2000) and Euclidean Geometry: A First Course by M. Solomonovich (2010). The steps taken in the study are: (1) reviewing the various references on the topic of Hyperbolic geometry. (2) representing the definitions and theorems on which the Hyperbolic geometry is based. (3) prepare all materials that have been collected in coherence to facilitate the reader in understanding it. This research succeeded in interpret the undefined terms of Hyperbolic geometry on Poincare disk model. The point is coincide point in the Euclid on circle . Then the point onl γ is not an Euclid point. That point interprets the point on infinity. Lines are categoried in two types. The first type is any open diameters of . The second type is any open arcs of circle. Half-plane in Poincare disk model is formed by Poincare line which divides Poincare field into two parts. The angle in this model is interpreted the same as the angle in Euclid geometry. The distance is interpreted in Poincare disk model defined by the cross-ratio as follows. The definition of distance from to is , where is cross-ratio defined by . Finally the study also is able to show that axioms of Hyperbolic geometry on the Poincare disk model consistent with respect to associated undefined terms.
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Balakan, Gülcan, and Oğuzhan Demirel. "The Formulas of Möbius-Bretschneider and Möbius-Cagnoli in the Poincaré Disc Model of Hyperbolic Geometry." Al-Mustansiriyah Journal of Science 32, no. 1 (2021): 31. http://dx.doi.org/10.23851/mjs.v32i1.932.
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Babalic, Elena Mirela, та Calin Iuliu Lazaroiu. "Generalizedα-Attractor Models from Elementary Hyperbolic Surfaces". Advances in Mathematical Physics 2018 (2018): 1–24. http://dx.doi.org/10.1155/2018/7323090.
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We consider generalizedα-attractor models whose scalar potentials are globally well-behaved and whose scalar manifolds are elementary hyperbolic surfaces. Beyond the Poincaré diskD, such surfaces include the hyperbolic punctured diskD⁎and the hyperbolic annuliA(R)of modulusμ=2logR>0. For each elementary surface, we discuss its decomposition into canonical end regions and give an explicit construction of the embedding into the Kerekjarto-Stoilow compactification (which in all three cases is the unit sphere), showing how this embedding allows for a universal treatment of globally well-behaved scalar potentials upon expanding their extension in real spherical harmonics. For certain simple but natural choices of extended potentials, we compute scalar field trajectories by projecting numerical solutions of the lifted equations of motion from the Poincaré half plane through the uniformization map, thus illustrating the rich cosmological dynamics of such models.
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BARBU, CATALIN. "Mathieu’s theorem in the Poincar´e disc model of hyperbolic geometry." Creative Mathematics and Informatics 20, no. 1 (2011): 16–19. http://dx.doi.org/10.37193/cmi.2011.01.09.
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Rakhmetullina, Zhenisgul, Indira Uvaliyeva, and Farida Amenova. "Differential equations of motion of a material point in the perpendicular plane to the plane of the gravitating disk." Indonesian Journal of Electrical Engineering and Computer Science 24, no. 3 (2021): 1307–14. https://doi.org/10.11591/ijeecs.v24.i3.pp1307-1314.
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This paper presents an analytical solution of the differential equations of motion of a material point in the plane perpendicular to the plane of the gravitating disk. The differential equations of the problem under study and the applied Gilden's method are described in the works of A. Poincaré. Differential equations refer to nonlinear equations. The analysis of methods for solving nonlinear differential equations was carried out. The methodology of applying the Gilden method to the solution of the differential equations under consideration can be applied in studies of the problem of the motion of celestial bodies in the “disk-material point” system in perpendicular planes. To identify the various properties of the gravitating disk, an analytical review of the state of the problem of the motion of a material point in the field of a gravitating disk is carried out. Summing up the presented review on the problem under study, a conclusion is made. The substantive formulation of the problem is described, which is formulated as follows: the study of the influence of disk-shaped bodies on the motion of a material point and methods for their solution.
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Demirel, Oğuzhan, Damla Topal, and Leyla Aslan. "The Beckman–Quarles Theorem in Hyperbolic Geometry." Journal of Mathematics 2021 (March 16, 2021): 1–4. http://dx.doi.org/10.1155/2021/5552198.
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In this paper, we present the counterpart of the Beckman–Quarles theorem in the Poincaré disc model of hyperbolic geometry to characterize the gyroisometries (hyperbolic isometries) with a single nonzero distance a ∈ 0,1 satisfying a 2 ∈ ℚ .
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BARBU, CATALIN, and LAURIAN-IOAN PISCORAN. "Some hyperbolic concurrency results in the Poincare disc." Carpathian Journal of Mathematics 28, no. 1 (2012): 9–15. http://dx.doi.org/10.37193/cjm.2012.01.19.
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Azarov, A. A., A. M. Gouskov, and G. Y. Panovko. "Precession of a rotor with a different number of discrete radial elastic damping supports." Bulletin of the National Engineering Academy of the Republic of Kazakhstan 93, no. 3 (2024): 256–66. http://dx.doi.org/10.47533/2024.1606-146x.65.
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A design model of a rotary system in the form of a cantilevered flexible shaft with a massive unbalanced disk at the free end is considered. In the plane of rotation of the disk, point elastically damped supports are discretely located with a clearance, stabilizing the vibrations of the rotor in the supercritical zone. With the help of the introduced precession indicator, the type and direction of the precession of the rotor during interaction with the supports is analyzed. The influence of a different number of supports on the evolution of the rotor precession in the supercritical zone after the Poincare-Andronov-Hopf bifurcation is investigated. The effect of the clearance and eccentricity of the disk mass at the contact of the disk with the supports on the precession of the rotor is analyzed.
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Ungar, Abraham A. "The Hyperbolic Pythagorean Theorem in the Poincaré Disc Model of Hyperbolic Geometry." American Mathematical Monthly 106, no. 8 (1999): 759–63. http://dx.doi.org/10.1080/00029890.1999.12005114.
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Ungar, Abraham A. "The Hyperbolic Pythagorean Theorem in the Poincare Disc Model of Hyperbolic Geometry." American Mathematical Monthly 106, no. 8 (1999): 759. http://dx.doi.org/10.2307/2589022.
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Nikolov, Svetoslav, and Valentin Nedev. "Bifurcation Analysis and Dynamic Behaviour of an Inverted Pendulum with Bounded Control." Journal of Theoretical and Applied Mechanics 46, no. 1 (2016): 17–32. http://dx.doi.org/10.1515/jtam-2016-0002.
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Abstract This paper presents an investigation on the behaviour of con- ventional inverted pendulum with an inertia disk in its free extreme. The system is actuated by means of torques applied to the disk by a DC mo- tor, mounted on the pendulum’s arm. Thus, the system is underactuated since the pendulum can rotate freely around its pivot point. The dynam- ical model is given with three ordinary nonlinear differential equations. Using Poincare-Andronov-Hopf’s theory, we find a new analytical formula for the first Lyapunov’s value at the boundary of stability. It enables one to study in detail the bifurcation behaviour of the above dynamic system. We check the validity of our analytical results on the first Lyapunov’s value by numerical simulations. Hence, we find some new results.
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Møller, Jesper, and Kateřina Helisová. "Power diagrams and interaction processes for unions of discs." Advances in Applied Probability 40, no. 2 (2008): 321–47. http://dx.doi.org/10.1239/aap/1214950206.
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We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.
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Møller, Jesper, and Kateřina Helisová. "Power diagrams and interaction processes for unions of discs." Advances in Applied Probability 40, no. 02 (2008): 321–47. http://dx.doi.org/10.1017/s0001867800002548.
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We study a flexible class of finite-disc process models with interaction between the discs. We let 𝒰 denote the random set given by the union of discs, and use for the disc process an exponential family density with the canonical sufficient statistic depending only on geometric properties of 𝒰 such as the area, perimeter, Euler-Poincaré characteristic, and the number of holes. This includes the quermass-interaction process and the continuum random-cluster model as special cases. Viewing our model as a connected component Markov point process, and thereby establishing local and spatial Markov properties, becomes useful for handling the problem of edge effects when only 𝒰 is observed within a bounded observation window. The power tessellation and its dual graph become major tools when establishing inclusion-exclusion formulae, formulae for computing geometric characteristics of 𝒰, and stability properties of the underlying disc process density. Algorithms for constructing the power tessellation of 𝒰 and for simulating the disc process are discussed, and the software is made public available.
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Zhang, Jun Hong, Zhen Peng He, Wen Peng Ma, Liang Ma, and Gui Chang Zhang. "Coupled Bending-Torsional Vibration Analysis of Rotor System with Two Asymmetric Disks." Applied Mechanics and Materials 130-134 (October 2011): 2335–39. http://dx.doi.org/10.4028/www.scientific.net/amm.130-134.2335.
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The dynamic equations derived based on the actual rotor system with two asymmetric disks. In the analysis, the eccentric, rubbing fault characteristics and internal damping effects is considered, and all the analysis is established based on nonlinear oil film force model and coupled bending-torsional differential equations. The Rugge-Kutta method is used to solve numerical model, the torsional displacement response, torsion angle and Poincare map are obtained. The results show torsion amplitudes with initial phase difference π / 2 is larger than initial phase difference of π and 0. In order to eliminate the rigid rolling component the relative torsional angle must be considered.
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Liu, Zhenxing, Zhansheng Liu, Yu Li, and Guanghui Zhang. "Dynamics response of an on-board rotor supported on modified oil-film force considering base motion." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 232, no. 2 (2016): 245–59. http://dx.doi.org/10.1177/0954406216682052.
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In an on-board rotor journal bearing system, hydrodynamic bearing forces are affected by the coupling motion between rotor and base. To study the system dynamic response, a model of the on-board rotor supported on hydrodynamic bearing force is proposed in this paper. Motion equations of rigid disk and elastic shaft are derived by applying the Lagrange principle and finite element method. Based on the short bearing assumption and Hahn boundary condition, a modified oil-film force model of the hydrodynamic bearing is derived by introducing the base motion velocity. Under sinusoidal excitations of the base translational motion, dynamic responses of the on-board rotor bearing system are calculated by a Newmark time-step integration algorithm. Results are presented by time history, fast Fourier transforms, orbits and Poincare maps, and are compared with the response dominated by Capone's oil-film force model. The influences of amplitude and frequency of base motion on system dynamic behavior are investigated. Differences after introducing base motion velocity into the oil-film force model show that the convective velocity of the bearings should not be ignored in some extreme cases.
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Barabanov, Nikita E., and Abraham A. Ungar. "Differential Geometry and Binary Operations." Symmetry 12, no. 9 (2020): 1525. http://dx.doi.org/10.3390/sym12091525.
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We derive a large set of binary operations that are algebraically isomorphic to the binary operation of the Beltrami–Klein ball model of hyperbolic geometry, known as the Einstein addition. We prove that each of these operations gives rise to a gyrocommutative gyrogroup isomorphic to Einstein gyrogroup, and satisfies a number of nice properties of the Einstein addition. We also prove that a set of cogyrolines for the Einstein addition is the same as a set of gyrolines of another binary operation. This operation is found directly and it turns out to be commutative. The same results are obtained for the binary operation of the Beltrami–Poincare disk model, known as Möbius addition. We find a canonical representation of metric tensors of binary operations isomorphic to the Einstein addition, and a canonical representation of metric tensors defined by cogyrolines of these operations. Finally, we derive a formula for the Gaussian curvature of spaces with canonical metric tensors. We obtain necessary and sufficient conditions for the Gaussian curvature to be equal to zero.
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Phadatare, Hanmant P., and Barun Pratiher. "Dynamic stability and bifurcation phenomena of an axially loaded flexible shaft-disk system supported by flexible bearing." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 234, no. 15 (2020): 2951–67. http://dx.doi.org/10.1177/0954406220911957.
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The present research studies the vibration and bifurcation analysis of a spinning rotor-disk-bearing system to carefully scrutinize the dynamic stability under extrinsic mass unbalance and pulsating axial load. The shaft is flexible and taken into account the geometrical nonlinearities due to large elastic deformation in bending. The rotor is supported by flexible bearings, which are modeled as an equivalent spring-damper system having linear and nonlinear stiffness elements. Equation of motion of the rotating system, which includes flexible shaft, rigid disk, and flexible bearing, is derived using extended Hamilton principle with the assumption of the Euler's beam theory. We studied initially the modal analysis to determine the modal parameters, i.e. natural frequency and mode shapes prior to the investigation of the dynamics of the system. Further, we developed the bifurcation diagram for steady-state solutions and to study the subsequent dynamic stability and verify with the findings solved numerically. The interactive behavior among the nonlinear shaft-bearing, axial load and an unbalance has been analyzed. Numerical simulation tools, i.e. frequency–response characteristics, time history, phase trajectories, and Poincaré map have been used to highlight the presence of nonlinear phenomena and its important role towards evaluating the system dynamics and subsequent stability. This current research showed that flexible bearings stabilize the system as a result of increasing the restoring force. Analyzing the bifurcation diagrams of pulsating axial load, we found that the system exhibits complex phenomena as multiple but stable periodic orbits leading to period doubling. The present system is highly vulnerable to catastrophic failure due to the S–N and Pitchfork bifurcation. The present research enables the notability of axial load and mechanical unbalance on the overall system behavior and stability in real working conditions.
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Wen, Chuanmei, Yuqi Li, Long Jin, and Dayong Yang. "Bifurcation and Stability Analysis of a Bolted Joint Rotor System Contains Multi-Discs Subjected to Rub-Impact Effect." Processes 10, no. 9 (2022): 1763. http://dx.doi.org/10.3390/pr10091763.
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In aero-engines, the rotor systems are frequently designed with multistage discs, in which the discs are fastened together through bolted joints. During operation, rotating machines are susceptible to rotor–stator rubbing faults. Those bolted joints are subjected to friction and impact forces during a rubbing event, leading to a dramatic change in mechanical properties at the contacting interfaces, influencing the rotor dynamics, which have attracted the attention of scholars. In the present work, a mathematical model, which considers the unbalance force, rotor dimensional properties, nonlinear oil-film force and rub-impact effect, is developed to study the bifurcation and stability characteristics of the bolted joint rotor system containing multi-discs subjected to the rub-impact effect. The time-domain waveforms of the system are obtained numerically by using the Runge–Kutta method, and a bifurcation diagram, time domain waveforms, spectrum plots, shaft orbits and Poincaré maps are adopted to reveal the rotor dynamics under the effect of the rub-impact. Additionally, the influences of rubbing position at the multi-discs on rotor dynamic properties are also examined through bifurcation diagrams. The numerical simulation results show that the segments of the rotating speeds for rubbing are wider and more numerous, and the middle disc is subjected to the rub-impact. When the rub-impact position is far away from disc 1, the rubbing force has little effect on the response of disc 1. The corresponding results can help to understand the bifurcation characteristics of a bolted joint rotor system containing multi-discs subjected to the rub-impact effect.
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Kim, Sitae, Sangwon Byun, and Junho Suh. "Effects of Tilting Pad Journal Bearing Design Parameters on the Pad-Pivot Friction and Nonlinear Rotordynamic Bifurcations." Applied Sciences 10, no. 16 (2020): 5406. http://dx.doi.org/10.3390/app10165406.
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This study numerically analyzes and investigates the effects of the bearing design parameters of a tilting pad journal bearing (TPJB) on the pad-pivot friction-induced nonlinear rotordynamic phenomena and bifurcations. The bearing parameters were set to the pad preload, pivot offset, spherical pivot radius, and bearing length to diameter (L/D) ratio. The Stribeck curve model (SCM) model was applied at the contact surface between the pad and the pivot, which varied to the boundary-mixed-fluid friction state depending on the friction condition. The rotor-bearing model was set up with a symmetrical five-pad TPJB system supporting a Jeffcott type rigid rotor. The fluid repelling force generated in the oil film between each pad and the shaft was calculated using a finite element method. The simulation recurrently conducted the transient numerical integration to obtain the Poincaré maps and phase states of the journal and pad with various bearing design variables, then the nonlinear properties of each condition were analyzed by expressing the bifurcation diagrams. As a result, the original findings of this study are: (1) The pad preload and pivot offset significantly influenced the emergence of Hopf bifurcations and the associated limit cycles. In contrast, (2) the pivot radius and L/D ratio contributed relatively less to the friction-induced instability. Resultantly, (3) all the effects diminished when the rotor operated under the larger mass eccentricity of the disc.
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NINNEMANN, HOLGER. "GUTZWILLER’S OCTAGON AND THE TRIANGULAR BILLIARD T*(2,3,8) AS MODELS FOR THE QUANTIZATION OF CHAOTIC SYSTEMS BY SELBERG’S TRACE FORMULA." International Journal of Modern Physics B 09, no. 13n14 (1995): 1647–753. http://dx.doi.org/10.1142/s0217979295000719.
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Two strongly chaotic systems are investigated with respect to quantization rules based on Selberg’s trace formula. One of them results from the action of a particular strictly hyperbolic Fuchsian group on the Poincaré disk, leading to a compact Riemann surface of genus g=2. This Fuchsian group is denoted as Gutzwiller’s group. The other one is a billiard inside a hyperbolic triangle, which is generated by the operation of a reflection group denoted as T*(2,3,8). Since both groups belong to the class of arithmetical groups, their elements can be characterized explicitly as 2×2 matrices containing entries, which are algebraic numbers subject to a particular set of restrictions. In the case of Gutzwiller’s group this property can be used to determine the geodesic length spectrum of the associated dynamical system completely up to some cutoff length. For the triangular billiard T*(2,3,8) the geodesic length spectrum is calculated by building group elements as products of a suitable set of generators and separating a unique representative for each conjugacy class. The presence of reflections in T*(2,3,8) introduces additional classes of group elements besides the hyperbolic ones, which correspond to periodic orbits of the dynamical system. Due to different choices of boundary conditions along the edges of the fundamental domain of T*(2,3,8), several quantum mechanical systems are associated to one classical system. It has been observed, that these quantum mechanical systems can be divided into two classes according to the behavior of their spectral statistics. This peculiarity is examined from the point of view of classical quantities entering quantization rules. It can be traced back to a subtle influence of the boundary conditions, which introduces contributions from non-periodic orbits for one of the two classes.
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Saeed, Nasser A., Emad Mahrous, Emad Abouel Nasr, and Jan Awrejcewicz. "Nonlinear Dynamics and Motion Bifurcations of the Rotor Active Magnetic Bearings System with a New Control Scheme and Rub-Impact Force." Symmetry 13, no. 8 (2021): 1502. http://dx.doi.org/10.3390/sym13081502.
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This article is dedicated to investigating the nonlinear dynamical behaviors of the 8-pole rotor active magnetic bearing system. The rub and impact forces between the rotating disc and the pole-legs are included in the studied model for the first time. A new control scheme based on modifying the 8-pole positions has been introduced. The proposed control methodology is designed such that four poles only are located in the horizontal and vertical directions (i.e., in +X,+Y,−X,−Y directions), while the other four poles are inserted in a way such that each pole makes 45° with two of the axes +X,+Y,−X,−Y. The control currents in the horizontal and vertical poles are suggested to be proportional to both the velocity and displacement of the rotor in the horizontal and vertical directions, respectively, while the control currents in the inclined poles are proposed to be dependent on the combination of both the displacement and velocity of the rotor in the horizontal and vertical directions. Accordingly, the whole-system mathematical model is derived. The derived discontinuous dynamical system is analyzed employing perturbation methods, Poincare maps, bifurcation diagrams, whirling orbits, and frequency spectrum. The obtained results demonstrated that the controller proportional control gain can play a significant role in changing the vibratory behaviors of the system, where the proposed control method can behave either as a cartesian control strategy or as a radial control one depending on the magnitude of the proportional gain. In addition, it is found that the rotor system can vibrate with periodic, periodic-n, quasiperiodic, or chaotic motion when the rub and/or impact forces occur. Moreover, it is reported for the first time that the rotor-AMB can oscillate symmetrically in X and Y directions either in full annular rub mode or quasiperiodic partial rub mode depending on the impact stiffness coefficient and the dynamic friction coefficient.
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Chi, Ian, Martin Fraas, and Tina Tan. "Poincaré disk as a model of squeezed states of a harmonic oscillator." Journal of Mathematical Physics 66, no. 6 (2025). https://doi.org/10.1063/5.0252902.
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Single-mode squeezed states exhibit a direct correspondence with points on the Poincaré disk. In this study, we delve into this correspondence and describe the motions of the disk generated by a quadratic Hamiltonian. This provides a geometric representation of squeezed states and their evolution. We discuss applications in bang-bang and adiabatic control problems involving squeezed states.
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Ушхо, А. Д. "On self-oscillations of one kinetic model." Вестник Адыгейского государственного университета, серия «Естественно-математические и технические науки», no. 2(321) (October 11, 2023). http://dx.doi.org/10.53598/2410-3225-2023-2-321-19-26.
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Методами качественной теории дифференциальных уравнений рассмотрено поведение кинетической модели, в частности, фазовый портрет на диске Пуанкаре. Приведен пример системы с устойчивым циклом. With the use of methods of qualitative theory of differential equations, the article examines the kinetic model behavior, in particular, the phase portrait on the Poincaré disk. An example of a system with a stable cycle is given.
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Lechtenfeld, Olaf, and Don Zagier. "A hyperbolic Kac-Moody Calogero model." Journal of High Energy Physics 2024, no. 6 (2024). http://dx.doi.org/10.1007/jhep06(2024)093.
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Abstract A new kind of quantum Calogero model is proposed, based on a hyperbolic Kac-Moody algebra. We formulate nonrelativistic quantum mechanics on the Minkowskian root space of the simplest rank-3 hyperbolic Lie algebra AE3 with an inverse-square potential given by its real roots and reduce it to the unit future hyperboloid. By stereographic projection this defines a quantum mechanics on the Poincaré disk with a unique potential. Since the Weyl group of AE3 is a ℤ2 extension of the modular group PSL(2,ℤ), the model is naturally formulated on the complex upper half plane, and its potential is a real modular function. We present and illustrate the relevant features of AE3, give some approximations to the potential and rewrite it as an (almost everywhere convergent) Poincaré series. The standard Dunkl operators are constructed and investigated on Minkowski space and on the hyperboloid. In the former case we find that their commutativity is obstructed by rank-2 subgroups of hyperbolic type (the simplest one given by the Fibonacci sequence), casting doubt on the integrability of the model. An appendix with Don Zagier investigates the computability of the potential. We foresee applications to cosmological billards and to quantum chaos.