Relevant bibliographies by topics / Centre manifold theory

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Centre manifold theory'

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Published: 4 June 2021
Last updated: 30 January 2023

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Journal articles on the topic "Centre manifold theory"

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Rendall, Alan D. "Cosmological Models and Centre Manifold Theory." General Relativity and Gravitation 34, no. 8 (August 2002): 1277–94. http://dx.doi.org/10.1023/a:1019734703162.

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ZHANG, CHUNRUI, and BAODONG ZHENG. "CODIMENSION ONE BIFURCATION OF EQUIVARIANT NEURAL NETWORK MODEL WITH DELAY." International Journal of Bifurcation and Chaos 20, no. 04 (April 2010): 1255–59. http://dx.doi.org/10.1142/s0218127410026459.

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In this paper, we consider double zero singularity of a symmetric BAM neural network model with a time delay. Based on the normal form approach and the center manifold theory, we obtain the normal form on the centre manifold with double zero singularity. Some numerical simulations support our analysis results.
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Lin, Xiaodong, Joseph W. H. So, and Jianhong Wu. "Centre manifolds for partial differential equations with delays." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 3-4 (1992): 237–54. http://dx.doi.org/10.1017/s0308210500021090.

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SynopsisA centre manifold theory for reaction-diffusion equations with temporal delays is developed. Besides an existence proof, we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order. As an illustration, this reduction procedure is applied to the Hutchinson equation with diffusion.
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LIU, L., Y. S. WONG, and B. H. K. LEE. "APPLICATION OF THE CENTRE MANIFOLD THEORY IN NON-LINEAR AEROELASTICITY." Journal of Sound and Vibration 234, no. 4 (July 2000): 641–59. http://dx.doi.org/10.1006/jsvi.1999.2895.

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WANG KAI-GE, WANG YU-LONG, and SUN YIN-GUAN. "APPLICATION OF CENTRE MANIFOLD THEORY IN GENERALIZED MAXWELL-BLOCH LASER EQUATIONS." Acta Physica Sinica 45, no. 1 (1996): 46. http://dx.doi.org/10.7498/aps.45.46.

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Psarros, N., G. Papaschinopoulos, and C. J. Schinas. "Semistability of two systems of difference equations using centre manifold theory." Mathematical Methods in the Applied Sciences 39, no. 18 (March 6, 2016): 5216–22. http://dx.doi.org/10.1002/mma.3904.

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Liu, Wei, and Yaolin Jiang. "Dynamics of a Modified Predator-Prey System to allow for a Functional Response and Time Delay." East Asian Journal on Applied Mathematics 6, no. 4 (October 19, 2016): 384–99. http://dx.doi.org/10.4208/eajam.141214.050616a.

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AbstractA modified predator-prey system described by two differential equations and an algebraic equation is discussed. Formulae for determining the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions are derived differential-algebraic system theory, bifurcation theory and centre manifold theory. Numerical simulations illustrate the results, which includes quite complex dynamical behaviour.
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Valls, Claudia. "Stability of some solutions for elliptic equations on a cylindrical domain." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2109 (June 10, 2009): 2647–62. http://dx.doi.org/10.1098/rspa.2009.0110.

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We study analytically a class of solutions for the elliptic equation where α >0 and ε is a small parameter. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every α >0, it contains solutions that are defined for large values of time and they are very close (of order O ( ε )) to a linear torus for long times (of order O ( ε −1 )). The proof uses the fact that the equation leaves invariant a smooth centre manifold and, for the restriction of the system to the centre manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.
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JI, J. C., X. Y. LI, Z. LUO, and N. ZHANG. "TWO-TO-ONE RESONANT HOPF BIFURCATIONS IN A QUADRATICALLY NONLINEAR OSCILLATOR INVOLVING TIME DELAY." International Journal of Bifurcation and Chaos 22, no. 03 (March 2012): 1250060. http://dx.doi.org/10.1142/s0218127412500605.

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The trivial equilibrium of a weakly nonlinear oscillator having quadratic nonlinearities under a delayed feedback control can change its stability via a single Hopf bifurcation as the time delay increases. Double Hopf bifurcation occurs when the characteristic equation has two pairs of purely imaginary solutions. An interaction of resonant Hopf–Hopf bifurcations may be possible when the two critical time delays corresponding to the two Hopf bifurcations have the same value. With the aid of normal form theory and centre manifold theorem as well as the method of multiple scales, the present paper studies the dynamics of a quadratically nonlinear oscillator involving time delay in the vicinity of the point of two-to-one resonances of Hopf–Hopf bifurcations. The ratio of the frequencies of two Hopf bifurcations is numerically found to be nearly equal to two. The two resonant Hopf bifurcations can generate two respective periodic solutions. Consequently, the centre manifold corresponding to these two solutions is determined by a set of four first-order differential equations under two-to-one internal resonances. It is shown that the amplitudes of the two bifurcating periodic solutions admit the trivial solution and two-mode solutions for the averaged equations on the centre manifolds. Correspondingly, the cumulative behavior of the original nonlinear oscillator exhibits the initial equilibrium and a quasi-periodic motion having two frequencies. Illustrative examples are given to show the unstable zero solution, stable zero solution, and stable two-mode solution of the nonlinear oscillator under the two-to-one resonant Hopf–Hopf interactions.
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Mielke, Alexander. "On Saint-Venant's problem for an elastic strip." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 110, no. 1-2 (1988): 161–81. http://dx.doi.org/10.1017/s0308210500024938.

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SynopsisThe equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.
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Dissertations / Theses on the topic "Centre manifold theory"

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Liu, Weishi. "Center manifold theory for smooth invariant manifolds." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/28762.

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Allahem, Ali Ibraheem. "Numerical investigation of chaotic dynamics in multidimensional transition states." Thesis, Loughborough University, 2014. https://dspace.lboro.ac.uk/2134/14058.

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Many chemical reactions can be described as the crossing of an energetic barrier. This process is mediated by an invariant object in phase space. One can construct a normally hyperbolic invariant manifold (NHIM) of the reactive dynamical system which is an invariant sphere that can be considered as the geometric representation of the transition state itself. The NHIM has invariant cylinders (reaction channels) attached to it. This invariant geometric structure survives as long as the invariant sphere is normally hyperbolic. We applied this theory to the hydrogen exchange reaction in three degrees of freedom in order to figure out the reason of the transition state theory (TST) failure. Energies high above the reaction threshold, the dynamics within the transition state becomes partially chaotic. We have found that the invariant sphere first ceases to be normally hyperbolic at fairly low energies. Surprisingly normal hyperbolicity is then restored and the invariant sphere remains normally hyperbolic even at very high energies. This observation shows two different energy values for the breakdown of the TST and the breakdown of the NHIM. This leads to seek another phase space object that is related to the breakdown of the TST. Using theory of the dividing surface including reactive islands (RIs), we can investigate such an object. We found out that the first nonreactive trajectory has been found at the same energy values for both collinear and full systems, and coincides with the first bifurcation of periodic orbit dividing surface (PODS) at the collinear configuration. The bifurcation creates the unstable periodic orbit (UPO). Indeed, the new PODS (UPO) is the reason for the TST failure. The manifolds (stable and centre-stable) of the UPO clarify these expectations by intersecting the dividing surface at the boundary of the reactive island (on the collinear and the three (full) systems, respectively).
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MacKenzie, Tony. "Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation." University of Southern Queensland, Faculty of Sciences, 2005. http://eprints.usq.edu.au/archive/00001466/.

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This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.
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Lichtner, Mark. "Exponential dichotomy and smooth invariant center manifolds for semilinear hyperbolic systems." Doctoral thesis, [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=981306659.

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Silva, Vinicius Barros da. "Bifurcação de Hopf e formas normais : uma nova abordagem para sistemas dinâmicos /." Rio Claro, 2018. http://hdl.handle.net/11449/180496.

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Orientador: Edson Denis Leonel
Resumo: Este estudo objetiva provar que sistemas dinâmicos de dimensão N, de codimensão um e satisfazendo as condições do teorema da bifurcação de Hopf, podem ser expressos em uma forma analítica simplificada que preserva a topologia do espaço de fases da configuração original, na vizinhança do ponto de equilíbrio. A esta forma simplificada é atribuído o nome de forma normal. Para tanto, foi utilizado a teoria da variedade central, necessária para reduzir a dimensão de sistemas à sua variedade bidimensional, e o teorema das formas normais, utilizando-se como método para determinar a forma simplificada da variedade central associada aos sistemas dinâmicos, atendendo as condições do teorema da bifurcação de Hopf. A partir da análise dos resultados aqui encontrados foi possível construir a prova matemática de que sistemas de dimensão N, atendendo as condições do teorema de Hopf, podem ser reescritos em uma expressão analítica geral e simplificada. Enfim, através deste estudo foi possível resumir todos os resultados aqui obtidos em um teorema geral que, além de reduzir a custosa tarefa de obtenção de formas normais, abrange sistemas N-dimensionais com ocorrência da bifurcação de Hopf.
Abstract: In this work we prove the following: consider a N-dimensional system that is reduced to its center manifold. If it is proved the system satisfies the conditions of Hopf bifurcation theorem, then the original system of differential equations is rewritten in a simpler analytical expression that preserves the phase space topology. This last is also known as the normal form. The center manifold is used to derive a reduced order expression, and the normal form theory is applied to simplify the form of the dynamics on the center manifold. The key results here allow constructing a general mathematical proof for the normal form of N-dimensional systems reduced to its center manifold. In the class of dynamical systems under Hopf bifurcations, the present work reduces the work done to obtain normal forms.
Mestre
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Kasnakoglu, Cosku. "Reduced order modeling, nonlinear analysis and control methods for flow control problems." Columbus, Ohio : Ohio State University, 2007. http://rave.ohiolink.edu/etdc/view?acc%5Fnum=osu1195629380.

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Arugaslan, Cincin Duygu. "Differential Equations With Discontinuities And Population Dynamics." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610574/index.pdf.

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In this thesis, both theoretical and application oriented results are obtained for differential equations with discontinuities of different types: impulsive differential equations, differential equations with piecewise constant argument of generalized type and differential equations with discontinuous right-hand sides. Several qualitative problems such as stability, Hopf bifurcation, center manifold reduction, permanence and persistence are addressed for these equations and also for Lotka-Volterra predator-prey models with variable time of impulses, ratio-dependent predator-prey systems and logistic equation with piecewise constant argument of generalized type. For the first time, by means of Lyapunov functions coupled with the Razumikhin method, sufficient conditions are established for stability of the trivial solution of differential equations with piecewise constant argument of generalized type. Appropriate examples are worked out to illustrate the applicability of the method. Moreover, stability analysis is performed for the logistic equation, which is one of the most widely used population dynamics models. The behaviour of solutions for a 2-dimensional system of differential equations with discontinuous right-hand side, also called a Filippov system, is studied. Discontinuity sets intersect at a vertex, and are of the quasilinear nature. Through the B&
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equivalence of that system to an impulsive differential equation, Hopf bifurcation is investigated. Finally, the obtained results are extended to a 3-dimensional discontinuous system of Filippov type. After the existence of a center manifold is proved for the 3-dimensional system, a theorem on the bifurcation of periodic solutions is provided in the critical case. Illustrative examples and numerical simulations are presented to verify the theoretical results.
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Marmo, Carlos Nehemy. "Bifurcações em PLLs de terceira ordem em redes OWMS." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-29012009-103841/.

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Este trabalho apresenta um estudo qualitativo das equações diferenciais nãolineares que descrevem o sincronismo de fase nos PLLs de 3ª ordem que compõem redes OWMS de topologia mista, Estrela Simples e Cadeia Simples. O objetivo é determinar, através da Teoria de Bifurcações, os valores ou relações entre os parâmetros constitutivos da rede que permitam a existência e a estabilidade do estado síncrono, quando são aplicadas, no oscilador mestre, duas funções de excitação muito comuns na prática: o degrau e a rampa de fase. Na determinação da estabilidade dos pontos de equilíbrio, sob o ponto de vista de Lyapunov, a existência de pontos de equilíbrio não-hiperbólicos não permite uma aproximação linear e, nesses casos, é aplicado o Teorema da Variedade Central. Essa técnica de simplificação de sistemas dinâmicos permite fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases e possibilitando determinar localmente suas estabilidades.
This work presents a qualitative study of the non-linear differential equations that describe the synchronous state in 3rd order PLLs that compose One-way masterslave time distribution networks with Single Star and Single Chain topologies. Using bifurcation theory, the dynamical behavior of third-order phase-locked loops employed to extract the syncronous state in each node is analyzed depending on constitutive node parameters when two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node. When parameter combinations result in non hyperbolic synchronous states, from Lyapunov point of view, the linear approximation does not provide any information about the local behavior of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behavior of the original system in the neighborhood of these points. Thus, the local stability can be determined.
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Garcia, Ignacio de Mateo. "Iterative matrix-free computation of Hopf bifurcations as Neimark-Sacker points of fixed point iterations." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2012. http://dx.doi.org/10.18452/16478.

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Klassische Methoden für die direkte Berechnung von Hopf Punkten und andere Singularitaten basieren auf der Auswertung und Faktorisierung der Jakobimatrix. Dieses stellt ein Hindernis dar, wenn die Dimensionen des zugrundeliegenden Problems gross genug ist, was oft bei Partiellen Diferentialgleichungen der Fall ist. Die betrachteten Systeme haben die allgemeine Darstellung f ( x(t), α) für t grösser als 0, wobei x die Zustandsvariable, α ein beliebiger Parameter ist und f glatt in Bezug auf x und α ist. In der vorliegenden Arbeit wird ein Matrixfreies Schema entwicklet und untersucht, dass ausschliesslich aus Produkten aus Jakobimatrizen und Vektoren besteht, zusammen mit der Auswertung anderer Ableitungsvektoren erster und zweiter Ordnung. Hiermit wird der Grenzwert des Parameters α, der zuständig ist für das Verlieren der Stabilität des Systems, am Hopfpunkt bestimmt. In dieser Arbeit wird ein Gleichungssystem zur iterativen Berechnung des Hopfpunktes aufgestellt. Das System wird mit einer skalaren Testfunktion φ, die aus einer Projektion des kritischen Eigenraums bestimmt ist, ergänzt. Da das System f aus einer räumlichen Diskretisierung eines Systems Partieller Differentialgleichungen entstanden ist, wird auch in dieser Arbeit die Berechung des Fehlers, der bei der Diskretisierung unvermeidbar ist, dargestellt und untersucht. Zur Bestimmung der Hopf-Bedingungen wird ein einzelner Parameter gesteuert. Dieser Parameter wird unabhängig oder zusammen mit dem Zustandsvektor in einem gedämpften Iterationsschritt neu berechnet. Der entworfene Algorithmus wird für das FitzHugh-Nagumo Model erprobt. In der vorliegenden Arbeit wird gezeigt, wie für einen kritischen Strom, das Membranpotential eine fortschreitende Welle darstellt.
Classical methods for the direct computation of Hopf bifurcation points and other singularities rely on the evaluation and factorization of Jacobian matrices. In view of large scale problems arising from PDE discretization systems of the form f( x (t), α ), for t bigger than 0, where x are the state variables, α are certain parameters and f is smooth with respect to x and α, a matrix-free scheme is developed based exclusively on Jacobian-vector products and other first and second derivative vectors to obtain the critical parameter α causing the loss of stability at the Hopf point. In the present work, a system of equations is defined to locate Hopf points, iteratively, extending the system equations with a scalar test function φ, based on a projection of the eigenspaces. Since the system f arises from a spatial discretization of an original set of PDEs, an error correction considering the different discretization procedures is presented. To satisfy the Hopf conditions a single parameter is adjusted independently or simultaneously with the state vector in a deflated iteration step, reaching herewith both: locating the critical parameter and accelerating the convergence rate of the system. As a practical experiment, the algorithm is presented for the Hopf point of a brain cell represented by the FitzHugh-Nagumo model. It will be shown how for a critical current, the membrane potential will present a travelling wave typical of an oscillatory behaviour.
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Marmo, Carlos Nehemy. "Sincronismo em redes mestre-escravo de via-única: estrela simples, cadeia simples e mista." Universidade de São Paulo, 2003. http://www.teses.usp.br/teses/disponiveis/3/3139/tde-18022004-233234/.

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Neste trabalho, são estudados os problemas de sincronismo de fase nas redes mestre-escravo de via única (OWMS), nas topologias Estrela Simples, Cadeia Simples e mista, através da Teoria Qualitativa de Equações Diferenciais, com ênfase no Teorema da Variedade Central. Através da Teoria das Bifurcações, analisa-se o comportamento dinâmico das malhas de sincronismo de fase (PLL) de segunda ordem que compõem cada rede, frente às variações nos seus parâmetros constitutivos. São utilizadas duas funções de excitação muito comuns na prática: o degrau e a rampa de fase, aplicadas pelo nó mestre. Em cada caso, discute-se a existência e a estabilidade do estado síncrono. A existência de pontos de equilíbrio não-hiperbólicos, não permite uma aproximação linear, e nesses casos é aplicado o Teorema da Variedade Central. Através dessa rigorosa técnica de simplificação de sistemas dinâmicos é possível fazer uma aproximação homeomórfica em torno desses pontos, preservando a orientação no espaço de fases. Desse modo, é possível determinar, localmente, suas estabilidades.
This work presents stability analysis of the syncronous state for three types of one-way master-slave time distribution network topologies: single star, single chain and both of them, mixed. Using bifurcation theory, the dynamical behavior of second-order phase-locked loops employed to extract the syncronous state in each node is analyzed in function of the constitutive parameters. Two usual inputs, the step and the ramp phase pertubations, are supposed to appear in the master node and, in each case, the existence and stability of the syncronous state are studied. For parameter combinations resulting in non hyperbolic synchronous states, the linear approximation does not provide any information, even about the local behaviour of the system. In this case, the center manifold theorem permits the construction of an equivalent vector field representing the asymptotic behaviour of the original system in the neighborhood of these points. Thus, the local stability can be determined.
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Books on the topic "Centre manifold theory"

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Dumortier, Freddy. Canard cycles and center manifolds. Providence, R.I: American Mathematical Society, 1996.

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Gérard, Iooss, ed. Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. London: Springer, 2011.

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1963-, Ruan Shigui, ed. Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models. Providence, R.I: American Mathematical Society, 2009.

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editor, Donagi Ron, Douglas, Michael (Michael R.), editor, Kamenova Ljudmila 1978 editor, and Roček M. (Martin) editor, eds. String-Math 2013: Conference, June 17-21, 2013, Simons Center for Geometry and Physics, Stony Brook, NY. Providence, Rhode Island: American Mathematical Society, 2014.

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Center for Mathematics at Notre Dame and American Mathematical Society, eds. Toplogy and field theories: Center for Mathematics at Notre Dame, Center for Mathematics at Notre Dame : summer school and conference, Topology and field theories, May 29-June 8, 2012, University of Notre Dame, Notre Dame, Indiana. Providence, Rhode Island: American Mathematical Society, 2014.

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Carr, J. Applications of Centre Manifold Theory. Springer London, Limited, 2012.

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Iooss, Gérard, and Mariana Haragus. Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems. Springer, 2018.

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Cattani, Eduardo, and Phillip Griffiths. Introduction to Kähler Manifolds. Edited by Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, Lê Dũng Tráng, Eduardo Cattani, Fouad El Zein, Phillip A. Griffiths, and Lê Dũng Tráng. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.003.0001.

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This chapter provides an introduction to the basic results on the topology of compact Kähler manifolds that underlie and motivate Hodge theory. This chapter consists of five sections which correspond, roughly, to the five lectures in the course given during the Summer School at the International Centre for Theoretical Physics (ICTP). The five topics under discussion are: complex manifolds; differential forms on complex manifolds; symplectic, Hermitian, and Kähler structures; harmonic forms; and the cohomology of compact Kähler manifolds. There are also two appendices. The first collects some results on the linear algebra of complex vector spaces, Hodge structures, nilpotent linear transformations, and representations of sl(2,ℂ) and serves as an introduction to many other chapters in this volume. The second contains a new proof of the Kähler identities by reduction to the symplectic case.
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Cattani, Eduardo, Fouad El Zein, Phillip A. Griffiths, and Lê Dung Tráng. Hodge Theory (MN-49). Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691161341.001.0001.

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This book provides a comprehensive and up-to-date introduction to Hodge theory—one of the central and most vibrant areas of contemporary mathematics—from leading specialists on the subject. The topics range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps. Of particular interest is the study of algebraic cycles, including the Hodge and Bloch–Beilinson Conjectures. Based on lectures delivered at the 2010 Summer School on Hodge Theory at the ICTP in Trieste, Italy, the book is intended for a broad group of students and researchers. The exposition is as accessible as possible and does not require a deep background. At the same time, the book presents some topics at the forefront of current research. The book is divided between introductory and advanced lectures. The introductory lectures address Kähler manifolds, variations of Hodge structure, mixed Hodge structures, the Hodge theory of maps, period domains and period mappings, algebraic cycles (up to and including the Bloch–Beilinson conjecture) and Chow groups, sheaf cohomology, and a new treatment of Grothendieck's algebraic de Rham theorem. The advanced lectures address a Hodge-theoretic perspective on Shimura varieties, the spread philosophy in the study of algebraic cycles, absolute Hodge classes (including a new, self-contained proof of Deligne's theorem on absolute Hodge cycles), and variation of mixed Hodge structures.
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Farb, Benson, and Dan Margalit. The Nielsen-Thurston Classification. Princeton University Press, 2017. http://dx.doi.org/10.23943/princeton/9780691147949.003.0014.

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This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.
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Book chapters on the topic "Centre manifold theory"

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Boettner, Reinhard. "Model Reduction and Stability of Nonlinear Dynamical Systems by means of Centre Manifold Theory." In Advances in Simulation, 159–62. New York, NY: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4684-6389-7_31.

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Mei, Zhen. "Center Manifold Theory." In Numerical Bifurcation Analysis for Reaction-Diffusion Equations, 129–50. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-662-04177-2_7.

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Rand, Richard H., and Dieter Armbruster. "Center Manifolds." In Perturbation Methods, Bifurcation Theory and Computer Algebra, 27–49. New York, NY: Springer New York, 1987. http://dx.doi.org/10.1007/978-1-4612-1060-3_2.

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Church, Kevin E. M., and Xinzhi Liu. "Computational Aspects of Centre Manifolds." In Bifurcation Theory of Impulsive Dynamical Systems, 111–37. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64533-5_6.

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Sideris, Thomas C. "Center Manifolds and Bifurcation Theory." In Atlantis Studies in Differential Equations, 155–98. Paris: Atlantis Press, 2013. http://dx.doi.org/10.2991/978-94-6239-021-8_9.

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Vanderbauwhede, A., and G. Iooss. "Center Manifold Theory in Infinite Dimensions." In Dynamics Reported, 125–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 1992. http://dx.doi.org/10.1007/978-3-642-61243-5_4.

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Church, Kevin E. M., and Xinzhi Liu. "Existence, Regularity and Invariance of Centre Manifolds." In Bifurcation Theory of Impulsive Dynamical Systems, 67–109. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64533-5_5.

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Mielke, Alexander. "The linear theory." In Hamiltonian and Lagrangian Flows on Center Manifolds, 17–26. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/bfb0097547.

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Oppo, Gian-Luca, and Antonio Politi. "Adiabatic Elimination for Laser Equations via Center Manifold Theory." In Coherence and Quantum Optics VI, 835–39. Boston, MA: Springer US, 1989. http://dx.doi.org/10.1007/978-1-4613-0847-8_151.

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Precup, Radu-Emil, Stefan Preitl, and Stefan Solyom. "Center Manifold Theory Approach to the Stability Analysis of Fuzzy Control Systems." In Lecture Notes in Computer Science, 382–90. Berlin, Heidelberg: Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-48774-3_44.

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Conference papers on the topic "Centre manifold theory"

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Khajepour, Amir, Farid Golnaraghi, and K. A. Morris. "Vibration Suppression of a Flexible Beam Using Center Manifold Theory." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0288.

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Abstract In this paper we develop a nonlinear control strategy based on modal coupling using the center manifold theory. As an example we use the technique for vibration suppression of a flexible beam. The controller in this case is a mass-spring-dashpot mechanism which is free to slide along the beam. The equations of the plant/controller are coupled and nonlinear, and the linearized equations of the system have two uncontrollable modes. As a result, the performance of the system can not be improved by linear control theory or by most conventional nonlinear control techniques. We use the normal forms method to simplify the center manifold equations and derive a relation which includes all system parameters. We then show that there exists a set of optimal controller parameters (feedback gains, controller damping and frequency) which maximized the energy dissipation. Finally we consider the stability and design issues, and use numerical simulation to verify the results.
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Wang, Deshi, Renbin Xiao, and Ming Yang. "The Attitude Stability for Longitudinal Motion of Underwater Vehicle." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21607.

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Abstract Although the equations describing the longitudinal motions of underwater vehicles are typically nonlinear, the linearized equations are still employed to design the depth controller by the traditional analysis methods in engineering for the sake of simplicity. The reduction of the nonlinearity loses the dynamics near the singular points, which may be responsible for the sudden climb or dive. The nonlinear systems limited in the longitudinal plane of the underwater vehicles are analyzed on center manifold through the bifurcation theory. It focuses on the case that single zero root in Jacobi matrix occurs at equilibrium points corresponding to nominal trajectory with varied angles of the elevator or the direction change of the flows. The center manifolds are calculated and one-dimensional bifurcation equations on the center manifolds are obtained and analyzed. Based on the transcritical bifurcation diagram, we have found the mechanism of the attitude stability loss as well as the abnormal trajectory of autonomous underwater vehicles. It gives good explainations to the practical climbing jump and diving fall and delivers the theoretical tools to design the controller and to design dynamics. Numerical simulation verifies the results.
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Pesheck, E., and C. Pierre. "A Global Methodology for the Modal Reduction of Large Nonlinear Systems Containing Quadratic and Cubic Nonlinearities." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-3952.

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Abstract A methodology is presented for the systematic modal reduction of structural systems which contain quadratic and cubic nonlinearities in displacement. The procedure is based on the center manifold approach for describing individual nonlinear modes, but it has been extended to account for simultaneous motion within several chosen modal coordinates. Motions of the reduced system are constrained to lie on high-dimensional manifolds within the phase space of the original system. Polynomial approximations of these manifolds are obtained through third order for arbitrary system parameters. Algorithms have been developed for automation of this procedure, and they are applied to an example system. Free and forced responses of the reduced system are discussed and compared to responses reduced through simple modal truncation. A more rigorous treatment of harmonic forcing is proposed, which will allow for the production of high-dimensional, time-dependent manifolds through a simple adaptation of the unforced procedure.
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Kasnakoglu, Cosku, and Andrea Serrani. "Analysis and Nonlinear Control of Galerkin Models Using Averaging and Center Manifold Theory." In 2007 American Control Conference. IEEE, 2007. http://dx.doi.org/10.1109/acc.2007.4282236.

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Sun, Chongyi, Fuxiang Quan, Kun-Zhi Liu, Yu Zhang, and Xi-Ming Sun. "Compressor Active Stability Control Based on Center Manifold Theorem and Parameter Optimization." In 2022 41st Chinese Control Conference (CCC). IEEE, 2022. http://dx.doi.org/10.23919/ccc55666.2022.9901745.

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Chen, Zhen, and Pei Yu. "Double-Hopf Bifurcation in an Oscillator With External Forcing and Time-Delayed Feedback Control." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85549.

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In this paper an oscillator with time delayed velocity feedback controls is studied in detail. The particular attention is focused on internal double-Hopf bifurcation with an external exciting force. Linear analysis is used to find the critical conditions under which a double-Hopf bifurcation occurs. Then center manifold theory is applied to obtain an ODE system described on a four-dimensional center manifold. Further, the technique of multiple-time scales is employed to find the approximate solutions of periodic and quasi-periodic motions. Finally, numerical simulation results are presented to verify the analytical predictions. Also, for some certain parameter values, numerical results show chaotic attractors.
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Wang Chao, Zhang Yao, and Wu ZhiGang. "Application of center manifold theory on analysis of voltage stability of single-machine infinite system with dynamic loads." In 7th IET International Conference on Advances in Power System Control, Operation and Management (APSCOM 2006). IEE, 2006. http://dx.doi.org/10.1049/cp:20062098.

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Sinha, S. C., Sangram Redkar, Eric A. Butcher, and Venkatesh Deshmukh. "Order Reduction of Nonlinear Time Periodic Systems Using Invariant Manifolds." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48445.

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The basic problem of order reduction of linear and nonlinear systems with time periodic coefficients is considered. First, the equations of motion are transformed using the Lyapunov-Floquet transformation such that the linear parts of new set of equations are time invariant. At this stage, the linear order reduction technique can be applied in a straightforward manner. A nonlinear order reduction methodology is also suggested through a generalization of the invariant manifold technique via Time Periodic Center Manifold Theory. A ‘reducibility condition’ is derived to provide conditions under which a nonlinear order reduction is possible. Unlike perturbation or averaging type approaches, the parametric excitation term is not assumed to be small. An example consisting of two parametrically excited coupled pendulums is given to show applications to real problems. Order reduction possibilities and results for various cases including ‘parametric’, ‘internal’, ‘true internal’ and ‘combination’ resonances are discussed.
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Fofana, M. S., and Dariusz Szwarc. "P-Bifurcation Aspects of a Modified Duffing Oscillator Near Equilibrium." In ASME 2001 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2001. http://dx.doi.org/10.1115/detc2001/vib-21596.

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Abstract A unified framework for the study of stability of solutions to delay differential equations perturbed deterministically and stochastically has been presented. We review the concepts of Hopf bifurcation, centre manifold, integral averaging method and pth–moment Lyapunov exponent, and then demonstrate their role in the stability study of a modified Duffing oscillator with time delay. Sufficient conditions ensuring Đ- and Þ–bifurcations and the changes in character of the probability density function are established for fixed and positive time delay.
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Wei, Heng, Jian-Wei Lu, Hang-Yu Lu, and Sheng-Yong Ye. "Bifurcation Characteristic and Energy Transfer of Vehicle Shimmy System Considering the Coupling of Vertical and Lateral Dynamics." In ASME 2021 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/imece2021-66827.

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Abstract Since the vehicle is a complex mechanical system with many subsystems, the influence of the dynamic coupling between the subsystems of the vehicle on shimmy should be taken seriously. Therefore, a 12 degrees-of-freedom dynamic model of vehicle shimmy system with consideration of the dynamic coupling between the vertical motion and the lateral motion of the vehicle is established. In particular, the influence of the vertical load of the tire on the nonlinear cornering force is also considered. Then, the dynamic stability of the shimmy system is discussed with the help of the system eigenvalues, and the influence of the damping of the front suspension on the dynamic stability is also examined from the viewpoint of the energy transfer. On this basis, the damping of the front suspension is selected as a bifurcation parameter, and the two-dimensional center manifold of the high dimensional shimmy system is obtained by means of the center manifold theory. Finally, the Hopf bifurcation characteristic of the shimmy system is analyzed, and the analytical solutions of the shimmy system are derived for different damping of the front suspension. The results show that the increase of the damping of the front suspension is beneficial to attenuate vehicle shimmy and improve the dynamic stability of the vehicle.
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