Relevant bibliographies by topics / Centre manifold theorem

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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 theorem'

Author: Grafiati
Published: 4 June 2021
Last updated: 4 February 2022

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Journal articles on the topic "Centre manifold theorem"

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Renardy, Michael. "A centre manifold theorem for hyperbolic PDEs." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 122, no. 3-4 (1992): 363–77. http://dx.doi.org/10.1017/s0308210500021168.

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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 (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 pape
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Luís, Rafael, and Elias Rodrigues. "Local Stability in 3D Discrete Dynamical Systems: Application to a Ricker Competition Model." Discrete Dynamics in Nature and Society 2017 (2017): 1–16. http://dx.doi.org/10.1155/2017/6186354.

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A survey on the conditions of local stability of fixed points of three-dimensional discrete dynamical systems or difference equations is provided. In particular, the techniques for studying the stability of nonhyperbolic fixed points via the centre manifold theorem are presented. A nonlinear model in population dynamics is studied, namely, the Ricker competition model of three species. In addition, a conjecture about the global stability of the nontrivial fixed points of the Ricker competition model is presented.
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ÇELİK, CANAN, and GÖKÇEN ÇEKİÇ. "BIFURCATION ANALYSIS OF A LOGISTIC PREDATOR–PREY SYSTEM WITH DELAY." ANZIAM Journal 57, no. 4 (2016): 445–60. http://dx.doi.org/10.1017/s1446181116000055.

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We consider a coupled, logistic predator–prey system with delay. Mainly, by choosing the delay time${\it\tau}$as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time${\it\tau}$passes some critical values. Based on the normal-form theory and the centre manifold theorem, we also derive formulae to obtain the direction, stability and the period of the bifurcating periodic solution at critical values of ${\it\tau}$. Finally, numerical simulations are investigated to support our theoretical results.
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KARAOGLU, E., and H. MERDAN. "HOPF BIFURCATION ANALYSIS FOR A RATIO-DEPENDENT PREDATOR–PREY SYSTEM INVOLVING TWO DELAYS." ANZIAM Journal 55, no. 3 (2014): 214–31. http://dx.doi.org/10.1017/s1446181114000054.

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AbstractThe aim of this paper is to give a detailed analysis of Hopf bifurcation of a ratio-dependent predator–prey system involving two discrete delays. A delay parameter is chosen as the bifurcation parameter for the analysis. Stability of the bifurcating periodic solutions is determined by using the centre manifold theorem and the normal form theory introduced by Hassard et al. Some of the bifurcation properties including the direction, stability and period are given. Finally, our theoretical results are supported by some numerical simulations.
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Fofana, M. S. "Dimensional reduction of nonlinear time delay systems." International Journal of Mathematics and Mathematical Sciences 2005, no. 2 (2005): 311–28. http://dx.doi.org/10.1155/ijmms.2005.311.

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Whenever there is a time delay in a dynamical system, the study of stability becomes an infinite-dimensional problem. The centre manifold theorem, together with the classical Hopf bifurcation, is the most valuable approach for simplifying the infinite-dimensional problem without the assumption of small time delay. This dimensional reduction is illustrated in this paper with the delay versions of the Duffing and van der Pol equations. For both nonlinear delay equations, transcendental characteristic equations of linearized stability are examined through Hopf bifurcation. The infinite-dimensiona
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Zhang, Huayong, Ge Pan, Tousheng Huang, Tianxiang Meng, Jieru Wang, and Hai Huang. "Bifurcation Dynamics and Pattern Formation of a Three-Species Food Chain System with Discrete Spatiotemporal Variables." Zeitschrift für Naturforschung A 74, no. 11 (2019): 945–59. http://dx.doi.org/10.1515/zna-2019-0086.

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AbstractThe bifurcation dynamics and pattern formation of a discrete-time three-species food chain system with Beddington–DeAngelis functional response are investigated. Via applying the centre manifold theorem and bifurcation theorems, the occurrence conditions for flip bifurcation and Neimark–Sacker bifurcation as well as Turing instability are determined. Numerical simulations verify the theoretical results and reveal many interesting dynamic behaviours. The flip bifurcation and the Neimark–Sacker bifurcation both induce routes to chaos, on which we find period-doubling cascades, invariant
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Carpintero, D. D., and J. C. Muzzio. "The Lyapunov exponents and the neighbourhood of periodic orbits." Monthly Notices of the Royal Astronomical Society 495, no. 2 (2020): 1608–12. http://dx.doi.org/10.1093/mnras/staa1227.

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ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to
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Zuo, Hong Kun, Quan Bao Ji, and Yi Zhou. "Hopf Bifurcation and Numerical Simulation in a Calcium Oscillation Model." Applied Mechanics and Materials 226-228 (November 2012): 510–15. http://dx.doi.org/10.4028/www.scientific.net/amm.226-228.510.

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Calcium oscillations play a very important role in providing the intracellular signaling, and many mathematical models have been proposed to describe calcium oscillations. The Shen-Larter model presented here is based on calcium-induced calcium release (CICR) and the inositol trisphosphate cross-coupling (ICC). Nonlinear dynamics of this model is investigated by using the centre manifold theorem and bifurcation theory, including the variation in classification and stability of equilibria with different parameter values. The results show that the appearance and disappearance of calcium oscillat
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Daogao, Wei, Jiang Tong, Chen Changhe, Jiang Yibin, Pan Ning, and Pan Zhijie. "Hopf bifurcation character of an interactive vehicle–road shimmy system under bisectional road conditions." Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 231, no. 3 (2016): 405–17. http://dx.doi.org/10.1177/0954407016640874.

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The adhesion coefficients of a bisectional road have significant coupled influences on the shimmy characteristics of the front wheels of a vehicle. A four-degree-of-freedom model for a representative sport utility vehicle was established. This model considered the adhesion coefficients of a bisectional road and the friction of the steering system of the suspension. The existence and stability of the system’s limit cycles were qualitatively determined using the Hopf bifurcation theorem and the centre manifold theory based on the model. The influences of the adhesion coefficients on the Hopf bif
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Dissertations / Theses on the topic "Centre manifold theorem"

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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 degr
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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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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-va
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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<br>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 m
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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
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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 lo
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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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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
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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 cad
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Books on the topic "Centre manifold theorem"

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Dumortier, Freddy. Canard cycles and center manifolds. American Mathematical Society, 1996.

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Gérard, Iooss, ed. Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. 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. 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. 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. American Mathematical Society, 2014.

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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, et al. 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 r
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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
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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 an
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Uro, Risto, Juliette J. Day, Rikard Roitto, and Richard E. DeMaris, eds. The Oxford Handbook of Early Christian Ritual. Oxford University Press, 2018. http://dx.doi.org/10.1093/oxfordhb/9780198747871.001.0001.

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Scholars of religion have long assumed that ritual and belief constitute the fundamental building blocks of religious traditions and that these two components of religion are interrelated and interdependent in significant ways. Generations of New Testament and early Christian scholars have produced detailed analyses of the belief systems of nascent Christian communities, including their ideological and political dimensions, but have by and large ignored ritual as an important element of early Christian religion and as a factor contributing to the rise and the organization of the movement. In r
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Book chapters on the topic "Centre manifold theorem"

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Chen, Yushu, and Andrew Y. T. Leung. "Centre Manifold Theorem and Normal Form Of Vector Fields." In Bifurcation and Chaos in Engineering. Springer London, 1998. http://dx.doi.org/10.1007/978-1-4471-1575-5_5.

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Rand, Richard H., and Dieter Armbruster. "Center Manifolds." In Perturbation Methods, Bifurcation Theory and Computer Algebra. 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. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64533-5_6.

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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. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-64533-5_5.

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Böhmer, Klaus. "On Hybrid Methods for Bifurcation and Center Manifolds for General Operators." In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/978-3-642-56589-2_4.

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Behrens, Stefan, Mark Powell, and Arunima Ray. "Context for the Disc Embedding Theorem." In The Disc Embedding Theorem. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198841319.003.0001.

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‘Context for the Disc Embedding Theorem’ explains why the theorem is the central result in the study of topological 4-manifolds. After recalling surgery theory and the proof of the s-cobordism theorem for high-dimensional manifolds, the chapter explains what goes wrong when trying to apply the same techniques in four dimensions and how to start overcoming these problems. The complete statement of the disc embedding theorem is provided. Finally the most important consequences to manifold theory are listed, including a proof of why Alexander polynomial one knots are topologically slice and the existence of exotic smooth structures on 4-dimensional Euclidean space.
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Kim, Min Hoon, Patrick Orson, JungHwan Park, and Arunima Ray. "Good Groups." In The Disc Embedding Theorem. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780198841319.003.0019.

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Good groups are defined in terms of whether capped gropes of height 1.5 contain certain types of immersed discs. The disc embedding theorem holds for 4-manifolds with good fundamental group. It is proven that the infinite cyclic group and finite groups are good, and that extensions and colimits of good groups are good. This shows that all elementary amenable groups are good. The proofs use grope height raising and contraction, together with an analysis of how fundamental group elements behave under these operations. A central open problem in the study of topological 4-manifolds is to determine precisely which groups are good.
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Tsuruyama, Tatsuaki. "Non-Linear Kinetic Analysis of Protein Assembly Based on Center Manifold Theory." In Kinetic Theory. InTech, 2018. http://dx.doi.org/10.5772/intechopen.70750.

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Tu, Loring W. "Overview." In Introductory Lectures on Equivariant Cohomology. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691191751.003.0001.

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This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Many topological and geometrical quantities can be expressed as integrals on a manifold. Integrals are vitally important in mathematics. However, they are also rather difficult to compute. When a manifold has symmetries, as expressed by a group action, in many cases the localization formula in equivariant cohomology computes the integral as a finite sum over the fixed points of the action, providing a powerful computational tool.
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Schlarb, Damien B. "Introduction." In Melville's Wisdom. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780197585566.003.0001.

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This chapter explains the book’s central arguments, provides background, and theorizes its approach to literature. It shows how Old Testament wisdom philosophy informs Melville’s response to the manifold crises of modernity, specifically the loss of religious certainty and biblical authority. It explicates the book’s argument that Melville’s response can help us understand the dynamics at play in this crisis by providing historical and contextual background: first, it inscribes Melville into a transnational theological conversation about biblical interpretation that lasts from the early eighteenth to the mid-nineteenth century; second, it outlines the American reception of new analytical methods such as higher criticism, surveys America’s intellectual infrastructure, and discusses how romantic literature fills the interpretive lacuna left by theological scholarship. Finally, it defines “wisdom” and “religious skepticism” and explains its approach to literary criticism as informed by a hermeneutic theory of contemplation (theology) and by postsecular as well as postcritical approaches (literary studies).
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Conference papers on the topic "Centre manifold theorem"

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Fofana, M. S., and Satish Bukkapatnam. "Analysis of Degenerate Bifurcation in Machining Using a Nonlinear Model." 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-21582.

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Abstract We present a two degrees of freedom model of machining dynamics that compactly captures the nonlinear effects of regeneration, tool wear and plowing under orthogonal machining. Extensive simulation experiments show that the solutions to the governing equations of the model bear distinct similarities to signals from our earlier experiments, as reflected by both visual state portraits as well as the values of the quantifiers of a steady state dynamic system. Governing equations of the model lead to nonlinear delay differential equations, which we reduce to ordinary differential equation
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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 J
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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 norm
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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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Yuan, Yuan Y. "Multiple Bifurcations of Synchronized Oscillators With Delays." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84584.

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The synchronized oscillator with two discrete time delays is considered. The local stability of the zero solution of this system is investigated by studying the distributions of the eigenvalues of the system. A complete bifurcation analysis is given by employing the center manifold theorem, normal form method and bifurcation theorem. It is shown that the trivial fixed point may lose stability via a transcritical/pitchfork bifurcation, Hopf bifurcation or Bogdanov-Takens bifurcation. Some numerical simulation examples are given for justify the theoretical results.
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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 t
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7

Fofana, M. S. "A Unified Framework for the Study of Periodic Solutions of Nonlinear Delay Differential Equations." 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-21617.

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Abstract Periodic solutions of delay differential equations (DDEs) splitting into stable and unstable branches are examined in an infinite-dimensional space for fixed and multiple time delays. The center manifold theorem and the classical Hopf bifurcation theorem for the study of periodic solutions of ordinary differential equations (ODEs) are employed to reduce the infinite-dimensional character of the DDEs to finite-dimensional ODEs. Using integral averaging method, the vector field of the ODEs is converted and averaged into amplitude a and phase φ relations. From these relations bifurcation
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8

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 w
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Han, Congying, and Alexandre N. Pechev. "Time-Varying Nonlinear Designs for Underactuated Attitude Control With Two Reaction Wheels." In ASME 2008 9th Biennial Conference on Engineering Systems Design and Analysis. ASMEDC, 2008. http://dx.doi.org/10.1115/esda2008-59387.

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Reaction wheels are commonly employed for high precisision and agile pointing in spacecraft. In this paper, we consider the realization of three-axis stabilization with only two reaction wheels installed along two principle axes. In practise, the total angular momentum of the whole spacecraft periodically varies with time because of the existence of in-orbit disturbances. With regard to the time varying system, firstly, a time varying control law is presented for velocity dumping based on center manifold theory. Then a continuous nonlinear feedback controller with a periodic time varying term
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