Relevant bibliographies by topics / Bifurcation

Academic literature on the topic 'Bifurcation'

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Published: 4 June 2021
Last updated: 7 July 2024

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Journal articles on the topic "Bifurcation"

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Liu, Ping, and Junping Shi. "A degenerate bifurcation from simple eigenvalue theorem." Electronic Research Archive 30, no. 1 (2021): 116–25. http://dx.doi.org/10.3934/era.2022006.

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<abstract><p>A new bifurcation from simple eigenvalue theorem is proved for general nonlinear functional equations. It is shown that in this bifurcation scenario, the bifurcating solutions are on a curve which is tangent to the line of trivial solutions, while in typical bifurcations the curve of bifurcating solutions is transversal to the line of trivial ones. The stability of bifurcating solutions can be determined, and examples from partial differential equations are shown to demonstrate such bifurcations.</p></abstract>
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Aizawa, H., K. Ikeda, M. Osawa, and J. M. Gaspar. "Breaking and Sustaining Bifurcations in SN-Invariant Equidistant Economy." International Journal of Bifurcation and Chaos 30, no. 16 (December 28, 2020): 2050240. http://dx.doi.org/10.1142/s0218127420502405.

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This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group [Formula: see text] [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics.
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Chien, C. S., Z. Mei, and C. L. Shen. "Numerical Continuation at Double Bifurcation Points of a Reaction–Diffusion Problem." International Journal of Bifurcation and Chaos 08, no. 01 (January 1998): 117–39. http://dx.doi.org/10.1142/s0218127498000097.

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We study double bifurcations of a reaction–diffusion problem, and numerical methods for the continuation of bifurcating solution branches. To ensure a correct reflection of the bifurcation scenario in discretizations and to reduce imperfection of bifurcations, we consider a preservation of multiplicities of the bifurcation points in the discrete problems. A continuation-Arnoldi algorithm is exploited to trace the solution branches, and to detect secondary bifurcations. Numerical results on the Brusselator equations confirm our analysis.
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Xu, Chaoqun, and Sanling Yuan. "Spatial Periodic Solutions in a Delayed Diffusive Predator–Prey Model with Herd Behavior." International Journal of Bifurcation and Chaos 25, no. 11 (October 2015): 1550155. http://dx.doi.org/10.1142/s0218127415501552.

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A delayed diffusive predator–prey model with herd behavior subject to Neumann boundary conditions is studied both theoretically and numerically. Applying Hopf bifurcation analysis, we obtain the critical conditions under which the model generates spatially nonhomogeneous bifurcating periodic solutions. It is shown that the spatially homogeneous Hopf bifurcations always exist and that the spatially nonhomogeneous Hopf bifurcations will arise when the diffusion coefficients are suitably small. The explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorems for partial functional differential equations.
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Li, Wei, Chunrui Zhang, and Mi Wang. "Analysis of the Dynamical Properties of Discrete Predator-Prey Systems with Fear Effects and Refuges." Discrete Dynamics in Nature and Society 2024 (May 11, 2024): 1–18. http://dx.doi.org/10.1155/2024/9185585.

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This paper examines the dynamic behavior of a particular category of discrete predator-prey system that feature both fear effect and refuge, using both analytical and numerical methods. The critical coefficients and properties of bifurcating periodic solutions for Flip and Hopf bifurcations are computed using the center manifold theorem and bifurcation theory. Additionally, numerical simulations are employed to illustrate the bifurcation phenomenon and chaos characteristics. The results demonstrate that period-doubling and Hopf bifurcations are two typical routes to generate chaos, as evidenced by the calculation of the maximum Lyapunov exponents near the critical bifurcation points. Finally, a feedback control method is suggested, utilizing feedback of system states and perturbation of feedback parameters, to efficiently manage the bifurcations and chaotic attractors of the discrete predator-prey model.
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Yan, Xiang-Ping, and Wan-Tong Li. "Global existence of periodic solutions in a simplified four-neuron BAM neural network model with multiple delays." Discrete Dynamics in Nature and Society 2006 (2006): 1–18. http://dx.doi.org/10.1155/ddns/2006/57254.

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We consider a simplified bidirectional associated memory (BAM) neural network model with four neurons and multiple time delays. The global existence of periodic solutions bifurcating from Hopf bifurcations is investigated by applying the global Hopf bifurcation theorem due to Wu and Bendixson's criterion for high-dimensional ordinary differential equations due to Li and Muldowney. It is shown that the local Hopf bifurcation implies the global Hopf bifurcation after the second critical value of the sum of two delays. Numerical simulations supporting the theoretical analysis are also included.
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Cai, Yongli, Zhanji Gui, Xuebing Zhang, Hongbo Shi, and Weiming Wang. "Bifurcations and Pattern Formation in a Predator–Prey Model." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850140. http://dx.doi.org/10.1142/s0218127418501407.

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In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.
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Zhai, Yanhui, Ying Xiong, Xiaona Ma, and Haiyun Bai. "Global Hopf Bifurcation Analysis for an Avian Influenza Virus Propagation Model with Nonlinear Incidence Rate and Delay." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/242410.

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The paper investigated an avian influenza virus propagation model with nonlinear incidence rate and delay based on SIR epidemic model. We regard delay as bifurcating parameter to study the dynamical behaviors. At first, local asymptotical stability and existence of Hopf bifurcation are studied; Hopf bifurcation occurs when time delay passes through a sequence of critical values. An explicit algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcation periodic solutions is derived by applying the normal form theory and center manifold theorem. What is more, the global existence of periodic solutions is established by using a global Hopf bifurcation result.
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SONG, YONGLI, JUNJIE WEI, and MAOAN HAN. "LOCAL AND GLOBAL HOPF BIFURCATION IN A DELAYED HEMATOPOIESIS MODEL." International Journal of Bifurcation and Chaos 14, no. 11 (November 2004): 3909–19. http://dx.doi.org/10.1142/s0218127404011697.

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In this paper, we consider the following nonlinear differential equation [Formula: see text] We first consider the existence of local Hopf bifurcations, and then derive the explicit formulas which determine the stability, direction and other properties of bifurcating periodic solutions, using the normal form theory and center manifold reduction. Further, particular attention is focused on the existence of the global Hopf bifurcation. By using the global Hopf bifurcation theory due to Wu [1998], we show that the local Hopf bifurcation of (1) implies the global Hopf bifurcation after the second critical value of the delay τ. Finally, numerical simulation results are given to support the theoretical predictions.
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Zhuang, Xiaolan, Qi Wang, and Jiechang Wen. "Numerical Dynamics of Nonstandard Finite Difference Method for Nonlinear Delay Differential Equation." International Journal of Bifurcation and Chaos 28, no. 11 (October 2018): 1850133. http://dx.doi.org/10.1142/s021812741850133x.

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In this paper, we study the dynamics of a nonlinear delay differential equation applied in a nonstandard finite difference method. By analyzing the numerical discrete system, we show that a sequence of Neimark–Sacker bifurcations occur at the equilibrium as the delay increases. Moreover, the existence of local Neimark–Sacker bifurcations is considered, and the direction and stability of periodic solutions bifurcating from the Neimark–Sacker bifurcation of the discrete model are determined by the Neimark–Sacker bifurcation theory of discrete system. Finally, some numerical simulations are adopted to illustrate the corresponding theoretical results.
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Dissertations / Theses on the topic "Bifurcation"

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Henderson, Michael E. Keller Herbert Bishop Keller Herbert Bishop. "Complex bifurcation /." Diss., Pasadena, Calif. : California Institute of Technology, 1985. http://resolver.caltech.edu/CaltechETD:etd-03262008-112516.

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Salih, Rizgar Haji. "Hopf bifurcation and centre bifurcation in three dimensional Lotka-Volterra systems." Thesis, University of Plymouth, 2015. http://hdl.handle.net/10026.1/3504.

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This thesis presents a study of the centre bifurcation and chaotic behaviour of three dimensional Lotka-Volterra systems. In two dimensional systems, Christopher (2005) considered a simple computational approach to estimate the cyclicity bifurcating from the centre. We generalized the technique to estimate the cyclicity of the centre in three dimensional systems. A lower bounds is given for the cyclicity of a hopf point in the three dimensional Lotka-Volterra systems via centre bifurcations. Sufficient conditions for the existence of a centre are obtained via the Darboux method using inverse Jacobi multiplier functions. For a given centre, the cyclicity is bounded from below by considering the linear parts of the corresponding Liapunov quantities of the perturbed system. Although the number obtained is not new, the technique is fast and can easily be adapted to other systems. The same technique is applied to estimate the cyclicity of a three dimensional system with a plane of singularities. As a result, eight limit cycles are shown to bifurcate from the centre by considering the quadratic parts of the corresponding Liapunov quantities of the perturbed system. This thesis also examines the chaotic behaviour of three dimensional Lotka-Volterra systems. For studying the chaotic behaviour, a geometric method is used. We construct an example of a three dimensional Lotka-Volterra system with a saddle-focus critical point of Shilnikov type as well as a loop. A construction of the heteroclinic cycle that joins the critical point with two other critical points of type planar saddle and axial saddle is undertaken. Furthermore, the local behaviour of trajectories in a small neighbourhood of the critical points is investigated. The dynamics of the Poincare map around the heteroclinic cycle can exhibit chaos by demonstrating the existence of a horseshoe map. The proof uses a Shilnikov-type structure adapted to the geometry of these systems. For a good understanding of the global dynamics of the system, the behaviour at infinity is also examined. This helps us to draw the global phase portrait of the system. The last part of this thesis is devoted to a study of the zero-Hopf bifurcation of the three dimensional Lotka-Volterra systems. Explicit conditions for the existence of two first integrals for the system and a line of singularity with zero eigenvalue are given. We characteristic the parameters for which a zero-Hopf equilibrium point takes place at any points on the line. We prove that there are three 3-parameter families exhibiting such equilibria. First order of averaging theory is also applied but we show that it gives no information about the possible periodic orbits bifurcating from the zero-Hopf equilibria.
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Binks, Douglas John. "Bifurcation phenomena in nematodynamics." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.306928.

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Taverner, S. "Bifurcation in physical systems." Thesis, University of Oxford, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.375327.

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Impey, M. D. "Bifurcation in Lapwood convection." Thesis, University of Bristol, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.234799.

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Arakawa, Vinicius Augusto Takahashi. "Um estudo de bifurcações de codimensão dois de campos de vetores /." São José do Rio Preto : [s.n.], 2008. http://hdl.handle.net/11449/94243.

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Orientador: Claudio Aguinaldo Buzzi
Banca: João Carlos da Rocha Medrado
Banca: Luciana de Fátima Martins
Resumo: Nesse trabalho são apresentados alguns resultados importantes sobre bifurcações de codimensão dois de campos de vetores. O resultado principal dessa dissertação e o teorema que d a o diagrama de bifurcação e os retratos de fase da Bifurcação de Bogdanov-Takens. Para a demonstracão são usadas algumas técnicas basicas de Sistemas Dinâmicos e Teoria das Singularidades, tais como Integrais Abelianas, desdobramentos de Sistemas Hamiltonianos, desdobramentos versais, Teorema de Preparação de Malgrange, entre outros. Outra importante bifurcação clássica apresentada e a Bifurca cão do tipo Hopf-Zero, quando a matriz Jacobiana possui um autovalor simples nulo e um par de autovalores imagin arios puros. Foram usadas algumas hipóteses que garantem propriedades de simetria do sistema, dentre elas, assumiuse que o sistema era revers vel. Assim como na Bifurcação de Bogdanov-Takens, foram apresentados o diagrama de bifurcao e os retratos de fase da Bifurcação Hopf-zero bifurcação reversível. As técnicas usadas para esse estudo foram a forma normal de Belitskii e o método do Blow-up polar.
Abstract: In this work is presented some important results about codimension two bifurcations of vector elds. The main result of this work is the theorem that gives the local bifurcation diagram and the phase portraits of the Bogdanov-Takens bifurcation. In order to give the proof, some classic tools in Dynamical System and Singularities Theory are used, such as Abelian Integral, versal deformation, Hamiltonian Systems, Malgrange Preparation Theorem, etc. Another classic bifurcation phenomena, known as the Hopf-Zero bifurcation, when the Jacobian matrix has a simple zero and a pair of purely imaginary eigenvalues, is presented. In here, is added the hypothesis that the system is reversible, which gives some symmetry in the problem. Like in Bogdanov-Takens bifurcation, the bifurcation diagram and the local phase portraits of the reversible Hopf-zero bifurcation were presented. The main techniques used are the Belitskii theory to nd a normal forms and the polar Blow-up method.
Mestre
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Jones, Mark C. W. "The bifurcation and secondary bifurcation of capillary-gravity waves in the presence of symmetry." Thesis, University of Bath, 1986. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.370986.

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Gaunersdorfer, Andrea, Cars H. Hommes, and Florian O. O. Wagener. "Bifurcation routes to volatility clustering." SFB Adaptive Information Systems and Modelling in Economics and Management Science, WU Vienna University of Economics and Business, 2000. http://epub.wu.ac.at/522/1/document.pdf.

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A simple asset pricing model with two types of adaptively learning traders, fundamentalists and technical analysts, is studied. Fractions of these trader types, which are both boundedly rational, change over time according to evolutionary learning, with technical analysts conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously in this model. Two mechanisms are proposed as an explanation. The first is coexistence of a stable steady state and a stable limit cycle, which arise as a consequence of a so-called Chenciner bifurcation of the system. The second is intermittency and associated bifurcation routes to strange attractors. Both phenomena are persistent and occur generically in nonlinear multi-agent evolutionary systems. (author's abstract)
Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"
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Fujihira, Takeo. "Hamiltonian Hopf bifurcation with symmetry." Thesis, Imperial College London, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.444087.

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Duka, E. D. "Bifurcation problems in finite elasticity." Thesis, University of Nottingham, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.384747.

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Books on the topic "Bifurcation"

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Hazewinkel, M., R. Jurkovich, and J. H. P. Paelinck, eds. Bifurcation Analysis. Dordrecht: Springer Netherlands, 1985. http://dx.doi.org/10.1007/978-94-009-6239-2.

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Kielhöfer, Hansjörg. Bifurcation Theory. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-0502-3.

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Waksman, Ron, and John A. Ormiston, eds. Bifurcation Stenting. Chichester, UK: John Wiley & Sons, Ltd, 2012. http://dx.doi.org/10.1002/9781444347005.

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Chen, Guanrong, David J. Hill, and Xinghuo Yu, eds. Bifurcation Control. Berlin, Heidelberg: Springer Berlin Heidelberg, 2003. http://dx.doi.org/10.1007/b79665.

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Kielhöfer, Hansjörg. Bifurcation Theory. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/b97365.

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Tanyi, G. E. Causality & bifurcation. Yaounde, Cameroon: Afromatics, 1987.

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Waksman, Ron. Bifurcation stenting. Chichester, West Sussex, UK: Wiley-Blackwell, 2012.

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Golubitsky, Martin, and John M. Guckenheimer, eds. Multiparameter Bifurcation Theory. Providence, Rhode Island: American Mathematical Society, 1986. http://dx.doi.org/10.1090/conm/056.

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Allgower, Eugene L., Klaus Böhmer, and Martin Golubitsky, eds. Bifurcation and Symmetry. Basel: Birkhäuser Basel, 1992. http://dx.doi.org/10.1007/978-3-0348-7536-3.

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Liebscher, Stefan. Bifurcation without Parameters. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-10777-6.

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Book chapters on the topic "Bifurcation"

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Borja, Ronaldo I. "Bifurcation." In Plasticity, 207–33. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-38547-6_9.

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Deimling, Klaus. "Bifurcation." In Nonlinear Functional Analysis, 378–425. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/978-3-662-00547-7_10.

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Zhou, Tianshou. "Bifurcation." In Encyclopedia of Systems Biology, 79–86. New York, NY: Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4419-9863-7_500.

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Zobitz, John M. "Bifurcation." In Exploring Modeling with Data and Differential Equations Using R, 251–62. Boca Raton: Chapman and Hall/CRC, 2022. http://dx.doi.org/10.1201/9781003286974-20.

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Chicone, Carmen. "Bifurcation." In Texts in Applied Mathematics, 623–91. Cham: Springer International Publishing, 2024. http://dx.doi.org/10.1007/978-3-031-51652-8_11.

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Barnett, William, and Gennady Cymbalyuk. "Bifurcation Analysis." In Encyclopedia of Computational Neuroscience, 366–71. New York, NY: Springer New York, 2015. http://dx.doi.org/10.1007/978-1-4614-6675-8_156.

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Lynch, Stephen. "Bifurcation Theory." In Dynamical Systems with Applications using MAPLE, 105–17. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4899-2849-8_8.

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Lynch, Stephen. "Bifurcation Theory." In Dynamical Systems with Applications Using Mathematica®, 135–53. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-61485-4_7.

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Lynch, Stephen. "Bifurcation Theory." In Dynamical Systems with Applications using Maple¿, 129–45. Boston: Birkhäuser Boston, 2009. http://dx.doi.org/10.1007/978-0-8176-4605-9_7.

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Arnold, Ludwig. "Bifurcation Theory." In Springer Monographs in Mathematics, 465–531. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998. http://dx.doi.org/10.1007/978-3-662-12878-7_9.

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Conference papers on the topic "Bifurcation"

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Lin, Guojian, Balakumar Balachandran, and Eyad H. Abed. "Bifurcation Behavior of a Supercavitating Vehicle." In ASME 2006 International Mechanical Engineering Congress and Exposition. ASMEDC, 2006. http://dx.doi.org/10.1115/imece2006-14052.

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In this effort, a numerical study of the bifurcation behavior of a supercavitating vehicle is conducted. The nonsmoothness of this system is due to the planing force acting on the vehicle. With a focus on dive-plane dynamics, bifurcations with respect to a quasi-static variation of the cavitation number are studied. The system is found to exhibit rich and complex dynamics including nonsmooth bifurcations such as the grazing bifurcation and smooth bifurcations such as Hopf bifurcations, cyclic-fold bifurcations, and period-doubling bifurcations, chaotic attractors, transient chaotic motions, and crises. The tailslap phenomenon of the supercavitating vehicle is identified as a consequence of the Hopf bifurcation followed by a grazing event. It is shown that the occurrence of these bifurcations can be delayed or triggered earlier by using dynamic linear feedback control aided by washout filters.
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Wu, Z. Q., and P. Yu. "Bifurcation Control of Ro¨ssler System." In ASME 2003 International Mechanical Engineering Congress and Exposition. ASMEDC, 2003. http://dx.doi.org/10.1115/imece2003-55035.

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In this paper, a new method is proposed for controlling bifurcations of nonlinear dynamical systems. This approach employs the idea used in deriving the transition variety sets of bifurcations with constraints to find the stability region of equilibrium points in parameter space. With this method, one can design, via a feedback control, appropriate parameter values to delay either static, or dynamic or both bifurcations as one wishes. The approach is applied to consider controlling bifurcations of the Ro¨ssler system. The uncontrolled Ro¨ssler has two equilibrium solutions, one of which exhibits static bifurcation while the other has Hopf bifurcation. When a feedback control based on the new method is applied, one can delay the bifurcations and even change the type of bifurcations. An optimal control law is obtained to stabilize the Ro¨ssler system using all feasible system parameter values. Numerical simulations are used to verify the analytical results.
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Ramuzat, A., H. Richard, and M. L. Riethmuller. "Unsteady Flows Within a 2D Model of Multiple Lung Bifurcations." In ASME 1999 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/imece1999-0363.

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Abstract In our environment, chronic pulmonary illness due to pollution effects or asthmatic problems are increasing. To identify the contribution of pollution effect on the alterations in breath patterns, a better understanding of the human pulmonary system is needed. As a result, fields to be investigated are mostly flows in the lungs at high breathing frequency and aerosol deposition in lung bifurcations under unsteady conditions. The respiration pattern has to be better understood and investigated, to have the possibility to get the most appropriate palliative treatment. Most of the medical treatments are based on aerosol deposition in the bronchial tree. The lung is a complex network of 23 successive generations of bifurcation (Weibel, 1963). The airways, from the trachea to the alveolar zone, divide by dichotomy and become shorter and narrower as they penetrate deeper into the lung. As a result, in vivo investigations of pulmonary flows are not possible, and in vitro experiments in models have to be performed. Flows in the bifurcating airways of the lung are modeled to determine velocities and pressure fields. A complete description of steady flow in a single 3D bifurcation has been previously performed by experimental and numerical modeling. As a result of this study, it has been shown that the first bifurcation influences the flow in the second and in the third bifurcation when the length of the second bifurcation is not sufficiently long. To extend the investigations to a system of three generations, an experimental study of steady and unsteady flows has been carried out on a 2D multiple bifurcations model.
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Pak, C. H., and Y. S. Choi. "On the Sensitivity of Non-Generic Bifurcation of Non-Linear Normal Modes." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34217.

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It is shown that a non-generic bifurcation of non-linear normal modes may occur if the ratio of linear natural frequencies is near r-to-one, r = 1, 3, 5 ·······. Non-generic bifurcations are explicitly obtained in the systems having certain symmetry, as observed frequently in literatures. It is found that there are two kinds of non-generic bifurcations, super-critical and sub-critical. The normal mode generated by the former kind is extended to large amplitude, but that by the latter kind is limited to small amplitude which depends on the difference between two linear natural frequencies and disappears when two frequencies are equal. Since a non-generic bifurcation is not generic, it is expected generically that if a system having a non-generic bifurcation is perturbed then the non-generic bifurcation disappears and generic bifurcation appear in the perturbed system. Examples are given to verify the change in bifurcations and to obtain the stability behavior of normal modes. It is found that if a system having a super-critical non-generic bifurcation is perturbed, then two new normal modes are generated, one is stable, but the other unstable, implying a saddle-node bifurcation. If the system having a sub-critical non-generic bifurcation is perturbed, then no new normal mode is generated, but there is an interval of instability on a normal mode, implying two saddle-node bifurcations on the mode. Application of this study is discussed.
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Burgner, Chris, Kimberly Turner, Nick Miller, and Steve Shaw. "Noise Squeezing Control for Bifurcation Sensing in MEMS." In ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2011. http://dx.doi.org/10.1115/detc2011-47461.

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This paper reports the development and implementation of a new method for tracking parameters at which dynamic bifurcations occur in MEMS. The underlying theory is developed for subcritical pitchfork bifurcations that occur near the subharmonic instability experienced near parametric resonance. The method relies on experimentally observed changes in response phase and amplitude just prior to the bifurcation, and these are used to forebode the bifurcation. These precursors are then employed in a feedback control scheme to stabilize a parametrically excited MEMS device at the edge of instability, making it highly sensitive to changes in device parameters. Implementation of the controller is shown through experimental validation. A comparison with the previous method of bifurcation detection for the same device shows that the new approach offers an improvement of over three orders of magnitude for the bifurcation point acquisition rate.
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Ikeda, Takashi. "Bifurcation Phenomena Caused by Two Nonlinear Dynamic Absorbers." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34714.

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The characteristics of two nonlinear vibration absorbers simultaneously attached to structures under harmonic excitation are investigated. The frequency response curves are theoretically determined using van der Pol’s method. It is found from the theoretical analysis that pitchfork bifurcations may appear on a part of the response curves which are stable in a system with one nonlinear dynamic absorber. Three steady-state solutions with different amplitudes appear just after the pitchfork bifurcation. After that, Hopf bifurcations may occur depending on the values of the system parameters, and amplitude- and phase-modulated motion including a chaotic vibration appears after the Hopf bifurcation. Lyapunov exponents are numerically calculated to prove the occurrence of a chaotic vibration. In addition, it is also found that only Hopf bifurcations, not pitchfork bifurcations, can occur even when the linear and nonlinear dynamic absorbers are combined.
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Hoi, Yiemeng, Bruce A. Wasserman, and David A. Steinman. "Normal Carotid Bifurcation Hemodynamics in Older Adults: Effect of Measured vs. Allometrically-Scaled Flow Rate Boundary Conditions." In ASME 2009 Summer Bioengineering Conference. American Society of Mechanical Engineers, 2009. http://dx.doi.org/10.1115/sbc2009-203639.

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A recent study demonstrated that certain geometric features of young adult carotid bifurcations are surrogate markers of disturbed flow [1]. However, as the carotid bifurcation ages, the bifurcation and flow waveform changes its geometry and dynamics, respectively [2, 3]. In addition to confounding systemic risk factors, such changes may interrupt homeostasis, potentially altering the likelihood of disturbed flow and atherosclerotic disease in the aging carotid bifurcation. Despite this, disturbed flow in aged bifurcations is typically inferred from image-based computational fluid dynamics (CFD) approaches using representative young and healthy adult flow waveforms.
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Guo, Siyu, and Albert C. J. Luo. "Period-1 to Period-2 Motions in a Discontinuous Oscillator." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22712.

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Abstract In this paper, grazing bifurcations on bifurcation trees in a discontinuous dynamical oscillator are discussed. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Thus, grazing bifurcations on a bifurcation tree of period-1 to period-2 motions varying spring stiffness are presented in a discontinuous oscillator with three domains divided by circular boundaries. The stability and bifurcations of period-1 and period-2 motions are discussed. From analytical predictions, periodic motions are simulated numerically. Stiffness effects on the periodic motions are discussed. Such studies will help one understand parameter effects in discontinuous dynamical systems, which can be applied for system design and control.
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Yamaguchi, Ryuhei, Kazuki Okada, and Tomohiko Ikeda. "Sinusoidal Oscillating Flow Through Asymmetrical Bifurcation." In ASME 1996 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 1996. http://dx.doi.org/10.1115/imece1996-1175.

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Abstract The respiratory system is characterized by many bifurcations. The bronchi are composed of symmetrical and asymmetrical bifurcations. It is generally considered that the gas mixing during the respiration are induced by the flow through the bronchi. In the present study, the axial and the transverse velocity components have been measured in the asymmetrical bifurcation by LDV. Consequently, the asymmetry of axial flow and the secondary flow around the flow divider have been clarified experimentally.
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Wang, Yuefang, Lefeng Lu¨, and Yingxi Liu. "On Multiple Hopf Bifurcations of Airflow Excited Vibration of a Translating String." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34451.

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This paper presents the stability and bifurcation of transverse motion of translating strings excited by a steady wind flowfield. The stability of the equilibrium configuration is presented for loss of stability and generation of limit cycles via the Hopf bifurcation. It is demonstrated that there are single, double and quadruple Hopf bifurcations in the parametric space that lead to the limit cycle motion. The method of Incremental Harmonic Balance is used to solve the limit cycle response of which the stability is determined by computation of the Floquet multipliers. For the forced vibration, it is pointed out that the periodic and quasi-periodic motions exist as parameters are changed. The quench frequency and the Neimark-Sacker (NS) bifurcation and flip bifurcation are obtained. The continuity software MATCONT is adopted and the Resonance 1:1, 1:3 and 1:4 as well as NS to NS bifurcations are presented. The bifurcation behavior reveals the complexity of the string’s motion response induce by aerodynamic excitations.
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Reports on the topic "Bifurcation"

1

Abed, Eyad H. Local Bifurcation Control,. Fort Belvoir, VA: Defense Technical Information Center, January 1987. http://dx.doi.org/10.21236/ada187435.

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Crawford, J., C. Kueny, B. Saphir, and B. Shadwick. Introduction to bifurcation theory. Office of Scientific and Technical Information (OSTI), November 1989. http://dx.doi.org/10.2172/5396551.

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Abed, E. H., J. H. Fu, H. C. Lee, and D. C. Liaw. Bifurcation Control of Nonlinear Systems. Fort Belvoir, VA: Defense Technical Information Center, January 1990. http://dx.doi.org/10.21236/ada444561.

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Wang, Hua O., and Eyad H. Abed. Bifurcation Control of Chaotic Dynamical Systems. Fort Belvoir, VA: Defense Technical Information Center, June 1992. http://dx.doi.org/10.21236/ada454958.

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Rosenblat, S. Bifurcation and Stability of Complex Flows. Fort Belvoir, VA: Defense Technical Information Center, October 1985. http://dx.doi.org/10.21236/ada161405.

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Marsden, J. E., and T. S. Ratiu. Stability, bifurcation, and control of Hamiltonian systems. Office of Scientific and Technical Information (OSTI), April 1993. http://dx.doi.org/10.2172/10142335.

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Marsden, J. E., and T. S. Ratiu. Stability, bifurcation, and control of Hamiltonian systems. Office of Scientific and Technical Information (OSTI), January 1993. http://dx.doi.org/10.2172/6642330.

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Mittelmann, Hans D. Continuation and Multi-Grid for Bifurcation Problems. Fort Belvoir, VA: Defense Technical Information Center, December 1993. http://dx.doi.org/10.21236/ada274965.

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Kim, Taihyun, and Eyad H. Abed. Stationary Bifurcation Control for Systems with Uncontrollable Linearization. Fort Belvoir, VA: Defense Technical Information Center, January 1999. http://dx.doi.org/10.21236/ada438515.

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Healy, Timothy J. Symmetry and Global Bifurcation in Nonlinear Solid Mechanics. Fort Belvoir, VA: Defense Technical Information Center, November 1987. http://dx.doi.org/10.21236/ada190521.

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