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Relevant bibliographies by topics / Heteroclinic and homoclinic bifurcations

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Academic literature on the topic 'Heteroclinic and homoclinic bifurcations'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 February 2022

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Journal articles on the topic "Heteroclinic and homoclinic bifurcations"

1

GENG, FENGJIE, DAN LIU, and DEMING ZHU. "BIFURCATIONS OF GENERIC HETEROCLINIC LOOP ACCOMPANIED BY TRANSCRITICAL BIFURCATION." International Journal of Bifurcation and Chaos 18, no. 04 (2008): 1069–83. http://dx.doi.org/10.1142/s0218127408020847.

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The bifurcations of generic heteroclinic loop with one nonhyperbolic equilibrium p1and one hyperbolic saddle p2are investigated, where p1is assumed to undergo transcritical bifurcation. Firstly, we discuss bifurcations of heteroclinic loop when transcritical bifurcation does not happen, the persistence of heteroclinic loop, the existence of homoclinic loop connecting p1(resp. p2) and the coexistence of one homoclinic loop and one periodic orbit are established. Secondly, we analyze bifurcations of heteroclinic loop accompanied by transcritical bifurcation, namely, nonhyperbolic equilibrium p1s

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XU, YANCONG, DEMING ZHU, and FENGJIE GENG. "CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS." International Journal of Bifurcation and Chaos 18, no. 12 (2008): 3689–701. http://dx.doi.org/10.1142/s0218127408022652.

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Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions a

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CAO, HONGJUN, and GUANRONG CHEN. "GLOBAL AND LOCAL CONTROL OF HOMOCLINIC AND HETEROCLINIC BIFURCATIONS." International Journal of Bifurcation and Chaos 15, no. 08 (2005): 2411–32. http://dx.doi.org/10.1142/s0218127405013393.

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A comprehensive resonant optimal control method is developed and discussed for suppressing homoclinic and heteroclinic bifurcations of a general one-degree-of-freedom nonlinear oscillator. Based on an adjustable phase shift, the primary resonant optimal control method is presented. By solving an optimization problem for the optimal amplitude coefficients to be used as the control parameters, the force term as the controller can be designed. Three kinds of resonant optimal control methods are compared. The control mechanism of the primary resonant optimal control method is to enlarge to the lar

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Geng, Fengjie, and Junfang Zhao. "Bifurcations of Orbit and Inclination Flips Heteroclinic Loop with Nonhyperbolic Equilibria." Scientific World Journal 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/585609.

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The bifurcations of heteroclinic loop with one nonhyperbolic equilibrium and one hyperbolic saddle are considered, where the nonhyperbolic equilibrium is supposed to undergo a transcritical bifurcation; moreover, the heteroclinic loop has an orbit flip and an inclination flip. When the nonhyperbolic equilibrium does not undergo a transcritical bifurcation, we establish the coexistence and noncoexistence of the periodic orbits and homoclinic orbits. While the nonhyperbolic equilibrium undergoes the transcritical bifurcation, we obtain the noncoexistence of the periodic orbits and homoclinic orb

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5

ALGABA, ANTONIO, MANUEL MERINO, and ALEJANDRO J. RODRÍGUEZ-LUIS. "HOMOCLINIC INTERACTIONS NEAR A TRIPLE-ZERO DEGENERACY IN CHUA'S EQUATION." International Journal of Bifurcation and Chaos 22, no. 06 (2012): 1250129. http://dx.doi.org/10.1142/s0218127412501295.

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In this work, we consider some degeneracies of homoclinic and heteroclinic connections organized by the triple-zero degeneracy, in Chua's equation. This allows us to numerically study the homoclinic-heteroclinic transition exhibited by the curve of Takens–Bogdanov bifurcations as it passes through the triple-zero degeneracy. Several codimension-two degenerate homoclinic and heteroclinic connections organized by the triple-zero bifurcation are involved in this transitional homoclinic-heteroclinic mechanism. In particular, we point out that the existence of a curve of T-points and a curve of Bel

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6

Zimmermann, Martín G., and Mario A. Natiello. "Homoclinic and Heteroclinic Bifurcations Close to a Twisted Heteroclinic Cycle." International Journal of Bifurcation and Chaos 08, no. 02 (1998): 359–75. http://dx.doi.org/10.1142/s0218127498000218.

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We study the interaction of a transcritical (or saddle-node) bifurcation with a codimension-0/codimension-2 heteroclinic cycle close to (but away from) the local bifurcation point. The study is motivated by numerical observations on the traveling wave ODE of a reaction–diffusion equation. The manifold organization is such that two branches of homoclinic orbits to each fixed point are created when varying the two parameters controlling the codimension-2 loop. It is shown that the homoclinic orbits may become degenerate in an orbit-flip bifurcation. We establish the occurrence of multi-loop homo

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7

SHUI, SHULIANG, JINGJING LI, and XUYANG ZHANG. "NONRESONANT BIFURCATIONS OF HETEROCLINIC LOOPS WITH ONE INCLINATION FLIP." International Journal of Bifurcation and Chaos 21, no. 01 (2011): 255–73. http://dx.doi.org/10.1142/s0218127411028404.

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Heteroclinic bifurcations in four-dimensional vector fields are investigated by setting up local coordinates near a heteroclinic loop. This heteroclinic loop consists of two principal heteroclinic orbits, but there is one stable foliation that involves an inclination flip. The existence, nonexistence, coexistence and uniqueness of the 1-heteroclinic loop, 1-homoclinic orbit, and 1-periodic orbit are studied. Also, the nonexistence, existence of the 2-homoclinic and 2-periodic orbit are demonstrated.

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Cao, Q. J., Y. W. Han, T. W. Liang, M. Wiercigroch, and S. Piskarev. "Multiple Buckling and Codimension-Three Bifurcation Phenomena of a Nonlinear Oscillator." International Journal of Bifurcation and Chaos 24, no. 01 (2014): 1430005. http://dx.doi.org/10.1142/s0218127414300055.

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In this paper, we investigate the global bifurcations and multiple bucklings of a nonlinear oscillator with a pair of strong irrational nonlinear restoring forces, proposed recently by Han et al. [2012]. The equilibrium stabilities of multiple snap-through buckling system under static loading are analyzed. It is found that complex bifurcations are exhibited of codimension-three with two parameters at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the correspon

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9

LIU, XINGBO, ZHENZHEN WANG, and DEMING ZHU. "BIFURCATION OF ROUGH HETEROCLINIC LOOP WITH ORBIT FLIPS." International Journal of Bifurcation and Chaos 22, no. 11 (2012): 1250278. http://dx.doi.org/10.1142/s0218127412502781.

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In this paper, heteroclinic loop bifurcations with double orbit flips are investigated in four-dimensional vector fields. We obtain the bifurcation equations by setting up a local coordinate system near the rough heteroclinic orbit and establishing the Poincaré map. By means of the bifurcation equations, we investigate the existence, coexistence and noncoexistence of periodic orbit, homoclinic loop and heteroclinic loop under some nongeneric conditions. The approximate expressions of corresponding bifurcation curves (or surfaces) are also given. An example of application is also given to demon

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10

AGLIARI, ANNA, GIAN-ITALO BISCHI, ROBERTO DIECI, and LAURA GARDINI. "GLOBAL BIFURCATIONS OF CLOSED INVARIANT CURVES IN TWO-DIMENSIONAL MAPS: A COMPUTER ASSISTED STUDY." International Journal of Bifurcation and Chaos 15, no. 04 (2005): 1285–328. http://dx.doi.org/10.1142/s0218127405012685.

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In this paper we describe some sequences of global bifurcations involving attracting and repelling closed invariant curves of two-dimensional maps that have a fixed point which may lose stability both via a supercritical Neimark bifurcation and a supercritical pitchfork or flip bifurcation. These bifurcations, characterized by the creation of heteroclinic and homoclinic connections or homoclinic tangles, are first described through qualitative phase diagrams and then by several numerical examples. Similar bifurcation phenomena can also be observed when the parameters in a two-dimensional param

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Dissertations / Theses on the topic "Heteroclinic and homoclinic bifurcations"

1

Manukian, Vahagn Emil. "Existence and stability of multi-pulses with applicatons to nonlinear optics." The Ohio State University, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=osu1117644269.

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Glendinning, P. A. "Homoclinic bifurcations." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355879.

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Drysdale, David. "Homoclinic bifurcations." Thesis, University of Oxford, 1994. http://ora.ox.ac.uk/objects/uuid:b21e4a5c-7e10-4abc-8727-07b894ad6f15.

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Previously obtained results from the study of homoclinic bifurcations in ordinary differential equations are presented. The standard technique of analysis involves the construction of a Poincaré map on a surface near to the homoclinic point. This map is the composition of an inside map, with behaviour linearized about the homoclinic point, together with an outside map, with behaviour linearized about the homoclinic orbit. The Poincaré map is then reduced to a one-dimensional map, involving the return time between successive visits to the Poincaré surface. These standard techniques in the conte

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4

Wagenknecht, Thomas. "Homoclinic bifurcations in reversible systems." [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=974857009.

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Li, Dongchen. "Heterodimensional cycles near homoclinic bifurcations." Thesis, Imperial College London, 2016. http://hdl.handle.net/10044/1/42543.

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In this thesis we study bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium (with a one-dimensional unstable manifold) in flows with dimension four or higher. Particularly, we show that heterodimensional cycles can be born from such bifurcations. A heterodimensional cycle consists of two saddle periodic orbits having different indices (dimensions of unstable manifolds), and two heteroclinic connections between those orbits. We find heterodimensional cycles for the flow as the suspension of heterodimensional cycles for a Poincaré map around the homoclinic loops. Especially

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Jukes, Alice Claire. "On homoclinic bifurcations with symmetry." Thesis, Imperial College London, 2007. http://hdl.handle.net/10044/1/1265.

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This thesis deals with the description of solutions of symmetric dynamical systems that lie in the neighbourhood of a homoclinic or heteroclinic cycle. Homoclinic and heteroclinic cycles are the main mechanism by which complicated behaviour is known to arise in dynamical systems. A starting point for studying the consequences of the existence of homoclinic and heteroclinic cycles is to focus on non-wandering dynamics, that is, solutions that remain in the neighbourhood of such cycles, both in the phase space and in parameter space. For general vector fields, such studies have been carried out

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7

Young, Todd Ray. "Saddle-node bifurcations with homoclinic orbits." Diss., Georgia Institute of Technology, 1995. http://hdl.handle.net/1853/29855.

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Zhang, Tiansi. "Problems of homoclinic flips bifurcation in four-dimensional systems." Lyon, École normale supérieure (sciences), 2007. http://www.theses.fr/2007ENSL0431.

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9

George, Daniel Pucknell. "Bifurcations and homoclinic orbits in piecewise linear ordinary differential equations." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.233083.

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10

Sottocornola, Nicola. "Classification des cycles homoclines forces par symetrie dans R^4." Phd thesis, Université de Nice Sophia-Antipolis, 2002. http://tel.archives-ouvertes.fr/tel-00001813.

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Dans cette thèse on classe les cycles homoclines robustes de<br /> $\mathbb{R}^4$ en présence de symétries. On se borne au cas<br /> ou le groupe de symétrie $G$ est fini et, sans perte de<br /> généralité, contenu dans le groupe orthogonal $O(4)$. On<br /> montre notamment qu'une famille infinie de cycles existe; on<br /> fournit les générateurs, une présentation ainsi qu'une<br /> étude détaillée de ses groupes de symétrie. La topologie<br /> des cycles est aussi étudiée.\\ Ces cycles peuvent<br /> appara\^{\i}tre par bifurcation à partir d'un équilibre<br /> trivial. Ceci permet de détermin

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Books on the topic "Heteroclinic and homoclinic bifurcations"

1

Doedel, Eusebius. On locating homoclinic and heteroclinic orbits. Cornell Theory Center, Cornell University, 1993.

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2

Thomsen, Klaus. C*-algebras of homoclinic and heteroclinic structure in expansive dynamics. American Mathematical Society, 2010.

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3

C*-algebras of homoclinic and heteroclinic structure in expansive dynamics. American Mathematical Society, 2010.

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4

Global aspects of homoclinic bifurcations of vector fields. American Mathematical Society, 1996.

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5

Freidman, M. J. Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems. National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Program, 1993.

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6

Friedman, M. J. Accurate computation and continuation of homoclinic and heteroclinic orbits for singular perturbation problems. George C. Marshall Space Flight Center, 1993.

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7

Hyberbolic [sic] periodic solutions, heteroclinic connections, and transversal homoclinic points in autonomous differential delay equations. American Mathematical Society, 1989.

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8

Floris, Takens, ed. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations: Fractal dimensions and infinitely many attractors. Cambridge University Press, 1993.

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9

Palis, Jacob. Hyperbolicity, stability and chaos at homoclinic bifurcations: Fractal dimensions and infinitely many attractors in dynamics. Cambridge University Press, 1995.

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1929-, McLeod J. Bryce, ed. Classical methods in ordinary differential equations: With applications to boundary value problems. American Mathematical Society, 2012.

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Book chapters on the topic "Heteroclinic and homoclinic bifurcations"

1

Wiggins, Stephen. "Homoclinic and Heteroclinic Motions." In Global Bifurcations and Chaos. Springer New York, 1988. http://dx.doi.org/10.1007/978-1-4612-1042-9_3.

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2

Han, Maoan, and Pei Yu. "Limit Cycles Near a Homoclinic or Heteroclinic Loop." In Normal Forms, Melnikov Functions and Bifurcations of Limit Cycles. Springer London, 2012. http://dx.doi.org/10.1007/978-1-4471-2918-9_8.

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3

Kuznetsov, Yuri A. "Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria." In Elements of Applied Bifurcation Theory. Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4757-2421-9_6.

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Kuznetsov, Yuri A. "Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria." In Elements of Applied Bifurcation Theory. Springer New York, 2004. http://dx.doi.org/10.1007/978-1-4757-3978-7_6.

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5

Rodríguez-Luis, A. J., E. Freire, and E. Ponce. "A Method for Homoclinic and Heteroclinic Continuation in Two and Three Dimensions." In Continuation and Bifurcations: Numerical Techniques and Applications. Springer Netherlands, 1990. http://dx.doi.org/10.1007/978-94-009-0659-4_13.

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Fowler, Andrew, and Mark McGuinness. "Homoclinic Bifurcations." In Chaos. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-32538-1_4.

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Coullet, P., E. Risler, and N. Vanderberghe. "Spatial Unfolding of Homoclinic Bifurcations." In Nonlinear PDE’s in Condensed Matter and Reactive Flows. Springer Netherlands, 2002. http://dx.doi.org/10.1007/978-94-010-0307-0_18.

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Viana, Marcelo. "Homoclinic Bifurcations and Strange Attractors." In Real and Complex Dynamical Systems. Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-015-8439-5_10.

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Homburg, Ale Jan. "Cascades of Homoclinic Doubling Bifurcations." 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_12.

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Akhmet, Marat, Mehmet Onur Fen, and Ejaily Milad Alejaily. "Homoclinic and Heteroclinic Motions in Economic Models." In Dynamics with Chaos and Fractals. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-35854-9_9.

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Conference papers on the topic "Heteroclinic and homoclinic bifurcations"

1

Assegaf, Ahmad F., Elvan Yuniarti, and Husin Alatas. "Bifurcation of heteroclinic to homoclinic connection of static fluxon in S/F/S long Josephson junction." In THE 5TH ASIAN PHYSICS SYMPOSIUM (APS 2012). AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4917103.

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Otsuka, Kenju. "Transition from homoclinic to heteroclinic chaos in coupled laser arrays." In Dallas - DL tentative, edited by Cyrus D. Cantrell and Charles M. Bowden. SPIE, 1991. http://dx.doi.org/10.1117/12.46786.

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YAGASAKI, K. "DETECTION OF HOMOCLINIC AND HETEROCLINIC BEHAVIOR IN HAMILTONIAN SYSTEMS WITH SADDLE-CENTERS." In Proceedings of the International Conference on Differential Equations. WORLD SCIENTIFIC, 2005. http://dx.doi.org/10.1142/9789812702067_0163.

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Zhang, Wei, Feng-Xia Wang, and Hong-Bo Wen. "Studies on Codimension-3 Degenerate Bifurcations of the Flexible Beam." 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-21586.

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Abstract We present the analysis of codimension-3 degenerate bifurcations of a simply supported flexible beam subjected to harmonic axial excitation. The equation of motion with quintic nonlinear terms and the parametrical excitation for the simply supported flexible beam is derived. The main attention is focused on the dynamical properties of the global bifurcations including homoclinic bifurcations. With the aid of normal form theory, the explicit expressions of normal form associated with a double zero eigenvalues and Z2-symmetry for the averaged equations are obtained. Based on the normal

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Algaba, A., C. García, M. Maestre, and M. Merino. "PERIODIC MOTIONS ASSOCIATED TO HOMOCLINIC AND TORUS BIFURCATIONS IN THE CHUA'S EQUATION." In Proceedings of the IEEE Workshop. WORLD SCIENTIFIC, 2000. http://dx.doi.org/10.1142/9789812792662_0007.

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Lenci, Stefano, and Giuseppe Rega. "Non-Smooth Dynamics and Non-Classical Bifurcations in Impulse-Impact Oscillators." In ASME 1999 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1999. http://dx.doi.org/10.1115/detc99/vib-8054.

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Abstract Some aspects of the nonlinear dynamics of an impulse-impact oscillator are investigated. After an initial description of the prototype mechanical model used to illustrate the results, attention is paid to the classical local and global bifurcations which are at the base of the changes of dynamical regime. Some non-classical phenomena due to the particular nature of the investigated system are then considered. At a local level, it is shown that periodic solutions may appear (or disappear) through a non-classical bifurcation which involves synchronization of impulses and impacts. Simila

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Dai, Liming, and Changping Chen. "Homoclinic Orbit Bifurcation of a Rotating Truncated Conical Shell." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-10605.

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This is a study on Homoclinic bifurcation and subharmonic bifurcation of a truncated conical shallow shell rotating around a single axle and excited by a transverse periodic load. A systematic numerical approach is used to study the nonlinear motion of the system. The conditions under which bifurcations occur are determined on the basis of the characteristics of the rotating shell. Hamilton’s singular distributions are also investigated in details.

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Ng, Leslie, and Richard Rand. "Bifurcations in a Mathieu Equation With Cubic Nonlinearities: Part II." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-32410.

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In a previous paper [6], the authors investigated the dynamics of the equation: d2xdt2+(δ+εcost)x+εAx3+Bx2dxdt+Cxdxdt2+Ddxdt3=0. We used the method of averaging in the neighborhood of the 2:1 resonance in the limit of small forcing and small nonlinearity. We found that a degenerate bifurcation point occurs in the resulting slow flow and some of the bifurcations near this point were looked at. In this work we present additional results concerning the bifurcations around this point using analytic techniques and AUTO. An analytic approximation for a heteroclinic bifurcation curve is obtained. Add

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Chen, Yushu, Liangqiang Zhou, Fangqi Chen, et al. "The Method for Solving Homoclinic∕Heteroclinic Orbits- The Undetermined Coefficient Method and its applications for a New Chaotic System." In PROCEEDINGS OF THE 2ND INTERNATIONAL SYMPOSIUM ON COMPUTATIONAL MECHANICS AND THE 12TH INTERNATIONAL CONFERENCE ON THE ENHANCEMENT AND PROMOTION OF COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE. AIP, 2010. http://dx.doi.org/10.1063/1.3452133.

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Zhang, Wei, and Qi-liang Wu. "Multi-Pulse Chaotic Dynamics of Four-Dimensional Non-Autonomous Nonlinear System for a Truss Core Sandwich Plate." In ASME 2014 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/detc2014-34133.

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In this paper, an extended high-dimensional Melnikov method is used to investigate global and chaotic dynamics of a simply supported 3D-kagome truss core sandwich plate subjected to the transverse and the in-plane excitations. Based on the motion equation derived by Zhang and the method of multiple scales, the averaged equation is obtained for the case of principal parametric resonance and 1:2 sub-harmonic resonance for the first-order mode and primary resonance for the second-order mode. From the averaged equation obtained, the system is simplified to a three order standard form with a double

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