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Academic literature on the topic 'Bifurcation from infinity'

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Published: 9 March 2023
Last updated: 10 March 2023

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Journal articles on the topic "Bifurcation from infinity"

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Krasnosel'skii, Alexander M., and Dmitrii I. Rachinskii. "Subharmonic bifurcation from infinity." Journal of Differential Equations 226, no. 1 (July 2006): 30–53. http://dx.doi.org/10.1016/j.jde.2005.09.011.

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S. Kozyakin, Victor, Alexander M. Krasnosel’skii, and Dmitrii I. Rachinskii. "Arnold tongues for bifurcation from infinity." Discrete & Continuous Dynamical Systems - S 1, no. 1 (2008): 107–16. http://dx.doi.org/10.3934/dcdss.2008.1.107.

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Korman, Philip. "An oscillatory bifurcation from infinity, and from zero." Nonlinear Differential Equations and Applications NoDEA 15, no. 3 (October 2008): 335–46. http://dx.doi.org/10.1007/s00030-008-7024-1.

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XUEJUN, GAO. "BIFURCATION BEHAVIORS OF THE TWO-STATE VARIABLE FRICTION LAW OF A ROCK MASS SYSTEM." International Journal of Bifurcation and Chaos 23, no. 11 (November 2013): 1350184. http://dx.doi.org/10.1142/s0218127413501848.

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Based on the stability and bifurcation theory of dynamical systems, the bifurcation behaviors and chaotic motions of the two-state variable friction law of a rock mass system are investigated by the bifurcation diagrams based on the continuation method and the Poincaré maps. The stick-slip of the rock mass is formulated as an initial values problem for an autonomous system of three coupled nonlinear ordinary differential equations (ODEs) of first order. The results of linear stability analysis indicate that there is an equilibrium position in the rock mass system. Furthermore, numerical results of nonlinear analysis indicate that the equilibrium position loses its stability from a sup-critical Hopf bifurcation point, and then the bifurcating periodic motion evolves into chaotic motion through a series of period-doubling bifurcations with the decreasing of the control parameter. The stick-slip and chaotic motions evolve into infinity in the end with some unstable periodic motions.
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WANG, HUAILEI, HAIYAN HU, and ZAIHUA WANG. "GLOBAL DYNAMICS OF A DUFFING OSCILLATOR WITH DELAYED DISPLACEMENT FEEDBACK." International Journal of Bifurcation and Chaos 14, no. 08 (August 2004): 2753–75. http://dx.doi.org/10.1142/s0218127404010990.

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This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.
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Gámez, José L., and Juan F. Ruiz-Hidalgo. "Bifurcation from Infinity and Resonance Results at High Eigenvalues in Dimension One." Journal of Function Spaces and Applications 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/284696.

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This paper is devoted to two different but related tags: firstly, the side of the bifurcation from infinity at every eigenvalue of the problem−u″(t)=λu(t)+g(t,u(t)),u∈H01(0,π), secondly, the solutions of the associated resonant problem at any eigenvalue. From the global shape of the nonlinearitygwe obtain computable integral values which will decide the behavior of the bifurcations and, consequently, the possibility of finding solutions of the resonant problems.
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Ward, James R. "A global continuation theorem and bifurcation from infinity for infinite-dimensional dynamical systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 126, no. 4 (1996): 725–38. http://dx.doi.org/10.1017/s0308210500023039.

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A general continuation theorem for isolated sets in infinite-dimensional dynamical systems is proved for a class of semiflows. This result is then used to prove the existence of continua of full bounded solutions bifurcating from infinity for systems of reaction—diffusion equations.
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Xing, Siyuan, and Albert C. J. Luo. "On an origami structure of period-1 motions to homoclinic orbits in the Rössler system." Chaos: An Interdisciplinary Journal of Nonlinear Science 32, no. 12 (December 2022): 123121. http://dx.doi.org/10.1063/5.0131970.

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In this paper, an origami structure of period-1 motions to spiral homoclinic orbits in parameter space is presented for the Rössler system. The edge folds of the origami structure are generated by the saddle-node bifurcations. For each edge, there are two layers to form the origami structure. On one layer of the origami structure, there is a pair of period-doubling bifurcations inducing periodic motions from period-1 to period-2 n motions [Formula: see text]. On such a layer, the unstable period-1 motion goes to the homoclinic orbits with a mapping eigenvalue approaching negative infinity. However, on the corresponding adjacent layers, no period-doubling bifurcations exist, and the unstable period-1 motion goes to the homoclinic orbit with a mapping eigenvalue approaching positive infinity. To determine the origami structure of the period-1 motions to homoclinic orbits, the implicit map of the Rössler system is developed through the discretization of the corresponding differential equations. The Poincaré mapping section can be selected arbitrarily. Before construction of the origami structure, the bifurcation diagram of periodic motions varying with one parameter is developed, and trajectories of stable periodic motions on the bifurcation diagram to homoclinic orbits are illustrated. Finally, the origami structures of period-1 motions to homoclinic orbits are developed through a few layers. This study provides the mathematical mechanisms of period-1 motions to homoclinic orbits, which help one better understand the complexity of periodic motions near the corresponding homoclinic orbit. There are two types of infinitely many homoclinic orbits in the Rössler system, and the corresponding mapping structures of the homoclinic orbits possess positive and negative infinity large eigenvalues. Such infinitely many homoclinic orbits are induced through unstable periodic motions with positive and negative eigenvalues accordingly.
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Li, Chunqiu, Desheng Li, and Zhijun Zhang. "Dynamic Bifurcation from Infinity of Nonlinear Evolution Equations." SIAM Journal on Applied Dynamical Systems 16, no. 4 (January 2017): 1831–68. http://dx.doi.org/10.1137/16m1107358.

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Llibre, J., and E. Ponce. "Hopf bifurcation from infinity for planar control systems." Publicacions Matemàtiques 41 (January 1, 1997): 181–98. http://dx.doi.org/10.5565/publmat_41197_11.

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Dissertations / Theses on the topic "Bifurcation from infinity"

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ZONGO, WEND BENEDO EMMANUEL. "BOUNDARY VALUE PROBLEMS FOR QUASI-LINEAR AND HIGHER-ORDER ELLIPTIC OPERATORS AND APPLICATION TO BIFURCATION AND STABILIZATION." Doctoral thesis, Università degli Studi di Milano, 2022. http://hdl.handle.net/2434/892091.

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In this thesis, we are interested in the study of nonlinear eigenvalue problem and the controllability of partial differential equations in a smooth bounded domain with boundary. The first part is devoted to the analysis of an eigenvalue problem for quasilinear elliptic operators involving homogeneous Dirichlet boundary conditions. We investigate the asymptotic behaviour of the spectrum of the related problem by showing on the one hand the bifurcation results from trivial solutions using the Krasnoselski bifurcation theorem and bifurcation from infinity using the Leray-Schauder degree on the other hand. We also prove the existence of multiple critical points using variational methods and the Krasnoselski genus. At last, we show a stabilization result for the damped plate equation with logarithmic decay of the associated energy. The proof of this result is achieved by means of a proper Carleman estimate for the fourth-order elliptic operators involving the so-called Lopatinskii-Šapiro boundary conditions and a resolvent estimate for the generator of the damped plate semigroup associated with these boundary conditions.
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Book chapters on the topic "Bifurcation from infinity"

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Le, Vy Khoi, and Klaus Schmitt. "Bifurcation from Infinity in Hilbert Spaces." In Global Bifurcation in Variational Inequalities, 79–102. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/978-1-4612-1820-3_5.

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Le, Vy Khoi, and Klaus Schmitt. "Bifurcation from Infinity in Banach Spaces." In Global Bifurcation in Variational Inequalities, 207–38. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/978-1-4612-1820-3_7.

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Freire, Emilio, Manuel Ordóñez, and Enrique Ponce. "Limit Cycle Bifurcation from a Persistent Center at Infinity in 3D Piecewise Linear Systems with Two Zones." In Trends in Mathematics, 55–58. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-55642-0_10.

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Zhao, Yulin, and Huaiping Zhu. "Bifurcation of Limit Cycles from a Non-Hamiltonian Quadratic Integrable System with Homoclinic Loop." In Infinite Dimensional Dynamical Systems, 445–79. New York, NY: Springer New York, 2012. http://dx.doi.org/10.1007/978-1-4614-4523-4_18.

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Walther, Hans-Otto. "Bifurcation from Homoclinic to Periodic Solutions by an Inclination Lemma with Pointwise Estimate." In Dynamics of Infinite Dimensional Systems, 459–70. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987. http://dx.doi.org/10.1007/978-3-642-86458-2_37.

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Tissandier, Alex. "A Leibnizian World." In Affirming Divergence, 88–116. Edinburgh University Press, 2018. http://dx.doi.org/10.3366/edinburgh/9781474417747.003.0005.

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This chapter uses concepts from Leibniz’s philosophy to provide an account of the metaphysical system Deleuze constructs in Difference and Repetition and Logic of Sense. This account has four key components. 1) An ideal continuum populated by reciprocally determined differential relations, from which individuals are produced. Leibniz’s infinitesimal calculus is the technique most suited to describe this continuum. 2) The singularities or events which populate the continuum and which eventually form the “predicates” which are included within individuals. An inverted version of Leibniz’s theory of infinite analysis, which Deleuze dubs ‘vice-diction’, allows us to describe how these singularities are distributed. 3) The relations of compossibility between singularities which allow the articulation of a structure prior to any logical relations of opposition or contradiction. In Leibniz, a divergence between singularities marks a bifurcation into two distinct possible worlds. In Deleuze, by contrast, divergent series resonate and communicate with one another. 4) An “ideal game” which presides over the actualisation of this pre-individual continuum through the genesis of individuals. In Leibniz this game is subject to the rules of a divine calculus in which God selects a “best of all possible worlds” whose harmony is guaranteed. Deleuze, however, will reject this theological constraint.
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Conference papers on the topic "Bifurcation from infinity"

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Gendelman, O. V. "Degenerate Bifurcation Scenarios in the Dynamics of Coupled Oscillators With Symmetric Nonlinearities." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84373.

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We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. Both the potential of the oscillator and of the coupling spring are adopted to be even-power polynomials with nonnegative coefficients. By defining the coupling parameter ε, the dynamics of this system at the limit ε → 0 and for finite ε is investigated. Bifurcation scenario of the nonlinear normal modes is revealed. The degeneracy in the dynamics is manifested by a ‘bifurcation from infinity’ where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by mechanism of successive cusp catastrophes with growth of the coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincare sections). For particular case of pure cubic nonlinearity of the oscillator and the coupling spring good agreement between quantitative analytical predictions and numerical results is observed.
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Arneodo, A., F. Argoul, and P. Richetti. "Symbolic dynamics in the Belousov-Zhabotinskii reaction: from Rössler’s intuition to experimental evidence for Shil’nikov homoclinic chaos." In Nonlinear Dynamics in Optical Systems. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/nldos.1990.is2.

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The Belousov-Zhabotinskii reaction has revealed most of the well-known scenarios to chaos including period-doubling, intermittency, quasiperiodicity, frequency locking, fractal torus …. However, although the data have been shown to display unambiguous features of deterministic chaos, the understanding of the nature and the origin of the observed behavior has been incomplete. In 1976, Rössler suggested an intuitive interpretation to explain chemical chaos. His feeling was that nonperiodic wandering trajectories might arise in chemical systems from a pleated slow manifold (Fig. 1a), if the flow on the lower surface of the pleat had the property of returning trajectories to a small neighborhood of an unstable focus lying on the upper surface. In this communication, we intend to revisit the terminology introduced by Rössler of “spiral-type”, “screw-type” and “funnel-type” strange attractors in terms of chaotic orbits that occur in nearly homoclinic conditions. According to a theorem by Shil’nikov, there exist uncountably many nonperiodic trajectories in systems which display a homoclinic orbit biasymptotic to a saddle-focus O, providing the following condition is fulfilled: ρ/λ < 1, where the eigenvalue of O are (−λ, ρ ± iω). This subset of chaotic trajectories is actually in one to one correspondance with a shift automorphism with an infinite number of symbols. Since homoclinic orbits are structurally unstable objects which lie on codimension-one hypersurfaces in the constraint space, one can reasonably hope to cross these hypersurfaces when following a one-parameter path. The bifurcation structure encountered near homoclinicity involves infinite sequences of saddle-node and period-doubling bifurcations. The aim of this paper is to provide numerical and experimental evidences for Shil’nikov homoclinic chaos in nonequilibrium chemical systems.
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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 equations of the form ℑ(a, φ) = 0 for the solution branches are derived.
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Lyubimov, D., T. Lyubimova, A. Sharifulin, and D. Volfson. "The Flows in Weightlessness Generated by Vibrations." In ASME/JSME 2004 Pressure Vessels and Piping Conference. ASMEDC, 2004. http://dx.doi.org/10.1115/pvp2004-3114.

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Vibration phenomena play an important role in many technical processes involving heated interfaces in the microgravity environment. The analysis of vibration effect on non-isothermal fluid in closed cavity is important for planning technological experiments in space. Control and optimization of these processes critically depend on the understanding of liquid response to the vibrations. With this aim the theoretical investigation for two kinds of problems are performed for infinite plane and cylindrical fluid layers. First of all we investigated simple case of the fluid response-thermal vibrational convection in a cylindrical fluid layer with rigid conducting boundaries. It is found that steady modes of thermal vibrational convection are subjected to various bifurcations. Bifurcations cause shay changes in heat transfer. The generalized Lorenz model is modified and used to conduct the analysis of bifurcations caused by the changing of the cavity shape and vibrational Rayleigh number. The shape of steady-state surface in space of {ψ, Rv, γ} is found, where ψ is the streamfunction of mean flow, Rv is the vibrational Rayleigh number, γ is the cavity curvature. The solution correctly illustrates the general view of steady states surface for the parameter values corresponding to the cavity with the curvature close to zero (thin cylindrical layer). The numerical solution of the vibrational convection equations is performed for plane and cylindrical fluid layers. The results of the analysis based on the generalized Lorenz model are compared with the data obtained by direct numerical simulation. It is shown that the steady-state surface is different from that in the Lorenz model. The bifurcation curve with extremum is found. Thus, bifurcations of complex shape could be observed. This is impossible in a Lorenz model. Second set of investigated problems is related to thermo-vibrational flows caused by oscillations of the boundaries. A general class of oscillations of the boundaries is shown to produce specific mechanisms, of mean transport of vorticity in a uniform fluid, and in addition heat/concentration transport in non-uniform fluid. We performed analytical and numerical study of the flow between two infinite cylinders when the axis of the inner one is subjected to high frequency, small amplitude and oscillations of circular polarization. This type of oscillation produces basic Couette-like mean flow in the gap between the cylinders through the diffusion of vorticity generated in the boundary layers near the rigid surfaces (so-called Schlichting mechanism). The linear stability analysis for this flow with and without radial temperature gradient is performed. Non linear regimes are studied numerically by finite difference method. The results of numerical and analytical simulation of fluid motion for both types vibration interactions are discussed as well.
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Jayaprakash, K. R., Alexander F. Vakakis, and Yuli Starosvetsky. "Strongly Nonlinear Spatially Periodic Traveling Waves in Granular Dimer Chains." In ASME 2012 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/detc2012-70298.

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In the present work we study the dynamics of spatially periodic traveling waves in granular 1:1 (each bead is followed and preceded by a bead of different mass and/or stiffness) dimer chain with no pre-compression. The dynamics of a 1:1 dimer chain is governed by a single parameter, the mass ratio of the two beads forming each dimer pair of the chain. In particular, we demonstrate numerically the formation of special families of traveling waves with spatially periodic waveforms that are realized in semi-infinite dimer chains with the application of an arbitrary impulse. These traveling waves were first observed in the form of oscillatory tails in the trail of the propagating primary pulse. The energy radiated by the propagating primary pulse manifests in the form of traveling waves of varying spatial periodicity depending on the mass ratio. These traveling waves depend only on the mass ratio and are rescalable with respect to any arbitrary applied energy. The dynamics of these families of traveling waves is systematically studied by considering finite dimer chains (termed the ‘reduced systems’) subject to periodic boundary conditions. We demonstrate that these waves may exhibit interesting bifurcations or loss of stability as the system parameter varies. In turn, these bifurcations and stability exchanges in infinite dimer chains are correlated to previous studies of pulse attenuation in finite dimer chains through efficient energy radiation from the propagating pulse to the far field, mainly in the form of traveling waves. Based on these results a new formulation of attenuation and propagation zones (stop and pass bands) in semi-infinite granular dimer chains is proposed.
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Huang, Jianzhe, Xilin Fu, Zhongliang Jing, and Siyuan Xing. "Discontinuous Dynamics and Bifurcation for Morphing Aircraft Switching on the Velocity Boundary." 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/detc2020-22008.

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Abstract The concept of morphing aircraft was developed many decades ago, and many researches on the morphing aircraft such as stability and control have been published. As the point of view of the dynamic theory, the dynamic system of the morphing aircraft is consisted with multiple subsystems, and each subsystem represents the morphing aircraft with specific structure to fulfill a particular flight task. The switching process from one structure to another for such a morphing aircraft is considered to be smooth and stable, and the switching time is also assumed to be infinitely small. In this paper, a morphing aircraft with high-speed structure, intermediate-structure and low-speed structure is studied. Such a morphing aircraft is set to switch between high-speed structure and low-speed structure when the speed of aircraft arrives a preset critical speed, and the analytical conditions for switchability is developed. If such a morphing aircraft cannot switch to a low-speed structure or high-speed structure at the moment when it arrives the critical speed, it will switch to an intermediate-structure and control to keep the speed remain constant. The analytical conditions for onset and vanish of such a morphing aircraft switching to the intermediate-structure are also provided. Mapping structure is defined to describe the periodic motions of such a morphing aircraft. The bifurcation scenario is calculated to show the complexity of such a hybrid dynamical system. A periodic motion is given to illustrate the flow of such a morphing aircraft switching on the velocity boundary.
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Butcher, Eric A., Venkatesh Deshmukh, and Ed Bueler. "Center Manifold Reduction of Periodic Delay Differential Systems." In ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/detc2007-34583.

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A technique for center manifold reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. Perturbation expansion converts the nonlinear response problem into solutions of a series of non-homogenous linear ordinary differential equations (ODEs) with time periodic coefficients. One set of linear non-homogenous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. Center manifold reduction on the map is then carried out. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation.
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Haelterman, M., S. Trillo, and S. Wabnitz. "Pulse pattern formation and chaos in fiber ring lasers." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/oam.1992.tuz3.

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Ring lasers containing a nonlinear dispersive fiber (eventually ion doped) should be described by an infinite dimensional map involving the nonlinear Schroedinger (NLS) equation. This includes, for example, sinchronously pumped modulational instability lasers and erbium doped soliton lasers. We show that, under proper hypotheses, spatial averaging procedures allow us to describe the periodic forcing and damping along the ring by means of a single partial derivative equation, in the form of a driven-damped NLS or forced Ginzburg–Landau equation. The model permits us to study the formation of dissipative temporal patterns from an injected signal (i.e., from modulational instability). The numerical integration of this equation, in agreement with that of the map, reveals the existence of regimes where either the stable emission of pulse trains or the chaotic emission of temporal structures may occur. We are able to determine the role of the different parameters, such as pump power, fiber dispersion, cavity detuning, and gain dispersion (for active fibers), in laser emission. A simple insight is given by means of linear stability analysis, whereas we describe the nonlinear depleted stage of the emission by means of a truncated modal expansion. The reduced description preserves the complexity of the original model and permits us to characterize the existence of Hopf bifurcations, leading to periodic emission of patterns as well as the chaotic regime of operation.
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Watanabe, T., H. Furukawa, and M. Suzuki. "Effects of Geometrical Structure on Flows Around a Rotating Disk in an Enclosure." In ASME/JSME 2007 5th Joint Fluids Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/fedsm2007-37067.

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Flows around a rotating disk in a cylindrical enclosure are typical models of flows found in fluid machinery and chemical reactors. They have their practical applications and draw engineering interests. When the radius of the disk is infinite, it is known that circular rolls, spiral rolls, turbulent spirals and turbulent spots appear. In this case, the parameters governing the flows are the Reynolds number based on the angular velocity of the disk and the axial gap between the disk surface and the end wall of the enclosure. We consider, in this paper, a more practical configuration. The disk has its thickness comparable with the axial height of the enclosure, and the radial gap between the disk rim and the side wall of the enclosure is not negligible. Vortex flows are driven by the centrifugal force around the disk rim, and they are expected to have effects on the entire flow. We performed numerical and experimental studies and investigated the unsteady three-dimensional behaviors. A new criterion to identify flow patterns is introduced and the Hopf bifurcation points from the axisymmetric flows to the three-dimensional flows are determined. The phase velocity of the spiral rolls are measured by a time-dependent analysis. The influence of the geometrical structure on the phase velocity is estimated. New types of flows are found, where bead-like vortices appear and spiral rolls with positive and negative front angles coexist.
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Khan, Abdul Muqtadir, and Jon Olson. "Impact of Hydraulic Fracture Fairway Development in Multi-Stage Horizontal Laterals - A Production Flow Simulation Study." In SPE Annual Technical Conference and Exhibition. SPE, 2021. http://dx.doi.org/10.2118/205859-ms.

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Abstract The vast shale gas and tight oil reservoirs cannot be economically developed without multi-stage hydraulic fracture treatments. Owing to the disparity in the density of natural fractures and the different in-situ stress conditions in these formations, micro-seismic fracture mapping has shown that hydraulic fracture treatments develop a range of large-scale fracture networks. The effect of these various fracture geometries on production is a subject matter in question. The fracture networks approximated with micro-seismic mapping are integrated with a commercial numerical production simulator that discretely models different network structures. Two fracture geometries have been broadly proposed, i.e., orthogonal and transverse. The orthogonal pattern represents a network with cross-cutting fractures orthogonal to each other, whereas transverse profile maps uninterrupted fractures achieving maximum depth of penetration into the reservoir. The response for a single stage is further investigated by comparing the propagation of each stage to be dendritic versus planar. A dendritic propagation is a bifurcation of the induced hydraulic fracture due to the intersection with the natural fracture (failure along the plane of weakness). For the same injected fracture treatment volume, the transverse network attains a higher penetration into the reservoir, achieves a higher stimulated reservoir volume (SRV), and produces around 40-65% more than the orthogonal network over a timespan of 10 years. The SRV will largely dictate the drainage area in a tight environment. The cumulative production rises until the pressure drawdown reaches the extent of the fracture fairway. For the orthogonal network, the unstimulated reservoir boundary is reached at a sooner time than the transverse network. It is found that by increasing the fracture spacing in both the networks from 100 ft to 150 ft, the relative production was enhanced in the orthogonal network by 41%, but when it was further increased to 200 ft- there was a minor drop (not increase) in the relative production (4.5%). For an infinite conductivity fracture, the width of the fracture has minimal effect on oil and gas production. For the dendritic pattern, the size of the SRV created due to the interaction between the induced and natural fractures largely depends on the length of natural fractures and the point of interaction (center, off-center, or extremity). Effect of length, distance of natural fracture from wellbore, and the point of interaction is evaluated. A novel approach for reservoir simulation is used, where porosity (instead of permeability) is used as a scaling parameter for the fracture width. The forward modeling effort, including the comparative fracture geometries setup, induced, and natural fracture interaction parametric study, is unique.
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