Academic literature on the topic 'Period-N bifurcations'
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Journal articles on the topic "Period-N bifurcations":
1
Yang, Fangyan, Yongming Cao, Lijuan Chen, and Qingdu Li. "Sequence of Routes to Chaos in a Lorenz-Type System." Discrete Dynamics in Nature and Society 2020 (January 23, 2020): 1–10. http://dx.doi.org/10.1155/2020/3162170.
Abstract:
This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.
2
van Kekem, Dirk L., and Alef E. Sterk. "Wave propagation in the Lorenz-96 model." Nonlinear Processes in Geophysics 25, no. 2 (April 27, 2018): 301–14. http://dx.doi.org/10.5194/npg-25-301-2018.
Abstract:
Abstract. In this paper we study the spatiotemporal properties of waves in the Lorenz-96 model and their dependence on the dimension parameter n and the forcing parameter F. For F > 0 the first bifurcation is either a supercritical Hopf or a double-Hopf bifurcation and the periodic attractor born at these bifurcations represents a traveling wave. Its spatial wave number increases linearly with n, but its period tends to a finite limit as n → ∞. For F < 0 and odd n, the first bifurcation is again a supercritical Hopf bifurcation, but in this case the period of the traveling wave also grows linearly with n. For F < 0 and even n, however, a Hopf bifurcation is preceded by either one or two pitchfork bifurcations, where the number of the latter bifurcations depends on whether n has remainder 2 or 0 upon division by 4. This bifurcation sequence leads to stationary waves and their spatiotemporal properties also depend on the remainder after dividing n by 4. Finally, we explain how the double-Hopf bifurcation can generate two or more stable waves with different spatiotemporal properties that coexist for the same parameter values n and F.
3
Honeycutt, Andrew, and Tony L. Schmitz. "Experimental Validation of Period-n Bifurcations in Milling." Procedia Manufacturing 5 (2016): 362–74. http://dx.doi.org/10.1016/j.promfg.2016.08.031.
4
Liu, Yun, and Xijuan Liu. "Bifurcations and Structures of the Parameter Space of a Discrete-Time SIS Epidemic Model." Journal of Mathematics 2022 (April 23, 2022): 1–14. http://dx.doi.org/10.1155/2022/2233452.
Abstract:
The dynamics of a discrete-time SIS epidemic model are reported in this paper. Three types of codimension one bifurcation, namely, transcritical, flip, Neimark–Sacker (N-S) bifurcations, and their intersection codimension two bifurcations including 1 : 2, 1 : 3, and 1 : 4 resonances are discussed. The necessary and sufficient conditions for detecting these types of bifurcation are derived using algebraic criterion methods. Numerical simulations are conducted not only to illustrate analytical results but also to exhibit complex behaviors which include period-doubling bifurcation in period − 2 , − 4 , − 8 , − 16 orbits, invariant closed cycles, and attracting chaotic sets. Especially, here we investigate the parameter space of the discrete model. We also investigate the organization of typical periodic structures embedded in a quasiperiodic region. We identify period-adding, Farey sequence of periodic structures embedded in this quasiperiodic region.
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SRINIVASAN, K. "MULTIPLE PERIOD DOUBLING BIFURCATION ROUTE TO CHAOS IN PERIODICALLY PULSED MURALI–LAKSHMANAN–CHUA (MLC) CIRCUIT." International Journal of Bifurcation and Chaos 18, no. 02 (February 2008): 541–55. http://dx.doi.org/10.1142/s021812740802046x.
Abstract:
In this paper, we study the effect of additional periodic forces in Murali–Lakshmanan–Chua (MLC) circuit. We show that the additional periodic forces of pulse type display novel dynamical features including multiple period doubling bifurcation route to chaos, followed by a rich variety of dynamical phenomena including enlarged periodic windows, attractor crises, distinctly modified bifurcation structures and so on. For certain types of periodic pulses, the circuit admits transcritical bifurcations preceding the onset of multiple period doubling bifurcations. We have characterized these dynamical behaviors using Lyapunov exponents, correlation dimension, Kaplan–Yorke dimension and power spectrum, which are found to be in good agreement with the experimental observations. We have also shown that the chaotic attractor becomes more complicated and their corresponding return maps are no longer simple for large n-periodic pulses, which has immense potential applications in secure communication. The above study facilitates one to generate any desired n-period doubling bifurcation behavior by applying n-periodic pulses to a chaotic system. Further, controlling and synchronization of chaos in this periodically pulsed MLC circuit have also been achieved by using suitable methods.
6
CHEN, XIANWEI, XIANGLING FU, and ZHUJUN JING. "COMPLEX DYNAMICS IN A PENDULUM EQUATION WITH A PHASE SHIFT." International Journal of Bifurcation and Chaos 22, no. 12 (December 2012): 1250307. http://dx.doi.org/10.1142/s0218127412503075.
Abstract:
Pendulum equation with a phase shift, parametric and external excitations is investigated in detail. By applying Melnikov's method, we prove the criteria of existence of chaos under periodic perturbation. Numerical simulations, including bifurcation diagrams of fixed points, bifurcation diagrams of the system in three- and two-dimensional spaces, homoclinic and heteroclinic bifurcation surfaces, Maximum Lyapunov exponents (ML), Fractal Dimension (FD), phase portraits, Poincaré maps are plotted to illustrate the theoretical analysis, and to expose the complex dynamical behaviors including the onset of chaos, sudden conversion of chaos to period orbits, interior crisis, periodic orbits, the symmetry-breaking of periodic orbits, jumping behaviors of periodic orbits, new chaotic attractors including two-three-four-five-six-eight-band chaotic attractors, nonchaotic attractors, period-doubling bifurcations from period-1, 2, 3 and 5 to chaos, reverse period-doubling bifurcations from period-3 and 5 to chaos, and so on.By applying the second-order averaging method and Melnikov's method, we obtain the criteria of existence of chaos in an averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 1, 2, 4, but cannot prove the criteria of existence of chaos in the averaged system under quasi-periodic perturbation for Ω = nω + ϵν, n = 3, 5 – 15, by Melnikov's method, where ν is not rational to ω. By using numerical simulation, we have verified our theoretical analysis and studied the effect of parameters of the original system on the dynamical behaviors generated under quasi-periodic perturbations, such as the onset of chaos, jumping behaviors of quasi-periodic orbits, interleaving occurrence of chaotic behaviors and nonchaotic behaviors, interior crisis, quasi-periodic orbits to chaotic attractors, sudden conversion of chaos to quasi-periodic behaviors, nonchaotic attractors, and so on. However, we did not find period-doubling and reverse period-doubling bifurcations. We found that the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the dynamics with a phase shift are different from the dynamics without phase shift.
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Paidoussis, M. P., G. X. Li, and R. H. Rand. "Chaotic Motions of a Constrained Pipe Conveying Fluid: Comparison Between Simulation, Analysis, and Experiment." Journal of Applied Mechanics 58, no. 2 (June 1, 1991): 559–65. http://dx.doi.org/10.1115/1.2897220.
Abstract:
A refined analytical model is presented for the dynamics of a cantilevered pipe conveying fluid and constrained by motion limiting restraints. Calculations with the discretized form of this model with a progressively increasing number of degrees of freedom, N, show that convergence is achieved with N = 4 or 5, which agrees with previously performed fractal dimension calculations of experimental data. Theory shows that, beyond the Hopf bifurcation, as the flow is increased, a pitchfork bifurcation is followed by a cascade of period doubling bifurcations leading to chaos, which is in qualitative agreement with observation. The numerically computed theoretical critical flow velocities are in excellent quantitative agreement (5–10 percent) with experimental values for the thresholds of the Hopf and period doubling bifurcations and for the onset of chaos. An approximation for the critical flow velocity for the loss of stability of the post-Hopf limit cycle is also obtained by using center manifold concepts and normal form techniques for a simplified version of the analytical model; it is found that the values obtained in this manner are approximately within 10 percent of those computed numerically.
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Kulenović, M. R. S., Connor O’Loughlin, and E. Pilav. "The Neimark–Sacker Bifurcation and Global Stability of Perturbation of Sigmoid Beverton–Holt Difference Equation." Discrete Dynamics in Nature and Society 2021 (November 26, 2021): 1–14. http://dx.doi.org/10.1155/2021/2092709.
Abstract:
We present the bifurcation results for the difference equation x n + 1 = x n 2 / a x n 2 + x n − 1 2 + f where a and f are positive numbers and the initial conditions x − 1 and x 0 are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.
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Zhao, Huitao, Yiping Lin, and Yunxian Dai. "A New Feigenbaum-Like Chaotic 3D System." Discrete Dynamics in Nature and Society 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/328143.
Abstract:
Based on Sprott N system, a new three-dimensional autonomous system is reported. It is demonstrated to be chaotic in the sense of having positive largest Lyapunov exponent and fractional dimension. To further understand the complex dynamics of the system, some basic properties such as Lyapunov exponents, bifurcation diagram, Poincaré mapping, and period-doubling route to chaos are analyzed with careful numerical simulations. The obtained results also show that the period-doubling sequence of bifurcations leads to a Feigenbaum-like strange attractor.
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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.
Abstract:
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.
Dissertations / Theses on the topic "Period-N bifurcations":
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Zhao, Yanqing. "Contributions à la détection précoce de chatter et à l’identification des bifurcations de période-N basée sur une approche de diagnostic cumulatif." Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0250.
Abstract:
Le diagnostic cumulatif des systèmes dynamiques nécessite la détection, l’identification et la caractérisation des dégradations naissantes. Son application à l'usinage à grande vitesse, par exemple, pourrait s’appuyer sur l’analyse des phénomènes de bifurcations de période-N pour détecter et identifier les chatters (broutages) naissants et améliorer la qualité des produits et des processus de fraisage. Jusqu'à présent, de nombrecuses méthodes efficaces ont été proposées pour détecter les broutages naissants et identifier les bifurcations de période-N. Cependant, ces méthodes peinent à mettre en œuvre ces tâches de manière fiable et précise. Le but de la présente thèse est de développer et mettre en œuvre des méthodes de détection de broutages naissants et d’identification de bifurcations de période-N dans une approche de diagnostic cumulatif temps réel. Afin de détecter les défauts de broutages naissants (early-chatter), nous avons proposé trois méthodes de détection et une méthode d’identification pour le diagnostic cumulatif. La première méthode peut être utilisée pour détecter à distance les broutages naissants. La deuxième méthode détecte rapidement les broutages naissants dans des conditions spécifiques de fonctionnement et de mesure. Mais dans la pratique, les conditions de fonctionnement et de mesure sont complexes et variables. Pour s'adapter aux différentes conditions de fonctionnement et de mesure, nous avons proposé une troisième méthode et cette dernière détecte de manière fiable les broutages naissants. On note également que dans les processus de fraisage, les broutages peuvent naître avec une bifurcation de type période-N ou de type Hopf. La qualité d'usinage sous un processus de bifurcation de type période-N est moins critique que celle de type Hopf. Ainsi, il est indispensable d’identifier précocement les bifurcations de type période-N pour améliorer l'efficacité d'usinage. Pour cela, nous avons développé une méthode d’identification du type et de la taille des bifurcations de période-N. Nous avons également prouvé l'efficacité des méthodes proposées, en utilisant deux modèles de processus de fraisage de référence. De plus, les méthodes proposées peuvent être utilisées pour le diagnostic de défaut d'autres systèmes dynamiques, tels que les systèmes de conversion d'énergie par modulation de largeur d'impulsion ou systèmes de paliers ou d’engrenageCumulative diagnosis of dynamic systems requires the detection, identification, and characterization of incipient degradations. Its application to high-speed machining, for instance, could rely on period-N bifurcations phenomena analysis to detect and identify early-chatters and improve the quality of milling products and processes. Up to now, many efficient methods were proposed to detect early-chatter and identify period-N bifurcations. But these methods are struggling to implement these tasks reliably and accurately due to the complex nonlinear characteristics of their dynamic behaviors, the noise, and the variation of their operating conditions. The present thesis aims to develop and implement methods of early-chatter detection and period-N bifurcations identification within a real-time cumulative diagnosis approach. Aimed at early-chatter detection, we proposed three detection methods and one identification method for the cumulative diagnosis. The first method can be used to detect early-chatters remotely. The second one detects early-chatter quickly under specific operating and measuring conditions. However, in practice, the operating and measuring conditions are complex and variable. To adapt to different operating and measuring conditions, we proposed a third method, and the latter detects early-chatter reliably. It is also noted that in milling processes, the early-chatter can give rise to a bifurcation of period-N or Hopf type. The machining quality under the bifurcation process of the period-N type is less critical than that under the Hopf bifurcation type. To improve machining productivity and ensure the required machining quality, we can mill the workpiece under the condition of period-N bifurcations. Thus, it is compulsory to identify the early period-N bifurcations for improving machining productivity. For that purpose, we developed a method for identifying the type and size of the period-N bifurcations. We also proved the effectiveness of the proposed methods, using two benchmark milling process models. Besides, the proposed methods can be used for fault diagnosis of other dynamic systems, such as the pulse energy conversion systems or bearing or gearing systems
Book chapters on the topic "Period-N bifurcations":
1
Kuznetsov, Yuri A. "Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Systems." In Elements of Applied Bifurcation Theory, 138–77. New York, NY: Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4757-2421-9_5.
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Kuznetsov, Yuri A. "Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems." In Elements of Applied Bifurcation Theory, 157–94. New York, NY: Springer New York, 2004. http://dx.doi.org/10.1007/978-1-4757-3978-7_5.
3
Kuznetsov, Yuri A. "Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems." In Elements of Applied Bifurcation Theory, 175–228. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-22007-4_5.
Conference papers on the topic "Period-N bifurcations":
1
Chen, Lihua, Ma Yepeng, and Wei Zhang. "Bifurcation Analysis of Piezoelectric Bi-Stable Plates." 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/detc2014-34484.
Abstract:
The complex nonlinear dynamic behaviors of the composite bi-stable plates with piezoelectric patch are analyzed. Based on the Vo n Karman hypothesis and Hamilton’s principle, the nonlinear dynamic model is derived. Temperature and piezoelectric effect are also considered in the model. Numerical simulations are performed to study the nonlinear vibration response of the composite bi-stable plate using the Runge-Kutta method. The analysis of the phase portrait, waveforms and bifurcation diagrams of numerical simulations shows that the period, multi-period and chaotic responses can be observed with the variation of the excitation in frequency and amplitude.
2
Luo, Albert C. J., and Yu Guo. "On Stable and Unstable Periodic Solutions of N-Dimensional Discrete Dynamical Systems." In ASME 2009 International Mechanical Engineering Congress and Exposition. ASMEDC, 2009. http://dx.doi.org/10.1115/imece2009-11441.
Abstract:
This paper presents a methodology to analytically predict the stable and unstable periodic solutions for n-dimensional discrete dynamical systems. The positive and negative iterative mappings of discrete maps are introduced for the mapping structure of the periodic solutions. The complete bifurcation and stability of the stable and unstable periodic solutions relative to the positive and negative mapping structures are presented. A discrete dynamical system with the Henon map is investigated as an example. The Poincare mapping sections relative to the Neimark bifurcation of periodic solutions are presented, and the chaotic layers for the discrete system with the Henon map are observed.
3
Guzman, Amador M., Maximiliano P. Beiza, and Paul F. Fischer. "Transition Scenario of Periodic and Quasiperiodic Flow Bifurcations in Symmetric Communicating Channels." In ASME/JSME 2007 5th Joint Fluids Engineering Conference. ASMEDC, 2007. http://dx.doi.org/10.1115/fedsm2007-37359.
Abstract:
The flow transition scenario in symmetric communicating channels has been investigated using direct numerical simulations of the mass and momentum conservation equations in the Reynolds numbers range of Re = [170–227]. The governing equations are solved for laminar and time-dependent transitional flow regimes by the spectral element method, using a periodic computational domain, for a periodic length of nL and an aspect ratio of r = aˆ / (2Lˆ) = 0.0405, where aˆ = 2a is the height of block within the channel, n an integer and Lˆ = L + 1 is the periodic length. Periodic computational domains with n = 1 and 2 are used in this investigation to determine the periodic length effect on the flow pattern characteristics. Numerical investigations with different domain meshes are carried out for determining the appropriate discretization for capturing transitional time-dependent flows. The numerical results show a transition scenario with two-flow Hopf bifurcations which develop as the pressure gradient is increased from a laminar to a time-dependent flow regime. The first Hopf bifurcation occurs to a critical Reynolds number of Rec1 and leads to a time-dependent periodic flow characterized by a fundamental frequency ω1. Further increases in the pressure gradient lead to successive quasi periodic flows after a second Hopf bifurcation B2 occurring to a critical Reynolds number Rec2 < Rec1, with two fundamental frequencies ω1 and ω2, and linear combinations of both frequencies—where the fundamental frequency ω1 increases continuously—and ω2 > ω1. This transition scenario is somewhat different from the Ruelle-Takens-Newhouse transition scenario obtained for symmetric wavy channels; in symmetric wavy channels, periodic and quasi periodic flow regimes develop as the Reynolds number increases. The friction factor for the symmetric communicating channel in the transitional regime is higher than the friction factor for the Poiseuille plane channel. The qualitative and quantitative behavior is compared to other channel geometries that also develop other transition scenarios.
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Luo, Albert C. J., and Arun Rajendran. "Dynamics of a Simplified van der Pol Oscillator Revisited." In ASME 2007 International Mechanical Engineering Congress and Exposition. ASMEDC, 2007. http://dx.doi.org/10.1115/imece2007-42687.
Abstract:
In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.
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Liu, Biyue. "A Numerical Study of Wall Shear Stress of Viscous Flows in Curved Atherosclerotic Tubes." In ASME 2002 International Mechanical Engineering Congress and Exposition. ASMEDC, 2002. http://dx.doi.org/10.1115/imece2002-32132.
Abstract:
Atherosclerosis is a disease of large- and medium-size arteries, which involves complex interactions between the artery wall and the blood flow. Both clinical observations and experimental results showed that the fluid shear stress acting on the artery wall plays a significant role in the physical processes which lead to atherosclerosis [1,2]. Therefore, a sound understanding of the effect of the wall shear stress on atherosclerosis is of practical importance to early detection, prevention and treatment of the disease. A considerable number of studies have been performed to investigate the flow phenomena in human carotid artery bifurcations or curved tubes during the past decades [3–8]. Numerical studies have supported the experimental results on the correlation between blood flow parameters and atherosclerosis [6–8]. The objective of this work is to understand the effect of the wall shear stress on atherosclerosis. The mathematical description of pulsatile blood flows is modeled by applying the time-dependent, incompressible Navier-Stokes equations for Newtonian fluids. The equations of motion and the incompressibility condition are ρut+ρ(u·∇)u=−∇p+μΔu, inΩ, (1)∇·u=0, inΩ (2) where ρ is the density of the fluid, μ is the viscosity of the fluid, u = (u1, u2, u3) is the flow velocity, p is the internal pressure, Ω is a curved tube with wall boundary Γ (see Figure 1). At the inflow boundary, fully developed velocity profiles corresponding to the common carotid velocity pulse waveform are prescribed u2=0,u3=0,u1=U(1+Asin(2πt/tp)), (3) where A is the amplitude of oscillation, tp is the period of oscillation; U is a fully developed velocity profile at the symmetry entrance plane. At the outflow boundary, surface traction force is prescribed as Tijnjni=0, (4)uiti=0 (5) where Tij=−pδij+μ(∂ui/∂xj+∂uj/∂xi) (6) is the stress tensor, n = (n1, n2, n3) is the out normal vector of the outlet boundary. On the wall boundary Γ, we assume that no slipping takes place between the fluid and the wall, no penetration of the fluid through the artery wall occurs: u|r=nHt, (7) where n = (n1, n2, n3) is the out normal vector of the wall boundary Γ. H is the function representing the location of the wall boundary. At initial time t = 0, H is input as shown in Figure 1. During the computation, H is updated by a geometry update condition based on the localized blood flow information. The initial condition is prescribed as u|t=0=u0,p|t=0=p0, where u0, p0 can be obtained by solving a Stokes problem: −μΔu0+∇p0=0,∇·u0=0, with boundary conditions (3)–(7) but zero in the right hand side of (7).
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Li, Ruo-ding, and Thomas Erneux. "Bifurcation to Standing and Traveling Waves in Large Laser Arrays." In Nonlinear Dynamics in Optical Systems. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nldos.1992.thb5.
Abstract:
Arrays of semiconductor diode lasers are promising devices for applications that require high optical power from a laser source (high-speed optical recording, high-speed printing, free-space communications, pumping of solid-state lasers) [1]. Experimental and numerical studies of arrays consisting of a small number of lasers have shown that they are unstable devices and may exhibit a large variety of spatio-temporal responses [2-5]. In order to control these instabilities by various external mechanisms (injection locking, periodic modulations), systematic bifurcation studies are needed. The laser equations are however stiff and accurate solutions for a large population of lasers require long computation times. Asymptotic methods based on the limit of weak coupling [6] also fail to provide simple phase equations because the semiconductor laser is not a limit cycle oscillator. We have recently reformulated the laser equations as a weakly perturbed system of coupled conservative oscillators which eliminate part of the stifness of the problem and allow an analytical study of the first Hopf bifurcation as the coupling strength is progressively increased. If N is even, the Hopf bifurcation is simple and corresponds to a transition from a nonuniform steady state to a time periodic standing wave solution [7]. However, if N is odd, bifurcation to periodic standing and traveling wave solutions are both possible. This multiple bifurcation problem is difficult analytically but can be simplified if we consider the limit N large.
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Arafat, Haider N., and Ali H. Nayfeh. "Modal Interactions in a Thermally Loaded Annular Plate." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48603.
Abstract:
We investigate the nonlinear forced vibrations of a thermally loaded annular plate with clamped-clamped immovable boundary conditions in the presence of a three-to-one internal resonance between the first and second axisymmetric modes. We consider the in-plane thermal load to be axisymmetric and excite the plate externally by a harmonic force near primary resonance of the second mode. We then use the nonlinear von Ka´rma´n plate equations to model the behavior of the system and apply the method of multiple scales to investigate its responses. We found that the response can be periodic oscillations consisting of both modes, with a large component from the first mode. Moreover, the periodic solutions may undergo Hopf bifurcations which lead to aperiodic oscillations of the plate.
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Szabó, Zsolt. "Quasi-Periodic Motions of Articulated Pipes Conveying Flowing Fluid." 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-21424.
Abstract:
Abstract In this paper two nice examples are investigated where a ‘chain’ of n = (1, 2) pieces of rigid pipes contains incompressible and frictionless flowing fluid. We give an overview about the linear and nonlinear analysis of the autonomous system, i.e. when the pipes contain steady flow. Assuming pulsatile flow, the system becomes time-periodic. The stability charts of the linearized system are generated applying a numerical method based on Chebyshev polynomials. Finally, we analyze the effect of the nonlinear part in some critical points of the obtained stability charts and the dynamic behaviour of the original nonlinear periodic system is simulated numerically. The results are shown in Poincaré maps and bifurcation diagrams.
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Otsuka, Kenju, and Jyh-Long Chern. "Factorial Dynamic Pattern Memory in Globally Coupled Lasers." In Nonlinear Dynamics in Optical Systems. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nldos.1992.thb1.
Abstract:
Recently, applicability of complex dynamics to information storage (memory) has been discussed in nonlinear optical systems. Spatial chaos memory was proposed in a bistable pixelsl1 and in a coupled bistable chain.2 Dynamic memory in a delayed feedback bistable system was demonstrated experimentally.3 On the other hand, Otsuka demonstrated that m a = (N − 1)! (N: number of oscillating modes) coexisting dynamical spatial patterns, i. e., antiphase periodic motions, can be selectively excited by applying seed signals to the modulated multimode lasers whose modes are globally coupled through spatial hole burning.4 In this paper, we discuss the detailed bifurcation scenario, featuring clustered states and chaotic itinerancy5 among destabilized clustered states. Also, factorial dynamical pattern memory associated with antiphase and clustered states as well as the effect of spontaneous emission on memory operation are demonstrated by numerical simulations.
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Samaranayake, S., Anil K. Bajaj, and O. D. I. Nwokah. "Resonant Vibrations in Weakly Coupled Nonlinear Structures With Cyclic Symmetry." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0052.
Abstract:
Abstract Periodic structures with cyclic symmetry are often used as idealized models of physical systems including bladed-disk assemblies, large space antennas, circular saws and disks, and flexible magnetic storage media. The present work considers a simplified model of such structures and investigates the nonlinear vibratory response to periodic excitations. The model consists of n identical particles, arranged in a circular ring, interconnected by extensional springs with linear and nonlinear stiffness characteristics, and hinged to the ground individually by nonlinear torsional springs. When the linear couping stiffness is O(ε), the cyclic system consists of n weakly coupled identical nonlinear oscillators so that all the oscillators are in internal resonance. The dynamic response of this system to resonant periodic excitation is studied using the method of averaging. The amplitude equations consist of a 2n system of first order ordinary differential equations that depend on the forcing amplitude and frequency, the modal damping, and the nature of the excitation. These equations are analyzed using local bifurcation theory, and the effects of coupling strength to nonlinearity parameter on the steady-state responses are determined. It is found that when only one particle is excited, the response for very weak coupling essentially remains localized. For larger coupling strength, both localized and extended responses are found over different excitation frequency intervals.