Showing papers on "Bifurcation diagram published in 2002"
TL;DR: A mathematical model of an idealized electrostatically actuated MEMS device is constructed and analyzed for the purpose of investigating the effects of the "pull-in" or "snap-down" instability, and variations in this bifurcation diagram for various dielectric profiles are studied, yielding insight into how this technique may be used to increase the stable range of operation.
Abstract: The "pull-in" or "snap-down" instability in electrostatically actuated microelec- tromechanical systems (MEMS) presents a ubiquitous challenge in MEMS technology of great im- portance. In this instability, when applied voltages are increased beyond a critical value, there is no longer a steady-state configuration of the device where mechanical members remain separate. This severely restricts the range of stable operation of many devices. In an attempt to reduce the effects of this instability, researchers have suggested spatially tailoring the dielectric properties of MEMS devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed and analyzed for the purpose of investigating this possibility. The pull-in instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in this bifurcation diagram for various dielectric profiles are studied, yielding insight into how this technique may be used to increase the stable range of operation of electrostatically actuated MEMS devices.
167 citations
TL;DR: In this paper, the authors discuss generalizations of various autonomous concepts of stability, instability, and invariance, and illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.
Abstract: There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the non-autonomous case. Inspired in part by the theory of pullback attractors, we discuss generalizations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.
101 citations
TL;DR: In this paper, the authors demonstrate how timeintegration of stochastic differential equations can be combined with continuum numerical bifurcation analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs).
Abstract: We demonstrate how time-integration of stochastic differential equations (i.e. Brownian dynamics simulations) can be combined with continuum numerical bifurcation analysis techniques to analyze the dynamics of liquid crystalline polymers (LCPs). Sidestepping the necessity of obtaining explicit closures, the approach analyzes the (unavailable in closed form) coarse macroscopic equations, estimating the necessary quantities through appropriately initialized, short bursts of Brownian dynamics simulation. Through this approach, both stable and unstable branches of the equilibrium bifurcation diagram are obtained for the Doi model of LCPs and their coarse stability is estimated. Additional macroscopic computational tasks enabled through this approach, such as coarse projective integration and coarse stabilizing controller design, are also demonstrated.
100 citations
TL;DR: In this article, the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface, is discussed.
Abstract: This paper discusses the bifurcation theory for the equations for traveling surface water waves, based on the formulation of Zakharov [58] and of Craig and Sulem [15] in terms of integro-differential equations on the free surface. This theory recovers the well-known picture of bifurcation curves of Stokes progressive wavetrains in two-dimensions, with the bifurcation parameter being the phase velocity of the solution. In three dimensions the phase velocity is a two-dimensional vector, and the resulting bifurcation equations describe two-dimensional bifurcation surfaces, with multiple intersections at simple bifurcation points. The integro-differential formulation on the free surface is posed in terms of the Dirichlet–Neumann operator for the fluid domain. This lends itself naturally to numerical computations through the fast Fourier transform and surface spectral methods, which has been implemented in Nicholls [32]. We present a perturbation analysis of the resulting bifurcation surfaces for the three-dimensional problem, some analytic results for these bifurcation problems, and numerical solutions of the surface water waves problem, based on a numerical continuation method which uses the spectral formulation of the problem in surface variables. Our numerical results address the problem in both two and three dimensions, and for both the shallow and deep water cases. In particular we describe the formation of steep hexagonal traveling wave patterns in the three-dimensional shallow water regime, and their transition to rolling waves, on high aspect ratio rectangular patterns as the depth increases to infinity.
96 citations
TL;DR: In this article, the authors introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes and investigate the capture of an incident gap soliton by these defects.
Abstract: Gap solitons are localized nonlinear coherent states that have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed v,0⩽v⩽c, they have been observed in experiments at speeds of approximately 50% of c. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as nonlinear defect modes. Using mathematical analysis and numerical simulations, we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap-soliton capture, reflection, and transmission can be obtained by simple conservation of energy and resonant energy transfer principles.
92 citations
TL;DR: Unprecedented agreement is demonstrated between a theoretical two-dimensional bifurcation diagram and the corresponding experimental stability map of an optically injected semiconductor laser over a large range of relevant injection parameter values.
Abstract: We demonstrate unprecedented agreement between a theoretical two-dimensional bifurcation diagram and the corresponding experimental stability map of an optically injected semiconductor laser over a large range of relevant injection parameter values. The bifurcation diagram encompasses both local and global bifurcations mapping out regions of regular, chaotic, and multistable behavior in considerable detail. This demonstrates the power of dynamical systems modeling for the quantitative prediction of nonlinear dynamics and chaos of semiconductor lasers.
91 citations
TL;DR: In this paper, the stability and Hopf bifurcation of a delay competition diffusion system were investigated and the existence and stability of the corresponding steady state solutions were discussed, and the stability of these solutions was analyzed by reducing the original system on the center manifold.
Abstract: This paper investigates the stability and Hopf bifurcation of a delay competition diffusion system. Firstly we discuss the existence and stability of the corresponding steady state solutions. Secondly our purpose is to give more detail information about the Hopf bifurcation of this system. We derive the basis of the eigenfunction subspace and then convert the existence of periodic solutions to the study of the existence of the implicit function. Finally, we analyze the stability of the periodic solutions by reducing the original system on the center manifold.
85 citations
01 Jul 2002
TL;DR: In this paper, the authors introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes and investigate the capture of an incident gap soliton by these defects.
Abstract: Gap solitons are localized nonlinear coherent states that have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed v,0⩽v⩽c, they have been observed in experiments at speeds of approximately 50% of c. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as nonlinear defect modes. Using mathematical analysis and numerical simulations, we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap-soliton capture, reflection, and transmission can be obtained by simple conservation of energy and resonant energy transfer principles.
78 citations
TL;DR: In this article, the authors compare analytical nonlinear dynamics methods with a two-dimensional brute-force bifurcation diagram for displaying safety margins in a 2-dimensional parameter space, where a simplified single-phase model is used to represent the case of ferroresonance between a transformer and circuit breaker grading capacitor.
Abstract: Catastrophic equipment failures continue to occur today due to ferroresonance even though this phenomenon has been extensively studied over the past 90 years. This paper is concerned with comparing analytical nonlinear dynamics methods with a two-dimensional brute-force bifurcation diagram for displaying safety margins in a two-dimensional parameter space. A simplified single-phase model is used to represent the case of ferroresonance between a transformer and circuit breaker grading capacitor. Comparisons are made between the analytical method and EMTP simulations of an actual ferroresonant event.
78 citations
TL;DR: The bifurcation diagram of a single-mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods and the behavior of the first connection is investigated as a function of the linewidth enhancement factor and the feedback phase.
Abstract: The bifurcation diagram of a single-mode semiconductor laser subject to a delayed optical feedback is examined by using numerical continuation methods. For this, we show how to cope with the special symmetry properties of the equations. As the feedback strength is increased, branches of modes and antimodes appear, and we have found that pairs of modes and antimodes are connected by closed branches of periodic solutions (bifurcation bridges). Such connections seem generically present as new pairs of modes and antimodes appear. We subsequently investigate the behavior of the first connection as a function of the linewidth enhancement factor and the feedback phase. Our results extend and confirm existing results and hypotheses reported in the literature. For large values of the linewidth enhancement factor (alpha=5-6), bridges break through homoclinic orbits. Changing the feedback phase unfolds the bifurcation diagram of the modes and antimodes, allowing different types of connections between modes.
73 citations
TL;DR: This paper will show that the addition of a slower Hebbian learning mechanism onto the Hopfield networks makes the resulting global dynamics to drive the network into a stable oscillatory regime, through a succession of intermittent and quasiperiodic regimes.
TL;DR: The roots of a variety of nonlinear phenomena observed numerically and experimentally such as, e.g., self-pulsations, excitability, hysteresis, or chaos are understood and located in the parameter plane.
Abstract: We investigate the dynamics of a multisection laser implementing a delayed optical feedback experiment where the length of the cavity is comparable to the length of the laser. First, we reduce the traveling-wave model with gain dispersion (a hyperbolic system of PDEs) to a system of ODEs describing the semiflow on a local center manifold. Then we analyze the dynamics of the system of ODEs using numerical continuation methods (AUTO). We explore the plane of the two parameters---feedback phase and feedback strength---to obtain a bifurcation diagram for small and moderate feedback strength. This diagram permits us to understand the roots of a variety of nonlinear phenomena observed numerically and experimentally such as, e.g., self-pulsations, excitability, hysteresis, or chaos, and to locate them in the parameter plane.
TL;DR: In this paper, the Hopf bifurcation behavior of a symmetric rotor/seal system was investigated using Muszynska's non-linear seal fluid dynamic force model.
TL;DR: A two-mass model was used to simulate irregular vocal fold vibrations and turbulent noise as an external random source, as well as random stiffness perturbation as an internal random source played important roles in the presence of irregular vocal Fold vibrations.
Abstract: The contribution of turbulent noise was modeled in symmetric vocal folds. A two-mass model was used to simulate irregular vocal fold vibrations. The threshold values of system parameters to produce irregular vibrations were decreased as a result of turbulent airflow. Periodic vibrations were then driven into the regions of irregular vibrations. Using nonlinear dynamics including Poincare map and Lyapunov exponents, irregular vibrations were demonstrated as chaos. For the deterministic vocal-fold model with noise free and steady airflow, a fine period-doubling bifurcation cascade was shown in a bifurcation diagram. However, turbulent noise added to the vocal-fold model would induce chaotic vibrations, broaden the regions of irregular vocal fold vibrations, and inhibit the fine period-doubling bifurcations in the bifurcation diagrams. The perturbations from neurological and biomechanical effects were simulated as a random variation of the vocal fold stiffness. Turbulent noise as an external random source, as well as random stiffness perturbation as an internal random source, played important roles in the presence of irregular vocal fold vibrations.
TL;DR: In this article, the topological structure of the coronal magnetic field arising from the interaction of two bipolar regions was studied, and four distinct, topologically stable states are possible.
Abstract: The Sun's atmosphere contains many diverse phenomena that are dominated by the coronal magnetic field. To understand these phenomena it is helpful to determine first the structure of the magnetic field, i.e., the magnetic topology. We study here the topological structure of the coronal magnetic field arising from the interaction of two bipolar regions, for which we find that four distinct, topologically stable states are possible. A bifurcation diagram is produced, showing how the magnetic configuration can change from one topology to another as the relative orientation and sizes of the bipolar regions are varied. The changes are produced either by a global separator bifurcation, a local double-separator bifurcation, a new, global separatrix quasi-bifurcation, or a new, global spine quasi-bifurcation.
TL;DR: In this paper, the surface stability of a thin solid elastic film subjected to surface interactions such as van der Waals forces due to the influence of another contacting solid is investigated and the characteristic wavelength of the bifurcation pattern is nearly independent of the precise nature and magnitude of the interaction and varies linearly with the film thickness.
Abstract: The surface stability of a thin solid elastic film subjected to surface interactions such as van der Waals forces due to the influence of another contacting solid is investigated. It is found that for nearly incompressible soft (shear modulus less than 10 MPa) films, the film surface is unstable and forms an undulating pattern without any concurrent mass transport. A complete stability/bifurcation diagram is obtained. A key new result uncovered in this analysis is that the characteristic wavelength of the bifurcation pattern is nearly independent of the precise nature and magnitude of the interaction and varies linearly with the film thickness, whenever the force of interaction attains a critical value. The rate of growth of perturbations is also analysed using a viscoelastic model and it is found that in nearly incompressible materials, the wavelength of the fastest growing perturbation is identical to that of the critical elastic bifurcation mode. These results provide a quantitative explanation for recent experiments. The present study is important in understanding problems ranging from adhesion and friction at soft solid interfaces, peeling of adhesives to the development of micro-scale pattern transfer technologies.
TL;DR: In this paper, the flow in a completely filled rotating cylinder driven by the counter-rotation of the top endwall is investigated both numerically and experimentally, and a supercritical symmetry-breaking Hopf bifurcation to a rotating wave state results.
Abstract: The flow in a completely filled rotating cylinder driven by the counter-rotation of the top endwall is investigated both numerically and experimentally. The basic state of this system is steady and axisymmetric, but has a rich structure in the radial and axial directions. The most striking feature, when the counter-rotation is sufficiently large, is the separation of the Ekman layer on the top endwall, producing a free shear layer that separates regions of flow with opposite senses of azimuthal velocity. This shear layer is unstable to azimuthal disturbances and a supercritical symmetry-breaking Hopf bifurcation to a rotating wave state results. For height-to-radius ratio of 0.5 and Reynolds number (based on cylinder radius and base rotation) of 1000, rotating waves with azimuthal wavenumbers 4 and 5 co-exist and are stable over an extensive range of the ratio of top to base rotation. Mixed modes and period doublings are also found, and a bifurcation diagram is determined. The agreement between the Navier-Stokes computations and the experimental measurements is excellent. The simulations not only capture the qualitative features of the multiple states observed in the laboratory, but also quantitatively replicate the parameter values over which they are stable, and produce accurate precession frequencies of the various rotating waves.
TL;DR: In this article, the authors examined axisymmetry in a swirling flow generated inside an enclosed cylindrical container by the steady rotation of one endwall and examined the two dimensionless parameters that govern these flows are the cylinder aspect ratio and a Reynolds number associated with the rotation of the endwall.
Abstract: The loss of axisymmetry in a swirling flow that is generated inside an enclosed cylindrical container by the steady rotation of one endwall is examined numerically. The two dimensionless parameters that govern these flows are the cylinder aspect ratio and a Reynolds number associated with the rotation of the endwall. This study deals with a fixed aspect ratio, height/radius = 2.5. At low Reynolds numbers the basic flow is steady and axisymmetric; as the Reynolds number increases the basic state develops a double recirculation zone on the axis, so-called vortex breakdown bubbles. On further increase in the Reynolds number the flow becomes unsteady through a supercritical Hopf bifurcation but remains axisymmetric. After the onset of unsteadiness, another two unsteady axisymmetric solution branches appear with further increase in Reynolds number, each with its own temporal characteristic: one is periodic and the other is quasi-periodic with a very low frequency modulation. Solutions on these additional branches are unstable to three-dimensional perturbations, leading to nonlinear modulated rotating wave states, but with the flow still dominated by the corresponding underlying axisymmetric mode. A study of the flow behaviour on and bifurcations between these solution branches is presented, both for axisymmetric and for fully three-dimensional flows. The presence of modulated rotating waves alters the structure of the bifurcation diagram and gives rise to its own dynamics, such as a truncated cascade of period doublings of very-low-frequency modulated states.
TL;DR: In this article, a temporally modulated Benney equation (TMBE) is derived and its solutions are investigated numerically, showing that the BE constitutes an accurate asymptotic reduction of the Navier-Stokes equations in the domain preceding the transition to its unbounded solutions.
Abstract: The two-dimensional spatiotemporal dynamics of falling thin liquid films on a solid vertical wall periodically oscillating in its own plane is studied within the framework of long-wave theory. A pertinent nonlinear evolution equation referred to as the temporally modulated Benney equation (TMBE) is derived and its solutions are investigated numerically. The bifurcation diagram of the Benney equation (BE) describing the film dynamics in the unforced regime is computed depicting the domains of linearly stable, linearly unstable bounded, and unbounded behaviors. The solutions obtained for film dynamics via the BE are compared to those documented for direct numerical simulations of the Navier–Stokes equations (NSE). The comparison demonstrates that the BE constitutes an accurate asymptotic reduction of the NSE in the domain preceding the transition to the regime of its unbounded solutions. It is found that periodic planar boundary excitation does not alter the fundamental unforced bifurcation structure and th...
TL;DR: In this article, a general class of first-order nonlinear delay-differential equations (DDEs) with reflectional symmetry is considered, and the bifurcations of the trivial equilibrium under some generic conditions on the Taylor coefficients of the DDE are analyzed.
07 Aug 2002
TL;DR: In this article, the dynamic voltage stability of a power system is analyzed and compared in terms of the reduced and unreduced Jacobian matrix of the system, whose eigenstructure matches well with the reduced one; and thus can be used for bifurcation analysis.
Abstract: The dynamic of a large class of power systems can be represented by parameter dependent differential-algebraic models of the form x/spl dot/ = f (x, y, p) and 0 = g(x, y, p). When the parameter p of the system (such as load of the system) changes, the stable equilibrium points may lose their dynamic stability at local bifurcation points. The systems will lose its stability at the feasibility boundary, which is caused by one of three different local bifurcations: the singularity induced bifurcation, saddle-node and Hopf bifurcation. In this paper, the dynamic voltage stability of a power system is introduced and analyzed. Both the reduced and unreduced Jacobian matrix of the system are studied and compared. It is shown that the unreduced Jacobian matrix, whose eigenstructure matches well with the reduced one; and thus can be used for bifurcation analysis. In addition, the analysis avoids the singularity induced infinity problem, which may happen at reduced Jacobian matrix analysis, and is more computationally attractive.
TL;DR: Hopf bifurcation is demonstrated in an interacting one-predator-two-prey model with harvesting of the predator at a constant rate and periodic solutions arise from stable stationary states when the harvest rate exceeds a certain limit.
TL;DR: A control method to create Hopf bifurcations in discrete-time nonlinear systems using the center manifold method, normal form technique and the Iooss's Hopfbifurcation theory are employed in the derivation of the control gain.
Abstract: Bifurcation characteristics of a nonlinear system can be manipulated by small controls. In this paper, we present a control method to create Hopf bifurcations in discrete-time nonlinear systems. The critical conditions for the Hopf bifurcations are discussed. The center manifold method, normal form technique and the Iooss’s Hopf bifurcation theory are employed in the derivation of the control gain. Numerical demonstration is provided.
TL;DR: The well-known Hurwitz-Routh criterion is generalized to critical cases and the problem of Hopf bifurcation of higher order for zeros of semistable polynomials is discussed.
TL;DR: In this paper, the authors presented a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate of thermo-mechanical coupling by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent.
Abstract: This paper presents a new approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection rectangular plate of thermo-mechanical coupling by utilizing the criteria of the fractal dimension and the maximum Lyapunov exponent. The governing partial differential equation of the simply supported rectangular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The numerical results indicate that large deflection motion of the thermo-mechanical coupling rectangular plate possesses many bifurcation points, chaotic motions and period-double phenomena under various lateral loads, bi-axial loads, thermo-mechanical coupling factors and aspect ratios. The modeling results of numerical simulation indicate that the chaotic motion may occur in the range of lateral loads Q =1.25 to 3.35 and near the bi-axial load η2=1.5. The dynamic motion of the thermal-couple plate is periodic if the aspect ratio is within a specific range or the thermo-mechanical coupling factor is within a specific range. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling plate in large deflection.
01 Feb 2002
TL;DR: In this paper, the qualitative behavior of a class of ratio-dependent predator-prey systems with delay at the equilibrium in the interior of the first quadrant is studied, and it is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions.
Abstract: Recently, ratio-dependent predator–prey systems have been regarded by some researchers as being more appropriate for predator–prey interactions where predation involves serious searching processes. Due to the fact that every population goes through some distinct life stages in real-life, one often introduces time delays in the variables being modelled. The presence of time delay often greatly complicates the analytical study of such models. In this paper, the qualitative behaviour of a class of ratio-dependent predator–prey systems with delay at the equilibrium in the interior of the first quadrant is studied. It is shown that the interior equilibrium cannot be absolutely stable and there exist non-trivial periodic solutions for the model. Moreover, by choosing delay as the bifurcation parameter we study the Hopf bifurcation and the stability of the periodic solutions.AMS 2000 Mathematics subject classification: Primary 34C25; 92D25. Secondary 58F14
TL;DR: In this article, the archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure.
TL;DR: In this article, bifurcation and chaos of an axially moving viscoelastic string are investigated, and the 1-term and 2-term Galerkin truncations are respectively employed to simplify the partial differential equation that governs the transverse motions of the string into a set of ordinary differential equations.
TL;DR: In this article, the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework is studied.
Abstract: In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.
TL;DR: In this article, a phenomenological model of a tumor interacting with the relevant cells of the immune system is proposed and analyzed, which has a simple formulation in terms of delay-differential equations (DDEs).
Abstract: A phenomenological model of a tumor interacting with the relevant cells of the immune system is proposed and analysed. The model has a simple formulation in terms of delay-differential equations (DDEs). The critical time-delay, for which a destabilising Hopf bifurcation of the relevant fixed point occurs, and the conditions on the parameters for such bifurcation are found. The bifurcation occurs for the values of the parameters estimated from real data. Local linear analyses of the stability is sufficient to qualitatively analyse the dynamics for small time-delays. Qualitative analyses justify the assumptions of the model. Typical dynamics for larger time-delay is studied numerically.