Showing papers on "Bifurcation diagram published in 1993"
TL;DR: In this article, the authors investigated Hopf-like bifurcation phenomena and chaotic behavior in cellular neural networks and found that the chaotic attractor found here has properties similar to the famous double scroll attractor.
Abstract: Bifurcation phenomena and chaotic behavior in cellular neural networks are investigated. In a two-cell autonomous system, Hopf-like bifurcation has been found, at which the flow around the origin, an equilibrium point of the system, changes from asymptotically stable to periodic. As the parameter grows further, by reaching another bifurcation value, the generated limit cycle disappears and the network becomes convergent again. Chaos is also presented in a three-cell autonomous system. It is shown that the chaotic attractor found here has properties similar to the famous double scroll attractor. >
277 citations
TL;DR: It is shown that the bifurcation diagram for the fiber coupler is similar to the b ifurcation diagrams for the stationary waves in a symmetric nonlinear planar waveguide.
Abstract: We find and analyze analytically and numerically two new families of coupled soliton states in nonlinear fiber couplers. The bifurcation diagram for the new types of soliton states is constructed. It is shown that the bifurcation diagram for the fiber coupler is similar to the bifurcation diagram for the stationary waves in a symmetric nonlinear planar waveguide. Physical reasons for this analogy are discussed.
211 citations
01 Jan 1993
TL;DR: In this article, the authors present a number of results mostly obtained in the last years and only published recently or even not yet published, which are meant to be a helpful guide in struggling through the different research papers they are based on.
Abstract: We present a number of results mostly obtained in the last years and only published recently or even not yet published. We do not give complete proofs but emphasize the different techniques and the precise way they have to be used. These notes are meant to be a helpful guide in struggling through the different research papers they are based on.
160 citations
TL;DR: In this article, the authors analyzed the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators, where the population is described by a Fokker-Planck equation for the distribution of phases, and they applied center-manifold reduction to obtain the amplitude equations for steady state and Hopf bifurcation from the equilibrium state with a uniform phase distribution.
Abstract: We analyze the nonlinear dynamics near the incoherent state in a mean-field model of coupled oscillators. The population is described by a Fokker-Planck equation for the distribution of phases, and we apply center-manifold reduction to obtain the amplitude equations for steady-state and Hopf bifurcation from the equilibrium state with a uniform phase distribution. When the population is described by a native frequency distribution that is reflection-symmetric about zero, the problem has circular symmetry. In the limit of zero extrinsic noise, although the critical eigenvalues are embedded in the continuous spectrum, the nonlinear coefficients in the amplitude equation remain finite in contrast to the singular behavior found in similar instabilities described by the Vlasov-Poisson equation. For a bimodal reflection-symmetric distribution, both types of bifurcation are possible and they coincide at a codimension-two Takens Bogdanov point. The steady-state bifurcation may be supercritical or subcritical and produces a time-independent synchronized state. The Hopf bifurcation produces both supercritical stable standing waves and supercritical unstable travelling waves. Previous work on the Hopf bifurcation in a bimodal population by Bonilla, Neu, and Spigler and Okuda and Kuramoto predicted stable travelling waves and stable standing waves, respectively. A comparison to these previous calculations shows that the prediction of stable travelling waves results from a failure to include all unstable modes.
134 citations
TL;DR: In this paper, the authors proposed a method for computing the locally closest bifurcation to the nominal parameters of a stable equilibrium in a saddle-node or Hopf bifurbation.
Abstract: Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated at a stable equilibrium. However, as the parameters λ vary from some nominal value λ0, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space of λ0 to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability to parameter variations and is important in applications. We propose computing a parameter vector λ* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ0. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ0 of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity to the closest voltage collapse instability of a simple electric power system.
130 citations
02 Jun 1993
TL;DR: In this article, it was shown that in the case of infinite reflux and an infinite number of trays, multiple steady states exist when the distillate flow varies non-monotonically along the continuation path of the bifurcation diagram.
Abstract: In this article we study multiple steady states in ternary homogeneous azeotropic distillation. We show that in the case of infinite reflux and an infinite number of trays, multiple steady states exist when the distillate flow varies non-monotonically along the continuation path of the bifurcation diagram with the distillate flow as the bifurcation parameter. We derive a necessary and sufficient condition for the existence of these multiple steady states based on the geometry of the distillation region boundaries. We also locate in the composition triangle the feed compositions that lead to these multiple steady states. We show that the prediction of the existence of multiple steady states in the case of infinite reflux and an infinite number of trays has relevant implications for columns operating at finite reflux and with a finite number of trays. Using numerically constructed bifurcation diagrams for specific examples, we show that these multiplicities tend to vanish for small columns and/or for low reflux flows.
122 citations
TL;DR: This analysis is motivated by a recent numerical study of bursting oscillations in an enzymatic system and found that the transition from oscillations to steady states is delayed even if the rate of change of the control parameter is extremely small.
Abstract: This paper investigates the slow passage through a supercritical Hopf bifurcation from a branch of slowly varying periodic solutions to a branch of slowly varying steady states. This analysis is motivated by a recent numerical study of bursting oscillations in an enzymatic system. It was found that the transition from oscillations to steady states is delayed even if the rate of change of the control parameter is extremely small.The delay due to the slow passage is characterized by determining the amplitude of the oscillations at the bifurcation point. Defining $\varepsilon $ as the rate of change of the bifurcation parameter, it is shown that the amplitude is an $O( \varepsilon^{1/4} )$ quantity as $\varepsilon \to 0$.In addition, a particular class of equations leading to relaxation oscillations is considered. It is assumed that frequency $\omega $ of the oscillations at the Hopf bifurcation can be controlled using a second parameter distinct from the bifurcation parameter. It is then shown that the ampl...
106 citations
TL;DR: In this paper, a method for determining the stability of general static capillary surfaces is illustrated by application to the liquid bridge, where axisymmetric bridges with fixed contact lines under gravity are parametrized by three quantities: bridge length L, bridge volume V, and Bond number B. The preferred diagram method gives stronger results than classical bifurcation theory based on properties of eigenvalues of the Jacobi equation for problems with a variational formulation.
Abstract: A method for determining the stability of general static capillary surfaces is illustrated by application to the liquid bridge. Axisymmetric bridges with fixed contact lines under gravity are parametrized by three quantities: bridge length L, bridge volume V, and Bond number B. The method delivers i) stability envelopes in the {L,V,B,} parameter space for constant pressure and constant volume disturbances (recovering classical and generating new results), ii) the number of unstable modes for any equilibrium (state of instability) once the stability of one equilibrium state is known, based on, iii) a demonstration that all known families of equilibria are ultimately connected. The state of instability of an equilibrium shape relative to its neighbors is immediate from a plot of volume V versus pressure p, a "preferred" bifurcation diagram. The preferred diagram method gives stronger results than classical bifurcation theory based on properties of eigenvalues of the Jacobi equation for problems with a variational formulation. Application to other capillary surfaces including drops and nonaxisymmetric shapes is discussed. In addition, motivated by general tangency considerations, an invariant wavenumber classification is introduced and used to label the numerous families of liquid bridge equilibria.
102 citations
TL;DR: In this paper, the existence of multiple steady states in ternary homogeneous azeotropic distillation with infinite reflux and an infinite number of trays has been studied.
Abstract: We study multiple steady states in ternary homogeneous azeotropic distillation. We show that in the case of infinite reflux and an infinite number of trays one can construct bifurcation diagrams on physical grounds with the distillate flow as the bifurcation parameter. Multiple steady states exist when the distillate flow varies nonmonotonically along the continuation path of the bifurcation diagram. We derive a necessary and sufficient condition for the existence of these multiple steady states based on the geometry of the distillation region boundaries. We also locate in the composition triangle the feed compositions that lead to these multiple steady states. We further note that most of these results are independent of the thermodynamic model used. We show that the prediction of the existence of multiple steady states in the case of infinite reflux and an infinite number of trays has relevant implications for columns operating at finite reflux and with a finite number of trays. Using numerically constructed bifurcation diagrams for specific examples, we show that these multiplicities tend to vanish for small columns and/or for low reflux flows. Finally, we comment on the effect of multiplicities on column design and operation for some specific examples
97 citations
TL;DR: In this paper, the bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurbcation curves.
Abstract: The bifurcation set in the three-dimensional parameter space of the periodically driven van der Pol oscillator has been investigated by continuation of local bifurcation curves. The two regions in which the driving frequency ω is greater or less than the limit cycle frequency ω0 of the nondriven oscillator are considered separately. For the case ω > ω0, the subharmonic region, the extent and location of the largest Arnol'd tongues are shown, as well as the period-doubling cascades and chaotic attractors that appear within most of them. Special attention is paid to the pattern of the bifurcation curves in the transitional region between low and large dampings that is difficult to approach analytically. In the case ω < ω0, the ultraharmonic region, a recurrent pattern of the bifurcation curves is found for small values of the damping d. At medium damping the structure of the bifurcation curves becomes involved. Period-doubling sequences and chaotic attractors occur.
86 citations
TL;DR: In this paper, the authors studied the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity and showed that this bifurcation is a generalized Hopf bifurbcation from the point at infinity.
TL;DR: In this article, the dynamics of the large-amplitude oscillation of the vocal folds is analyzed using the two-mass model. And the main results are discussed relative to previous analytical works; it is shown that they disprove the previous oscillation theory based on the existence of a glottal negative differential resistance.
Abstract: The dynamics of the large‐amplitude oscillation of the vocal folds is analyzed using the two‐mass model. First, the equilibrium positions are determined in the case of a rectangular prephonatory glottis, and the existence of two equilibrium positions besides the rest position is shown. Their stability is examined and a bifurcation diagram is derived with a normalized subglottal pressure and a coupling coefficient as control parameters. Phase plane plots are shown to illustrate the results. The cases of convergent and divergent prephonatory glottis are then briefly considered. The main results are finally discussed relative to previous analytical works; it is shown that they disprove the previous oscillation theory based on the existence of a glottal negative differential resistance.
TL;DR: In this paper, the authors considered Kolmogorov's problem of viscous incompressible fluid motions on two-dimensional tori and computed numerically bifurcation diagrams with the Reynolds number as a splitting parameter.
Abstract: We consider Kolmogorov's problem of viscous incompressible fluid motions on two dimensional tori. The problem is a bifurcation problem with two parameters, the Reynolds number and the aspect ratio. Varying the aspect ratio as a splitting parameter, we compute numerically bifurcation diagrams with the Reynolds number as a bifurcation parameter. As the aspect ratio changes, we observe turning points and secondary bifurcation points appear or disappear. Furthermore, Hopf bifurcation points are also found when the aspect ratio of the torus satisfies a certain condition. This paper is an improved and enlarged version of the report [26]. Some errors in [17, 26] are corrected here.
TL;DR: In this paper, the authors studied the local bifurcation of critical periods of periodic orbits in the neighborhood of a non-degenerate center of a vector field with a homogeneous nonlinearity of the third degree.
Abstract: In this paper we study the local bifurcation of critical periods of periodic orbits in the neighborhood of a nondegenerate centre of a vector field with a homogeneous nonlinearity of the third degree. We show that at most three local critical periods bifurcate from a weak linear centre of finite order or from the linear isochrone and at most two local critical periods from the nonlinear isochrone. Moreover, in both cases, there are perturbations with the maximum number of critical periods.
TL;DR: In this article, the morphogenesis of secondary vortices is investigated for the axisymmetric flow of a viscoelastic fluid, confined between two independently rotating, infinitely long cylinders, in the region near the onset of instability of the purely azimuthal Couette flow.
Abstract: The morphogenesis of secondary vortices is investigated for the axisymmetric flow of a viscoelastic fluid, confined between two independently rotating, infinitely long cylinders, in the region near the onset of instability of the purely azimuthal Couette flow. The Oldroyd-B constitutive equation is used to model viscoelasticity. Three characteristic regions in the parameter space, corresponding to three distinct solution families have been investigated where the onset of instability is due primarily to inertia, both inertia and elasticity, and exclusively elasticity, respectively. The secondary flow corresponds to a steady Taylor vortex in the first case, but to a time-periodic one when elasticity becomes important (Hopf bifurcation). Degenerate Hopf bifurcation theory in the presence of symmetries ( O (2) × S 1 ) has been used to show the existence of two different time-periodic solution families, each following either one of two possible patterns, the rotating wave or the standing wave. Through a computer-aided nonlinear analysis, all of the steady and time-periodic bifurcating solutions are shown to be supercritical, implying that one and only one is stable. These results are consistent with the conclusions of time-dependent numerical simulations which have demonstrated an exchange of stabilities from the rotating to the standing wave pattern emerging after the bifurcation, as the elasticity of the fluid increases.
TL;DR: In this article, the existence of localized structures is discussed within the framework of the pattern selection problem for a model for the chlorine dioxide-iodine-malonic acid reaction that represents a key to the understanding of the recently obtained Turing structures.
TL;DR: In this article, the authors focus on the characteristics of these different transitions to MMOs and try to construct a bifurcation diagram, showing that the transition from a simple periodic, period doubled, or chaotic attractor arising from a Feigenbaum route constitutes an interior crisis.
Book•
01 Oct 1993
TL;DR: In this paper, the authors present a monograph for researchers and graduate students in the areas of Hamiltonian systems, singularity theory, and equivariant maps, which will appeal to both researchers and students.
Abstract: The monograph will appeal to researchers and graduate students in the areas of symplectic maps, Hamiltonian systems, singularity theory and equivariant ...
TL;DR: In this article, weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third-order set of ordinary differential equations, and the equations are shown to have three codimension-two bifurcation points: a Takens-Bogdanov bifurecation, at which a gluing bifuration is created; a point at which the gluingbifury is replaced by a pair of homoclinic explosions between which there are Lorenz-like chaotic trajectories; and a new type of bif
TL;DR: In this article, the Bonhoeffer-van der Pol equation was used as a prototype model for excitable systems and the effect of application of a periodic stimulus was investigated for a number of biological and physiological applications.
TL;DR: In this article, a family of two-dimensional maps is given that models (strictly) dissipative oscillators and shows essential features of the bifurcation pattern found.
Abstract: Periodically driven strictly dissipative nonlinear oscillators in general possess a recurring bifurcation structure in parameter space. It consists of slightly modified versions of a basic pattern of bifurcation curves that was found to be essentially the same for many different oscillators. The periodic orbits involved in these bifurcation scenarios also possess common topological properties characterized in terms of their torsion numbers and the way they are connected when parameters are varied. In this paper, this typical bifurcation structure of periodically driven strictly dissipative oscillators will be presented and discussed in terms of examples from Duffing’s equation. Furthermore a family of two-dimensional maps is given that models (strictly) dissipative oscillators and shows essential features of the bifurcation pattern found.
TL;DR: In this paper, the authors present an algorithm for the computation of stable manifolds of equilibrium points, and describe the Hopf bifurcations for equilibria in parametrized families of vector fields.
Abstract: We present several topics involving the computation of dynamical systems. The emphasis is on work in progress and the presentation is informal — there are many technical details which are not fully discussed. The topics are chosen to demonstrate the various interactions between numerical computation and mathematical theory in the area of dynamical systems. We present an algorithm for the computation of stable manifolds of equilibrium points, describe the computation of Hopf bifurcations for equilibria in parametrized families of vector fields, survey the results of studies of codimension two global bifurcations, discuss a numerical analysis of the Hodgkin and Huxley equations, and describe some of the effects of symmetry on local bifurcation.
TL;DR: In this paper, the equation of motion of small, heavy, rigid spherical particles in a periodic Stuart vortex flow is studied as a four-dimensional nonlinear dynamical system with a parametric space of five dimensions.
Abstract: The equation of motion of small, heavy, rigid spherical particles in a periodic Stuart vortex flow is studied as a four‐dimensional nonlinear dynamical system with a parametric space of five dimensions. The five parameters are a scaled Reynolds number, the Stokes number, the fluid‐to‐particle density ratio, the vorticity distribution in the flow, and a gravitational parameter. Depending on the values of these parameters, heavy particles may either settle or remain indefinitely suspended against gravity. When suspension occurs, suspended particles asymptotically collect along periodic, quasiperiodic, or chaotic open trajectories located above or below the vortices. The nature of these asymptotic paths is investigated using the standard tools of power spectrum and bifurcation diagram (Poincare sections). Furthermore, the basins of attraction in the physical and parametric spaces are also computed for both types of suspensions (above and below the vortices). In addition to the two types of upper and lower asymptotic orbits, the dynamical system of this study also exhibits the phenomenon of intermittency, whereby a particle remains suspended alternately above and below the vortices. Apart from open trajectories, closed orbits encircling one or several vortices are also observed in this work.
TL;DR: In this article, a general method for deriving equations for bifurcation hypersurfaces in terms of rate constants and other experimentally controllable parameters is developed, which can be used to obtain better rate constant values and confirm mechanisms from experimental data.
Abstract: Chemical mechanisms with oscillations or bistability undergo Hopf or saddle‐node bifurcations on parameter space hypersurfaces, which intersect in codimension‐2 Takens–Bogdanov bifurcation hypersurfaces. This paper develops a general method for deriving equations for these hypersurfaces in terms of rate constants and other experimentally controllable parameters. These equations may be used to obtain better rate constant values and confirm mechanisms from experimental data. The method is an extension of stoichiometric network analysis, which can obtain bifurcation hypersurface equations in special (h,j) parameters for small networks. This paper simplifies the approach using Orlando’s theorem and takes into consideration Wegscheider’s thermodynamic constraints on the rate constants. Large realistic mechanisms can be handled by a systematic method for approximating networks near bifurcation points using essential extreme currents. The algebraic problem of converting the bifurcation equations to rate constants is much more tractable for the simplified networks, and agreement is obtained with numerical calculations. The method is illustrated using a seven‐species model of the Belousov–Zhabotinskii system, for which the emergence of Takens–Bogdanov bifurcation points is explained by the presence of certain positive and negative feedback cycles.
TL;DR: In this article, the stability information can be derived from the analysis of the bifurcation phenomena on the equilibrium states of the BWRs, which can be used to obtain qualitative and global information on the stability of a nonlinear system.
Abstract: This paper presents a new approach using the bifurcation theory for the stability analysis of BWRs In this approach, the dependencies of the equilibrium states on the parameters that have a large influence on the stability are investigated topological over a wide range of phase space The stability information can be derived from the analysis of the bifurcation phenomena on the equilibrium states This investigation enabled us to obtain qualitative and global information on the stability of a nonlinear system The new approach was applied to the analysis of the stability associated with in-phase power oscillation (core reactivity stability) The loss of linear stability took place at a lower reactor power as the coolant flow rate decreased, and this instability occurs at the Hopf bifurcation point The sensitivity analysis of the stability boundary for the various parameters revealed that the channel hydrodynamics heavily play a significant role in the stability The Hopf bifurcation analysis pr
TL;DR: In this article, a low-order model confirms that these pulsating waves appear via a pitchfork-Hopf-gluing bifurcation sequence from the steady state.
TL;DR: In this paper, the gap and turnstile structures for cantori with non-degenerate anti-integrable limits with arbitrarily many wells per period were computed and the results imply a rich bifurcation diagram for families of maps containing degenerate antiintegrably limits.
01 Jan 1993
TL;DR: In this article, a general course on stability and bifurcation of dissipative systems based upon energetic considerations is presented, which provides an unified framework for the study of quasi-static evolutions and of stability or bifurbation problems in a variety of interesting applications.
Abstract: This paper presents a general course on stability and bifurcation of dissipative systems based upon energetic considerations For time-independent systems, it provides an unified framework for the study of quasi-static evolutions and of stability or bifurcation problems in a variety of interesting applications The theoretical presentation is complemented by a number of simple analytical examples
Journal Article•
TL;DR: In this article, the Coulomb model was used to simulate the dynamics of a forced oscillator with Coulomb friction dependent on both displacement and velocity, and period-doubling bifurcations were observed for the oscillator.
Abstract: In some parameter ranges, the dynamics of a forced oscillator with Coulomb friction dependent on both displacement and velocity is reducible to the dynamics of a one-dimensional map In numerical simulations, period-doubling bifurcations are observed for the oscillator In this bifurcation procedure, the map arising from the Coulomb model may not have ‘standard’ form The bifurcation sequence of the Coulomb model is compared to that of the standard one-dimensional maps to see if it exhibits ‘universal’ behavior All observed components of the bifurcation sequence fit the universal sequence, although some universal events are not witnessed