Bifurcation Formula Research Articles

Application of Cholette’s model in non-ideal mixing CSTR: a simulation study on dynamic behavior

Abstract Non-ideal mixing phenomena are widely found in industrial chemical reactors. In this work we derived the bifurcation formulas for a non-adiabatic CSTR with an irreversible exothermic first order reaction with the non-ideal mixing effect. This is investigated via dynamic behavior simulations based on Chollete’s model. The results show that the non-ideal mixing parameter n (the fraction of the feed entering the perfect mixing zone) determines the variation between six classified regions and dominates the dynamic behavior patterns in the steady-state response diagram. On the other hand, the phase portraits of examples verify the formulas derived in this work. We note that the non-ideal mixing effect has significant importance in CSTR design and control steps. For example, in the safe operating region for an ideal mixing CSTR, non-linear dynamics are obtained by the system under non-ideal mixing conditions (n ≠ 1). The present study has significance and help for chemical reactor design and CSTR control.

Lyapunov Quantities in the Problem about Local Bifurcations of Non-Autonomous Periodic Dynamical Systems

The article discusses the problems about the main scenarios of dynamical systems’ local bifurcations described by non-autonomous differential equations with periodic right-hand side. New formulae for calculating Lyapunov quantities are proposed. The proposed formulae are obtained on the basis of the general operator method for studying local bifurcations and do not require a transition to normal forms and the usage of the theorems about the central manifold. The indicated method allowed for obtaining new bifurcation formulae for studying the main scenarios of local bifurcations. The paper shows how these bifurcation formulae lead to new formulae for calculating Lyapunov quantities in the problems about bifurcations of forced oscillations and Andronov–Hopf bifurcations.

On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation

We study the bifurcation of periodic travelling waves of the capillary–gravity Whitham equation. This is a nonlinear pseudo-differential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillary–gravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov–Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and, therefore, positive and convex; the threshold value for this to happen turns out to be $$T = \frac{4}{\pi ^2}$$, not the critical Bond number $$\frac{1}{3}$$), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in $$L^2(\mathbb {S})$$ (but not in $$L^\infty $$) towards one of the two constant solution curves.

Bifurcation Formulas and Algorithms of Constructing Central Manifolds of Discrete Dynamical Systems

Ones of the main questions in theory of local bifurcations and its applications are questions about direction of bifurcations (sub- or supercriticality) and on stability of the solutions arising in neighborhood of a nonhyperbolic equilibrium point or cycle dynamic system. We consider problems of local bifurcations in dynamical systems with discrete time. New features are proposed to orientation of bifurcations and properties stability of bifurcation solutions for problems on basic scenarios of bifurcations. We also propose new algorithms for constructing central manifolds of the corresponding problems, allowing to obtain new bifurcation formulas, in particular, formulas to calculate Lyapunov quantities. Proposed algorithms and formulas are based on the common operator method the study of problems on local bifurcations and allow under the new conditions effective qualitative analysis of bifurcations in terms of the initial equations.

Бифуркационные формулы и алгоритмы построения центральных многообразий дискретных динамических систем

Бифуркационные формулы и алгоритмы построения центральных многообразий дискретных динамических систем

Hopf bifurcation formulae and applications to the Genesio-Tesi system

Hopf bifurcation formulae and applications to the Genesio-Tesi system

Hopf bifurcation and global stability of a diffusive Gause-type predator–prey models

This paper mainly provides Hopf bifurcation formulas for a general Gause type predator–prey system with diffusion and Neumann boundary condition by using the center manifold theory and normal form method, where the spectral and stability analysis around an equilibrium is addressed, and our results can be applied to the case without diffusion. As an application of these results, we give a complete and rigorous analysis of the global dynamics of a diffusive predator–prey model with herd behavior, especially, the Hopf bifurcation and its direction, and the stability of the bifurcating periodic solutions.

Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator–Prey System with Beddington–DeAngelis Functional Response

In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator–prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter [Formula: see text]. Especially, we generalize the geometric criterion in [Beretta & Kuang, 2002] by introducing the condition [Formula: see text] which can be assured by the condition [Formula: see text], and adopting the technique of lifting to define the function [Formula: see text] for alternatively determining stability switches at the zeroes of [Formula: see text]s. Afterwards, by the Poincaré normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results.

Global Bifurcation for the Whitham Equation

We prove the existence of a global bifurcation branch of 2π -periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Holder class C α , α < 1/2. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a ‘highest’, cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.

Stable periodic oscillations in a two-stage cancer model of tumor and immune system interactions

This paper presents qualitative and bifurcation analysis near the degenerate equilibrium in a two-stage cancer model of interactions between lymphocyte cells and solid tumor and contributes to a better understanding of the dynamics of tumor and immune system interactions. We first establish the existence of Hopf bifurcation in the 3-dimensional cancer model and rule out the occurrence of the degenerate Hopf bifurcation. Then a general Hopf bifurcation formula is applied to determine the stability of the limit cycle bifurcated from the interior equilibrium. Sufficient conditions on the existence of stable periodic oscillations of tumor levels are obtained for the two-stage cancer model. Numerical simulations are presented to illustrate the existence of stable periodic oscillations with reasonable parameters and demonstrate the phenomenon of long-term tumor relapse in the model.

Numerical investigation of bifurcations of equilibria and Hopf bifurcations in disease transmission models

One of the general SIRS disease transmission model is considered under the assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. A combination of analytical and numerical techniques is used to show that (for some parameters) the bifurcations of equilibria can occur and also asymptotically orbitally stable periodic solutions with asymptotic phase can arise through Hopf bifurcations. The investigation is based on computer simulation of bifurcation manifolds in the parameter space. Hopf bifurcations are investigated on the base of center manifold theory by the computation of bifurcation parameters and the approximation of Hopf-bifurcating cycles by bifurcation formulas. This method finds the limit cycle to a good approximation and also its stability. For computer simulations the necessary computer oriented algorithms were developed and encoded by C++. Some results of computer simulations are presented and numerical evidence of existence of bifurcations of equilibria and Hopf bifurcations for the considered model is provided.

Hopf bifurcation formula for first order differential-delay equations

This work presents an explicit formula for determining the radius of a limit cycle which is born in a Hopf bifurcation in a class of first order constant coefficient differential-delay equations. The derivation is accomplished using Lindstedt’s perturbation method.

Corridor Stability in the Dendrinos Model of Regional Factor Movements

Geographical AnalysisVolume 27, Issue 4 p. 360-368 Free Access Corridor Stability in the Dendrinos Model of Regional Factor Movements Thomas Lux, Thomas Lux Thomas Lux holds a post-doctoral position in the department of economics and social sciences, University of Bamberg.Search for more papers by this author Thomas Lux, Thomas Lux Thomas Lux holds a post-doctoral position in the department of economics and social sciences, University of Bamberg.Search for more papers by this author First published: October 1995 https://doi.org/10.1111/j.1538-4632.1995.tb00916.x AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat LITERATURE CITED Arthur, W. B. (1988). “ Urban Systems and Historical Path Dependence.” In Cities and Their Vital Systems, edited by J. H. Ausubel and R. Herman. Washington: National Academy Press. Arthur, W. B. (1991). “Silicon Valley Locational Clusters: When Do Increasing Returns Imply Monopoly? Mathematical Social Sciences 19, 235– 51. David, P. A., and J. L. Rosenbloom (1990). “Marshallian Factor Market Externalities and the Dynamics of Industrial Localization. Journal of Urban Economics 28, 349– 70. Dendrinos, D. S. (1982). “On the Dynamic Stability of Interurban/Regional Labor and Capital Movements. Journal of Regional Science 22, 529– 40. Dendrinos, D. S., and G. Haag (1984). “Toward a Stochastic Dynamical Theory of Location: Empirical Evidence. Geographical Analysis 16, 287– 300. Dendrinos, D. S., and H. Mullaly (1984). “Interurban Population and Capital Accumulations and Structural Stability. Applied Mathematics and Computation 14, 11– 24. Dendrinos, D. S., and H. Mullaly (1985). Urban Evolution: Studies in the Mathematical Ecology of Cities. Oxford: University Press. Guckenheimer, J., and P. Holmes (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. New York: Springer. Krugman, P. (1991). “History and Industry Location: The Case of the Manufacturing Belt. American Economic Review, Papers and Proceedings 81, 80– 83. Orishimo, I. (1987). “An Approach to Urban Dynamics. Geographical Analysis 19, 200– 209. Perko, L. (1991). “ Differential Equations and Dynamical Systems.” New York: Springer. Ponce-Nunez, E. J., and E. Gamero. (1991). “ Generating Hopf Bifurcation Formulae with MAPLE.” Paper, University of Sevilla. Zhang, W.-B. (1988). “Population Oscillations in a Nonlinear Migration Model. Geographical Analysis 20, 156– 75. Volume27, Issue4October 1995Pages 360-368 ReferencesRelatedInformation

Application of higher order bifurcation formulas

The conclusions of the Hopf bifurcation theorem are exact in an unspecified neighborhood of the critical value of the bifurcation parameter at which stability change takes place. It is shown that when the bifurcation parameter is far from its critical value and/or the system is highly nonlinear, it is necessary to use higher order bifurcation formulas to ensure good results. A brief description of the method and the applied algorithm is presented. Results that show the improvements obtained by the application of this approach to a chemical reactor are included. >

Degenerate Hopf bifurcations via feedback system theory: Higher-order harmonic balance

In this work we present some higher-order Hopf bifurcation formulas to analyze the rich dynamic behavior of chemical systems under the failure of some hypotheses of the classical Hopf theorem. The treatment is done entirely in an equivalent formulation in the so-called “frequency domain”. This approach gives a natural interpretation about some Hopf degeneracies involving limit points on the periodic branch and multiple periodic solutions. Moreover, we calculate the expression for the second curvature coefficient, extending previous results using this formulation. This coefficient is equivalent to one of the higher-order coefficients in the Hopf bifurcation normal form used previously by other researchers.

A class of bifurcation solutions of almost-periodic parametric vibration systems

A class of bifurcation solutions of almost-periodic (a. p. for short) parametric vibration systems is studied by Liapunov-Schmidt reduction. The bifurcation diagrams and formulas are given.

Degenerate Hopf Bifurcation and Nerve Impulse. Part II

The bifurcation from equilibrium of periodic solutions of the Hodgkin and Huxley equations for the nerve impulse is studied. In earlier work singularity theory techniques were used to establish that these equations have a branch of periodic solutions undergoing two Hopf bifurcations, and the equations were conjectured to be equivalent to a member of a one-parameter family of generalized Hopf bifurcation problems. Here the invariants for equivalence to this family and the value of the modal parameter are computed (see [W. W. Farr et al., “Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem,” SIAM J. Math. Anal., 20 (1989), pp. 13–30]). The value of this parameter determines the type of bifurcation, and in this way it is decided which of the proposed bifurcation diagrams are actually to be found. Thus a topological description of periodic orbits of the Hodgkin and Huxley equations near the equilibrium solution is obtained. In this way, a periodic solution branch is found that does not arise thro...

Local feedback stabilization and bifurcation control, I. Hopf bifurcation

Local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases. Sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple, nonzero imaginary eigenvalues. The conditions show, in particular, that generically these nonlinear critical systems can be stabilized locally, even if the critical modes are uncontrollable. The analysis also yields a direct method for computing stabilizing feedback controls. Use is made of bifurcation formulae which require only a series expansion of the vector field.

Recursive bifurcation formulae for normal form and center manifold theory

Applicable recursive formulae are obtained for n-dimensional bifurcation problems. These formulae give a reduced system, corresponding to the center manifold theorem, in Brjuno's normal form, up to an arbitrary order. Previous formulae by I.D. Hsu-N.D. Kazarinoff and B. Hassard-Y.H. Wan can be essentially included as particular cases.

Hopf Bifurcation to Repetitive Activity in Nerve

Various nerve axons and other excitable systems exhibit repetitive activity (e.g., trains of propagated impulses) in response to a spatially localized, time-independent stimulus. If the stimulus is too weak, or in some cases too strong, one finds rather a spatially nonuniform, steady response which attenuates with distance from the input site. We consider these stimulus-response properties for a qualitative model of nerve conduction, the FitzHugh–Nagumo (parabolic partial differential) equations. For each stimulus amplitude there is a unique steady state solution. At critical stimulus values this steady solution loses stability and a branch of time periodic (spatially nonuniform) solutions appears via Hopf bifurcation. We associate this with the onset of repetitive activity. We derive analytic bifurcation formulae and evaluate these for the case of a cubic nonlinearity to determine regions in parameter space where the steady solution is unstable. We interpret physiologically the effect of stimulus form; either voltage or current input, and either spatially localized or spatially uniform stimulus.