Imperfect Bifurcation Research Articles

Coupled Lorenz oscillators

A model system is derived consisting of two Lorenz oscillators, linearly coupled. When the temperature differences for both are equal, coexisting stable, steady convective states exist for a range of Rayleigh numbers. These correspond to co-current and counter-current flows. When the system jumps from the former to the latter, there is flow reversal in one oscillator and a reduction in the heat transport in the system. When the temperature differences are unequal, there is an imperfect secondary bifurcation showing similar jump behavior.

Coupled stationary bifurcations in non-flux boundary value problems

AbstractCoupled stationary bifurcations in nonlinear operator equations for functions, which are defined on a real interval with non-flux boundary conditions at the ends, are analysed in the framework of imperfect bifurcation theory. The bifurcation equations resulting from a Lyapunov–Schmidt reduction possess a natural structure which can be obtained by taking real parts of a diagonal action in ℂ2 of the symmetry group 0(2). A complete unfolding theory is developed and bifurcation equations are classified up to codimension two. Structurally stable bifurcation diagrams are given and their dependence on the wave numbers of the unstable modes is clarified.

Structures of steady states of an enzyme reaction in a CSTR

Structures at steady states have been investigated for an enzyme reaction in a continuous stirred tank reactor that has a random bi-uni reaction mechanism, using the imperfect bifurcation theory via singularities. The analysis has shown that two types of singular points exist. One of these points has the two types of transition states characterized by the hysteresis and double-limit points. It is obtained when the derivative of the steady-state equation with respect to the bifurcation parameter does not vanish. When the derivative vanishes, the other type of singular point is obtained. This point has the two transition states of hysteresis and bifurcation points.

Finite dimensional approximation of bifurcation problems in presence of symmetries

We give here a brief summary of the results of [4], [14]. The aim of our work was not only the finite dimensional approximation of the specific problem studied in [3], but we also wanted to show how to adapt to discrete Problems and how to use, in this case, some classical mathematical tools (e.g. the Splitting lemma of Gromoll-Meyer-Magnus, the singularity theory, etc...). Such considerations are developed in [5]. In particular, we prove in [5] that in most cases (e.g. in the examples given in [6], [7], [9], [10]), the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of [12]. Another way to use universal unfoldings in the approxima-tion of bifurcation problems is suggested by [2].

Imperfect Bifurcation with a Slowly-Varying Control Parameter

We consider a general class of imperfect bifurcation problems described by the following first order nonlinear differential equation:\[ y_i = ky^p + \lambda (t)y + \delta , \] where $k = 1$ or $-1$ and $p = 2$ or 3 are fixed quantities. The solution depends on the values of the “imperfection” parameter $\delta (0 < \delta \ll 1)$ and the time-dependent control parameter $\lambda (t) = \lambda _0 + \varepsilon t$ and $0 < \varepsilon \ll 1$). If $\delta = \varepsilon = 0$, this equation admits at $\lambda = 0$ a bifurcation from the basic state $y = 0$ to nonzero steady states. In the first part of the paper, we analyze the perturbation of the bifurcation solutions produced both by the small imperfection $(\delta \ne 0)$ and the slow variation of $\lambda (\varepsilon \ne 0)$. We show that $\lambda = 0$ does not correspond to the transition between the two branches of slowly-varying steady states. This transition appears at a larger value of $\lambda = \lambda _1 $. Provided that $\delta $ is sufficiently ...

Thermal convection under external modulation of the driving force. I. The Lorenz model.

Thermal convection between horizontal plates is considered for a situation in which the driving force varies periodically in time. This variation may come from changes in the temperature or the plates or from a vertical oscillation of the cell, causing variation of the gravitational force. Truncation of the Boussinesq equations leads to a three-mode model which is a generalization of the Lorenz model to the case of external modulation. Similar models have previously been introduced by Finucane and Kelly for stress-free horizontal boundaries and by Gresho and Sani for the rigid-boundary case. The threshold behavior is that of a parametrically driven damped oscillator, whose bifurcations are studied numerically, as well as analytically in certain limits. It is found that in general the modulation stabilizes the conducting state. For stress-free horizontal boundaries the threshold shifts predicted by the model coincide with the results of Rosenblat and Herbert in the limit of low frequency and agree well with Venezian's results for small modulation amplitude, both obtained using the full Boussinesq equations. For rigid boundaries the results agree well with numerical calculations of Rosenblat and Tanaka. The nonlinear behavior of the model is also studied, and the convective contribution to the heat current evaluated. The Lorenz model is shown to reproduce, either exactly or to a good approximation, most previous theoretical results on modulated convection, and the model can be studied simply for a wide range of parameters. The above discussion refers to a laterally infinite system. For a real finite system, sidewall effects are shown to cause a rounding of the convective threshold in the presence of modulation, particularly at low frequencies. A calculation of these effects is carried out within the framework of the Lorenz truncation, and the resulting imperfect bifurcation of the model is studied numerically. In a companion paper (immediately following this one) quantitative experimental results are presented and compared to the predictions of the Lorenz model.

Thermal convection under external modulation of the driving force. II. Experiments.

Experimental data are presented on the temperature response of a Rayleigh-B\'enard system to sinusoidal modulation of the heat current supplied to the lower plate near the onset of convection. Quantitative results are obtained for the average convective current as a function of the average Rayleigh number and of the amplitude and frequency of the modulation. Results are also presented on the temporal behavior of the response, above the convective threshold. The data are interpreted in terms of a previously proposed model which is a generalization of the Lorenz equations to periodic external driving. An important effect included in the model is the dynamic thermal mismatch between the sidewalls and the fluid which leads to an imperfect bifurcation between the conductive and convective states. As a result, the upward shift of the convective threshold caused by modulation in the ideal system is masked by the presence of sidewalls. In our model, this effect is controlled by a single adjustable parameter which turns out to agree in order of magnitude with previous experimental and theoretical estimates pertaining to a lower frequency range. The detailed comparison between experiment and theory shows good qualitative agreement, and some quantitative discrepancies in the parameter range explored. Suggestions are made for additional experiments to test a larger parameter range and in particular to minimize the sidewall effect in order to approximate the ideal threshold behavior more closely.

Transition to the convective regime in a vertical slot

The break-down of the conductive regime in a laterally heated vertical slot is shown to be associated with the inward penetration of nonlinear convective effects from the ends of the slot. This penetration can occur in two quite distinct ways, depending on the Prandtl number of the fluid and the aspect ratio of the slot. At finite and small Prandtl numbers it takes the form of an imperfect bifurcation at a critical value of the Rayleigh number, leading to the establishment of a multiple-roll state throughout the slot. At larger Prandtl numbers there is a different transition process in which convective effects gradually penetrate inwards as the Rayleigh number increases. In this case the solution of an eigenvalue problem provides a quantitative measure of the transition, which leads to the so-called ‘convective regime’ in the slot.

Viscous flow through a rotating square channel

Fully developed flow of an incompressible Newtonian fluid driven by a pressure gradient through a square channel that rotates about an axis perpendicular to the channel roof is analyzed here with the aid of the penalty/Galerkin/finite element method. Coriolis force throws fast-moving fluid in the channel core in the direction of the cross product of the mean fluid velocity with the channel’s angular velocity. Two vortex cells form when convective inertial force is weak. Asymptotic limits of rectilinear flow and geostrophic plug flow are approached when viscous force or Coriolis force dominates, respectively. A flow structure with an ageostrophic, virtually inviscid core is uncovered when Coriolis and convective inertial forces are both strong. This ageostrophic two-vortex structure becomes unstable when the strength of convective inertial force increases past a critical value. The two-vortex family of solutions metamorphoses into a family of four-vortex solutions at an imperfect bifurcation composed of a pair of turning points.

Imperfection sensitivity of interacting hopf and steady-state bifurcations and their classification

The bifurcation problem of interacting time-periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory. A complete classification, up to symmetry-covariant contact equivalence and codimension three, of generic perturbations of interacting Hopf and steady-state bifurcations is presented. The sensitivity of the bifurcation diagrams to imperfections is analyzed. Normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined. A variety of phenomena, such as gaps in Hopf branches, periodic motions not stably connected to steady states and the formation of islands, is discovered, which one can expect to find in perturbed evolution equations on pure geometric grounds. Implications for physical systems are discussed.

Classification and Unfolding of Sequential Bifurcations

Golubitsky and Schaeffer have developed an extensive theory of imperfect bifurcation by adapting the determinacy and unfolding theorems of singularity/catastrophe theory to singularities with a distinguished parameter $\lambda $. In this paper we adapt their methods and results to “sequential bifurcations” of the form $a(u,\lambda ) = 0$, $b(x,u,\lambda ) = 0$. Here $\lambda $ is a bifurcation parameter, x represents the final “output” state of the system, and u is interpreted as a “hidden” variable. Such problems arise when a bifurcating process in $(u,\lambda )$ is coupled in sequence with a u-dependent bifurcation process in $(x,\lambda )$. Assuming that the first process is independent of x, it is inappropriate to treat the coupled system as a simple bifurcation of $(x,u)$ with $\lambda $. Instead, it is necessary to develop a version of the theory which preserves the special “intermediate” role of u—much as the Golubitsky–Schaeffer theory preserves the special role of $\lambda $. Such a theory is developed here. As well as finding determinacy and unfolding criteria, we classify all sequential bifurcation problems of codimension 4 or less when x and u are one-dimensional. We exhibit the possible bifurcation diagrams for codimension 2 or less (those of higher codimension are omitted for reasons of space) and give analytic conditions for the occurrence of a given bifurcation in the classification. We discuss applications of these ideas to suitable systems of chemical reactions. The presence of “hidden” or intermediate variables u has a strong effect on the expected bifurcation phenomenology, and hence on the inferences that may be made from theoretical results: (a) Multiple limit points can occur in a persistent (structurally stable) fashion. (b) Eliminating a hidden variable can change the codimension of a bifurcation problem (because some perturbations of the resulting problem, that contribute to the codimension, may be incompatible with the elimination step). (c) Bifurcation diagrams that are ordinarily considered inequivalent may become equivalent if a hidden variable is present. The effect of (a) is to introduce new types of persistent diagram. The likelihood of seeing a given diagram in a parametrized family of bifurcation problems is affected by (b). And (c) implies that different ways of eliminating hidden variables from equations may produce apparently different observable consequences.

Bifurcation theoretic analysis of discrete-continuum coupling in atomic and molecular systems

A bifurcation theoretic approach for problems involving bound-continuum transition is pointed out. Models are used to illustrate the technique. The practical utility of such a method is discussed utilizing the concept of imperfect bifurcations.

Applications of nonelementary catastrophe theory

In this paper, a sequel to Stewart [37], [38] we describe some of the recent applications of nonelementary catastrophe theory, especially the Golubitsky-Schaeffer approach to imperfect bifurcation. It is in these applications that the importance of Catastrophe Theory for the "hard" sciences becomes clearest, in that several of the applications make decisive contributions to problems that have been open for some time-even several decades. These include the phenomenon of mode-jumping in the buckling plate, the Benard Problem, and the effects of finite cylinders in the Taylor vortex problem. Other applications discussed include chemical reactions, nerve impulse transmission, and reaction-diffusion equations. "Nonelementary" is used here in the sense of Stewart [37], [38] to refer to extensions of the theory of elementary catastrophes that are applicable to problems not of gradient type. According to an arguably more useful distinction made by Zeeman [44], many become "elementary" in his sense. For applications of the original elementary catastrophes see Poston and Stewart 1251, Stewart [34]-[36], and Zeeman [43].

Flame Stabilization in a Layered Medium

Abstract We analyze the steady-state structure of a l1ame stabilized on a burner which is fed by a nonuniform mixture. Our diffusional-thermal model is based on the assumption of large overall activation energy, and the nonuniformity in the feed is treated as a perturbation of the uniform case. The flame shape for a given inflow enthalpy profile depends on Lewis number, inflow velocity, and conductance of the burner, and for sufficiently small Lewis number, we exhibit the onset of cellular flame formation. Unlike previously encountered flame instabilities, the transition to a cellular name occurs here in a gradual, rather than an abrupt, fashion through an "imperfect bifurcation."

Nonelementary catastrophe theory

This tutorial paper is intended to complement the preceding one on Elementary Catastrophe Theory, and to emphasize that many of the supposed limitations of the elementary catastrophes may be overcome by suitably modifying the mathematical setting. The main generalization discussed here is the Imperfect Bifurcation Theory of Golubitsky and Schaeffer Catastrophe Theory with a distinguished parameter. The effects of symmetry are also discussed, leading to some important results on degenerate Hopf bifurcation. Illustrative examples of applications of these ideas are included, but are kept simple to illuminate the mathematical ideas involved. Surveys of applications in the physical sciences of these and related ideas may be found in Stewart (1981, 1982) and in the forthcoming sequel to this tutorial paper. Applications of Non-elementary Catastrophe Theory, to appear in this journal.

An organizing center for optical bistability and self-pulsing

The problem of self-pulsing in optically bistable systems is discussed within the framework of imperfect bifurcation theory. The joint appearance of a hysteresis cycle in the cw-transmission curve and of transitions to self-pulsing is described as an interaction between steady-state and Hopf bifurcations induced by varying the incident field intensity. The bifurcation equations for the most degenerate case are shown to be determined by a corank-two and codimension-four polynomial normal form. This form can be extracted from analytical and numerical studies on the Maxwell-Bloch equations, and acts as an organizing center for bistable switching and the self-pulsing mechanism. The structurally stable unfolded bifurcation diagrams are analyzed. Besides describing correctly and in a comprehensive way all bifurcations to self-pulsing that have so far been observed, a number of new generic transitions are predicted. These include self-pulsing from the low transmission branch and transitions leading to the formation of islands with self-pulsing behavior.

Classification of Z(2)-Equivariant Imperfect Bifurcations with Corank 2

Classification of Z(2)-Equivariant Imperfect Bifurcations with Corank 2

Singularities of dispersion relations

SynopsisGeneric singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.

Analytic singularities in dispersive wave phenomena based on topological singularities of dispersion relations

The connection between topological singularities of dispersion relations and analytic singularities with respect to frequency of spectral densities and wavefunctions in dispersive media due to time harmonic sources is demonstrated. The topological singularities are discussed within the framework of imperfect bifurcation theory, regarding the frequency as a distinguished bifurcation parameter and the wavenumbers as bifurcation variables, and using a recently obtained classification of topological singularities up to codimension four which is given in terms of a list of normal forms. To each normal form corresponds an analytic singularity, governed by characteristic exponents which are tabulated.

Imperfect Bifurcations and Spatial Structures in Dissipative Systems

Abstract One-dimensional reaction-diffusion equations associated with the trimolecular model of Prigogine and Lefever ("Brusselator") are analyzed. A physical description of possibilities of keeping con-centrations of initial components constant is discussed. It is shown that the problem considering diffusion of initial components gives rise to an imperfect bifurcation problem. The diffusion equa-tions have been solved numerically by a continuation procedure for the fixed and zero flux boundary conditions. The analysis indicates that the models including diffusion of all reacting components do not admit an occurence of trivial solutions. These models, as a result, also exclude the pos-sibility of primary bifurcations. The models which consider diffusion of the initial components suppress the number of possible solutions of governing equations. These models may also predict both symmetric and asymmetric states. Apparently this type of models is more suitable for predic-tion of patterns of spatial organization in growth. Since the number of possible profiles is strongly reduced this principle may lead to a more deterministic way of an evolution process. Symmetric profiles occuring on an isola cannot be reached by an evolution process unless a large perturbation is imposed on the system.