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... He imposes on the reaction term f (u) some technical conditions and avoids computational difficulties when takes into account only this solution. In the works of Ambrosetti and Rabinowitz [1], Aronson and Weinberger [2], Crandal and Rabinowitz [3], [4], Keller and Cohen [9], Simpson and Cohen [18], Sattinger [14], [15], [16] and Turner [19], to quote but a few, the authors take the advantage of a linear term in the full reaction term f (u). Consequently, these analyses are useless for our purposes. ...

... Thus, (17) and (3) imply (15). More than that, dJ * (u+εv,z) dε ε=0 = 0 means the first equation in the system (15). ...

... Thus, (17) and (3) imply (15). More than that, dJ * (u+εv,z) dε ε=0 = 0 means the first equation in the system (15). This completes the proof. ...

We consider a nonlinear, second-order, two-point boundary value problem that models some reaction-diffusion processes. When the reaction term has a particular form, f (u) = u 3 , the problem has a unique positive solution that satisfies a conserved integral condition. We study the bifurcation of this solution with respect to the length of the interval and it turns out that solution bifurcates from infinity. In the first part, we obtain the numerical approximation to the positive solution by direct (variational) methods, while in the second part we consider indirect numerical methods. In order to obtain directly accurate numerical approximations to this positive solution, we characterize it by a variational problem involving a conditional extremum. Then we carry out some numerical experiments by usual finite elements method. MSC 2000. 34B18, 34C23, 65L10, 65L60.

... The Jacobian of the reduced equations is evaluated and it is proved that the eigenvalues of the Jacobian of the reduced equations determine, to lowest nonvanishing order, the stability of the bifurcating solutions. This is done in $7. Theorem 7.2 is a general result which is independently of interest in itself and has as an immediate corollary a number of previous results on the stability of bifurcating solutions due to Sattinger [27], C randall and Rabinowitz [3], and McLeod and Sattinger [ 131. (S ee also L. Nirenberg [17], pp. 102-110). ...

... In this section we relate the stability of the bifurcating solutions to the eigenvalues of the Jacobian of the reduced bifurcation equations. In one respect the results of this section can be viewed as an extension to multiple eigenvalues of previous results of Sattinger [27], McLeod and Sattinger [13], and Crandall and Rabinowitz [3]. (See also L. Nirenberg [17], pp. ...

... Thus, we have shown that supercritical solutions are stable and subcritical solutions are unstable. (See [27], [3], [17]). The reader may go on to the more general case when several of the derivatives y'(O), . . . ...

... Interesting questions about traveling fronts, for instance, are about the existence of traveling waves, their monotonicity for space, stability and its convergence rate to a traveling wave. About these problems, for the continuous Allen-Cahn model, we refer the reader to [1,8,12,15,3,14], for example. The lattice system (2) arises in chemical reaction theory [7,9,11] and biology [2,10]. ...

... Combining this with (14), one has ...

The existence and stability of the Allen-Cahn equation discretized in space and time are studied in a finite spatial interval. If a parameter is less than or equals to a critical value, the zero solution is the only stationary solution. If the parameter is larger than the critical value, one has a positive stationary solution and this positive stationary solution is asymptotically stable.

... We derive a theoretical result that the spatio-temporal pattern formation of model (1.5) may appear if and only if the chemotactic parameter χ < 0 and offer a review of abstract global bifurcation theory in the next section. By utilizing the user-friendly version of Crandall-Rabinowitz bifurcation theory proposed by Shi and Wang [32], we introduce and prove our first main theoretical result in Section 3. Together with linearized stability in [33,34], the results on perturbation of simple eigenvalues and spectrum theory, we, in some cases, show that the positive nonconstant steady states of (1.6) possess local stability in Section 4. In Section 5, a numerical example is given to describe the feasibility of our main theoretical results. Finally, some conclusions are performed to summarize our main analytic results in Section 6. ...

... Together with linearized stability in [33,34], the results on perturbation of simple eigenvalues and spectrum theory, we, in this section, show that the bifurcating branches, arise from the homogeneous patterns (u * , v * ), possess the local stability in some situations. ...

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall–Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.

... We derive a theoretical result that the spatio-temporal pattern formation of model (1.5) may appear if and only if the chemotactic parameter χ < 0 and offer a review of abstract global bifurcation theory in the next section. By utilizing the user-friendly version of Crandall-Rabinowitz bifurcation theory proposed by Shi and Wang [32], we introduce and prove our first main theoretical result in Section 3. Together with linearized stability in [33,34], the results on perturbation of simple eigenvalues and spectrum theory, we, in some cases, show that the positive nonconstant steady states of (1.6) possess local stability in Section 4. In Section 5, a numerical example is given to describe the feasibility of our main theoretical results. Finally, some conclusions are performed to summarize our main analytic results in Section 6. ...

... Together with linearized stability in [33,34], the results on perturbation of simple eigenvalues and spectrum theory, we, in this section, show that the bifurcating branches, arise from the homogeneous patterns (u * , v * ), possess the local stability in some situations. ...

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction-diffusion predator-prey model with prey-taxis and double Beddington-DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall-Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.

... Thus, we will use the method on linearized stability and perturbation of simple eigenvalues in [41,42] and spectrum theory to study the local stability of the nonconstant positive solutions. ...

In this paper, the pattern formations in a general predator−prey model with ratio-dependent predator influence and prey-taxis is investigated. We research the local stability of the positive equilibrium and gives a priori estimates for any posi- tive solutions. By an abstract bifurcation theorem, we get a branch of nonconstant solutions that bifurcate from the positive equilibrium. Furthermore, the local sta- bility of the bifurcating solutions is discussed by the method of perturbation of simple eigenvalues and spectrum theory. These results indicate that repulsive prey- taxis can destabilize the positive equilibrium and induced the emergence of spatial patterns.

... The code allows us to switch branches at these points and to inspect the instability-causing modes (v, f y , Y). Note that since the equations governing these modes are linear (and therefore homogeneous), the bifurcations detected are necessarily degenerate: they are 'vertical bifurcations' [27], in which the bifurcating branch of solutions has a constant bifurcation parameter (μ = μ c ) and is parametrized by amplitude (or by some norm). This is not a problem for the numerical bifurcation detection. ...

Motivated by applications of soft-contact problems such as guidewires used in medical and engineering applications, we consider a compressed rod deforming between two parallel elastic walls. Free elastica buckling modes other than the first are known to be unstable. We find the soft constraining walls to have the effect of sequentially stabilizing higher modes in multiple contacts by a series of bifurcations, in each of which the degree of instability (the index) is decreased by one. Further symmetry-breaking bifurcations in the stabilization process generate solutions with different contact patterns that allow for a classification in terms of binary symbol sequences. In the hard-contact limit, all these bifurcations collapse into highly degenerate ‘contact bifurcations’. For any given wall separation at most a finite number of modes can be stabilized and eventually, under large enough compression, the rod jumps into the inverted straight state. We chart the sequence of events, under increasing compression, leading from the initial straight state in compression to the final straight state in tension, in effect the process of pushing a rod through a cavity. Our results also give new insight into universal features of symmetry-breaking in higher mode elastic deformations. We present this study also as a showcase for a practical approach to stability analysis based on numerical bifurcation theory and without the intimidating mathematical technicalities often accompanying stability analysis in the literature. The method delivers the stability index and can be straightforwardly applied to other elastic stability problems.

... предполагается периодической по переменной t с периодом T = 2π/ω, ω > 0 (предположение о существовании составного решения является очевидным выводом из общей методики исследования эволюционных гидродинамических уравнений [28]- [30]). Cлагаемое v † (x, t) можно ассоциировать с нестационарным вихревым течением в исследуемой гидродинамической системе, описываемой с помощью редуцированных уравнений Барнетта, представляющих собой УНС с дополнительным операторным слагаемым F (2) j . ...

... Since B(z, z) = 0 if and only if z = 0 (see [8, p. 85 6. Stability. Stability will be defined as in [21] or [22] The operator, ^'(w*, A*), is a symmetric perturbation of a self-adjoint operator (see [14, p. 226]) and is therefore itself a self-adjoint operator with only real eigenvalues (see Kato,[11,p. 287]). If p^ / p°x, then I -p£A has a negative eigenvalue. ...

The author studies the buckling of a thin elastic shallow spherical cap, simply supported at its edge, and subjected to a constant centrally directed external pressure. Previously obtained experimental evidence indicates that a buckled state of a spherical shell may possess various degrees of symmetry. The most symmetric buckled state observed is that of a circular dimple, while the least symmetric state observed is that of a pear-shaped dimple with one plane of reflectional symmetry. The author shows that when the cap is shallow, there exist stable axisymmetric solutions. In addition, he shows the existence of a variety of other unstable solutions which bifurcate from the unbuckled state of the shell. 25 Refs.

In this paper, a general reaction–diffusive predator–prey system with prey-taxis subject to the homogeneous Neumann boundary condition is considered. Firstly, we investigate the local stability of the unique positive equilibrium by analyzing the characteristic equation and study a priori estimates of positive solutions by the iterative technique. And then, choosing the prey-tactic sensitivity coefficient as bifurcation parameter, we proved that a branch of nonconstant solutions can bifurcate from the unique positive equilibrium when the prey-tactic sensitivity is repulsive. Moreover, we find the stable bifurcating solutions near the bifurcation point by the spectrum theory under some suitable conditions. Our results show that prey-taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns.

In this paper, we present a criterion for pitchfork bifurcation of smooth vector fields based on a topological argument. Our result expands Rajapakse and Smale's result \cite{RS2} significantly. Based on our criterion, we present a class of families of non-symmetric vector fields undergoing a pitchfork bifurcation.

This chapter presents a survey of bifurcation theory. Bifurcation is a term used in several parts of mathematics. It generally refers to a qualitative change in the objects being studied because of a change in the parameters on which they depend. Much of the motivation for studying bifurcation is provided by varied phenomena in the physical sciences, which can be formulated easily. There is a considerable literature on bifurcation, both for particular physical problems and for the general theory. Several examples of the former are also described in the chapter.

This chapter focuses on the stability of solutions bifurcating from steady or periodic solutions. The method of Lyapounov and Schmidt allows the obtainment of branches of solutions of the functional equation F(x,λ) = 0 in a neighborhood of a point, say (0,0), where the Freenet derivative Fx(0,0) is a Fredholm operator. In this case, x lies in a Banach space X, and the parameter λ lies in a Banach space λ. If, in particular, Fx (0,0) has a one-dimensional null space, the solutions of F(x,λ) = 0 near (0,0) can be expressed in the form {x(α,λ),λ} where α is a real parameter defined in terms of a suitable linear functional d* as α = d*x and a solution is obtained if and only if the bifurcation equation φ(α,λ) = 0 is satisfied. This chapter discusses the proof of the equation sgn σ(α,λ) = sgn φ α(α,λ) for the case of the Lyapounov–Schmidt bifurcation. It discusses how to obtain the analogous result for the bifurcations from a nonconstant periodic solution originally treated by Poincaré. The chapter describes a case where the system is time dependent and periodic. It also presents the proof for the bifurcation of periodic solutions from a steady-state solution obtained by E. Hopf.

Experimental evidence, the principle of exchange of stabilities and numerical computations by Ritz’s and Galerkin’s methods suggested that some steady flows become unstable owing to the appearance of new steady or time periodic flows—called secondary flows. This is why branching theory of solutions of the Navier-Stokes equations is closely connected with hydrodynamic stability theory.

We study the 2D Doi--Onsager models with general potential kernel, with
special emphasis on the classical Onsager kernel. Through application of
topological methods from nonlinear functional analysis, in particular the
Leray--Schauder degree theory, we obtain the uniqueness of the trivial solution
for low temperatures as well as the local bifurcation structure of the
solutions.

This chapter reviews recent progress in bifurcation theory. To begin with, bifurcation theory deals with the analysis of branch points of nonlinear functional equations in a vector space, usually a Banach space. The subject of bifurcation is an important topic for applied mathematics in as much as it arises naturally in any physical system described by a nonlinear set of equations depending on a set of parameters. When discussing applied problems, stability considerations are unavoidable. In fact, the phenomenon of bifurcation is intimately associated with the loss of stability. A complete resolution of the branching problem therefore requires an analysis of the stability of the bifurcating solutions. When the nonlinear equations are invariant under a transformation group, the invariance is inherited by the bifurcation equations, and this fact may aid considerably in their analysis. Thereafter, the branching problem in the case of higher multiplicities is considerably more complicated. The assumption of some kind of group invariance is a natural one, and the importance of this aspect of bifurcation theory has attracted the attention of a number of workers in recent years. Moreover, the appearance of cellular solutions can be described mathematically as the bifurcation of doubly periodic solutions.

In this contribution we consider model equations of increasing complexity which exhibit bifurcation in a continuum of points. In spite of the wealth of solutions present there is - under certain simple geometrical conditions - a selection principle yielding only a finite number of stable solutions. The geometrical conditions mentioned guarantee essentially “supercritical” bifurcation for the continuum as an entity and thus reflect at this level the now classical result for simple eigenvalue bifurcation [2], [7].

This chapter highlights the nonlinear bifurcation problem and buckling of an elastic plate subjected to unilateral conditions in its plane. Many concrete situations of buckling of elastic structures arise in engineering sciences. The corresponding problems have been studied since a long time, when the prescribed conditions are bilateral; they are described by an eigenvalue problem for a system of partial differential equations and boundary conditions. The chapter discusses the buckling of an elastic plate, when a crack is expanded at the boundary. Along the crack, the horizontal displacement is limited by the support. Under some assumptions, the critical load can be described and the bifurcated solutions exhibited, but the number of these solutions remains unknown. This chapter, using von Karman's theory, solves a problem of linear elasticity in the place of the plate with unilateral boundary conditions so the horizontal displacement is given by a variational inequality. In teis chapter, the physical problem is described and some results are summarized. Variational formulation and functional formulation for the vertical displacement is also elaborated in the chapter.

Résumé. — Dans cet article, on étudie numériquement, à l'aide d'un schéma obtenu par discrétisation en temps du problème d'évolution associé, un problème elliptique non linéaire qui entre dans le cadre des problèmes de bifurcation. Des résultats de convergence montrent que pour certaines valeurs du pas de discrétisation en temps, les solutions stables du problème elliptique sont encore stables pour le schéma, tandis que pour d'autres valeurs du pas, ce sont des solutions instables du problème elliptique qui deviennent stables pour le schéma. Des essais numériques en dimension un et deux complètent cette étude, Abstract. — In this paper we investigate numerical analysis of a non linear elliptic problem which belows to a class of bifurcation problems, by use of a time discrétisation of the associated évolution équation. Results of convergence show thatfor some values of the step of time discrétisation, stable solutions ofthe elliptic problem are still stable for the scheme, whilefor other values ofthe step, stable solutions of the scheme are instable solutions for the elliptic problem. Numerical calculations in dimension one and two complete this work.

Previous experiments have established that when a light fluid is accelerated with a dense fluid in the direction of the latter, the plane interface is unstable, but when a dense fluid is accelerated with a light fluid in the direction of the latter, the plane interface is stable. An attempt is made to study the incompressible inviscid irrotational hydrodynamic problem, with variable interface, as a problem in nonlinear partial differential equations. (LCL)

This chapter focuses on the principle of exchange of stability. The principle of exchange of stability is generally employed in the context of a family of evolution equations for which bifurcation occurs. It refers to a qualitative relationship between the shape of the bifurcating curve of solutions and their stability. This chapter presents a version of the principle of exchange of stability in the context of Hopf bifurcation, that is, the bifurcation of a family of time periodic solutions from a family of equilibrium solutions. It discusses some preliminaries on bifurcation. The chapter presents the results for the time periodic case, first in the simpler context of Hopf's work and then for an evolution equation in a Banach space.

This paper is devoted to some of the results in bifurcation theory obtained by topological methods in the last 25 years. The cases of one and several parameters will be reviewed, with “necessary” and sufficient conditions for bifurcation, both local and global, and the structure of the bifurcation set will be studied. The case of equivariant bifurcation will be considered, with a special application to the case of abelian groups.

We study pattern formations in a predator–prey model with prey-taxis. It is proved that a branch of nonconstant solutions can bifurcate from the positive equilibrium only when the chemotactic is repulsive. Furthermore, we find the stable bifurcating solutions near the bifurcation point under suitable conditions. Copyright © 2014 John Wiley & Sons, Ltd.

There exist some connections between the stability properties of fixed points of maps and the corresponding topological degree. A sufficient condition for asymptotic stability based on this idea is obtained. This condition is used to find new results on the stability of periodic solutions of a class of differential equations of dissipative type.

The non-local nature of the behaviour of dissipative structure as the size of the space domain increases is studied. A class of systems is isolated, for which the continuum of inhomogeneous solutions, branching from the homogeneous solution, can be continued without limit as the size of the space domain increases; it is shown that, in general, this is not true. An example of a system is given, for which all possible types of successive bifurcations of dissipative structures are realized as the size of the space domain increases: stiff birth, stiff death, and secondary branching. All the predicted effects are modelled numerically.

The behavior of a slightly helically deformed Tokamak equilibrium with a small finite resistivity is analyzed, with special emphasis on bifurcated equilibria due to current shrinkage and disruption. The peculiarity of the analysis lies in the assumption that the behavior is constrained by the invariance of the helical flux and an integral quantity expressed in terms of the magnetic field, the vector potential of the magnetic field, and the plasma volume. The treatment is restricted to the case of a cylindrical Tokamak with an initially uniform axial magnetic field much greater than the initial poloidal field. It is shown that a helically deformed Tokamak equilibrium with peaked or constricted current surrounded by a cold plasma is resistively unstable and tends to blow up through the excitation of a nonlinearly explosive radial compressible mode.

A one-component quasilinear kinetic system with nonlocal spatial coupling is considered. The long-range inhibition effect combined with a short-range activation is shown to provide a possibility of spontaneous dissipative structure formation. The effective nonlocality can be due to diffusion of an intermediate.

We give short, elementary, constructive proofs of the basic theorems concerning the bifurcation of equilibrium and periodic
solutions, from a trivial solution of an ordinary differential equation, as a parameter in the equation passes through a critical
value. We begin by considering a finite dimensional equation, formulated to be analogous to an abstract Navier-Stokes equation
in Hilbert space. Later in the paper we consider generalizations, including generalizations to partial differential equations,
and especially to the Navier-Stokes equations.

The paper contains an exposition of variational and topological methods of investigating general nonlinear operator equations in Banach spaces. Application is given of these methods to the proof of solvability of boundary-value problems for nonlinear elliptic equations of arbitrary order, to the problem of eigenfunctions, and to bifurcation of solutions of differential equations. Results are presented of investigations of the properties of generalized solutions of quasilinear elliptic equations of higher order.

A model for the spatiotemporal activity of neuronal nets is proposed. Stationary periodic spatial patterns are discussed from the point of view of bifurcation theory. Existence of spatial patterns on the whole line is established by the implicit function theorem. Singularity theory is used to study the local structure of the bifurcation equations. A Poincare-Lindstedt series is developed to establish the form of the periodic stationary states and their stability. The biological relevance of these patterns is briefly discussed.

Hopf bifurcation at a degenerate stationary pitchfork. - In: Nonlinear analysis. 10. 1986. S. 339-351.

The set of solutions of the equation A(u, ?)=0 in the case of general position consists of smooth curves. Solutions of a quasilinear elliptic equation with large instability index are constructed and an estimate obtained for the number of turning points of the solution curves.

This chapter discusses the stability of bifurcating solutions. The steady-state bifurcation and the Hopf bifurcation are examined with the aim of studying conditions where the stability of bifurcating solutions might be possible.

This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered Banach spaces. By means of iterative techniques and by using topological tools, fixed point theorems for completely continuous maps in ordered Banach spaces are deduced, and particular attention is paid to the derivation of multiplicity results. Moreover, solvability and bifurcation problems for fixed point equations depending nonlinearly on a real parameter are investigated.
In order to demonstrate the importance of the abstract results, there are given some nontrivial applications to nonlinear elliptic boundary value problems. But, of course, the abstract techniques and results of this paper apply also to a variety of other problems which are not considered here.
This paper presents in a unified manner most of the recent work in this field. In addition, by making consequent use of the fixed point index for compact maps, short and simple proofs are obtained for most of the “classical” results contained in M. A. Krasnosel’skii’s book [11].

The appearance of secondary motions in a viscous fluid field can be understood to some extent as a bifurcation phenomenon with exchange of stability between the basic and the secondary flow. This article summarizes the main mathematical results of bifurcation and stability in hydrodynamic stability theory so far obtained. A unified functional-analytic approach is presented which tries to accentuate the ideas and to avoid technicalities. Besides the general results on the existence, the number of solutions and their qualitative behavior, the constructive analytical methods are emphasized. The Taylor and the Benard models are studied in detail. In the latter case, all possible solutions of regular cell pattern are classified. Stability and instability and their exchange at the point of bifurcation are studied.

Several examples are presented which illustrate some capabilities and some limitations of the method of matched asymptotic expansions for solving evolution equations. The results are listed according to spectral properties of the linear problem resulting near a known steady state of the system. When the linear problem is stable, it is shown that the solution can be written as a (finite) sum of terms, each responding on a different time scale. When the linear problem is unstable, it is shown that the method can be used to determine initial data which excite only decaying modes, and, in the case of bifurcation of new steady states, to construct the new states as well as the transients to them.

A number of apparently disparate problems from engineering, meteorology, geophysics, fluid mechanics and applied mathematics are considered under the unifying heading of natural convection. After a review of the mathematical framework that serves to delineate these problems, the heuristic approach to Bénard and Rayleigh convection is discussed with special attention to buoyancy and surface tension. Then consideration is given to some aspects of scaling, and the nondimensionalization of equations to a given problem. The thermohydrodynamic description of a Newtonian fluid is presented, and the Boussinesq-Oberbeck model. This is followed by a treatment of the linear stability problem, and a description of the basic ideas of Landau and Hopf concerning the bifurcation of secondary solutions. Quantitative, though approximate, estimations are given for quantities belonging to the nonlinear steady convective regime: flow velocity and temperature distribution. Higher-order, though steady, bifurcations are discussed, as well as the transition to turbulence, along with such time-dependent phenomena as relaxation oscillations. The paper concludes with an Appendix showing a simple application of the Leray-Schauder topological degree of a mapping.

This paper presents both a general review on developmental biomechanics and a concrete proposition for the computation of a symmetry breaking instability of a model of biological development in terms of self-organization theory. The necessary biological and physical facts taken from the literature are described and discussed in the context of a unified statement of the problems for mathematical modeling of pattern formation. This is then applied to planar cell polarization of the Drosophila wing. In this way the process is modeled by an elasto-polarization equation. In terms of this statement, the mechanical specificity (interaction with basal plate) of wing planar cell polarization is characterized. Some aspects of modeling somite formation as well as other developmental processes are also concerned.

A previous application of the theory, to the nonlinear boundary-value problem for steady flows of a viscous fluid in a bounded domain, is first retraced in order to verify a general theorem concerning the indices of multiple solutions. Then, in §5, the bearing of the theorem on questions of the bifurcation of steady flows is discussed, and the conclusion is drawn that a transcritical form of bifurcation is virtually universal in practice. In §6 it is proved that a flow represented by a solution with index i = – 1 is necessarily unstable, and hence it appears that lack of uniqueness generally implies the existence of an unstable steady flow.

The magnetic configuration in the confinement zone of a tokamak admits a magnetic flux-bifurcation for critical values of the edge parameters and given injected power when the initial L-state is so defined as to be associated with a stationary magnetic entropy. Assuming the ITER89-P scaling of the energy confinement time in the L state, it is shown that the power is changing along the bifurcation threshold, for fixed values of the edge parameters, according to the law P=Cn0.77B0.97 (where n is the average density and B is the toroidal magnetic field), where the numerical value of the constant C follows from ITER89-P and is consistent with observations of the L–H transition. It is also shown that the observed independence of the threshold with respect to the current is a consequence of the fact that, at the threshold point, the ratio between the auxiliary power and the Ohmic power P[Omega] in the confinement region is constant when the current is changing and that the value of P[Omega] at the threshold is a very slow function of the current.

The method of investigating bifurcations developed in [1 and 2] is applicable to many hydrodynamic problems. In the present paper it is applied to investigate the origin of convection in a horizontal fluid layer heated from below.Secondary stationary flows are of particular interest in the convection problem since the loss of stability is associated with these flows: “the principle of the change in stability” is not only valid here but has been proved rigorously [3]. It has also been proved that secondary stationary flows are generated by branching off from the state of rest [4 and 5].The problem under consideration is invariant relative to the group of motions of a horizontal plane.The single solution invariant relative to this whole group is the rest solution. When this solution is unstable, it is natural to expect the occurrence of solutions invariant relative to some subgroup of the group of motions. If the mentioned subgroup is generated by a pair of translations (in perpendicular directions), we arrive at doubly-periodic solutions (Section 1), and if invariance relative to rotation through a certain angle is required in addition, we arrive at solutions of hexagonal type (Section 2). As is known, precisely these latter are realized in convection experiments [6]. Deductions on the existence of doubly-pertodic convection flows are elucidated in Theorem 1.1, and the existence of solutions of hexagonal convection type is asserted in Theorem 1.2. The applied method has slight connection with the boundary conditions. Only for definiteness is it assumed that the boundaries of the layer are solid walls on each of which the temperature is specified.

This paper describes a method for the investigation of some properties of branching solutions of the Navier-Stokes boundary-value problem. Questions arising in this context are closely connected to problems in hydrodynamic stability. Many of the nonlinear aspects of this theory have been investigated thoroughly (cf. J. T. Stuart [11]). However, it was only in recent times that a mathematically rigorous basis was layed for these results, beginning with the use of topological methods by Velte [13] and Iudovich [3] and the application of the method of Schmidt-Lyapunov by kirchGässner [7] and Iudovich [4].

The static state of a horizontal layer of fluid heated from below may become unstable. If the layer is infinitely large in horizontal extent, the Boussinesq equations admit many different steady solutions. A systematic method is presented here which yields the finite-amplitude steady solutions by means of successive approximations. It turns out that not every solution of the linear problem is an approximation to the non-linear problem, yet there are still an infinite number of finite amplitude solutions. A similar procedure has been applied to the stability problem for these steady finite amplitude solutions with the result that three-dimensional solutions are unstable but there is a class of two-dimensional flows which are stable. The problem has been treated for both rigid and free boundaries.

In this paper we generalize the method of energy to discuss the stability of thermally-driven convective flows governed by the Boussinesq equations. The energy method as applied to non-convective flows has the striking advantage that it may be applied to difference (in contrast to perturbation) motions and can, therefore, accommodate effects of inertially non-linear disturbances. SEI~IN [1] has exploited this special feature of the method to obtain sufficient conditions for the stability of non-convective viscous flows in bounded and unbounded regions. The sufficiency conditions can be expressed as a Reynolds number estimate and also lead to a uniqueness theorem for steady bounded flows and to a variational algorithm for improving the Reynolds number estimate. The results are important because they apply for arbitrary disturbances in regions of unspecified geometry. The energy method has a long history in classical hydrodynamics, and relevant references can be found in SEmUN'S paper. Recently CONRAD & Cm~NAI.E [2] have used the method to discuss the stability of timedependent laminar motions. The simplified Boussinesq system, treated in this paper, involves a coupling of internal (thermal) energy to kinetic energy by the action of buoyancy. In these flows density differences established by thermally imposed temperature gradients can induce fluid motion by driving less dense (typically hotter) fluid elements against the direction of the gravity vector and, in the process, liberating internal energy. In the simplified (Boussinesq) system density differences are ignored except as they induce buoyant forces. Also neglected is the effect of the variation of thermal properties and the effects of viscous dissipation on the temperature distribution (see CHA~DRASEKHAR [3], pp. 16-- 18, for full discussion). The generalization developed in this paper proceeds from the observation that suitable energy equations for the integrated motion may be formed from a difference motion. The quadratic and bilinear integrals which appear in these equations can be easily estimated and lend themselves naturally to the formulation of a variational problem.

Am Beispiel der Strömung in einem horizontalen Rohr mit von unten erwärmter Wand wird gezeigt, wie man mit Hilfe der Theorie des topologischen Abbildungsgrades von Leray und Schauder im überkritischen Bereich der Rayleigh-Zahl auf die Existenz einer von der Grundströmung verschiedenen, stationären Strömung (thermische Konvektionsströmung) schließen kann. Unterhalb der kritischen Rayleigh-Zahl existiert genau eine stationäre Lösung der Navier-Stokesschen Bewegungsgleichungen, nämlich die Grundströmung. Beim Überschreiten der kritischen Rayleigh-Zahl verzweigt sich diese stationäre Lösung, wobei die Grundströmung gleichzeitig instabil wird.
Die Methode des Abbildungsgrades ist auf ähnliche Strömungsbeispiele mit „zellularer Instabilität” anwendbar wie z.B. die Strömung zwischen rotierenden Zylindern (Taylor-Wirbel) oder die von unten erwärmte Flüssigkeitsschicht (Benard-Zellen).

An investigation is made of the instability of stationary flow between two rotating cylinders for any gap distance. Previous work on this problem, viz. the Taylor theory, has been done with the restriction of small gaps. Moreover, in contrast to the recent work ofS. Chandrasekhar [2], who has also analysed this problem, the method used is independent of the basic flow and is therefore valid for all flows between two coaxial cylinders. In fact, it can even be used for the analysis of the flow in a curved channel.
The major item of physical interst is the critical Reynoldsnumber, i. e. the value of the Reynoldsnumber where small disturbances are amplified for the first time. The value of this parameter is determined by the smallest positive eigen value of the boundary value problem. An existence proof is made for this eigenvalue for any wavenumber and for all possible cases of cylindrical flow, with the exception of the case where the cylinders rotate in opposite directions. The results are depicted in a convenient form, where the critical Reynoldsnumber is the dependent variable and the gap distance and the angular velocity ratio of the two cylinders are the independent variables respectively.
A comparison with experiment is made for the case where the ratio of the two radii is 2: 1 and the outer cylinder is at rest. The agreement with the theory is good; the noticeable error being approximately 1%.

O11 the stability of the Boussinesq equations

JOSEPH, D. D., O11 the stability of the Boussinesq equations. Arch. Rational Mech. Anal. 20,
59-71 (1965).

On the origin of convection

- V I Yudovich

- J. Leray