Sufficient Conditions For Bifurcation Research Articles

Bifurcation for nonlinear elliptic problems with singular potentials

We study the bifurcation phenomena for a class of nonlinear elliptic problems with Hardy term and subcritical nonlinearity. Sufficient conditions for bifurcation are established. The existence of a positive solution is also obtained for an asymptotically linear problem.

On Von Kármán Equations and the Buckling of a Thin Circular Elastic Plate

Abstract We shall be concerned with the buckling of a thin circular elastic plate simply supported along a boundary, subjected to a radial compressive load uniformly distributed along its boundary. One of the main engineering concerns is to reduce deformations of plate structures. It is well known that von Kármán equations provide an established model that describes nonlinear deformations of elastic plates. Our approach to study plate deformations is based on bifurcation theory. We will find critical values of the compressive load parameter by reducing von Kármán equations to an operator equation in Hölder spaces with a nonlinear Fredholm map of index zero. We will prove a sufficient condition for bifurcation by the use of a gradient version of the Crandall-Rabinowitz theorem due to A.Yu. Borisovich and basic notions of representation theory. Moreover, applying the key function method by Yu.I. Sapronov we will investigate the shape of bifurcation branches.

The index bundle and bifurcation from infinity of solutions of nonlinear elliptic boundary value problems

We present three criteria for bifurcation from infinity of solu- tions of general boundary value problems for nonlinear elliptic systems of partial differential equations. Our sufficient conditions for bifurcation are computable, via the Atiyah-Singer family index theorem, from the coefficients of derivatives of leading order of the linearized differential operators and do not involve the analysis of the asymptotic derivative at infinity.

Helical bifurcation and tearing mode in a plasma—a description based on Casimir foliation

The relation between the helical bifurcation of a Taylor relaxed state (a Beltrami equilibrium) and a tearing mode is analyzed in a Hamiltonian framework. Invoking an Eulerian representation of the Hamiltonian, the symplectic operator (defining a Poisson bracket) becomes non-canonical, i.e. the symplectic operator has a nontrivial cokernel (dual to its nullspace), foliating the phase space into level sets of Casimir invariants. A Taylor relaxed state is an equilibrium point on a Casimir (helicity) leaf. Changing the helicity, equilibrium points may bifurcate to produce helical relaxed states; a necessary and sufficient condition for bifurcation is derived. Tearing yields a helical perturbation on an unstable equilibrium, producing a helical structure approximately similar to a helical relaxed state. A slight discrepancy found between the helically bifurcated relaxed state and the linear tearing mode viewed as a perturbed, singular equilibrium state is attributed to a Casimir element (named ‘helical flux’) pertinent to a ‘resonance singularity’ of the non-canonical symplectic operator. While the helical bifurcation can occur at discrete eigenvalues of the Beltrami parameter, the tearing mode, being a singular eigenfunction, exists for an arbitrary Beltrami parameter. Bifurcated Beltrami equilibria appearing on the same helicity leaf are isolated by the helical-flux Casimir foliation. The obstacle preventing the tearing mode to develop in the ideal limit turns out to be the shielding current sheet on the resonant surface, preventing the release of the ‘potential energy’. When this current is dissipated by resistivity, reconnection is allowed and tearing instability occurs. The Δ′ criterion for linear tearing instability of Beltrami equilibria is shown to be directly related to the spectrum of the curl operator.

The family index theorem and bifurcation of solutions of nonlinear elliptic BVP

We obtain some new bifurcation criteria for solutions of general boundary value problems for nonlinear elliptic systems of partial differential equations. The results are of different nature from the ones that can be obtained via the traditional Lyapunov–Schmidt reduction. Our sufficient conditions for bifurcation are derived from the Atiyah–Singer family index theorem and therefore they depend only on the coefficients of derivatives of leading order of the linearized differential operators. They are computed explicitly from the coefficients without the need of solving the linearized equations. Moreover, they are stable under lower order perturbations.

A sufficient condition for bifurcation in random dynamical systems

Some properties of the random Conley index are obtained, and then a sufficient condition for the existence of abstract bifurcation points for both discrete-time and continuous-time random dynamical systems is presented. This stochastic bifurcation phenomenon is demonstrated by a few examples.

Periodic Bifurcation For Semilinear Differential Equations With Lipschitzian Perturbations in Banach Spaces

Abstract Let A : D(A) → E, D(A) ⊂ E; be an infinitesimal generator either of an analytic compact semigroup or of a contractive C0-semigroup of linear operators acting in a Banach space E. In this paper we give both necessary and sufficient conditions for bifurcation of T- periodic solutions for the equation ẋ = Ax + f(t, x) + εg(t, x, ε) from a k-parameterized family of T-periodic solutions of the unperturbed equation corresponding to ε = 0. We show that by means of a suitable modification of the classical Mel’nikov approach we can construct a bifurcation function and to formulate the conditions for the existence of bifurcation in terms of the topological index of the bifurcation function. To do this, since the perturbation term g is only Lipschitzian we need to extend the classical Lyapunov- Schmidt reduction to the present nonsmooth case.

Asymptotic Resonance, Interaction of Modes and Subharmonic Bifurcation

We study the existence of small amplitude oscillations near elliptic equilibria of autonomous systems, which mix different normal modes. The reference problem is the Fermi-Pasta-Ulam β-model: a chain of nonlinear oscillators with nearest-neighborhood interaction. We develop a new bifurcation approach that locates secondary bifurcations from the unimodal primary branches. Two sufficient conditions for bifurcation are given: one involves only the arithmetic properties of the eigenvalues of the linearized system (asymptotic resonance), while the other takes into account the nonlinear character of the interaction between normal modes (nonlinear coupling). Both conditions are checked for the Fermi-Pasta-Ulam problem.

The necessary and sufficient condition for bifurcation in the von K�rm�n equations

The paper is devoted to the study of bifurcation in the von Karman equations with two parameters $ \alpha, \beta\in \mathbb{R}_{+} $ that describe the behaviour of a thin round elastic plate lying on an elastic base under the action of a compressing force. The problem appears in the mechanics of elastic constructions. We prove the necessary and sufficient condition for bifurcation at points of the set of trivial solutions. Our proof is based on reducing the von Karman equations to an operator equation in Banach spaces with a nonlinear Fredholm map of index 0 and applying the Crandall-Rabinowitz theorem on simple bifurcation points or a finite-dimensional reduction and degree theory.

Bifurcation Theorems via Second-Order Optimality Conditions

We present a new approach to bifurcation study that relies on the theory of second-order optimality conditions for abnormal constrained optimization problems developed earlier by the first author. This theory does not subsume the “primal” description of the feasible set in terms of tangent vectors or in any other way. As a result, we obtain new sufficient conditions for bifurcation, which are to some extent complementary with respect to the known bifurcation theory.

On Sufficient Conditions for Bifurcation

A new approach to bifurcation study is presented, relying on the theory of second order optimality conditions for abnormal constrained optimization problems developed earlier by the first author. This theory does not subsume the "primal" description of the feasible set (in terms of tangent vectors, or in any other way). As a result, the new sufficient conditions for bifurcation are obtained. These conditions are to some extent complementary with respect to the known bifurcation theory. Application to controllability problems is also discussed.

Bifurcation in a Quasilinear Variational Problem on a One-Dimensional Manifold

Necessary and sufficient conditions for bifurcation in a variational problem on a one-dimensional closed manifold are obtained. In local coordinates, this problem corresponds to a semilinear 4th-order differential equation and can be regarded as a model equation of problems in the theory of shells. Bibliography: 5 titles.

Bifurcation to pear-shaped equilibria of pressurized spherical membranes

We study the problem of non-spherical, axisymmetric equilibria of an inflated, spherical membrane. We model the membrane as a two-dimensional elastic body characterized by a general class of strain-energy functions, and we consider a general class of loading devices, including (soft) pressure control and (hard) control of the total mass of gas enclosed by the membrane as special cases. Employing tools of modern bifurcation theory, we illuminate the precise necessary and sufficient conditions for bifurcation from the spherical state to an axisymmetric “pear-shaped” state, and we perform a local post-bifurcation analysis. In particular, for a large class of physically reasonable strain-energy functions, we demonstrate the existence of an isola bifurcation (closed loop of non-spherical solutions), which is consistent with experimental observations.

A local bifurcation theorem for 𝐶¹-Fredholm maps

The Krasnosel’skii Bifurcation Theorem is generalized to C 1 {C^1} -Fredholm maps. Let X X and Y Y be Banach spaces, F : R × X → Y F:R \times X \to Y be C 1 {C^1} -Fredholm of index 1 and F ( λ , 0 ) ≡ 0 F(\lambda ,0) \equiv 0 . If I ⊆ R I \subseteq R is a closed, bounded interval at whose endpoints ∂ F ∂ x ∂ F ∂ x ( λ , 0 ) \frac {{\partial F}}{{\partial x}}\frac {{\partial F}}{{\partial x}}(\lambda ,0) is invertible, and the parity of ∂ F ∂ x ( λ , 0 ) \frac {{\partial F}}{{\partial x}}(\lambda ,0) on I I is -1, then I I contains a bifurcation point of the equation F ( λ , x ) = 0 F(\lambda ,x) = 0 . At isolated potential bifurcation points, this sufficient condition for bifurcation is also necessary.

Bifurcation and stability results for an equation arising in population genetics

We consider a semilinear elliptic boundary value problem, which arises in population genetics. Although there is no obvious corresponding linearized problem, we establish, by using Implicit Function Theorem methods, necessary and sufficient conditions forbifurcation to occur from a branch of trivial solutions. The stability of the bifurcating solutions is also investigated.

Necessary and sufficient conditions for bifurcation from the continuous spectrum

Necessary and sufficient conditions for bifurcation from the continuous spectrum