A bifurcation problem for the equation $$\Delta u+\lambda u-\alpha u^{+}+\beta u^{-}+g(\lambda ,u)=0$$ in a bounded domain in ${}^N$ with mixed boundary conditions, given nonnegative functions $\alpha ,\beta \in L_{\infty }$ and a small perturbation $g$ is considered. The existence of a global bifurcation between two given simple eigenvalues ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}$ of the Laplacian is proved under some assumptions about the supports of the functions $\alpha ,\beta$. These assumptions are given by the character of the eigenfunctions of the Laplacian corresponding to ${\lambda }^{\left(1\right)},{\lambda }^{\left(2\right)}$.

In this paper, we consider the Swift–Hohenberg equation with perturbed boundary conditions. We do not a priori know the eigenfunctions for the linearized problem since the $\mathrm{SO}\left(2\right)$ symmetry of the problem is broken by perturbation. We show that how the neutral stability curves change and, as a result, how the bifurcation diagrams change by the perturbation of the boundary conditions.

While alternans in a single cardiac cell appears through a simple period-doubling bifurcation, in extended tissue the exact nature of the bifurcation is unclear. In particular, the phase of alternans can exhibit wave-like spatial dependence, either stationary or travelling, which is known as discordant alternans. We study these phenomena in simple cardiac models through a modulation equation proposed by Echebarria-Karma. As shown in our previous paper, the zero solution of their equation may lose...

The paper deals with the analysis of generalized von Kármán equations which describe stability of a thin circular clamped viscoelastic plate of constant thickness under a uniform compressive load which is applied along its edge and depends on a real parameter, and gives results for the linearized problem of stability of viscoelastic plates. An exact definition of a bifurcation point for the generalized von Kármán equations is given. Then relations between the critical points of the linearized problem...

We consider parameter-dependent cocycles generated by nonautonomous difference equations. One of them is a discrete-time cardiac conduction model. For this system with a control variable a cocycle formulation is presented. We state a theorem about upper Hausdorff dimension estimates for cocycle attractors which includes some regulating function. We also consider the existence of invariant measures for cocycle systems using some elements of Perron-Frobenius theory and discuss the bifurcation of parameter-dependent...

The object of this paper is to find the solution of the differential equation of the buckling problem of anisotropic cylindrical shells with shear load in case of torsion of a long tube. The critical values of the shear load and the total torque are also found. The corresponding results for the isotropic case are deduced as a special case.