Bifurcation diagram

About: Bifurcation diagram is a research topic. Over the lifetime, 8379 publications have been published within this topic receiving 139664 citations.

Stability of a thin elastic film interacting with a contactor

Abstract: The surface stability of a thin solid elastic film subjected to surface interactions such as van der Waals forces due to the influence of another contacting solid is investigated. It is found that for nearly incompressible soft (shear modulus less than 10 MPa) films, the film surface is unstable and forms an undulating pattern without any concurrent mass transport. A complete stability/bifurcation diagram is obtained. A key new result uncovered in this analysis is that the characteristic wavelength of the bifurcation pattern is nearly independent of the precise nature and magnitude of the interaction and varies linearly with the film thickness, whenever the force of interaction attains a critical value. The rate of growth of perturbations is also analysed using a viscoelastic model and it is found that in nearly incompressible materials, the wavelength of the fastest growing perturbation is identical to that of the critical elastic bifurcation mode. These results provide a quantitative explanation for recent experiments. The present study is important in understanding problems ranging from adhesion and friction at soft solid interfaces, peeling of adhesives to the development of micro-scale pattern transfer technologies.

Symmetry breaking and overturning oscillations in thermohaline-driven flows

Abstract: The bifurcation structure of thermohaline-driven flows is studied within one of the simplest zonally averaged models which captures thermohaline transport: a Boussinesq model of surface-forced thermohaline flow in a two-dimensional rectangular basin. Under mixed boundary conditions, i.e. prescribed surface temperature and fresh-water flux, it is shown that symmetry breaking originates from a codimension-two singularity which arises through the intersection of the paths of two symmetry-breaking pitchfork bifurcations. The physical mechanism of symmetry breaking of both the thermally and salinity dominated symmetric solution is described in detail from the perturbation structures near bifurcation. Limit cycles with an oscillation period in the order of the overturning time scale arise through Hopf bifurcations on the branches of asymmetric steady solutions. The physical mechanism of oscillation is described in terms of the most unstable mode just at the Hopf bifurcation. The occurrence of these oscillations is quite sensitive to the shape of the prescribed fresh-water flux. Symmetry breaking still occurs when, instead of a fixed temperature, a Newtonian cooling condition is prescribed at the surface. There is only quantitative sensitivity, i.e. the positions of the bifurcation points shift with the surface heat transfer coefficient. There are no qualitative changes in the bifurcation diagram except in the limit where both the surface heat flux and fresh-water flux are prescribed. The bifurcation structure at large aspect ratio is shown to converge to that obtained by asymptotic theory. The complete structure of symmetric and asymmetric multiple equilibria is shown to originate from a codimension-three bifurcation, which arises through the intersection of a cusp and the codimension-two singularity responsible for symmetry breaking.

"crossroad area–spring area" transition (i) parameter plane representation

Abstract: This paper is devoted to the bifurcation structure of a parameter plane related to one- and two-dimensional maps. Crossroad area and spring area correspond to a characteristic organization of fold and flip bifurcation curves of the parameter plane, involving the existence of cusp points (fold codimension-two bifurcation) and flip codimension-two bifurcation points. A transition "mechanism" (among others) from one area type to another one is given from a typical one-dimensional map.