Bifurcation diagram

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

Spatial and temporal structure in systems of coupled nonlinear oscillators

Abstract: The spatial and temporal bifurcations of a ring of coupled, discrete-time, nonlinear oscillators are studied. The model displays many of the phenomena observed in diffusively coupled, nonlinear, chemical oscillators which can possess complex dynamics when isolated. The low-order bifurcation diagram of the discrete-time model may be computed analytically and shows how in-phase and out-of-phase solutions arise and undergo further bifurcations to quasiperiodic or chaotic states. Spatial bifurcations (pattern formation) accompany the temporal bifurcations and the results indicate how some of these processes occur. The phase diagram possesses self-similar scaling features associated with the higher-order periodic states. The model should prove useful in identifying the analogous phenomena in physical systems.

A simple method for the calculation of postcritical branches

Abstract: The practical behaviour of problems exhibiting bifurcation with secondary branches cannot be studied in general by using standard path‐following methods such as arc‐length schemes. Special algorithms have to be employed for the detection of bifurcation and limit points and furthermore for branch‐switching. Simple methods for this purpose are given by inspection of the determinant of the tangent stiffness matrix or the calculation of the current stiffness parameter. Near stability points, the associated eigenvalue problem has to be solved in order to calculate the number of existing branches. The associated eigenvectors are used for a perturbation of the solution at bifurcation points. This perturbation is performed by adding the scaled eigenvector to the deformed configuration in an appropriate way. Several examples of beam and shell problems show the performance of the method.

Mathematical Modeling of Electrostatic MEMS with Tailored Dielectric Properties

Abstract: The "pull-in" or "snap-down" instability in electrostatically actuated microelec- tromechanical systems (MEMS) presents a ubiquitous challenge in MEMS technology of great im- portance. In this instability, when applied voltages are increased beyond a critical value, there is no longer a steady-state configuration of the device where mechanical members remain separate. This severely restricts the range of stable operation of many devices. In an attempt to reduce the effects of this instability, researchers have suggested spatially tailoring the dielectric properties of MEMS devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed and analyzed for the purpose of investigating this possibility. The pull-in instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in this bifurcation diagram for various dielectric profiles are studied, yielding insight into how this technique may be used to increase the stable range of operation of electrostatically actuated MEMS devices.

Bifurcations, chaos, and crises in voltage collapse of a model power system

Abstract: Bifurcations occurring in power system models exhibiting voltage collapse have been the subject of several recent studies. Although such models have been shown to admit a variety of bifurcation phenomena, the view that voltage collapse is triggered by possibly the simplest of these, namely by the (static) saddle node bifurcation of the nominal equilibrium, has been the dominant one. The authors have recently shown that voltage collapse can occur "prior" to the saddle node bifurcation. In the present paper, a new dynamical mechanism for voltage collapse is determined: the boundary crisis of a strange attractor or synonymously a chaotic blue sky bifurcation. This determination is reached for an example power system model akin to one studied in several recent papers. More generally, blue sky bifurcations (not necessarily chaotic) are identified as important mechanisms deserving further consideration in the study of voltage collapse. >