Bifurcation diagram

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

Nonlocal problems in MEMS device control

Abstract: Perhaps the most ubiquitous phenomena associated with electrostatically actuated MEMS devices is the `pull-in' voltage instability. In this instability, when applied voltages are increased beyond a certain critical voltage there is no longer a steady-state configuration of the device where mechanical members remain separate. This instability severely restricts the range of stable operation of many devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed for the purpose of analyzing various schemes proposed for the control of the pull-in instability. This embedding of a device into a control circuit gives rise to a nonlinear and nonlocal elliptic problem which is analyzed through a variety of asymptotic, analytical, and numerical techniques. The pull-in voltage instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in various capacitive control schemes are shown to give rise to variations in the bifurcation diagram and hence to effect the pull-in voltage and pull-in distance.

Capillary Surfaces: Stability from Families of Equilibria with Application to the Liquid Bridge

Abstract: A method for determining the stability of general static capillary surfaces is illustrated by application to the liquid bridge. Axisymmetric bridges with fixed contact lines under gravity are parametrized by three quantities: bridge length L, bridge volume V, and Bond number B. The method delivers i) stability envelopes in the {L,V,B,} parameter space for constant pressure and constant volume disturbances (recovering classical and generating new results), ii) the number of unstable modes for any equilibrium (state of instability) once the stability of one equilibrium state is known, based on, iii) a demonstration that all known families of equilibria are ultimately connected. The state of instability of an equilibrium shape relative to its neighbors is immediate from a plot of volume V versus pressure p, a "preferred" bifurcation diagram. The preferred diagram method gives stronger results than classical bifurcation theory based on properties of eigenvalues of the Jacobi equation for problems with a variational formulation. Application to other capillary surfaces including drops and nonaxisymmetric shapes is discussed. In addition, motivated by general tangency considerations, an invariant wavenumber classification is introduced and used to label the numerous families of liquid bridge equilibria.

Stability, instability, and bifurcation phenomena in non-autonomous differential equations

Abstract: There is a vast body of literature devoted to the study of bifurcation phenomena in autonomous systems of differential equations. However, there is currently no well-developed theory that treats similar questions for the non-autonomous case. Inspired in part by the theory of pullback attractors, we discuss generalizations of various autonomous concepts of stability, instability, and invariance. Then, by means of relatively simple examples, we illustrate how the idea of a bifurcation as a change in the structure and stability of invariant sets remains a fruitful concept in the non-autonomous case.

Normalizable, Integrable, and Linearizable Saddle Points for Complex Quadratic Systems in \Bbb C 2

Abstract: In this paper, we consider complex differential systems in the neighborhood of a singular point with eigenvalues in the ratio 1 : −λ with λ e \Bbb R+*. We address the questions of orbital normalizability, normalizability (i.e., convergence of the normalizing transformation), integrability (i.e., orbital linearizability), and linearizability of the system. As for the experimental part of our study, we specialize to quadratic systems and study the values of λ for which these notions are distinct. For this purpose we give several tools for demonstrating normalizability, integrability, and linearizability. Our main interest is the global organization of the strata of those systems for which the normalizing transformations converge, or for which we have integrable or linearizable saddles as λ and the other parameters of the system vary. Many of the results are valid in the larger context of polynomial or analytic vector fields. We explain several features of the bifurcation diagram, namely, the existence of a continuous skeleton of integrable (linearizable) systems with sequences of holes filled with orbitally normalizable (normalizable) systems and strata finishing at a particular value of λ. In particular, we introduce the Ecalle-Voronin invariants of analytic classifcation of a saddle point or the Martinet-Ramis invariants for a saddle-node and illustrate their role as organizing centers of the bifurcation diagram.