Showing papers on "Bifurcation diagram published in 1976"

The Hopf Bifurcation Theorem.

Abstract: : The Hopf Bifurcation Theorem concerns the bifurcation of a family of time periodic solutions of an evolution equation from a family of equilibrium solutions. Both finite and infinite dimensional versions of this theorem are given in this paper. Symmetry properties and the linearized stability of these solutions are also studied. (Author)

Successive Bifurcations Leading to Stochastic Behavior

Abstract: A model of interacting normal modes in a nonlinear, dissipative system is constructed in order to analyze speculations by Ruelle and Takens. The first bifurcation leads to a periodic state. The second bifurcation leads to phaselocking, if the first mode is sufficiently energetic. A third bifurcation leads to stochastic behavior. Possible relevance of these phenomena for physical systems is discussed.

Bifurcation of solutions with crystalline symmetry

Abstract: We consider the BBGKY equation for the single particle probability density in a hard sphere system. We investigate whether there is bifurcation from the fluid phase to functions which have crystalline symmetry. We find that as the density of the fluid increases from zero, there is bifurcation in one, two, and three dimensions. The bifurcation is shown to be characteristic of metastability and in general it does not occur at the equilibrium coexistence of two phases. The direction of branching and the stability of solutions near bifurcation is also discussed.

General class of nonlinear bifurcation problems from a point in the essential spectrum: application to shock wave solutions of kinetic equations

Abstract: An abstract class of bifurcation problems is investigated from the essential spectrum of the associated Frechet derivative. This class is a very general framework for the theory of one-dimensional, steady-profile traveling- shock-wave solutions to a wide family of kinetic integro-differential equations from nonequilibrium statistical mechanics. Such integro-differential equations usually admit the Navier--Stokes system of compressible gas dynamics or the MHD systems in plasma dynamics as a singular limit, and exhibit similar viscous shock layer solutions. The mathematical methods associated with systems of partial differential equations must, however, be replaced by the considerably more complex Bifurcation Theory setting. A hierarchy of bifurcation problems is considered, starting with a simple bifurcation problem from a simple eigenvalue. Sections are entitled as follows: introduction and background from mechanics; the mathematical problem: principal results; a generalized operational calculus, and the derivation of the generalized Lyapunov--Schmidt equations; and methods of solution for the Lyapunov--Schmidt and the functional differential equations. 1 figure. (RWR)

On the Dynamical Behavior of the Two-Temperature Feedback Nuclear Reactor Model

Abstract: A concise summary of the Hopf-Friedrichs bifurcation theory and the algebraic methods of application are given. Included in this summary are algebraic expressions which, when nonzero, completely reduce the existence, direction of bifurcation, change in the period of oscillation, and stability to an algebraic problem. This theory is then applied to the nuclear reactor model with two-temperature feedbacks to prove the bifurcation of stable periodic orbits in all parameters of the system. These bifurcation results and their relation to a study of the reactor dynamics are then discussed to give a perspective of the use of this bifurcation theory.

Numerical bifurcation and secondary bifurcation: a case history*

Abstract: Publisher Summary This chapter presents a case history of numerical bifurcation and secondary bifurcation. Secondary transitions or secondary bifurcations occur frequently in many nonlinear stability problems. In hydrodynamic stability, secondary bifurcations are called secondary transitions and in elastic stability, they are called secondary bucklings. The chapter describes some recent progress that has been made in the theory and numerical computation of secondary bifurcation by considering two problems for the nonlinear buckling of elastic plates. In the buckled circular plate, the edge of a circular elastic plate is clamped and compressed by a uniform, radial edge thrust. For sufficiently small thrusts, the plate deforms by uniformly contracting in its plane to a circle of smaller radius. This equilibrium state is called the unbuckled state.

Secondary or Direct Bifurcation of a Steady Solution of the Navier-Stokes Equations into an Invariant Torus

Abstract: Publisher Summary This chapter discusses secondary or direct bifurcation of a steady solution of the navier-stokes equations into an invariant torus, basic flow perturbed equations, known results about the first bifurcation into a periodic solution, Poincare map and its properties, bifurcation into periodic solutions, various theorems, and lemmas.

Hopf Bifurcation and the Method of Averaging

Abstract: The method of averaging* provides an algorithm for preparing a bifurcation problem, that is, putting it into a normal form. Once this is done, one may more readily determine certain qualitative features of the bifurcation, by means of the implicit function theorem (or contraction mapping principle) and the center manifold theorem.