by Courant Institute of Mathematical Sciences, New York University in New York .
|The Physical Object|
|Number of Pages||63|
Hydromagnetic equilibria are obtained for a variety of situations which differ little from that of a zero pressure uniform axial magnetic field. Criteria for ascertaining the stability of these equilibria are found by means of an energy by: The hydromagnetic energy principle is applied to the derivation of necessary and sufficient conditions for the hydromagnetic stability of a linear pinch with distributed plasma current (a diffuse linear pinch). The results are quite general in that the axial and azimuthal components of the magnetic field, which determine the structure of the Cited by: The relationship between this formalism and that in which the equilibrium problem is reduced to solving a partial differential equation for the stream function is established. The necessary and sufficient condition for the stability of helically invariant systems with respect to localized displacements is by: Abstract. We analyze some cases in which the Bifurcation Function obtained by applying the Liapunov-Schmidt method to a nonlinear O.D.E. problem can be used to discuss the stability properties of the equilibrium points.
Vanderbauwhede A. () Stability of bifurcating equilibria and the principle of reduced stability. In: Salvadori L. (eds) Bifurcation Theory and Applications. Lecture Notes in . Hydromagnetic stability of plane Poiseuille and Couette flow of viscoelastic fluid Nabil T M Eldabe, M F El-Sabbagh and M A-S El-Sayed \(Hajjaj\)-On the Hydromagnetic Stability of a Nonuniformly Rotating Fluid A Brahme-Couette flow of an incompressible fluid in a porous channel with mass transfer N Niranjana, M Vidhya and A Govindarajan-Recent. Abstract. The course deals with the stability theory (Parts I,II) and its connection with the bifurcation theory (Part III). Some more recent approaches to stability problems are described, for example by methods which make use of suitable one parameter families of Liapunov functions. This book covers the central role that bifurcations play in nonlinear phenomena, explaining mechanisms of how stability is gained or lost. It emphasizes practical and computational methods for analyzing dynamical systems. A wide range of phenomena between equilibrium and chaos is explained and illustrated by examples from science and engineering.
Note that the calculated lines of neutral stability of the mode m = 0 hardly differ for P m = 1 and P m = 1 0 − 5. The eigenvalues along the line of neutral stability of the axisymmetric modes, therefore, appear to scale with R e and H a for P m → 0. In both cases the magnetic field simply suppresses the axisymmetric mode as predicted by Eq. An expansion technique is employed to study the stability condition for toroidal hydromagnetic configurations in which the curvature of the system, the rotational transform per unit length, and the pressure which is contained in the system can be treated as small. The most general condition which is necessary for stability with respect to localized perturbations is obtained as a ratio of two. 3. Equilibria and local stability In , we proved that the system is positively invariant and that all trajectories starting in the positive octant are uniformly bounded in it. Our goal in this section, is to identify the equilibria together with their local stability. Equilibria To simplify the analysis, we deﬁne the parameters β1. A simple ``single‐term'' stability criterion is obtained for hydromagnetic equilibria with multipolar fields. It replaces an earlier criterion in which it is necessary to evaluate the ratio of two infinite determinants.