Finite and infinite H-plane bifurcation waveguide with anisotropic plasma medium.
H-plane bifurcation of parallel plate waveguide filled with anisotropic plasma
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Bifurcation and stability for a nonlinear parabolic partial differential equation
Theorems are developed to support bifurcation and stability of nonlinear parabolic partial differential equations in the solution of the asymptotic behavior of functions with certain specified properties.
Chafee, N.↗
Investigation of two bifurcated-duct inlet systems from Mach 0 to 2.0 over a wide range of angles of attack
A 15.354 percent/scale lightweight fighter type inlet/forebody was tested over a Mach number range of 0 to 2.0. Model configurations consisted of side mounted normal shock and fixed overhead ramp type inlets. Each configuration consisted of two inlets ducted (bifurcated) to supply a single engine face. The normal shock inlet variables included a boundary layer splitter bleed system, alternate boundary layer splitter plates, alternate upper and lower cowl lip shapes, and a blow-in-door (auxiliary inlet) in one lower lip. The only variable of the fixed overhead ramp inlet was the boundary layer bleed flow. Reynolds numbers ranged from 7.6 x 1 million to 19.5 x 1 million/m. Angle of attack ranged from -10 to 35 deg and angle of sideslip from -8 to 8 deg. Test measurements included engine face total pressure recovery, steady state distortion, dynamic distortion, and surface static pressures on the forebody and inlet surfaces.
Latham, E. A.↗
Bifurcations from stationary to periodic solutions in a low-order model of forced, dissipative barotropic flow
The considered investigation is concerned with periodic solutions in the context of a forced, dissipative, barotropic spectral model truncated to three complex coefficients with constant forcing on only the intermediate scale. It is found that determining a periodic solution of this three-coefficient model also reduces to finding the algebraic roots of a real polynomial. In the derivation of this polynomial, a class of hydrodynamic spectral systems is described for which a periodic solution might be similarly specified. The existence of periodic solutions of the three-coefficient model is controlled by the roots of the stability polynomial of the basic stationary solution, which represents the simplest response to the constant forcing. When the forcing exceeds a critical value, the basic solution becomes unstable. Owing to the nature of the roots of the stability polynomial at critical forcing, bifurcation theory guarantees the existence of a periodic solution.
Mitchell, K. E.↗
Thermal bifurcation in the upper solar photosphere inferred from heterodyne spectroscopy of OH rotational lines
Low noise high spectral resolution observations of two pure rotation transitions of OH from the solar photosphere were obtained. The observations were obtained using the technique of optically null-balanced infrared heterodyne spectroscopy, and consist of center-to-limb line profiles of a v=1 and a v=0 transition near 12 microns. These lines should be formed in local thermodynamic equilibrium (LTE), and are diagnostics of the thermal structure of the upper photosphere. The v=0 R22 (24.5)e line strengthens at the solar limb, in contradiction to the predictions of current one dimensional photospheric models. Data for this line support a two dimensional model in which horizontal thermal fluctuations of order + or - 800K occur in the region Tau (sub 5000) approximately .001 to .01. This thermal bifurcation may be maintained by the presence of magnetic flux tubes, and may be related to the solar limb extensions observed in the 30 to 200 micron region.
Deming, D.↗
Circular waveguide bifurcation for asymmetric modes
An analytical solution for a circular waveguide bifurcation with incident TE(11)- and TM(11)-modes is presented using the residue calculus method. It is shown how the coupling between the TE(1n)- and TM(1n)-modes can be taken into account by expressing the coupling factor as a contour integral, which is the key step enabling the residue calculus method to be applied. Graphical results for the scattering matrix parameters of the junction are given for a range of waveguide radii of interest in the design of dual-mode coaxial prime focus feeds for paraboloidal antennas.
Schilling, H. W.↗
Thermal bifurcation in the upper photosphere inferred from heterodyne spectroscopy of OH rotational lines
Low-noise (S/N greater than 100), high spectral resolution observations of two pure rotation transitions of OH from the solar photosphere are used to make inferences concerning the thermal structure and inhomogeneity of the upper photosphere. It is found that the v = O R22(24.5)e line strengthens at the solar limb, in contradiction to the predictions of current one-dimensional photospheric models. The results for this line support a two-dimensional model in which horizontal thermal fluctuations in the upper photosphere are of the order plus or minus 800 K. This thermal bifurcation may be maintained by the presence of magnetic flux tubes and may be related to the solar limb extensions observed in the 30-200-micron region.
Deming, D.↗
Ayres' bifurcated solar model
The assumptions of Ayres' model of the upper solar atmosphere are examined. It is found that the bistable character of his model is postulated - through the assumptions concerning the opacity sources and the effect of mechanical waves, which are allowed to destroy the CO molecules but not to heat the gas. The neglect of cooling by metal lines is based on their reduced local cooling rate, but it ignores the increased depth over which this cooling occurs. Thus, the bifurcated model of the upper solar atmosphere consists of two models, one cold at the temperature minimum, with a kinetic temperature of 2900 K, and the other hot, with a temperature of 4900 K.
Kalkofen, W.↗
Hopf bifurcation with dihedral group symmetry - Coupled nonlinear oscillators
The theory of Hopf bifurcation with symmetry developed by Golubitsky and Stewart (1985) is applied to systems of ODEs having the symmetries of a regular polygon, that is, whose symmetry group is dihedral. The existence and stability of symmetry-breaking branches of periodic solutions are considered. In particular, these results are applied to a general system of n nonlinear oscillators coupled symmetrically in a ring, and the generic oscillation patterns are described. It is found that the symmetry can force some oscillators to have twice the frequency of others. The case of four oscillators has exceptional features.
Golubitsky, Martin↗
Application of the Thurston bifurcation solution strategy to problems with modal interaction
The solution of bifurcation problems with closely-spaced critical points is achieved by first separating the singular part of the equation system encountered during a Newton iteration and carrying the Taylor expansion of the reduced system out to higher order. This separation is accomplished by transforming the equation system into an equivalent system in which some of the original unknowns are replaced with an equal number of modal amplitude coefficients. This method was used to continue the analysis of two significant example problems well past multiple fiburcation points, allowing a detailed examination of postbuckling behavior in the presence of modal interaction.
Rankin, C. C.↗
Application of bifurcation theory to axial flow compressor instability
When a compression system goes unstable, the mode of response depends on the operating and system parameters, such as throttle setting and B parameter. Previous numerical work on the model developed by Moore and Greitzer (1986) has provided a limited picture of the parametric effects. Applying bifurcation theory to a single-harmonic version of the model has supplied much more complete information, defining the boundaries of each mode of response in the parameter space. Specifically, this is shown in a plot of B vs. throttle setting, which compares well with the corresponding map produced experimentally. The importance of the shape of the rotating stall characteristic is emphasized. The analysis shows a qualitative difference between classic surge and deep surge.
Mccaughan, F. E.↗
Bifurcations of relative equilibria
The characteristics of equivariant dynamical systems near relative equilibria (RE: group orbits which are invariant in the flow of an equivariant vector field) are investigated analytically, with a focus on the dynamics and bifurcation (B) behavior. The principles of Lie-group theory are reviewed; the decomposition of the vector field is explained; and particular attention is given to the Bs of RE occurring when an eigenvalue passes through zero, Hopf Bs of RE, the classification of generic secondary steady-state and Hopf Bs with symmetry group O(2), Bs of the zero solution of the Kuramoto-Shivashinsky equation, and possible generic steady-state Bs in the two-dimensional Benard problem. In the latter case, it is shown that the primary generic Bs are to two types of equilibria (hexagons and rolls), while the secondary Bs result in trajectories which are either equilibria or rotating waves.
Krupa, Martin↗
Stability and bifurcation analysis of oscillators with piecewise-linear characteristics - A general approach
A general approach is developed for determining the periodic solutions and their stability of nonlinear oscillators with piecewise-smooth characteristics. A modified harmonic balance/Fourier transform procedure is devised for the analysis. The procedure avoids certain numerical differentiation employed previously in determining the periodic solutions, therefore enhancing the reliability and efficiency of the method. Stability of the solutions is determined via perturbations of their state variables. The method is applied to a forced oscillator interacting with a stop of finite stiffness. Flip and fold bifurcations are found to occur. This led to the identification of parameter ranges in which chaotic response occurred.
Noah, S. T.↗
The Eckhaus and the Benjamin-Feir instability near a weakly inverted bifurcation
We investigate how the criteria for two prototype instabilities in one dimensional pattern forming systems, namely for the Eckhaus instability and for the Benjamin-Feir instability, change as one goes from a continuous bifurcation, to a spatially periodic or spatially and/or time periodic state, to the corresponding weakly inverted, i.e., hysteretic, cases. We also give the generalization to two dimensional patterns in systems with anisotropy as they arise from hydrodynamic instabilities in nematic liquid crystals.
Brand, Helmut R.↗
Upwind schemes and bifurcating solutions in real gas computations
The area of high speed flow is seeing a renewed interest due to advanced propulsion concepts such as the National Aerospace Plane (NASP), Space Shuttle, and future civil transport concepts. Upwind schemes to solve such flows have become increasingly popular in the last decade due to their excellent shock capturing properties. In the first part of this paper the authors present the extension of the Osher scheme to equilibrium and non-equilibrium gases. For simplicity, the source terms are treated explicitly. Computations based on the above scheme are presented to demonstrate the feasibility, accuracy and efficiency of the proposed scheme. One of the test problems is a Chapman-Jouguet detonation problem for which numerical solutions have been known to bifurcate into spurious weak detonation solutions on coarse grids. Results indicate that the numerical solution obtained depends both on the upwinding scheme used and the limiter employed to obtain second order accuracy. For example, the Osher scheme gives the correct CJ solution when the super-bee limiter is used, but gives the spurious solution when the Van Leer limiter is used. With the Roe scheme the spurious solution is obtained for all limiters.
Suresh, Ambady↗
Terrestrial superrotation - A bifurcation of the general circulation
When sufficiently large zonally asymmetric tropical heating is introduced in a two-level model of global atmospheric flow, its general circulation becomes strongly superrotating. The nature of the superrotating solutions is studied by examining momentum and heat budgets for a range of values of thermal forcing. Changes in the transport of zonal momentum by transient eddies appear to play the key role in the transition to superrotation. The dramatic bifurcation of the solutions of this model may help explain the maintenance and variability of the zonal mean flow in the tropics.
Suarez, Max J.↗
Stability and bifurcation of quasiparallel Alfven solitons
The inverse scattering transformation (IST) is used to study the one-parameter and two-parameter soliton families of the derivative nonlinear Schroedinger (DNLS) equation. The two-parameter soliton family is determined by the discrete complex eigenvalue spectrum of the Kaup-Newell scattering problem and the one-parameter soliton family corresponds to the discrete real eigenvalue spectrum. The structure of the IST is exploited to discuss the existence of discrete real eigenvalues and to prove their structural stability to perturbations of the initial conditions. Also, though the two-parameter soliton is structurally stable in general, it is shown that a perturbation of the initial conditions may change the two-parameter soliton into a degenerate soliton which, in turn, is structurally unstable. This degenerate, or double pole, soliton may bifurcate due to a perturbation of the initial conditions into a pair of one-parameter solitons. If the initial profile is on compact support, then this pair of one-parameter solitons must be compressive and rarefactive respectively. Finally, the Gelfand-Levitan equations appropriate for the double pole soliton are solved.
Hamilton, R. L.↗
Multi-layer holographic bifurcative neural network system for real-time adaptive EOS data analysis
Optical data processing techniques have the inherent advantage of high data throughout, low weight and low power requirements. These features are particularly desirable for onboard spacecraft in-situ real-time data analysis and data compression applications. The proposed multi-layer optical holographic neural net pattern recognition technique will utilize the nonlinear photorefractive devices for real-time adaptive learning to classify input data content and recognize unexpected features. Information can be stored either in analog or digital form in a nonlinear photorefractive device. The recording can be accomplished in time scales ranging from milliseconds to microseconds. When a system consisting of these devices is organized in a multi-layer structure, a feed forward neural net with bifurcating data classification capability is formed. The interdisciplinary research will involve the collaboration with top digital computer architecture experts at the University of Southern California.
Liu, Hua-Kuang↗