Analytical Bifurcation Research Articles

Imperfect transcritical and pitchfork bifurcations

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and perturbed diffusive logistic equation.

Quasi-steady formation of bubbles and drops viewed as processes that break bifurcation

Instabilities occurring during the quasi-steady formation of bubbles and drops at a circular orifice through a thin plate, which separates a cylindrical upper vessel of quiescent liquid from a lower air chamber maintained at constant pressure, are considered in connection with the Rayleigh-Taylor instability of a flat meniscus. The control parameters of the problem are the Eötvös number Eö, and the excess-pressure number Δ, which characterizes the pressure difference between gas and liquid across the orifice. The Rayleigh-Taylor case i.e., Δ = 0, can be viewed as a perfect bifurcating problem. A subcritical bifurcation emerges from the critical point, Eöc = 5 B783186, beyond which the flat meniscus is unstable for axisymmetric perturbations. Bubbles (Δ > 0), and drops (Δ < 0) appear as solutions that break bifurcation. When an appropriate measure of their magnitude ε is introduced, it can be shown analytically that the equilibrium surface at the critical point is a cusp; its intermediate sheet is stable, while its two upper and lower sheets are unstable. The analytical bifurcation set onto the control parameters plane is valid only around the critical point.

Ikeda Hopf bifurcation revisited

A large variety of passive optical systems subject to a delayed feedback have appeared in the literature and are described mathematically by the same class of scalar delay differential equations (DDEs). These equations include Ikeda DDE and their solutions are determined in terms of a control parameter distinct from the delay. We concentrate on the first Hopf bifurcation generated by a fixed delay and determine a general expression for its direction of bifurcation. We then examine our result in the two limits of small and large delays. For small delays, we show that a Hopf bifurcation to nearly sinusoidal oscillations is possible provided that the feedback rate is sufficiently high (bifurcation from infinity). For large delays, we complement the early work by Chow et al. [Proc. Roy. Soc. Edinburgh A 120 (1992) 223–229] and Hale and Huang [J. Diff. Equ. 114 (1994) 1–23] by comparing analytical and numerical bifurcation diagrams as the oscillations progressively change from sine to square-wave.

Morphometry of human coronary arterial trees.

Some groups, including ours, have been generating arterial tree models using constrained constructive optimization (CCO). Arterial trees have been grown to arbitrary resolution without input of anatomical data. We performed this study to learn about the shortcomings that might have resulted from neglecting the anatomical data in CCO models. In a total of 450 segments obtained from 4 human cast hearts, the ratio ofbifurcating daughter segment radii (O < Sbif = r(2)/r(1) < 1) was examined, which corresponds to the split of the total flow of the mother segment. For any complete bifurcation, where the radii of the parent segments and the radii of daughters were known, the area expansion ratio was computed (Aexp = [r(1)2 + r(2)2]/r(parent)2). The bifurcating ratio was found to be distributed in a nonnormal fashion, with a median of 0.76. The average area expansion ratio Aexp, characterizing the change of cross-sectional area of the vasculature from proximal to distal, was 0.93+/-0.26. The 'rate of branching' (d(i)/(d(0)) was defined by the segment diameter relative to the diameter of the root segment. Averaging the rate of branching over segments within each bifurcation level resulted in a decreasing function of bifurcation level. This article provides new experimental data on branching geometry of coronary arteries (i.e., the trees evaluated in this study are purely delivering rather than conveying). Based on these facts, we suggest that the analytical bifurcation law in CCO might be replaced by the bifurcation rule obeyed on a stochastic basis only.

Morphometry of human coronary arterial trees

Background Some groups, including ours, have been generating arterial tree models using constrained constructive optimization (CCO). Arterial trees have been grown to arbitrary resolution without input of anatomical data. We performed this study to learn about the shortcomings that might have resulted from neglecting the anatomical data in CCO models. Methods In a total of 450 segments obtained from 4 human cast hearts, the ratio of bifurcating daughter segment radii (0 < Sbif = r(2)/r(1) < 1) was examined, which corresponds to the split of the total flow of the mother segment. For any complete bifurcation, where the radii of the parent segments and the radii of daughters were known, the area expansion ratio was computed (Aexp = [r(1)2 + r(2)2]/ r(parent)2). Results The bifurcating ratio was found to be distributed in a nonnormal fashion, with a median of 0.76. The average area expansion ratio Aexp, characterizing the change of cross-sectional area of the vasculature from proximal to distal, was 0.93 ± 0.26. The ‘rate of branching’ (d(i)/(d(0)) was defined by the segment diameter relative to the diameter of the root segment. Averaging the rate of branching over segments within each bifurcation level resulted in a decreasing function of bifurcation level. Conclusions This article provides new experimental data on branching geometry of coronary arteries (i.e., the trees evaluated in this study are purely delivering rather than conveying). Based on these facts, we suggest that the analytical bifurcation law in CCO might be replaced by the bifurcation rule obeyed on a stochastic basis only. Anat. Rec. 251:50–59, 1998. © 1998 Wiley-Liss, Inc.

Analytical stability analysis of periodic systems by Poincaré mappings with application to rotorcraft dynamics

Apoint mappinganalysis is employed to investigate the stability of periodic systems. The method is applied to simplified rotorcraft models. The proposed approach is based on a procedure to obtain an analytical expression for the period-to-period mapping description of system's dynamics, and its dependence on system's parameters. Analytical stability and bifurcation conditions are then determined and expressed as functional relations between important system parameters. The method is applied to investigate the parametric stability of flapping motion of a rotor and the ground resonance problem encountered in rotorcraft dynamics. It is shown that the proposed approach provides very accurate results when compared with direct numerical results which are assumed to be an “exact solution” for the purpose of this study. It is also demonstrated that the point mapping method yields more accurate results than the widely used classical perturbation analysis. The ability to perform analytical stability studies of systems with multiple degrees-of-freedom is an important feature of the proposed approach since most existing analysis methods are applicable to single degree-of-freedom systems. Stability analysis of higher dimensional systems, such as the ground resonance problems, by perturbation methods is not straightforward, and is usually very cumbersome.

STABILITY ANALYSIS OF PERIODIC SYSTEMS BY TRUNCATED POINT MAPPINGS

An approach is presented deriving analytical stability and bifurcation conditions for systems with periodically varying coefficients. The method is based on a point mapping(period to period mapping) representation of the system's dynamics. An algorithm is employed to obtain an analytical expression for the point mapping and its dependence on the system's parameters. The algorithm is devised to derive the coefficients of a multinominal expansion of the point mapping up to an arbitrary order in terms of the state variables and of the parameters. Analytical stability and bifurcation condition are then formulated and expressed as functional relations between the parameters. To demonstrate the application of the method, the parametric stability of Mathieu's equation and of a two-degree of freedom system are investigated. The results obtained by the proposed approach are compared to those obtained by perturbation analysis and by direct integration which we considered to the “exact solution”. It is shown that, unlike perturbation analysis, the proposed method provides very accurate solution even for large valuesof the parameters. If an expansion of the point mapping in terms of a small parameter is performed the method is equivalent to perturbation analysis. Moreover, it is demonstrated that the method can be easily applied to multiple-degree-of-freedom systems using the same framework. This feature is an important advantage since most of the existing analysis methods apply mainly to single-degree-of-freedom systems and their extension to higher dimensions is difficult and computationally cumbersome.

Dynamic strain aging and plastic instabilities

A constitutive model proposed by McCormick [(1988) Theory of flow localization due to dynamic strain ageing. Acta. Metall. 36, 3061–3067] based on dislocation-solute interaction and describing dynamic strain aging behavior, is analyzed for the simple loading case of uniaxial tension. The model is rate dependent and includes a time-varying state variable, representing the local concentration of the impurity atoms at dislocations. Stability of the system and its post-instability behavior are considered. The methods used include analytical and numerical stability and bifurcation analysis with a numerical continuation technique. Yield point behavior and serrated yielding are found to result for well defined intervals of temperature and strain rate. Serrated yielding emerges as a branch of periodic solutions of the relaxation oscillation type, similar to frictional stick-slip. The distinction between the temporal and spatial (loss of homogeneity of strain) instability is emphasized. It is found that a critical machine stiffness exists above which a purely temporal instability cannot occur. The results are compared to the available experimental data.

Bifurcation Analysis of Cellular Interfaces in Unidirectional Solidification of a Dilute Binary Mixture

Analytical bifurcation theory is used to study interface patterns in the unidirectional solidification of a dilute binary mixture. Armbruster and Dangelmayr’s classification of bifurcation equations equivariant under the diagonal action in $\mathbb{C}^2 $ of the symmetry group $O( 2 )$ is used to show that the local bifurcation structure of the model long-wave evolution equation derived by Riley and Davis is equivalent to that of a codimension-one normal form. xamination of the bifurcation structure of the normal form then leads to a description of the development of three-dimensional interface patterns during unidirectional solidification. The general methodology underlying this study is also discussed, and conditions under which the method can be applied to other evolution equations are derived and clearly stated.

Chemically driven convection in a porous medium

AbstractThis paper is focused on the analysis of interaction of free convection and exothermic chemical reaction. As a consequence of the chemical reaction, free convection effects can result. It is difficult to perform an analytical bifurcation analysis of the full nonlinear governing equations; however, Fourier expansion combined with a Galerkin approximation results in a small set of ordinary nonlinear differential equations (initial‐value problem) that are amenable to analysis. Conditions for branching of the solution can be determined in an analytical way. A continuation algorithm makes it possible to calculate the branches of stability. The results of the approximative analysis are supported by the numerical integration of the full governing nonlinear equations.

Traveling waves in multi-phase flows

Equations of motion based on the theory of interacting and interpenetrating continua are used to study pattern forming instabilities in two-dimensional dispersed two-phase flows such as a suspension of particles in a fluidized bed and a bubbly gas-liquid flow. By making use of analytical and numerical bifurcation techniques it is shown that the main instability sequence in a vertically suspended configuration consists of the transition from a homogeneous flow through a one-dimensional concentration wave train to a two-dimensional traveling wave.