Instabilities,bifurcations and catastrophes

Two independently developed bifurcational theories for the instability of a slowly evolving system, such as a stellar mass, are correlated. Applied to the fracture of a mechanically stressed perfect crystal, they predict a sharp cusp on the failure stress locus.     

Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.     

Instabilities in buoyant flows under localized heating

We study, from the numerical point of view, instabilities developed in a fluid layer with a free surface in a cylindrical container which is nonhomogeneously heated from below. In particular, we consider the case in which the applied heat is localized around the origin. An axisymmetric basic state appears as soon as a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. We find different kinds of instabilities: extended patterns growing on the whole domain, which include those known as targets, and spiral waves. Spirals are present even for infinite Prandtl number. Localized structures both at the origin and at the outer part of the cylinder may appear either as Hopf or stationary bifurcations. An overview of the developed instabilities as functions of the dimensionless parameters is presented in this article.     

Instabilities developed in stratified flows over pronounced obstacles

In the present work we study numerical and experimentally the flow of a two-layer stratified fluid over a topographic obstacle. The problem reflects a wide number of oceanographic and meteorological situations, where the stratification plays an important role. We identify the different instabilities developed by studying the pycnocline deformation due to a pronounced obstacle. The numerical simulations were made using the model caffa3D.MB which works with a numerical model of Navier-Stokes equations with finite volume elements in curvilinear meshes. The experimental results are contrasted with numerical simulations. Linear stability analysis predictions are checked with particle image velocimetry (PIV) measurements.     

Instabilities in laser-produced carbon plasma expanding in a nonuniform magnetic field

The laser-produced carbon plasma expanding in an ambient atmosphere in the presence of an inhomogeneous magnetic field has been studied by emission spectroscopy and fast photography. A double-peak structure is observed in the temporal profile of CII and CIII transition. A sudden increase in delay observed in the second peak when the plasma expands in the concave region of a magnetic field is attributed to Rayleigh–Taylor instability in a magnetic field. An estimate of the growth rate of the instability inferred using intensity and velocity profile of the expanding plasma is reported. Received: 26 August 1999 / Revised version: 3 January 2000 / Published online: 20 September 2000     

Computation of fluid flows in non-inertial contracting, expanding, and rotating reference frames

We present the method for computation of fluid flows that are characterized by the large degree of expansion/contraction and in which the fluid velocity is dominated by the bulk component associated with the expansion/contraction and/or rotation of the flow. We consider the formulation of Euler equations of fluid dynamics in a homologously expanding/contracting and/or rotating reference frame. The frame motion is adjusted to minimize local fluid velocities. Such approach allows to accommodate very efficiently large degrees of change in the flow extent. Moreover, by excluding the contribution of the bulk flow to the total energy the method eliminates the high Mach number problem in the flows of interest. An important practical advantage of the method is that it can be easily implemented with virtually any Eulerian hydrodynamic scheme and adaptive mesh refinement (AMR) strategy.We also consider in detail equation invariance and existence of conservative formulation of equations for special classes of expanding/contracting reference frames. Special emphasis is placed on extensive numerical testing of the method for a variety of reference frame motions, which are representative of the realistic applications of the method. We study accuracy, conservativity, and convergence properties of the method both in problems which are not its optimal applications as well as in systems in which the use of this method is maximally beneficial. Such detailed investigation of the numerical solution behavior is used to define the requirements that need to be considered in devising problem-specific fluid motion feedback mechanisms.     

Saddle-point solutions and grazing bifurcations in an impacting system

This paper focuses on the intricate relationship between smooth and nonsmooth phenomena in an impacting system. In particular a boundary saddle-point solution, that is born in a nonsmooth fold, is analysed. Accessible boundary saddle-point solutions play a key role in determining the global dynamics of a system and here we will show how grazing bifurcations can affect their existence.     

The trajectory scaling function for period doubling bifurcations in flows

We extend the trajectory scaling function as defined for maps to flows whose dynamics is governed by ordinary differential equations. The results are obtained for the Duffing oscillator and are expected to be the same for other dissipative flows as well.     

Spontaneous linearization and periodic solutions in Hopf and symmetric bifurcations

We show that for a rather large class of symmetries, symmetric polynomial evolution equations — and therefore symmetric bifurcation equations — exhibit spontaneous linearization; i.e. although the equations are nonlinear, their asymptotic solutions are governed by linear equations. The same mechanism leads to periodicity of such solutions. This is exemplified in the cases of SO(2) and SU(2) symmetries, corresponding to standard Hopf and quaternionic bifurcations.     

The effect of Lagrangian chaos on locking bifurcations in shear flows

The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.     

On the bifurcations of non-resonant solutions in autonomous systems

The behaviour of a non-linear system represented by a set of autonomous ordinary differential equations containing two real parameters is considered. At a certain critical value of these parameters it is assumed that the linearized system has two pairs of imaginary eigenvalues which are not even nearly commensurable. This non-resonant case is analyzed via a perturbation technique, and asymptotic formulate type results are obtained.     

Two-parameter sliding bifurcations of periodic solutions in a dry-friction oscillator

This paper is concerned with numerical continuation and analytical investigations of sliding bifurcations in Filippov systems. In particular, a methodology developed for the continuation of grazing bifurcations in impacting systems is used to continue sliding bifurcations in Filippov systems. A dry-friction oscillator is investigated from a sliding bifurcations point of view and a complex two-parameter bifurcation diagram of sliding bifurcations is presented. A number of codimension-two sliding bifurcation points that act as organising centres for codimension-one sliding bifurcations are revealed. Two representative codimension-two points are analysed and unfolded, and the analysis is used to explain the dynamics of the dry-friction oscillator in the neighbourhood of these points.     

A simplified model for flows with eddies in symmetrically expanding channels

We derive a simplified model for two-dimensional (2D) channel flows with recirculated regions at moderate Reynolds numbers based on an extension of the boundary layer (BL) theory and averaging across the channel. The model reproduces symmetry-breaking bifurcations and resulting flow structures accurately. Analytical estimates for the decay rates toward the parabolic profile before and after a sudden change in the walls agree well with the full numerical simulations. A seemingly chaotic steady flow is also discovered in a channel with periodic expansions and contractions.     

Slip effects on streamline topologies and their bifurcations for peristaltic flows of a viscous fluid

We discuss the effects of the surface slip on streamline patterns and their bifurcations for the peristaltic transport of a Newtonian fluid. The flow is in a two-dimensional symmetric channel or an axisymmetric tube. An exact expression for the stream function is obtained in the wave frame under the assumptions of long wavelength and low Reynolds number for both cases. For the discussion of the particle path in the wave frame, a system of nonlinear autonomous differential equations is established and the methods of dynamical systems are used to discuss the local bifurcations and their topological changes. Moreover, all types of bifurcations and their topological changes are discussed graphically. Finally, the global bifurcation diagram is used to summarize the bifurcations.     

Nonlocal effects in flows of wormlike micellar solutions

The flow curve of wormlike micelles usually exhibits a stress plateau sigma* separating high and low viscosity branches, leading to shear-banded flows. We study the flow of semidilute wormlike micellar systems in a confined geometry: a straight microchannel. We characterize their local rheology thanks to particle image velocimetry. We show that flow curves cannot be described by a simple constitutive equation linking the local shear stress to the local shear rate. We demonstrate the existence of nonlocal effects in the flow of wormlike micellar systems and make use of a theoretical framework allowing the measurement of correlation lengths.     

Existence of multiple attractors and the nature of bifurcations in a discontinuous logistic map

We present the analytical investigations on a logistic map with a discontinuity at the centre. An explanation for the bifurcation phenomenon in discontinuous systems is presented. We establish that whenever the elements of ann-cycle (n > 1) approach the discontinuities of thenth iterate of the map, a bifurcation other than the usual period-doubling one takes place. The periods of the cycles decrease in an arithmetic progression, as the control parameter is varied. The system also shows the presence of multiple attractors. Our results are verified by numerical experiments as well.