Secondary Bifurcation Point Research Articles

Bifurcation effects in thermodynamics of Markovian copolymers away from the critical point

Numerical computations of inhomogeneous free-energy extremals on the basis of the thermodynamic model of Markovian copolymers have shown that they exhibit two qualitatively different scenarios of behavior with varying temperature. The first occurs in systems with close points of the trivial and nontrivial spinodals, while the second scenario takes place when the turning point of an extremal is sufficiently far away from the point of the nontrivial spinodal. In this case, as the binodal is approached, the model exhibits a secondary bifurcation point, i.e., the intersection point with the homogeneous nontrivial extremal issuing from the point of the trivial spinodal. At this point, the homogeneous nontrivial extremal becomes unstable with respect to periodic perturbations of frequency q* corresponding to a bifurcation of inhomogeneous extremals from the trivial state (point of the nontrivial spinodal). This bifurcation is studied for lamellar, hexagonal, and body-centered cubic structures. The evolution of the extremals is numerically computed until an absolutely stable state corresponding to the binodal is reached.

Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves

The existence question for two-dimensional symmetric steady waves travelling on the surface of a deep ocean beneath a heavy elastic membrane is analyzed as a problem in bifurcation theory. The behaviour of the two-dimensional cross-section of the membrane is modelled as a thin (unshearable), heavy, hyperelastic extensible rod, and the fluid beneath is supposed to be in steady two-dimensional irrotational motion under gravity. When the wavelength has been normalized to be 2 , and when gravity and the density of the undeformed membrane are prescribed, there are two free parameters in the problem: the speed of the wave and the drift velocity of the membrane. It is observed that the problem, when linearized about uniform horizontal flow, has at most two independent solutions for any values of the parameters. When the linearized problem has only one normalized solution, it is shown that the full nonlinear problem has a sheet of solutions consisting of a family of curves bifurcating from simple eigenvalues. Here one of the problem’s parameters is used to index a family of bifurcation problems in which the other is the bifurcation parameter. When the linearized problem has two solutions, with wave numbers k and l such that maxfk;lg=minfk;lg = 2 Z, it is shown that there are three two-dimensional sheets of bifurcating solutions. One consists of “special” solutions with minimal period 2=k ; another consists of “special” solutions with minimal period 2=l ; and the third, apart from those on the curves where it intersects the “special” sheets, consists of “general” solutions with minimal period 2 . The two sheets of “special” solutions are rather similar to those that occur when the linearized problem has only one solution. However, points where the first sheet or the second sheet intersects the third sheet are period-multiplying (or symmetry-breaking) secondary bifurcation points on primary branches of “special” solutions. This phenomenon is analogous to that of Wilton ripples, which arises in the classical water-wave problem when the surface tension has special values. In the case of Wilton ripples, the coefficient of surface tension and the wave speed are the problem’s two parameters. In the present context, there are two speed parameters, meaning that the membrane elasticity does not need to be highly specified for this symmetry-breaking phenomenon to occur. 2010 Mathematics Subject Classification: 35R35, 74B20, 74F10, 76B07, 37G40.

Large deflections of spatial variable-arc-length elastica under terminal forces

This paper aims to study the large deflections of variable-arc-length elastica subjected to the terminal forces (e.g., axial force and torque). Based on Kirchhoff's rod theory and with help of Euler parameters, the set of nonlinear governing differential equations which free from the effect of singularity are established together with boundary conditions. The system of nonlinear differential equations is solved by using the shooting method with high accuracy integrator, seventh-eighth order Runge-Kutta with adaptive step-size scheme. The error norm of end conditions is minimized within the prescribed tolerance (<TEX>$10^{-5}$</TEX>). The behavior of VAL elastica is studied by two processes. One is obtained by applying slackening first. After that keeping the slackening as a constant and then the twist angle is varied in subsequent order. The other process is performed by reversing the sequence of loading in the first process. The results are interpreted by observing the load-deflection diagram and the stability properties are predicted via fold rule. From the results, there are many interesting aspects such as snap-through phenomenon, secondary bifurcation point, loop formation, equilibrium configurations and effect of variable-arc-length to behavior of elastica.

Non-existence of a secondary bifurcation point for a semilinear elliptic problem in the presence of symmetry

We give two sufficient conditions for a branch consisting of non-trivial solutions of an abstract equation in a Banach space not to have a (secondary) bifurcation point when the equation has a certain symmetry. When the nonlinearity f is of Allen–Cahn type (for instance f(u)=u−u3), we apply these results to an unbounded branch consisting of non-radially symmetric solutions of the Neumann problem on a disk D⊂R2Δu+λf(u)=0in D,∂νu=0on ∂D and emanating from the second eigenvalue. We show that the maximal continuum containing this branch is homeomorphic to R×S1 and that its closure is homeomorphic to R2.

SECONDARY INSTABILITY AND MODE JUMPING OF DEEP THERMOELASTICALLY BUCKLED COMPOSITE LAMINATES

An analytic method is presented to study the secondary instability (bifurcation) and mode jumping of thermomechanically deep buckled composite laminates. Unlike most ad hoc approaches, the governing partial differential equations (PDEs) and constitutive relations are rigorously derived from an asymptotically correct, geometrically nonlinear theory. A novel and simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The generalized Galerkin method is used to solve boundary value problems corresponding to antisymmetric angle-ply and cross-ply composite plates. The variety of possible modal interactions is expressed in an explicit and concise form by transforming the coupled nonlinear governing equations into a system of nonlinear ordinary differential equations (ODEs). The comparison between the present method and the finite element analysis (FEA) shows a pretty good match in their numerical results in the primary postbuckling region. While the FEA may lose its convergence when solution comes close the secondary bifurcation point, the analytic approach has the capability of exploring deeply into the post-secondary buckling realm and capture the mode jumping phenomenon for various combinations of plate configurations and in-plane boundary conditions. Qualitatively different propagations of buckling pattern are observed before and after mode jumping. Free vibration along the stable primary postbuckling and the jumped equilibrium paths are also studied.

Postbuckling and mode jumping analysis of composite laminates using an asymptotically correct, geometrically non-linear theory

An analytic method is presented in this paper to study the postbuckling and mode jumping behavior of bi-axially compressed composite laminates. The governing partial differential equations (PDEs) are derived rigorously from an asymptotically correct, geometrically non-linear theory. A novel and relatively simpler solution approach is developed to solve the two coupled fourth-order PDEs, namely, the compatibility equation and the dynamic governing equation. The generalized Galerkin method is used to solve boundary value problems corresponding to antisymmetric angle-ply and cross-ply composite plates, respectively. The variety of possible modal interactions is expressed in an explicit and concise form by transforming the coupled non-linear governing equations into a system of non-linear ordinary differential equations (ODEs). The comparison between the present method and the finite element analysis (FEA) shows a pretty good match in their numerical results in the primary postbuckling region. While the FEA may lose its convergence when solution comes close to the secondary bifurcation point, the analytic approach has the capability of exploring deeply into the post-secondary buckling realm and capture the mode jumping phenomenon for various combinations of plate configurations and in-plane boundary conditions. Free vibration along the stable primary postbuckling and the jumped equilibrium paths are also studied.

Secondary bifurcations in systems with all-to-all coupling. Part II.

In a recent paper Dias and Stewart studied the existence, branching geometry, and stability of secondary branches of equilibria in all-to-all coupled systems of differential equations, that is, equations that are equivariant under the permutation action of the symmetric group S N . They consider the most general cubic order system of this type. Primary branches in such systems correspond to partitions of N into two parts p, q with p + q = N. Secondary branches correspond to partitions of N into three parts a, b, c with a + b + c = N. They prove that except in the case a = b = c secondary branches exist and are (generically) globally unstable in the cubic order system. In this work they realized that the cubic order system is too degenerate to provide secondary branches if a = b = c. In this paper we consider a general system of ordinary differential equations commuting with the permutation action of the symmetric group S 3n on R 3n . Using singularity theory results, we find sufficient conditions on the coefficients of the fifth order truncation of the general smooth S 3n -equivariant vector field for the existence of a secondary branch of equilibria near the origin with S n × S n × S n symmetry of such system. Moreover, we prove that under such conditions the solutions are (generically) globally unstable except in the cases where two tertiary bifurcations occur along the secondary branch. In these cases, the instability result holds only for the equilibria near the secondary bifurcation points. We show an example where stability between tertiary bifurcation points on the secondary branch occurs.

POST-BUCKLING OF AN ELASTIC COLUMN WITH VARIOUS ROTATIONAL END RESTRAINTS

In this technical note, the post-buckling behavior of a simply supported elastic column with various rotational end conditions of the supports is investigated. The compressive force is applied at the tip of the column. The characteristic equation for solving the critical loads is obtained from the boundary value problem of linear systems. In the post-buckling state, a set of nonlinear differential equations with boundary conditions is established and numerically solved by the shooting method. The interesting features associated with this problem such as the limit load point, snap-through phenomenon and the secondary bifurcation point will be highlighted herein.

Dynamic analysis of modal shifting and mode jumping in thermally buckled plates

Both analytical and finite element investigations are performed for the various static and dynamic aspects of the mode jumping phenomenon of a simply-supported rectangular plate heated deeply into the post-buckling regime. For the analytical method, the von Kármán plate equation is reduced to a system of non-linear ODEs by expressing the transverse deflection as a series of linear buckling modes. The ODEs, combined with the non-linear algebraic constraint equations obtained from in-plane boundary conditions, are then solved numerically under the parametric variation of the temperature. The results are checked by the finite element method, where a hybrid static–dynamic scheme is implemented. The contribution of each assumed (buckling) mode component is studied systematically. Characterized by the strong geometrical non-linearity, the secondary bifurcation point of the thermally loaded plate with fixed in-plane boundary conditions occurs far beyond the primary buckling point, and the jump behavior cannot be predicted correctly without sufficient assumed modes. Stationary bifurcation analysis indicates that while the post-buckling deflection before mode jumping is composed of pure symmetric modes, additional pure antisymmetric modes will appear after the occurrence of the snapping and they play the role of destabilizing the equilibrium. Furthermore, by monitoring natural frequencies and modal shapes, we find that a mode shifting phenomenon (the exchanging of vibration modes) exists in the primary post-buckling regime. Breaking of the symmetry of the dynamic modes is also found. By introducing a linear temperature sweeping scheme, transient analysis is performed to capture the snapping phenomenon dynamically, which occurs with moderate heating ratio. Comparison between the analytic and finite element results shows good agreement.

Hopf Bifurcation Subject to a Large Delay in a Laser System

Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities start from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.

Hopf Bifurcation Subject to a Large Delay in a Laser System

Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities are starting from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.

On the onset of convective instabilities in cylindrical cavities heated from below. I. Pure thermal case

Three-dimensional steady flows are simulated in a circular cylindrical cavity of aspect ratio A=H/D, where H is the height and D the diameter of the cavity. The cavity is heated from below and its sidewalls are considered to be adiabatic. The effect of the geometry of the cavity on the onset of convection and on the structure and symmetries of the flow is analyzed. The nonlinear evolution of the convection beyond its onset is presented through bifurcation diagrams for two typical aspect ratios A=0.5 and A=1. Axisymmetric (m=0) and asymmetric (m=1 and m=2) azimuthal modes [exp (imφ)] are observed. For A=0.5, the axisymmetric solution loses its stability to a three-dimensional solution at a secondary bifurcation point. Better understanding of the mechanisms leading to this instability is obtained by analyzing the energy transfer between the basic state and the critical mode. To study the influence of the Prandtl number on the flow pattern and on the secondary bifurcation, three values of the Prandtl number are investigated: Pr=0.02 (liquid metal), Pr=1 (transparent liquids), and Pr=6.7 (water).

Buckling Mode Interaction in Fixed-End Column with Central Brace

The postbuckling behavior of an elastic fixed-end column with an elastic brace at the center is investigated. Attention is focused on those of brace stiffness near its threshold value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. We show that, for the brace stiffness greater than the threshold value, there are precisely two secondary bifurcation points on each primary postbuckling path bifurcating from one of the least two classical buckling loads, and the corresponding secondary postbuckling paths connect all of these secondary bifurcation points in a loop. For the brace stiffness less than the threshold value, no secondary bifurcation occurs. The asymptotic expansions of the primary and secondary postbuckling paths are constructed. The stability analysis indicates that, when the brace stiffness goes beyond its threshold value, the primary postbuckling path with a node in the center becomes unstable from stable by means of the secondary bifurcation (i.e., secondary buckling occurs).

Secondary buckling of an elastic column with spring-supports at clamped ends

The postbuckling behavior of an elastic column with spring supports of equal stiffness of extensional type at both clamped ends is studied. Attention is focused on those of spring stiffnesses near the critical value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. By using the Liapunov-Schmidt-Koiter approach, we show that there are precisely two secondary bifurcation points on each primary postbuckling state for the spring stiffness greater than the critical value. The bifurcation takes place at one of the two least buckling loads. The corresponding secondary postbuckling states connect all the secondary bifurcation points in a loop. For the spring stiffness less than the critical value, no secondary bifurcation occurs. Asymptotic expansions of the primary and secondary postbuckling states are constructed. The stability analysis indicates that the primary postbuckling state for the spring stiffness greater than the critical value is bifurcating from the first buckling load and becomes unstable from a stable state via the secondary bifurcation, i.e., secondary buckling occurs.

Global analysis of secondary bifurcation of an elastic bar

In a three dimensional framework of finite deformation configurations, this paper investigates the secondary bifurcation of a uniform, isotropic and linearly elastic bar under compression in a large range of parameters. The governing differential equations and finite dimensional equations of this problem are discussed. It is found that, for a bar with two ends hinged, usually many secondary bifurcation points appear on the primary branches which correspond to the maximum bending stiffness. Results are shown on parameter charts. Secondary modes and branches are also calculated with numerical methods.

Mode interactions in reaction-driven convection in a porous medium

Convective-mode interaction occurring in reaction-driven convection in a porous medium is analyzed using local bifurcation theory. The system considered is a porous medium in the shape of a rectangular box with nonflux boundary conditions at the side walls and the bottom. From linear stability analysis it is found that vertical mode 1 (with one roll in the vertical direction) was always the first to destabilize. Hence, mode interactions occurring among various horizontal modes and the first vertical mode are considered. Center manifold theory is used to derive formulae for evaluating the coefficients of the amplitude equations of the normal form for a double zero singularity, directly from the original system of PDEs. Three different kinds of bicritical points are considered. For the case of interaction between mode 1 and mode 2 solutions, pure mode 1 solution does not exist and the presence of a secondary Hopf bifurcation and hence, oscillatory convection is predicted. From the simple zero and limit point interaction, it is shown that when the convective solution branches from the middle (unstable) conduction branch, a secondary Hopf bifurcation point exists on the convective branch leading to stable oscillatory convective solutions. Similar analysis done for mode 2 and mode 3 interaction shows that three types of solutions, pure mode 2, pure mode 3 along with a mixed mode 2 3 solution exist. These results indicate that mode interactions occurring in reaction-driven convection are profoundly different from those in Lapwood or Benard convection. A simple lower-dimensional, discretized version of the original model was developed and shown to have the same local behavior as the infinite-dimensional model. Finally, global bifurcation diagrams were calculated for the full system of equations using the orthogonal collocation method. It was found that even though the collocation method breaks the symmetry present in the problem, spurious solutions can be avoided by taking enough terms in the direction where symmetry is present.

The Computation of One-Parameter Families of Bifurcating Elastic Surfaces

The problem of constructing the middle surface of a deformed elastic shell from its first and second fundamental forms $\hat{a} _{\alpha \beta } $ and $\hat{b} _{\alpha \beta } $ is considered. The undeformed shell is a spherical cap of radius R and thickness h with an angular width $2\theta _0 $, where$0 < \theta _0 < \pi/2$. The cap is subjected to a constant uniform load $\lambda $ and is simply supported at its edge. The authors seek to compute the one-parameter families of buckled states that branch from the unbuckled state of the shell. This is accomplished in two steps..First, a finite element method is used to solve the governing shell equations, a pair of fourth-order nonlinear partial differential equations (PDEs). A solution of this system is a curvature potential w, a stress potential f, and the load $\lambda $. Using Liapunov– Schmidt reduction, it can be shown that solutions possessing a variety of symmetries bifurcate from the unbuckled state of the shell. In the work presented here, these local branches will be numerically continued. Solution branches are parameterized in terms of a pseudo-arc-length parameter $\rho $ (i.e., $( \lambda , f, w) = ( \lambda ( \rho ), f_\rho , w_\rho ) $), enabling them to be tracked around turning points. The second step in the solution process is to solve numerically for the parameterization $\hat{\bf X} _\rho $ corresponding to the middle surface of the buckled shell $\hat{\mathcal{S}} _\rho $. This is done by integrating the PDEs of $\hat{\mathcal{S}} _\rho $. The coefficients in these differential equations involve the first and second fundamental forms of the deformed shell SP, which can be computed from $( \lambda ( \rho ), f_\rho ,w_\rho )$. A number of bifurcation diagrams corresponding to the first three branch points of a spherical cap of size $\theta _0 = 12.85^\circ $ are presented. For this example, a secondary bifurcation point was found connecting two distinct nonaxisymmetric solution branches. Computer graphics are used to display images of various buckled surfaces that branch from the unbuckled state of the shell.

Secondary bifurcations in plane Poiseuille flow with three interacting modes

A degenerate secondary bifurcation point in a plane channel flow is investigated. In the absence of any symmetry assumption between the two channel halves the analysis of the corresponding three-dimensional normal form reveals the presence of mixed-mode equilibria with more complicated spatial structures. It is further shown that the dynamics of the normal form contains trajectories between unstable equilibria for open sets of parameter values.

GROUP THEORETICAL MODE INTERACTIONS WITH DIFFERENT SYMMETRIES

In two-parameter systems two symmetry breaking bifurcation points of different types coalesce generically within one point. This causes secondary bifurcation points to exist. The aim of this paper is to understand this phenomenon with group theory and the inner-connectivity of irreducible representations of supergroup and subgroups. Colored pictures of examples are included.

Bifurcation diagrams in Kolmogorov's problem of viscous incompressible fluid on 2-D flat tori

We consider Kolmogorov's problem of viscous incompressible fluid motions on two dimensional tori. The problem is a bifurcation problem with two parameters, the Reynolds number and the aspect ratio. Varying the aspect ratio as a splitting parameter, we compute numerically bifurcation diagrams with the Reynolds number as a bifurcation parameter. As the aspect ratio changes, we observe turning points and secondary bifurcation points appear or disappear. Furthermore, Hopf bifurcation points are also found when the aspect ratio of the torus satisfies a certain condition. This paper is an improved and enlarged version of the report [26]. Some errors in [17, 26] are corrected here.