Multiple Bifurcation Points Research Articles

Stability of thermally-induced martensitic transformations in Bi-atomic lattices

A well-known feature of some metallic alloys, such as NiTi, is their thermally induced, stress-dependent shape memory behavior. These alloys' remarkable properties are due to one or more martensitic transformations near room temperature, in which the crystalline configuration changes from a higher symmetry austenite (cubic lattice), to a lower symmetry martensite (rhombohedral, orthorhombic, tetragonal or monoclinic lattice) with decreasing temperature. In contrast to existing phenomenological approaches, the present work constructs a continuum energy density function W(F;θ) (as a function of a uniform deformation gradient and temperature) of a perfect periodic bi-atomic lattice from temperature dependent atomic potentials. Of interest in this work are the equilibrium solutions and their stability as functions of temperature for crystals under an applied pressure. Although the full problem is solved numerically, a post-bifurcation asymptotic analysis is necessary to guide the numerical solution near multiple bifurcation points. For the particular choice of a Morse-type pair potential, two stable cubic phases are predicted, one that corresponds to austentite (CsCl structure), which is stable at higher temperatures, and one that has a denser packing (NaCI structure), which is stable at lower temperatures. Theses stable portions overlap at intermediate temperatures, which is suggestive of a hysteretic temperature-induced martensitic transformation. Lower symmetry crystals, such as orthorhombic, monoclinic, and rhombohedral structures, are also predicted.

Stability of thermally-induced martensitic transformations in bi-atomic crystals

Some of the most interesting, and technologically important solid–solid transformations are the first order diffusionless transformations that occur in certain equiatomic, ordered, bi-atomic crystals. These displacive transformations include thermally-induced, reversible, proper martensitic transformations as seen in shape memory alloys such as NiTi (where group–subgroup relationships exist between the symmetry groups of the crystal phases) and the reconstructive martensitic transformations seen in certain ionic compounds such as CsCl (where no group–subgroup relationship exists between the phases). In contrast to existing continuum mechanics approaches, the present work constructs a continuum energy density function W( F ,θ) (as a function of a uniform deformation gradient and temperature) of a perfect periodic bi-atomic crystal from temperature-dependent atomic pair-potentials. Equilibrium solutions and their stability are examined as a function of temperature for crystals under no external stress. The full problem is solved numerically, and an asymptotic theory is employed to guide the numerical solution near multiple bifurcation points. Using pair-potentials and enforcing constrained kinematics (uniform deformation of a cubic CsCl-type crystal), lower symmetry crystals, such as orthorhombic, monoclinic, and rhombohedral structures are predicted. The first two of these are unstable within the chosen temperature window for our particular case, while the third is stable for higher temperatures. In addition, a hysteretic transformation was discovered in which the CsCl structure is stable at high temperatures and the NaCl structure is stable at low temperatures. These two cubic phases are connected by an unstable rhombohedral path corresponding to the transformation mechanism proposed by Buerger (1951). The CsCl–NaCl transformation suggested by the numerical results is a reconstructive transformation with a group–nonsubgroup relationship between the symmetry groups of the two phases.

Modified stiffness iteration to pinpoint multiple bifurcation points

A modified stiffness iteration to precisely compute a bifurcation point with multiple zero eigenvalues is presented. The iteration combines the direct pinpointing iteration for simple bifurcation points with a procedure to modify the tangent stiffness matrix based on pertinent congruent transformation to reduce the multiple eigenvalues to single ones. Two different transformation matrices are employed to modify the stiffness matrix: One is a non-orthogonal transformation matrix amplifying the values of the entries in a certain row and a column of the stiffness matrix. The other is an orthogonal one, which is chosen with reference to the symmetry of the system under consideration. The use of the non-orthogonal transformation matrix is a key idea termed “stiffness amplification method” in this paper, whereas the orthogonal matrix is based on block-diagonalization, which is becoming popular in group-theoretic analysis of symmetric structures. Extensive numerical examples show the robustness and usefulness of the proposed iteration method for pinpointing multiple bifurcation points.

Stationary and drifting localized structures near a multiple bifurcation point

Localized states embedded in a patterned background are found in numerical simulations of spontaneous pattern formation in a spin-1/2 atomic system with optical feedback. In the vicinity of a parameter region with bistability between two homogeneous states large amplitude peaks as well as dark holes exist as stable localized states on a hexagonal background. Moreover, resonant interaction between oscillatory and stationary inhomogeneous modes produces a nonstationary background which may force the localized states to drift.

Conjugate gradient type methods for semilinear elliptic problems with symmetry

We study block conjugate gradient methods in the context of continuation methods for bifurcation problems. By exploiting symmetry in certain semilinear elliptic differential equations, we can decompose the problems into small ones and reduce computational cost. On the other hand, the associated centered difference discretization matrices on the subdomains are nonsymmetric. We symmetrize them by using simple similarity transformations and discuss some basic properties concerning the discretization matrices. These properties allow the discrete pure mode solution paths branching from a multiple bifurcation point [0, λ m, n ] of the centered difference analogue of the original problem to be represented by the solution path branching from the first simple bifurcation point (0, μ 1,1) of the counterpart of the reduced problem. Thus, the structure of a multiple bifurcation is preserved in discretization, while its treatment is reduced to those for simple bifurcation of problems on subdomains. In particular, we can adapt the continuation-Lanczos algorithm proposed in [1] to trace simple solution paths. Sample numerical results are reported.

Formula omitted] Resonance in non-Boussinesq convection

Numerical and laboratory experiments on non-Boussinesq convection have revealed a possibly stable pattern with square symmetry but with upflow in isolated plumes and downflow in connected sheets, a topology normally associated with a hexagonal pattern. The appearance of this pattern can be interpreted as the sum of two different pairs of orthogonal rolls aligned at 45° and with wave numbers in the ratio 2 :1 . We examine the interaction between these modes by expanding about the multiple bifurcation point at which they are both marginal. The relevant amplitude equations are written down and are shown to account for several of the features found in the experiments: the new pattern can be the only stable solution near onset and can occur subcritically. These features of the solution depend crucially on the lack of Boussinesq symmetry, which induces quadratic resonance between the two sets of rolls, in close analogy with the 2:1 resonance problem in non-Boussinesq convection with O(2) symmetry.

Synaptic organizations and dynamical properties of weakly connected neural oscillators. I. Analysis of a canonical model.

We study weakly connected networks of neural oscillators near multiple Andronov-Hopf bifurcation points. We analyze relationships between synaptic organizations (anatomy) of the networks and their dynamical properties (function). Our principal assumptions are: (1) Each neural oscillator comprises two populations of neurons; excitatory and inhibitory ones; (2) activity of each population of neurons is described by a scalar (one-dimensional) variable; (3) each neural oscillator is near a nondegenerate supercritical Andronov-Hopf bifurcation point; (4) the synaptic connections between the neural oscillators are weak. All neural networks satisfying these hypotheses are governed by the same dynamical system, which we call the canonical model. Studying the canonical model shows that: (1) A neural oscillator can communicate only with those oscillators which have roughly the same natural frequency. That is, synaptic connections between a pair of oscillators having different natural frequencies are functionally insignificant. (2) Two neural oscillators having the same natural frequencies might not communicate if the connections between them are from among a class of pathological synaptic configurations. In both cases the anatomical presence of synaptic connections between neural oscillators does not necessarily guarantee that the connections are functionally significant. (3) There can be substantial phase differences (time delays) between the neural oscillators, which result from the synaptic organization of the network, not from the transmission delays. Using the canonical model we can illustrate self-ignition and autonomous quiescence (oscillator death) phenomena. That is, a network of passive elements can exhibit active properties and vice versa. We also study how Dale's principle affects dynamics of the networks, in particular, the phase differences that the network can reproduce. We present a complete classification of all possible synaptic organizations from this point of view. The theory developed here casts some light on relations between synaptic organization and functional properties of oscillatory networks. The major advantage of our approach is that we obtain results about all networks of neural oscillators, including the real brain. The major drawback is that our findings are valid only when the brain operates near a critical regime, viz. for a multiple Andronov-Hopf bifurcation.

Synaptic organizations and dynamical properties of weakly connected neural oscillators. II. Learning phase information.

This is the second of two articles devoted to analyzing the relationship between synaptic organizations (anatomy) and dynamical properties (function) of networks of neural oscillators near multiple supercritical Andronov-Hopf bifurcation points. Here we analyze learning processes in such networks. Regarding learning dynamics, we assume (1) learning is local (i.e. synaptic modification depends on pre- and postsynaptic neurons but not on others), (2) synapses modify slowly relative to characteristic neuron response times, (3) in the absence of either pre- or postsynaptic activity, the synapse weakens (forgets). Our major goal is to analyze all synaptic organizations of oscillatory neural networks that can memorize and retrieve phase information or time delays. We show that such network have the following attributes: (1) the rate of synaptic plasticity connected with learning is determined locally by the presynaptic neurons, (2) the excitatory neurons must be long-axon relay neurons capable of forming distant connections with other excitatory and inhibitory neurons, (3) if inhibitory neurons have long axons, then the network can learn, passively forget and actively unlearn information by adjusting synaptic plasticity rates.

Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces

Numerical algorithms for mapping boundaries of manipulator workspaces are developed and illustrated. Analytical criteria for boundaries of workspaces for both manipulators having the same number of input and output coordinates and redundantly controlled manipulators with a larger number of inputs than outputs are well known, but reliable numerical methods for mapping them have not been presented. In this paper, a numerical method is first developed for finding an initial point on the boundary. From this point, a continuation method that accounts for simple and multiple bifurcation of one-dimensional solution curves is developed. Second order Taylor expansions are derived for finding tangents to solution curves at simple bifurcation points of continuation equations and for characterizing barriers to control of manipulators. A recently developed method for tangent calculation at multiple bifurcation points is employed. A planar redundantly controlled serial manipulator is analyzed, determining both the exterior boundary of the accessible output set and interior curves that represent local impediments to motion control. Using these methods, more complex planar and spatial Stewart platform manipulators are analyzed in a companion paper.

Dynamical behaviors for a three-dimensional differential equation in chemical system

In this paper, a mathematical model of chemical system is investigated, the conditions of existence and local stability and bifurcation for the steady-states are obtained, the periodic solutions of the Hopf type are considered, the multiple Hopf bifurcation points exist if one parameter varies, and a technique for studying the Hopf bifurcation value is given here.

Dextrous Workspaces of Manipulators, Part III: Calculation of Continuation Curves at Bifurcation Points

ABSTRACT An efficient numerical method is presented to obtain all tangent vectors at bifurcation points of continuation curves that define the boundary of manipulator workspaces. The method developed is based on analytical criteria presented in Part I of this paper [1] and computational methods presented in Part II [2]. As observed in the latter paper, difficulties in mapping boundaries of accessible output sets include (1) systematically finding an initial point on the boundary and (2) efficiently calculating tangent vectors at multiple bifurcation points. The first difficulty has been solved by Haug, Luh, Adkins, and Wang [3]. The second difficulty is resolved in this paper.

Dynamics of a parametrically excited double pendulum

The 2:2 mode interaction of a parametrically excited double pendulum is explored in the excitation frequency/excitation amplitude plane. To determine the bifurcation structure at small amplitudes of oscillation, the method of averaging combined with centre manifold reduction is used. The full equations are solved numerically to extend the bifurcation set to larger amplitudes of response. Numerical centre manifold reduction is employed to derive two maps, valid near two multiple bifurcation points which organise the dynamical phenomena of the mode interaction region. Iteration of these maps shows the existence of global bifurcations. These results are discussed in the light of numerical integrations which show that a whole range of interesting behaviour occurs including torus doubling, torus ‘gluing’ and chaos.

Low-intensity chaotic operations of a laser with a saturable absorber

We study the dynamics of a laser with a saturable absorber for parameter values close to a multiple bifurcation point. We show that different chaotic attractors exist in a small vicinity of the zero-intensity steady state near the laser threshold.

A direct theory of viscous fluid flow in pipes II. Flow of incompressible viscous fluid in curved pipes

This paper is a companion to Part I under the same title and is concerned with the development of an analytical solution (via a direct formulation) for flow of an incompressible viscous fluid in curved pipes. Although a major portion of the analysis is presented for circular pipes of elliptical cross section, detailed calculations are carried out only for pipes of circular cross section. These calculations include the friction loss factor and the velocity contours, both of which are presented over a range of Dean number th a t significantly goes beyond the corresponding range of recent numerical solutions and the available experimental data. The results obtained show very favourable agreements with existing experimental data and several recent numerical solutions of the same problem based on the Navier—Stokes equations. Also included (for pipes of circular cross section) is a comparison of the multiple solutions and bifurcation points, as predicted by the present analytical solution, with corresponding available inform ation from several recent num erical solutions of the problem.

Nonlinear dynamics in Langmuir circulations with O(2) symmetry

A direct comparison is made between the dynamics obtained by weakly nonlinear theory and full numerical simulations for Langmuir circulations in a density-stratified layer having finite depth and infinite horizontal extent. In one limit, the mathematical formulation employed is analogous to that of double-diffusion phenonema with the flux of one diffusing quantity fixed at the boundaries of the layer. These problems have multiple bifurcation points, but their amplitude equations have no intrinsic (nonlinear) degeneracies, in contrast to ‘standard’ double-diffusion problems. The symmetry of the physical problem implies invariance with respect to translations and reflections in the horizontal direction normal to the applied wind stress (so-called O(2) symmetry). A multiple bifurcation at a double-zero point serves as an organizing centre for dynamics over a wide range of parameter values. This double zero, or Takens–Bogdanov, bifurcation leads to doubly periodic motions manifested as modulated travelling waves. Other multiple bifurcation points appear as double-Hopf bifurcations. It is believed that this paper gives the first quantitative comparison of dynamics of double-diffusive type predicted by rationally derived amplitude equations and by full nonlinear partial differential equations. The implications for physically observable natural phenomena are discussed. This problem has been treated previously, but the earlier numerical treatment is in error, and is corrected here. When the Stokes drift gradient due to surface waves is not constant, the analogy with the common formulations of double-diffusion problems is compromised. Our bifurcation analyses are extended here to include the case of exponentially decaying Stokes drift gradient.

Mode interaction in a double pendulum

The results of a theoretical and experimental study of a parametrically excited double pendulum with Z 2⊕ Z 2 symmetry are presented. Local analysis of the equations at multiple bifurcation points demonstrates the existence of global bifurcations. These suggest the presence of torus-doubling and chaos, both of which are observed experimentally and numerically.

Multiple bifurcation of rotating pipe flow

Viscous rotating Hagen-Poiseuille flow is shown to contain a multiple bifurcation point at which two Hopf bifurcations coalesce. The double Hopf bifurcation has nonsemisimple 1:1 resonance. The full problem in the neighborhood of this point may be reduced to a four-dimensional system of weakly nonlinear amplitude equations using an expansion of the center manifold type. An efficient numerical procedure to find the coupling coefficients in the amplitude equations is employed which avoids the explicit need to compute adjoint eigenfunctions. Numerical investigation of the amplitude equations has lead to the discovery of several new states of the fluid motion in this system, including stable rotating traveling and modulated traveling waves, a Feigenbaum period-doubling cascade, and chaotic trajectories. Homoclinic loops are found leading to dynamics which may be associated with “bursting” events in the physical problem under consideration.

Finite element analysis of bifurcation in nonlocal strain softening solids

Progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise (e.g. ill-posedness and multiple bifurcation points). This characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters, for instance nonlocal damage constitutive relations. The issue of possible (stable or unstable) equilibrium paths, multiple localization zones, and of the detection of bifurcation points has, however, never been addressed in the context of nonlocal constitutive laws. We extend here the eigenmode analysis and perturbation method proposed by De Borst to the study of the bifurcation and post-bifurcation response of discrete nonlocal strain softening solids. Numerical applications on beams show that bifurcation and instability may occur in the post-peak regime. As opposed to the case of local constitutive relations, the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent.

Chaotic combustion of solids and high-density fluids near points of strong resonance

The loss of stability of steady, planar combustion waves propagating through solids and high-density fluids occurs via Hopf bifurcation for sufficiently large values of a parameter related to the activation energy of the reaction. Near multiple eigenvalues corresponding to points of strong resonance, the dynamical behaviour of an asymptotic model derived from the original set of conservation equations is governed by an ordinary dynamical system for the amplitudes of the eigenmodes of the linearized problem. An analysis of these amplitude equations then completely determines the local bifurcation structure. It is shown that in certain parameter régimes, these equations admit finite-amplitude, chaotic solutions following the loss of stability of a secondary bimodal solution branch of entrained time-periodic combustion waves. Thus, in contrast to direct numerical integration of the original system of conservation equations far above the neutral stability boundary that marks the transition from steady to non-steady burning, it is shown that multi-dimensional chaotic flame propagation can be predicted from nonlinear stability theory in the neighbourhood of a multiple bifurcation point.

Analysis of the multiple hope bifurcation phenomena in a CSTR with two consecutive reactions—the singlularity theory approach

The singularity theory was applied to the analysis of the Hopf bifurcation phenomena by finding multiple Hopf bifurcation points. For a CSTR with two consecutive reactions, two third-order swallow-tail catastrophes were obtained. As the combined results obtained from the static bifurcation of the Hopf bifurcation, it was found that the parameter space could be divided into regions having different numbers of turning points and Hopf bifurcation points. In total 34 bifurcation diagrams were obtained, each of which had different dynamic features and some of which showed the maximum 17 different behaviors of phase plane trajectories for a range of the Damköhler number.