Structurally unstable equilibrium

In this chapter we study non-rough or structurally unstable equilibrium states on the stability boundaries, i.e. those which have at least one characteristic exponent on the imaginary axis in the complex plane. As for the rest of the characteristic exponents we will assume that they lie in the open left-half-plane. 

The basis of stability theory for systems with structurally unstable equilibrium states was developed by Lyapunov. His works and numerous subsequent studies on various aspects of stability in critical cases, as well as of bifurcation phenomena accompanying the loss of stability of equilibrium states had became the foundation on which the principal notions in the theory of nonlinear oscillations had spawned in the twenties and thirties. 

The most general definition of a template is as a structure-directing agent. In surfactant solutions the final templated polymers can be either discrete nanoparticles or mesostructured bulk materials as a consequence of polymerization, respectively, in the non-continuous or continuous domains of the template. Thermodynamically stable media, such as microemulsions, equilibrium vesicles, or lyotropic mesophases are especially useful as templates because of their structural definition and reproducible morphologies. The mesostructure of a thermodynamically stable template is defined by composition and temperature, but this same feature makes the structure unstable to changes in temperature, pH, or concentration. The aim of template synthesis is to transfer the self-organized template structure into a mechanically and chemically stable, durable, and processable material. 

It is clear that the joints were not sufficiently stiff to provide sway restraint. Although the first and third storey heights were to contain partitions the second storey height would have contained only non-load bearing partitions. As it happened none of the partitions were completed at the time of collapse. The structure was in a state of unstable equilibrium and it needed only a small disturbance to precipitate collapse. The failure of one of the joints on the second floor therefore triggered a collapse of the whole building. 

We remark that an equilibrium state, or a periodic orbit, may be arbitrarily degenerate. It is therefore logical to begin our study with the simplest structurally unstable systems which Andronov and Leontovich called systems of first degree of non-roughness. 

The primary scope of this book will focus on the analysis of the internal bifurcations within the class of systems with simple dynamics, such as Morse-Smale systems. Furthermore, we will restrict our study mostly to bifurcations of codimension-one. The reason for this restriction is that some bifurcations of higher codimension turn out to be boundary bifurcations in many cases, such as when the normal forms for the equilibrium states are three-dimensional. Nevertheless, we will examine some codimension-two cases which are concerned with equilibrium states and the loss of stability of periodic orbits. Meanwhile, let us start our next section with a discussion of some questions related to structurally unstable heteroclinic connections. 

The heteroclinic cycles including the saddles whose unstable manifolds have different dimensions were first studied in [34, 35]. This study mostly focused on systems with complex dynamics. Let us, however, discuss here a case where the dynamics is simple. Let a three-dimensional infinitely smooth system have two equilibrium states 0 and O2 with real characteristic exponents, respectively, 7 > 0 > Ai > A2 and 772 > 1 > 0 > (i.e. the unstable manifold of 0 is onedimensional and the unstable manifold of O2 is two-dimensional). Suppose that the two-dimensional manifolds (Oi) and W 02) have a transverse intersection along a heteroclinic trajectory To (which lies neither in the corresponding strongly stable manifold, nor in the strongly unstable manifold). Suppose also that the one-dimensional unstable separatrix of Oi coincides with the one-dimensional stable separatrix of ( 2j so that a structurally unstable heteroclinic orbit F exists (Fig. 13.7.24). The additional non-degeneracy assumptions here are that the saddle values are non-zero and that the extended unstable manifold of Oi is transverse to the extended stable manifold of O2 at the points of the structurally unstable heteroclinic orbit F. 

With the microfocus instrument it is possible to combine the weak Raman scattering of liquid water with a water-immersion lens on the microscope and to determine spectra on precipitates in equilibrium with the mother liquor. Unique among characterization tools, Raman spectroscopy will give structural information on solids that are otherwise unstable when removed from their solutions. 

Fluids on the Earth s surface that are in hydrostatic equilibrium may be stable or unstable depending on their thermal structure. In the case of freshwater (an incompressible fluid), density decreases with temperature above ca. 4°C. Warm water lying over cold water is said to be stable. If warm water underlies cold, it is buoyant it rises and is unstable. The buoyant force, F, on the parcel of fluid of unit volume and density p is ... 

The large energy differences between the global minimum structure of C2v symmetry and the other isomers indicate that equilibrium sulfur vapor will contain only minute amounts of the latter, even at very high temperatures. However, under non-equilibrium conditions as in electrical discharges or by illumination with a laser as in Raman spectroscopy unstable isomers may be formed and detected. 

Truss Stress Analysis The computation of member forces in an arbitrary plane truss is now examined. There exist some simple counting tests that may determine if a given truss is unstable. Failing that, one must attempt to compute the equilibrium state given some external forces in the process, one obtains values for all member forces. In this example, all truss members are identical in terms of material and area, grown in a developmental space where units are measured in meters EA is set to 1.57 x 104 N, corresponding to a modulus of elasticity for steel and a cylindrical member of diameter 1 cm. Consider a general truss with n joints and m beams external forces are applied at joints and the member forces are computed. Let the structure forces be... 

Phenoxyl with such a structure recombines with the rate constant close to that of the diffusionally controlled reaction. 2,4,6-Trisubstituted phenoxyls form unstable dimers. The latter dissociate back to phenoxyls. The values of the formed bonds lie between 30 and 120 kJ mol-1 [3], The rate constants and equilibrium constants of dimerization for a few phenoxyls are presented in Table 15.9. 

The iron-carbon solid alloy which results from the solidification of iron blastfurnace metal is saturated with carbon at the metal-slag temperature of about 2000 K, which is subsequently refined by the oxidation of carbon to produce steel containing less than 1 wt% carbon, the level depending on the application. The first solid phases to separate from liquid steel at the eutectic temperature, 1408 K, are the (f.c.c) /-phase Austenite together with cementite, Fe3C, which has an orthorhombic structure, and not the thermodynamically stable carbon phase which is to be expected from the equilibrium diagram. Cementite is thermodynamically unstable with respect to decomposition to iron and carbon from room temperature up to 1130 K... 

Solvates and hydrates can be unstable when removed from solution, and are not usually desired as the solid form of the final API. The water or solvent molecules often lie along a crystal axis and can diffuse out of the crystal along these channels to achieve equilibrium with the surrounding vapour phase. In some instances this weakens the crystal structure and may cause fragmentation. 

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