Instability driving constant

The same equations, albeit with damping and coherent external driving field, were studied by Drummond et al. [104] as a particular case of sub/second-harmonic generation. They proved that below a critical pump intensity, the system can reach a stable state (field of constant amplitude). However, beyond the critical intensity, the steady state is unstable. They predicted the existence of various instabilities as well as both first- and second-order phase transition-like behavior. For certain sets of parameters they found an amplitude self-modula-tion of the second harmonic and of the fundamental field in the cavity as well as new bifurcation solutions. Mandel and Erneux [105] constructed explicitly and analytically new time-periodic solutions and proved their stability in the vicinity of the transition points. 

A more complicated variation of the EC scheme, largely studied by voltammetry, is the situation where reaction (12.3.28) is reversible, but the product Y is unstable and undergoes a fast following reaction (Y X). This instability of Y tends to drive reaction (12.3.28) to the right, so the observed behavior resembles that of the Ei-Cj scheme. In this case, the 0/R couple mediates the reduction of species Z, with the ultimate production of species X, and the process is called redox catalysis. By selecting a mediator couple whose lies positive of that of the Z/Y couple and noting changes in the cyclic voltammetric response with v and the concentration of Z, one can find the rate constant for the decomposition of Y to X, even if it is too rapid to measure by direct electrochemistry of Z (i.e., as an EC reaction) (8, 9). 

Fig. 2. Solute distribution and transport phenomena at the interface of a growing crystal (a) Instability of the crystal-liquid interface and formation of a nonplanar pattern (schematically), (b) Faceted growth. It is assumed that the solute concentration in the liquid far from the interface (Cq) is constant due to forced and natural convection (stirring) whereas a thin solute diffusion layer (S) is quiet and possesses a solute distribution profile depending on the crystallization process type (a) interfacial control or a surface reaction (interfacial kinetics), Ce < C RJ C a difference between Cj and Cj is responsible for the driving force to buUd up the crystal surface (c) diffusion control Cj < Cj, providing a driving force for bulk diffusion in the liquid (b) mixed control.

Biochemistry s hidden asymmetry was discovered by Louis Pasteur in 1857. Nearly 150 years later, its true origin remains an unsolved problem, but we can see how such a state might be realized in the framework of dissipative structures. First, we note that such an asymmetry can arise only under far-from-equilibrium conditions at equilibrium the concentrations of the two enantiomers will be equal. The maintenance of this asymmetry requires constant catalytic production of the preferred enantiomer in the face of interconversion between enantiomers, called racemization. (Racemization drives the system to the equilibrium state in which the concentrations of the two enantiomers will become equal.) Second, following the paradigm of order through fluctuations, we will presently see how, in systems with appropriate chiral autocatalysis, the thermodynamic branch, which contains equal amounts of L- and D-enantiomers, can become unstable. The instability is accompanied by the bifurcation of asymmetric states, or states of broken symmetry, in which one enantiomer dominates. Driven by random fluctuations, the system makes, a transition to one of the two possible states. 

When a uniform sol of Pbl2 in agar is allowed to age it is observed to form mottled patterns of precipitate and precipitate free domains. The driving force for this instability of the uniform sol has been conjectured to be due to the competitive growth of the particles due to the particle radius dependence of the equilibrium constant as a result of surface tension [13]. The simplest mathematical model of this mechanism is... 

In the opposite direction, the constant miniaturization of electronic components, driven by Moore s law, necessitated that both lateral sizes and different layer thicknesses in electronic components be decreased. This soon became a driving force for the investigation of the lower limit of film thicknesses, which could be deposited using sol-gel precursors by, for instance, spin coating, dip coating, spray coating, and so on. In early literature, several issues were identified such as microstructural instability and the so-called dead layer effects, which inhibited any functional properties to be observed below a critical film thickness. 

In Sects. 5.5 and 5.6, we have introduced two 2-DOF models for the lead screw drives. In this section, the equations of motion of these models are transformed into matrix form and linearized with respect to their respective steady-sliding equilibrium point. These equations are then used in the next sections to study the local stability of the equilibrium point and the role of mode coupling instability mechanism. In this chapter, for simplicity, the coefficient of friction, n, is taken as a constant. 

---