Existence of oscillations

We require that the stationary state should lie on the section of the g(a, 0) nullcline along which a is decreasing. Thus 0SS, which is simply the quotient 

must lie between 0A and 0C. The requirement for relaxation oscillations is, therefore, 

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The non existence of oscillation in the proposed discretization allied with the robustness of the computer code indicate that the association of these techniques seems to be adequate to solve the mathematical model of gasoil cracking. Other discretization methods, such as global polynomial approximation [17], could bring about oscillatory profiles through the bed length. 

Salnikov specifically reported multiple singular points and a limit cycle establishing the existence of oscillations in chemical reactions. Bilous and Amundson (1955) referred to Salnikov s (1948) paper as the first work where periodic phenomenon in reaction systems was discussed. They also indicated that a reaction A -> B in CSTR is irreversible, exothermic, and kinetically first order. Considering mass balance and heat balance equations it is known that at the steady states, the heat consumption... 

Orban et al. (1982-2) discovered that in a CSTR within an extremely narrow range of flow rates and input concentrations a system containing Br03, Br" and Mn(II) or Ce(III) exhibits oscillations in the potential of either a Pt redox or Br- selective electrode. Existence of oscillations was predicted by the model calculations of Bar-Eli [Bar-Eli in Vidal and Pacault (1981) 228-239]. The bromate oscillators such as the B-Z reaction were derived from this fundamental system by adding feedback species which enlarges the region of critical space in which oscillations occur. 

The problems of simultaneously treating spatial distributions of both temperature and concentration are currently the concern of the chemical engineer in his treatment of catalyst particles, catalyst beds, and tubular reactors. These treatments are still concerned with systems that are kineticaliy simple. The need for a unified theory of ignition has been highlighted by contemporary studies of gas-phase oxidations, many features being revealed that neither thermal theory, nor branched-chain theory for that matter, can resolve alone. A successful theoretical basis for such reactions necessarily involves the treatment of both the enorgy balance and mass balance equations. Such equations are invariably coupled and cannot be solved independently of each other. However, much information is offered by the phase-plane analj s of the syst (e.g. stability of equilibrium solutions, existence of oscillations) without the need for a formal solution of the balance equations. 

For a conservative oscillator, the modulus of the wave function is a constant and, according to Equation 9.113, the square of this modulus in the linear case is equal to twice the total energy of the system. As the probability of existence of oscillations in any range of phase angles is equal to 1, one must have for the normalization function in this case... 

In addition to multiple steady states also existence of oscillations of various types has been observed in the model. Continuation methods (Kubicek and Marek, 1983) can be used to locate positions of limit points (multiple solutions), Hopf bifurcation points (origin of oscillations) and period doubling bifurcation points. Fig. 2 shows an example of the results of such computations, using the continuation software CONT (Kohout et al.. 

In 1930, London [1,2] showed the existence of an additional type of electromagnetic force between atoms having the required characteristics. This is known as the dispersion or London-van der Waals force. It is always attractive and arises from the fluctuating electron clouds in all atoms that appear as oscillating dipoles created by the positive nucleus and negative electrons. The derivation is described in detail in several books [1,3] and we will outline it briefly here. 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

Aside from merely calculational difficulties, the existence of a low-temperature rate-constant limit poses a conceptual problem. In fact, one may question the actual meaning of the rate constant at r = 0, when the TST conditions listed above are not fulfilled. If the potential has a double-well shape, then quantum mechanics predicts coherent oscillations of probability between the wells, rather than the exponential decay towards equilibrium. These oscillations are associated with tunneling splitting measured spectroscopically, not with a chemical conversion. Therefore, a simple one-dimensional system has no rate constant at T = 0, unless it is a metastable potential without a bound final state. In practice, however, there are exchange chemical reactions, characterized by symmetric, or nearly symmetric double-well potentials, in which the rate constant is measured. To account for this, one has to admit the existence of some external mechanism whose role is to destroy the phase coherence. It is here that the need to introduce a heat bath arises. 

The principal effect of the presence of a smooth wall, compared to a free surface, is the occurrence of a maximum in the density near the interface due to packing effects. The height of the first maximum in the density profile and the existence of additional maxima depend on the strength of the surface-water interactions. The thermodynamic state of the liquid in a slit pore, which has usually not been controlled in the simulations, also plays a role. If the two surfaces are too close to each other, the liquid responds by producing pronounced density oscillations. 

Lattice models for bulk mixtures have mostly been designed to describe features which are characteristic of systems with low amphiphile content. In particular, models for ternary oil/water/amphiphile systems are challenged to reproduce the reduction of the interfacial tension between water and oil in the presence of amphiphiles, and the existence of a structured disordered phase (a microemulsion) which coexists with an oil-rich and a water-rich phase. We recall that a structured phase is one in which correlation functions show oscillating behavior. Ordered lamellar phases have also been studied, but they are much more influenced by lattice artefacts here than in the case of the chain models. 

To determine the likely possibility of oscillations occurring in a new or an existing column, or even sections of a column, the original article is recommended. 

Theory of Bifurcations.—In the preceding sections we have reviewed a few more important points of the existing topological methods in the theory of oscillation, assuming that the topological configuration or the phase portrait remains fixed. 

In the first place, the difference between the (NA) systems and the (A) ones is that for the first there exists always a periodic solution with period 2w (or a rational fraction of 2ir), whereas for the second, the period of oscillation (if it exists) is determined by the parameters... 

This condition is thus the necessary and sufficient condition for the existence of a stable stationary solution (oscillation) of the differential equation (6-127). 

Assume that we have a pendulum (Fig. 6-14) provided with a piece of soft iron P placed coaxially with a coil C carrying an alternating current that is, the axis of the coil coincides with the longitudinal axis OP of the pendulum at rest. If the coil is excited, one finds that the pendulum in due course begins to oscillate, and th oscillations finally reach a stationary amplitude. It is important to note that between the period of oscillation of the pendulum and the period of the alternating current there exists no rational ratio, so that the question of the subharmonic effect is ruled out. 

From the standpoint of the classical (analytical) theory with which we were concerned in this review, the situation is obviously absurd since each of these two equations is linear and of a dissipative type (since h > 0) trajectories of both of these equations are convergent spirals tending to approach a stable focus. However, if one carries out a simple analysis (see Reference 6, p. 608), one finds that change of equations for = 0, results in the change of the focus in a quasi-discontinuous manner, so that the trajectory can still be closed owing to the existence of two nonanalytic points on the -axis. If, however, the trajectory is closed, this means that there exists a stationary oscillation and in such a case the system (6-197) is nonlinear, although, from the standpoint of the differential equations, it is linear everywhere except at the two points at which the analyticity is lost. 

These phenomena are being actively studied at the present time, and constitute a new chapter in the theory of oscillations that is known as piecewise linear oscillations. There exists already a considerable literature on this subject in the theory of automatic control systems11-34 but the situation is far from being definitely settled. One can expect that these studies will eventually add another body of knowledge to the theory of oscillations, that will be concerned with nonanalytic oscillatory phenomena. 

Optimal flow method, 261 Optimization non-constrained, 286 of functionals, 305 Ordinary value, 338 Orthogonalization, Schmidt," 65 Osaki, S., 664 Oscillation hysteresis, 342 Oscillations autoperiodic, 372 discontinuous theory, 385 heteroperiodic, 372 piecewise linear, 390 relaxation asymptotic theory, 388 relaxation, 383 Oscillatory circuit, 380 "Out field, 648 existence of, 723... 

Abnormally low atomic heats were explained by Richarz on the assumption of a diminution of freedom of oscillation consequent on a closer approximation of the atoms, which may even give rise to the formation of complexes. This is in agreement with the small atomic volume of such elements, and with the increase of atomic heat with rise of temperature to a limiting value 6 4, and the effect of propinquity is seen in the fact that the molecular heat of a solid compound is usually slightly less than the sum of the atomic heats of the elements, and the increase of specific heat with the specific volume when an element exists in different allotropic forms. 

Meiron (12) and Kessler et al. (13) have shown that numerical studies for small surface energy give indications of the loss-of-existence of the steady-state solutions. In these analyses numerical approximations to boundary integral forms of the freeboundary problem that are spliced to the parabolic shape far from the tip don t satisfy the symmetry condition at the cell tip when small values of the surface energy are introduced. The computed shapes near the tip show oscillations reminiscent of the eigensolution seen in the asymptotic analyses. Karma (14) has extended this analysis to a model for directional solidification in the absence of a temperature gradient. 

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