Spontaneously Oscillating Models

This work is centred around the study of the response to periodic forcing of systems that, when autonomous, had a stable limit cycle surrounding an unstable steady state in their phase plane. For the sake of simplicity—and since many of the fundamental phenomena are the same—we studied two-dimensional systems. We chose two examples of isothermal reactor models the first is an autocatalytic homogeneous Brusselator (Glansdorff and Prigog-ine, 1971)  

The second model results from a bimolecular surface reaction, A + B — products, with competitive Langmuir-Hinshelwood kinetics, which occurs in a heterogeneous differential reactor with perfectly mixed gas phase. The reaction is first order in both adsorbed A and B, and two vacant sites are required in the reaction mechanism. If the reaction products desorb immediately, the 

We also studied the forced nonisothermal CSTR with the A — B reaction [Uppal et al., 1974]  

The forcing variable is the coolant temperature jc2c as in Sincic and Bailey (1977) and more recently in Mankin and Hudson (1984). In eq. (10) jti is a dimensionless reactant concentration while x2 is a dimensionless reactor temperature. These equations hold at the limit of infinite reaction activation energy. All models were thus chosen so that extensive simulation results existed in the literature, and they cover a wide range of lumped reactor types. 

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F. Ritort, Spontaneous relaxation in generalized oscillator models for glassy dynamics. J. Phys. Chem. B 108, 6893-6900 (2004). 

In the present paper we study common features of the responses of chemical reactor models to periodic forcing, and we consider accurate methods that can be used in this task. In particular, we describe an algorithm for the numerical computation and stability analysis of invariant tori. We shall consider phenomena that appear in a broad class of forced systems and illustrate them through several chemical reactor models, with emphasis on the forcing of spontaneously oscillating systems. 

The observation of oscillations in heterogeneous catalytic reactions is an indication of the complexity of catalyst kinetics and makes considerable demands on the theories of the rates of surface processes. In experimental studies the observed fluctuations may be in catalyst temperature, surface species concentrations, or most commonly because of its accessibility, in the time variation of the concentrations of reactants and products in contact with the catalyst. It is now clear that spontaneous oscillations are primarily due to non-linearities associated with the rates of surface reactions as influenced by adsorbed reactants and products, and the large number of experimental studies of the last decade have stimulated a considerable amount of theoretical kinetic modelling to attempt to account for the wide range of oscillatory behaviour observed. 

Spontaneous oscillations are a widespread phenomenon in nature. They have been studied for a large number of experiments, including electrochemical systems such as the oxidation of metals and organic materials [Miller and Chen (2006)]. Electrochemical systems exhibiting instabilities often behave like activator-inhibitor systems. In these systems the electrode potential is an essential variable and takes on the role of either activator or the inhibitor. If certain conditions are met, an activator-inhibitor system generates oscillations [Krischer (2001)]. In this section we present experimental data of electric potential self-oscillations on the electrode of IPMC which results in the oscillating actuation of the material. Furthermore, we also present a physical model to predict these oscillations. 

The term x E represents changes in the index of refraction due to changes in optical intensity, such as those due to spontaneous emission, and its inclusion in the traditional laser oscillator model for a laser leads to a new result for the laser linewidth ... 

There exist two major approaches to the theoretical description of the time and frequency gated spontaneous emission (TFG SE). In the first approach, the TFG SE spectrum is defined as the rate of emission of photons of a certain frequency within a definite time interval. The influence of the measuring device is not taken into account in this formulation. Starting from this definition, one obtains an ideal (bare) TFG SE spectrum, which is not guaranteed to be positive, however. For instance, for certain parameters of the Brownian oscillator model, the spectrum can attain negative values. Moreover, the time and frequency resolutions of this ideal spectrum are not limited by the fundamental time-frequency uncertainty principle. This underlines the necessity to develop a more comprehensive theory, in which both a spectrometer and a time-gating device enter the description from the outset. 

There a different kind of differential-equation model of excitability which might turn out to have significantly different properties [103-109]. Such media have three fixed points instead of just one and make the transition from excitability to spontaneous oscillation by a saddle-node bifurcation through infinite period ( SNIPER ), e.g., as revealed by the period and amplitude of the Belousov-Zhabotinsky reagent s bulk oscillations [24, Figure 1 110-114], In such media spontaneous oscillation and excitability are indeed mutually exclusive alternatives as commonly supposed, rather than independent, typically coexisting properties of the medium (as in the most profusely... 

During the period of instability, the system will move spontaneously away from the stationary state. For the present model there is only ever one stationary state, so there is no other resting state for the system to move to. The concentrations of A and B must vary continuously in time. They eventually tend to a periodic oscillatory motion around the unstable state. We thus see oscillations over the range of conditions described by (2.20). 

FIGURE 17.21 In spontaneous nuclear fission, the oscillations of the heavy nucleus in effect tear the nucleus apart, thereby forming two or more smaller nuclei of similar mass. This picture is based on the liquid drop model of the nucleus. 

Spontaneous nuclear fission takes place when the natural oscillations of a heavy nucleus cause it to break into two nuclei of similar mass (Fig. 17.21). In terms of the liquid drop model, we can think of the nucleus as distorting into a dumbbell shape and then breaking into two smaller droplets. An example is the disintegration of americium-244 into iodine and molybdenum ... 

Fig. 12.7 (a) Pressure variations obtained from the single-nephron model fora = 32 and T = 16 s. (b) Corresponding phase plot. With these parameters the model displays chaotic oscillations resembling the behavior observed for spontaneously hypertensive rats [13]. 

Thus starting from the simplest pseudo-spin model of proton dynamics in the hydrogen bond, we have studied a possibility of spontaneous tunnel oscillations of the polarization of a short hydrogen-bonded chain. The phenomenon can be affected by two reasons (a) the coherent motion of protons along the hydrogen bonds and (b) the coherent motion of protons around heavy backbone atoms. 

The natural line width of the spectral line is a significant result of the dissipative quantum process which accompanies the spontaneous emission of an atom. We will treat this emission process in a dissipative two-state model. We consider the two states of the atom as the zeroth and the first occupation number state of a linearly damped oscillator. In this case, the spontaneous emission of a photon is the consequence of the transition from the first occupations number state to the equilibrium state of the damped oscillator. In this model, the spectrum density of the emitted photon follows from Equation (92)... 

These processes correspond, respectively, to spontaneous reproduction of the prey, reproduction of the predator mediated by prey consumption, and predator death. In fact one obtains Eqs. (3.68)-(3.69) with a2 = oi, but this is just a consequence of measuring the amount of chemicals in moles or molecules, and it expresses the fact that in (3.70) the consumption of each molecule of P leads to a new molecule of Z. Changing to other units such as mass makes the coefficients of the nonlinear terms to become different as in (3.68)-(3.69). In biological settings, common units for either P and Z are either mass or number of individuals, and in these units it is no longer true that ingestion of one unit of P produces one unit of new Z. Lotka presented his model as a chemical scheme that would produce persistent chemical oscillations. 

The first models for the mitotic oscillator specifically based on the interaction between cyclin and cdc2 kinase also relied on positive feedback. The experimental basis for autocatalytic regulation of cdc2 kinase stems primarily from observations showing that catalytic amounts of active MPF promote the transition from inactive to active MPF, which consists of a complex between cyclin and the active form of cdc2 kinase spontaneous activation of this transition, however, does not normally... 

In the model proposed by Hyver Le Guyader (1990), two systems of equations are considered. In the first version, inactive p34 (i.e. cdc2 kinase) transforms into active p34, either spontaneously or in an auto-catalytic manner active p34 then combines with cyclin to yield active MPF. This situation is described by four differential equations, of a polynomial nature, in which the highest nonlinearities are of the quadratic type. In a second version of this model, governed by three kinetic equations of a similar form, the authors consider the effect of an activation of MPF by MPF itself as well as cyclin, and show that oscillations develop when the degradation of cyclin is brought about by the formation of a complex between cyclin and MPF. That study was the first to show the occurrence of sustained oscillations in a model based on the interactions between cyclin and cdc2 kinase. The type of kinetics considered for these interactions remained, however, remote from the actual kinetics of phosphorylation-dephosphorylation cycles. 

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