Oscillations and excitability

MA and IMA are the malonic and the iodomalonic acid, respectively. For the CDIMA case in the usual experimental conditions, it is appropriate to assume that the only dynamical variables are [I-] and [C1C 2 ], the other concentrations staying essentially constant. In addition, the two first reactions in (3.44) follow the law of mass action, whereas the third one proceeds at a rate which is empirically given 

With these approximations, and after proper rescaling to render the variables dimensionless, an appropriate model for the CDIMA reaction is 

Despite the importance of the chlorite-iodide systems in the development of nonlinear chemical dynamics in the 1980s, the Belousov-Zhabotinsky(BZ) reaction remains as the most intensively studied nonlinear chemical system, and one displaying a surprising variety of behavior. Oscillations here were discovered by Belousov (1951) but largely unnoticed until the works of Zhabotinsky (1964). Extensive description of the reaction and its behavior can be found in Tyson (1985), Murray (1993), Scott (1991), or Epstein and Pojman (1998). There are several versions of the reaction, but the most common involves the oxidation of malonic acid by bromate ions BrOj in acid medium and catalyzed by cerium, which during the reaction oscillates between the Ce3+ and the Ce4+ state. Another possibility is to use as catalyst iron (Fe2+ and Fe3+). The essentials of the mechanisms were elucidated by Field et al. (1972), and lead to the three-species model known as the Oregonator (Field and Noyes, 1974). In this 

The first two reactions describe the consumption of bromide Br, whereas the last three ones model the buildup of HBrC 2 and Ce4+ that finally leads to bromide recovery, and then to a new cycle. By assuming that the bromate concentration [A] remains constant as well as [B], and noting that P enters only as a passive product of the dynamics, the law of mass action leads to 

We follow Tyson (1985) and bring the model to dimensionless form by rescaling time and concentrations  

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HOPF BIFURCATIONS, THE GROWTH OF SMALL OSCILLATIONS, RELAXATION OSCILLATIONS, AND EXCITABILITY... 

Goldbeter, A. (1980) Models for oscillations and excitability in biochemical systems. 

Noise-induces oscillations and excitable systems. These systems represent a separate class exhibiting synchronization. Without noise, these systems are stable and do not oscillate. In the presence of noise the dynamics is similar to the dynamics of noisy self-... 

Similar complex oscillatory phenomena have been observed in a closely related model containing two regulated enzyme reactions coupled in a different manner (Li, Ding Xu, 1984). An additional indication of the generality of the results obtained in the multiply regulated biochemical system is given by the study of the model for the synthesis of cAMP in Dictyostelium amoebae. In addition to simple periodic oscillations and excitability (see chapter 5), this realistic model based on experimental observations also predicts the appearance of more complex oscillatory phenomena in the form of birhythmicity, bursting and chaos (chapter 6). 

Goldbeter, A. 1980. Models for oscillations and excitability in biochemical systems. In Mathematical Models in Molecular and Cellular Biology. L. A. Segel, ed. Cambridge Univ. Press, Cambridge, pp. 248-91. 

Linear stability analysis provides one, rather abstract, approach to seeing where spatial patterns and waves come from. Another way to look at the problem has been suggested by Fife (1984), whose method is a bit less general but applies to a number of real systems. In Chapter 4, we used phase-plane analysis to examine a general two variable model, eqs. (4.1), from the point of view of temporal oscillations and excitability. Here, we consider the same system, augmented with diffusion terms a la Fife, as the basis for chemical wave generation ... 

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... 

As already mentioned, electronically resonant, two-pulse impulsive Raman scattering (RISRS) has recently been perfonned on a number of dyes [124]. The main difference between resonant and nom-esonant ISRS is that the beats occur in the absorption of tlie probe rather than the spectral redistribution of the probe pulse energy [124]. These beats are out of phase with respect to the beats that occur in nonresonant ISRS (cosinelike rather tlian sinelike). RISRS has also been shown to have the phase of oscillation depend on the detuning from electronic resonance and it has been shown to be sensitive to the vibrational dynamics in both the ground and excited electronic states [122. 124]. 

Fig. 4.65. Different spectral features of tanf for a strong model oscillator at 1000 cm" on a metal substrate. The TO mode (1000 cm" ), Berreman effect (1050 cm" ), and excitation ofa surface wave (1090 cm" ) are seen for different 1150 thicknesses - 1, 5, 10, 50,100, 500, and 1000 nm.

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. 

Figure 10.7 The population of excited energy states relative to that of the ground state for the CO molecule as predicted by the Boltzmann distribution equation. Graph (a) gives the ratio for the vibrational levels while graph (b) gives the ratio for the rotational levels. Harmonic oscillator and rigid rotator approximations have been used in the calculations. The dots represent ratios at integral values of v and 7. which are the only allowed values.

The main goal of this report is to present a phenomenon of highly general nature manifested in various dynamical systems. We present the occurrence of peculiar quantization by the parameter of intensity of the excited oscillations and we show that given unchanging conditions it is possible to excite oscillations with a strictly defined discrete set of amplitudes the rest of the amplitudes being forbidden . The realization of oscillations with a specific amplitude from the permitted discrete set of amplitudes is determined by the initial conditions. The occurrence of this unusual property is predetermined by the new general initial conditions, i.e. the nonlinear action of the external excited force with respect to the coordinate of the system subjected to excitation. 

The performed analysis shows that the continuous wave having a frequency much larger than the frequency of a given oscillator can excite in it oscillations with a frequency close to its natural frequency and an amplitude belonging to a discrete set of possible stable amplitudes. 

The method developed of entering energy in oscillation processes and the excitation of quantized oscillations in dynamic macro-systems finds and will find in the future applications in the solving of important practical problems in the creation of new methods and mechanisms for the excitation and the sustaining of continuous oscillations and different energy transformations which could be grouped in the following way ... 

Damgov, V. N. Quantized Oscillations and Irregular Behaviour of a Class of Kick-Excited Self-Adaptive Dynamical Systems. Progress of Theoretical Physics Suppl, Kyoto, Japan, No. 139, P. 344 (2000)... 

Damgov, V. N. and Popov I. Discrete Oscillations and Multiple Attractors in Kick-Excited Systems. Discrete Dynamics in Nature and Society, Vol. 4, P. 99 (2000)... 

Fig. 2 Surface plasmon resonance (SPR) principle. Surface plasmons are excited by the light energy at a critical angle (9) causing an oscillation and the generation of an evanescent wave. Under this condition a decrease in the reflected light intensity is observed. The angle 9 depends on the dielectric medium close to the metal surface and therefore is strongly affected by molecules directly adsorbed on the metal surface. This principle allows the direct detection of the interaction of the analyte and the antibody...

The decrease in reflectivity at the SPR angle (2sp) is due to absorption of the incident light at this particular angle of incidence. At this angle the incident light is absorbed and excites electron oscillations on the metal surface. 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

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