Oscillating systems

Some autocatalytic chemical reactions such as the Brusselator and the Belousov-Zhabotinsky reaction schemes can produce temporal oscillations in a stirred homogeneous solution. In the presence of even a small initial concentration inhomogeneity, autocatalytic processes can couple with diffusion to produce organized systems in time and space. 

A well-known oscillating reaction scheme is the Brusselator system, representing a trimolecular model given by 

The net reaction is A I B - E I F. This reaction scheme has been developed by the Brussels School of Thermodynamics, and consists of a trimolecular collision and an autocatalytic step. This reaction may take place in a well-stirred medium leading to oscillations, or the diffusions of the components A and B may be considered. In the latter case, the system may produce Turing structures. 

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Zaikin A N and Zhabotinsky A M 1970 Concentration wave propagation in two-dimensional liquid-phase self-oscillating system Nature 225 535-7... 

The back-aiid-forth motion of a child s swing is an example of an oscillating system, oscillator, for short. Set into motion, these systems oscillate at a frequency determined by properties of the oscillator. For exam-... 

Autocatalysis can cause sustained oscillations in batch systems. This idea originally met with skepticism. Some chemists believed that sustained oscillations would violate the second law of thermodynamics, but this is not true. Oscillating batch systems certainly exist, although they must have some external energy source or else the oscillations will eventually subside. An important example of an oscillating system is the circadian rhythm in animals. A simple model of a chemical oscillator, called the Lotka-Volterra reaction, has the assumed mechanism ... 

Since the idea that all matters are composed of atoms and molecules is widely accepted, it has been a long intention to understand friction in terms of atomic or molecular interactions. One of the models proposed by Tomlinson in 1929 [12], known as the independent oscillator model, is shown in Fig. 13, in which a spring-oscillator system translates over a corrugating potential. Each oscillator, standing for a surface atom, is connected to the solid substrate via a spring of stiffness k, and the amplitude of the potential corrugation is. ... 

Electrochemical oscillation during the Cu-Sn alloy electrodeposition reaction was first reported by Survila et al. [33]. They found the oscillation in the course of studies of the electrochemical formation of Cu-Sn alloy from an acidic solution containing a hydrosoluble polymer (Laprol 2402C) as a brightening agent, though the mechanism of the oscillatory instability was not studied. We also studied the oscillation system and revealed that a layered nanostructure is formed in synchronization with the oscillation in a self-organizational manner [25, 26]. 

The oscillation of membrane current or membrane potential is well-known to occur in biomembranes of neurons and heart cells, and a great number of experimental and theoretical studies on oscillations in biomembranes as well as artificial membranes [1,2] have been carried out from the viewpoint of their biological importance. The oscillation in the membrane system is also related to the sensing and signal transmission of taste and olfaction. Artificial oscillation systems with high sensitivity and selectivity have been pursued in order to develop new sensors [3-8]. 

The oscillations observed with artificial membranes, such as thick liquid membranes, lipid-doped filter, or bilayer lipid membranes indicate that the oscillation can occur even in the absence of the channel protein. The oscillations at artificial membranes are expected to provide fundamental information useful in elucidating the oscillation processes in living membrane systems. Since the oscillations may be attributed to the coupling occurring among interfacial charge transfer, interfacial adsorption, mass transfer, and chemical reactions, the processes are presumed to be simpler than the oscillation in biomembranes. Even in artificial oscillation systems, elementary reactions for the oscillation which have been verified experimentally are very few. 

The oscillation at a liquid liquid interface or a liquid membrane is the most popular oscillation system. Nakache and Dupeyrat [12 15] found the spontaneous oscillation of the potential difference between an aqueous solution, W, containing cetyltrimethylammo-nium chloride, CTA+CK, and nitrobenzene, NB, containing picric acid, H" Pic . They explained that the oscillation was caused by the difference between the rate of transfer of CTA controlled by the interfacial adsorption and that of Pic controlled by the diffusion, taking into consideration the dissociation of H Pic in NB. Yoshikawa and Matsubara [16] realized sustained oscillation of the potential difference and pH in a system similar to that of Nakache and Dupeyrat. They emphasized the change of the surface potential due to the formation and destruction of the monolayer of CTA" Pic at the interface. It is... 

Nelson, R.W. J.A. Schur. (1980). Assessment of effectiveness of geologic oscillation systems PATHS groundwater hydrologic model. Battelle, Pacific Northwest Laboratory, Richland, WA. 

McCarley, R. W. Massaquoi, S. G. (1986). A limit cycle mathematical model of the REM sleep oscillator system. Am. J. Physiol 251, R1011-R29. 

A. N. Malakhov, Fluctuations in Self-Oscillating Systems, Science, Moscow, 1968, in Russian. 

When this oscillating system is perturbed by a pulse of an analyte such as vitamin B6, it undergoes a change in its amplitude or period (amplitude for this vitamin) that is proportional to the concentration and can be used to construct a calibration plot. 

Figure 11 shows typical CL oscillating responses of this system as perturbed by vitamin B6 pulses, which decrease the oscillation amplitude. Arrowheads indicate the times at which analyte pulses were introduced. Zone A corresponds to the oscillating steady state zone B to the response of the oscillating system to vitamin B6 perturbations and zone C to the recovery following each perturbation (second response cycle), which was the measured parameter. This... 

The excitation of oscillations with a quasi-natural system frequency and numerous discrete stationary amplitudes, depending only on the initial conditions (i.e. discretization of the processes of absorption by the system of energy, coming from the high-frequency source). A new in principle property is the possibility for excitation of oscillations with the system s natural frequency under the influence of an external high-frequency force on unperturbed linear and conservative linear and non-linear oscillating systems. 

The phenomenon of continuous oscillation excitation with an amplitude belonging to a discrete set of stationary amplitudes has been demonstrated on the basis of a common model - an oscillator under wave influence. It is shown that the conditions necessary for the manifestation of this phenomenon are realized in a natural way in an oscillator system interacting with a continuous fall wave. 

The opportunity of creation of oscillating system in the structure with braking potential field, which were made by the distributed potentials and accelerating potential, is shown. The particle in such the field will make fourfold process of braking and accelerating. 

First model for oscillating system was proposed by Volterra for prey-predator interactions in biological systems and by Lotka for autocatalytic chemical reactions. Lotka s model can be represented as... 

Chirikov, B. V. (1979), A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep. 52, 263. 

Voth, G. A. (1986), On the Relationship of Classical Resonances to the Quantum Mechanics of Coupled Oscillator Systems, J. Phys. Chem. 90, 3624. 

Wx or Wf Let us suppose that/is the control parameter. In this case the JE and GET, Eqs. (40) and (41), are valid for the work, Eq. (96). How large is the error that we make when we apply the JE using Wx instead This question has been experimentally addressed by Ciliberto and co-workers [97, 98], who measured the work in an oscillator system with high precision (within tenths of fesT). As shown in Eq. (99), the difference between both works is mainly a boundary term, A xf). Fluctuations of this term can be a problem if they are on the same order as fluctuations of Wx itself. For a harmonic oscillator of stiffness constant equal to k, the variance of fluctuations mfx are equal to k8(x ), that is, approximately on the order of k T due to the fluctuation-dissipation relation. Therefore, for experimental measurements that do not reach such precision, Wx or Wf is equally good. 

An oscillating system is formed by the interplay of the three protein classes and the activity of this system makes up the specific biochemical functions of the individual phases of the cycle. The activity of the cychn-dependent protein kinases (CDKs) is central to the oscillating system. These create a signal that initiates downstream biochemical processes and thus determines the individual phases of the cycle. CDK activity is also the starting point for intrinsic and external control mechanisms. 

Following an idea of Zhabotinsky [4] that chemical oscillating systems could expediently be treated as a black box and that the relevant mathematical semi-phenomenological model has to focus on the basic reactions only neglecting those less important, Vasiliev, Romanovsky and Yakhno [5] suggested a concept of the basic model. These simplified models of an extended active medium could be obtained either by a reduction of pre-existing... 

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. 

A quasi-periodic solution to a system of ODEs is characterized by at least two frequencies that are incommensurate (their ratio is an irrational number) (Bohr, 1947 Besicovitch, 1954). Several such frequencies may be present on high-order tori, but for the two-dimensional forced systems we examine, we may have no more than two distinct frequencies (a two-torus, T2). A quasi-periodic solution is typically bora when a pair of complex conjugate FMs of a periodic trajectory leave the unit circle at some angle , where /2ir is irrational. Such a solution is also expected when we periodically perturb an autonomously oscillating system with a frequency incommensurate to its natural frequency. 

The detailed study of these bifurcations (which we shall describe in subsequent publications) is important for several reasons. It will be helpful in understanding some of the mechanisms that synchronize chemical oscillators, by breaking the pattern of higher-dimensional tori that exist when they are weakly coupled. It also provides a very convenient setting for studying on practically any oscillating system phenomena that are, by comparison, less frequently observed in autonomous systems. 

Both transducers are self-excited oscillating systems (usually piezoelectric crystals— see Section 6.3.1) which act alternately as transmitters and receivers, the receiving pulses being used to trigger the transmitted pulses in a feedback arrangement. A pulse from transmitter Tt is directed downstream over a path length Lp to transducer T2 (Fig. 6.3). The downstream elapsed time is ... 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

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