Linear dynamic phenomena

If the peaks that exhibit detector overload do not need to be quantitated, then we can just ignore this overload phenomenon. Otherwise we should inject a smaller amount or use a detector (or detector setting) with a larger linear dynamic range. 

Notice that the normal form is simpler than either of the examples we have considered in that no quadratic term exists. This is because quadratic non-linearities are not important for the existence of bistability it is the cubic term that is crucial for this particular phenomenon. Thus, the normal form retains only the features necessary for the universal dynamical phenomenon in question. 

Actually, a similar phenomenon to this steady-state ion suppression in shotgun lipidomics is also present in any method developed with LC-MS for quantitative analysis of lipid mixtures. For example, if it is intended to quantify a species of a minor lipid class in the presence of other abundant species [24], the amount of total lipids that can be loaded onto a column are capped by the upper limit of the linear dynamic range of the most abundant species in the mixture under the experimental conditions. The loaded amount of total lipids to expand the linear dynamic range of the minor component in the method cannot be increased greatly if there is a need for quantification of major components as well. Of course, the minor species can be analyzed separately with a pre-isolated fraction or with a saturated concentration of the abundant species to increase the dynamic range for quantification of the minor components. 

The chemical transformation of a rock by a pervading convecting fluid in disequilibrium with it is modelled by a non-linear partial differential equation of hyperbolic type. Starting from a continuous concentration profile, the solution may become discontinuous within a finite time in some cases. We show that this is due to the instability, in the sense of the second principle of thermodynamics, of a range of the possible concentrations with respect to this dynamical phenomenon. 

The bubble formed in stable cavitation contains gas (and very small amount of vapor) at ultrasonic intensity in the range of 1-3 W/cm2. Stable cavitation involves formation of smaller bubbles with non linear oscillations over many acoustic cycles. The typical bubble dynamics profile for the case of stable cavitation has been shown in Fig. 2.3. The phenomenon of growth of bubbles in stable cavitation is due to rectified diffusion [4] where, influx of gas during the rarefaction is higher than the flux of gas going out during compression. The temperature and pressure generated in this type of cavitation is lower as compared to transient cavitation and can be estimated as ... 

An intense femtosecond laser spectroscopy-based research focusing on the fast relaxation processes of excited electrons in nanoparticles has started in the past decade. The electron dynamics and non-linear optical properties of nanoparticles in colloidal solutions [1], thin films [2] and glasses [3] have been studied in the femto- and picosecond time scales. Most work has been done with noble metal nanoparticles Au, Ag and Cu, providing information about the electron-electron and electron-phonon coupling [4] or coherent phenomenon [5], A large surface-to-volume ratio of the particle gives a possibility to investigate the surface/interface processes. 

What will happen further away from equilibrium when the linear relationship between cause and effect breaks down is clear from the above example but for other instances is the subject of much research and speculation. With nonlinear relationships, the scope of phenomena becomes nearly unpredictable. Nonlinear dynamics [9,10] may well provide the clue to the phenomenon of macroscopic complexity [11], a rapidly expanding field of science, defined by some [12] as quickly becoming a field of "perplexity."... 

In Section IV.B.4 we have shown that the quadratic dynamic susceptibilities of a superparamagnetic system display temperature maxima that are sharper than those of the linear ones. If the maximum occurs as well at the temperature dependence of the signal-to-noise ratio, this should be called the nonlinear stochastic resonance. However, before discussing this phenomenon, one has to define what should be taken as the signal-to-noise ratio in a nonlinear case. 

After the kinetic model for the network is defined, a simulation method needs to be chosen, given the systemic phenomenon of interest. The phenomenon might be spatial. Then it has to be decided whether in addition stochasticity plays a role or not. In the former case the kinetic model should be described with a reaction-diffusion master equation [81], whereas in the latter case partial differential equations should suffice. If the phenomenon does not involve a spatial organization, the dynamics can be simulated either using ordinary differential equations [47] or master equations [82-84]. In the latter case but not in the former, stochasticity is considered of importance. A first-order estimate of the magnitude of stochastic fluctuations can be obtained using the linear noise approximation, given only the ordinary differential equation description of the kinetic model [83-85, 87]. 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

Autocatalysis is a distinctive phenomenon while in ordinary catalysis the catalyst re-appears from the reaction apparently untouched, additional amounts of catalyst are actively produced in an autocatalytic cycle. As atoms are not interconverted during chemical reactions, this requires (all) the (elementary or otherwise essential) components of autocatalysts to be extracted from some external reservoir. After all this matter was extracted, some share of it is not introduced in and released as a product but rather retained, thereafter supporting and speeding up the reaction(s) steadily as amounts and possibly also concentrations of autocatalysts increase. At first glance, such a system may appear doomed to undergo runaway dynamics ( explosion ), but, apart from the limited speeds and rates of autocatalyst resupply from the environment there are also other mechanisms which usually limit kinetics even though non-linear behavior (bistability, oscillations) may not be precluded ... 

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