Nonlinear phenomena linearized equations

Remark. From the linear integro-differential equation for P(y, t) we have derived a nonlinear equation for y(t). Thus the essentially linear master equation may well correspond to a physical process that in the laboratory would be regarded as a nonlinear phenomenon inasmuch as its macroscopic equation is nonlinear. This is not paradoxical provided one bears in mind that the distinction between linear and nonlinear is defined for equations. It is wrong to apply it to a physical phenonemon, unless one has agreed upon a specific mathematical description of it. Newton s equations for the motion of the planets are nonlinear, but the Liouville equation of the solar system is linear. This connection between linear and nonlinear equations is not a matter of approximation the linear Liouville equation is rigorously equivalent with the nonlinear equations of motion of the particles. Generally any linear partial... 

Instability is a nonlinear phenomenon. However, the dynamic behavior of nuclear reactors can be assumed to be linear for small perturbations around steady-state conditions. This allows the reactor stability to be studied and the threshold of instability in nuclear reactors to be predicted by using a linear model and solving linearized equations. Linear stability analyses in the frequency domain have been... 

L. Mandelstam and N. Papalexi performed an interesting experiment of this kind with an electrical oscillatory circuit. If one of the parameters (C or L) is made to oscillate with frequency 2/, the system becomes self-excited with frequency/ this is due to the fact that there are always small residual charges in the condenser, which are sufficient to produce the cumulative phenomenon of self-excitation. It was found that in the case of a linear oscillatory circuit the voltage builds up beyond any limit until the insulation is ultimately punctured if, however, the system is nonlinear, the amplitude reaches a stable stationary value and oscillation acquires a periodic character. In Section 6.23 these two cases are represented by the differential equations (6-126) and (6-127) and the explanation is given in terms of their integration by the stroboscopic method. 

The apparent dispersion coefficient in Equation 10.8 describes the zone spreading observed in linear chromatography. This phenomenon is mainly governed by axial dispersion in the mobile phase and by nonequilibrium effects (i.e., the consequence of a finite rate of mass transfer kinetics). The band spreading observed in preparative chromatography is far more extensive than it is in linear chromatography. It is predominantly caused by the consequences of the nonlinear thermodynamics, i.e., the concentration dependence of the velocity associated to each concentration. When the mass transfer kinetics is fast, the influence of the apparent axial dispersion is small or moderate and results in a mere correction to the band profile predicted by thermodynamics alone. 

Evaporated PDA(12-8) film was used as a nonlinear optical medium in a layered guided wave directional coupler. The directional coupling phenomenon happens in two adjacent waveguide by periodical energy transfer. The theory of linear directional coupler was exactly established [11]. It can be reduced to coupled mode equations ... 

Instead of using this equation, in the literature, there are few models proposed by which the frequency or Strouhal number of the shedding is fixed. Koch (1985) proposed a resonance model that fixes it for a particular location in the wake by a local linear stability analysis. Upstream of this location, flow is absolutely unstable and downstream, the flow displays convective instability. Nishioka Sato (1973) proposed that the frequency selection is based on maximum spatial growth rate in the wake. The vortex shedding phenomenon starts via a linear instability and the limit cycle-like oscillations result from nonlinear super critical stability of the flow, describ-able by Eqn. (5.3.1). 

Another point to be noted is the fact that the mesophase temperature ranges, i.e., ATlq= Ti - Tm, are broader for the random copolyesters than for the ordered sequence ones. This phenomenon is particularly evident when a copolyester is composed of linear and nonlinear units, as shown for the copolymers of Equations 20 and 22. 

In anal3rtical chemistry, developii a calibration curve or modelling a phenomenon often requires the use of a mathematical fitting procedure. Probably the most familiar of these procedures is linear least-squares fitting [1]. Criteria other than least-squares for defining the best fit have been developed for linear parameters when the data possibly contain outliers [2,3]. Sometimes, the model equation to be fit is nonlinear in the parameters. This requires appeal to other fitting methods [4]. 

For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. Extended nonequilibrium thermodynamics is mainly concerned with the nonlinear region and deriving the evolution equations with the dissipative flows as independent variables, beside the usual conserved variables. Typical nonequilibrium variables, such as flows, gradients of intensive properties may contribute to the rate of entropy production. When the relaxation time of these variables differs from the observation time they act as constant parameters. The phenomenon becomes complex when the observation time and the relaxation time are of the same order, and the description of system requires additional variables. Polymer solutions are highly relevant systems for analyses beyond the local-equilibrium theory. 

The nonlinear study of bifurcations of the elastic equilibrium of a straight bar involves, in a way, a change of the physical point of view, mostly due to the mathematical difficulties related to the direct approach of the Bernoulli-Euler (B.-E.) equation. This aspect gave rise to various models describing the same phenomenon, such as Kirchhoff s pendulum analogy [1], as well as to different methods of calculus, such as Thompson s potential energy method [2], [3]. In this paper, we use the linear equivalence method (LEM) to a B.-E. type model, thus deducing an approach for the critical and postcritical behavior of the cantilever bar. 

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