Weakly nonlinear system

D. D. Weiner, J. F. Spina, Sinusoidal Analysis and Modelling of Weakly Nonlinear Systems, Van Nostrand Reinhold, New York, NY, 1980. 

The method presented in this chapter is based on nonlinear frequency response and the concept of higher-order frequency response functions, which have been proven as very convenient tools for analyzing weakly nonlinear systems. The basics for their application lay in the facts that ... 

A model of a weakly nonlinear system can be replaced by an indefinite sequence of linear models of different orders. In the frequency domain, these linear models are defined as frequency response functions (FRFs) of different orders, which can be estimated from different harmonics of the FR. 

On the other hand, the response of a weakly nonlinear system, for which the system nonlinearity has a polynomial form (or can be developed in a Taylor series) can be represented in the form of a Volterra series ... 

For weakly nonlinear systems, the contributions of the higher harmonics and higher FRFs decrease with the increase of their order. Different harmonics of the output can be estimated directly by harmonic analysis of the output signal. On the other hand, as can be seen from Equation (11.13) to Equation (11.16), the first-order function Gi(cd) corresponds to the dominant term of the first harmonic, the second-order functions G2(dominant terms of the second harmonic and the DC component, respectively, the third-order function Gs(ft), ft), to) to the dominant term of the third harmonic, etc. This fact enables estimation of different FRFs from the harmonics of the output obtained for different input amplitudes, as shown by Lee [53]. 

For nonlinear systems we have introduced the Newton-Picard method as an alternative to methods more commonly used in chemical engineering literature. It is found that for the weakly nonlinear System C, Broyden s method is fastest and that the Newton-Picard method needs a CPU time equal to the dynamic simulation. For the more strongly nonlinear System D, Broyden s method is again the fastest but here the Newton-Picard also reduces the CPU time as compared to the dynamic simulation by a factor two. The main drawback of the Newton-Picard method is the lengthy first iteration, which arises from the construction of the Jacobian in order to determine a basis for the slowly converging subspace. 

Wherever applicable, we use the method of averaging [55, 56] to study the behavior of the equations of motion as a weakly nonlinear system. There are a few variations on the basic theorem for the first-order periodic averaging [56-58]. A slightly modified version of the theorem proven in [57] - suitable for otur subsequent analyses - is presented below which establishes the error estimate of the solution of the averaged system, with respect to the that of the original differential equation. 

The method of first-order averaging applies to weakly nonlinear systems. This requirement is fulfilled here since friction and damping coefficients are small quantities. The first step is to convert (4.7) to a nondimensional form. Let... 

This information is used for the following formal weakly nonlinear analysis of the bifurcating solution in the system (6.3.9)-(6.3.15). 

Weakly coupled systems of nonlinear elliptic boundary value problems (with K. Zygour-akis). Nonlin. Anal. 6, 555-569 (1982). 

This is normally a fairly weak interaction that will only take place in strongly nonlinear systems. 

Composite Sorption Magnitude and Isotherm Nonlinearity. Accurate assessment of the extent to which the global isotherm for a system is nonlinear is important for accurate portrayal of sorption processes in that system. From a practical point of view, the extrapolation of linear approximations of weakly nonlinear or near-linear sorption isotherms to concentration ranges beyond which they are valid can result in significant errors in projections of contaminant fate and transport (1). From a conceptual point of view, observations of isotherm nonlinearity over specific concentration ranges may be employed in conjunction with models such as the DRM to probe and evaluate the extent to which multiple sorption mechanisms are operative in a particular system. 

The convolution defined in (4.2.1) is a linear operation applied to the input function x(t). Nonlinear systems transform the input signal into the output signal in a nonlinear fashion. A general nonlinear transformation can be described by the Volterra series. It forms the basis for the theory of weakly nonlinear and time-invariant systems [Marl, Schl] and for general analysis of time series [Kanl, Pril]. In quantum mechanics, the Volterra series corresponds to time-dependent perturbation theory, and in optics it leads to the definition of nonlinear susceptibilities [Bliil]. 

First, we consider a 1+1 system (2D film with ID surface) since some important features of the nonhnear evolution of the film instability can be studied in this case. In order to study the possibility of pattern formation one first performs a weakly nonlinear analysis of stationary solutions of eq.(5) near the instabihty threshold. A characteristic feature of the system described by eq.(5) is the presence of the zero mode, cr = 0, corresponding to k = 0 (see Fig.2) and associated with the conservation of mass. The nonlinear interaction between the zero mode and the unstable mode can substantially affect the system behavior near the instabihty threshold [15, 16] and must be taken into account in weakly nonlinear analysis. 

Being generally weakly nonlinear, adsorption systems make good candidates for investigation by means of nonlinear FR. As it will be shown further in the text, the nonlinear FR method enables identification of the kinetic mechanism and estimation of both equilibrium and kinetic parameters from the same experimental data. 

Recently, the one-dimensional evaporation and condensation problem between the two plane interfaces has been studied in [4] and [5] under the same assumptions as made above the former paper for the case of moderate values of the concentration of the inert gas and the latter for the case of its small values. Unfortunately, these two cases cannot be combined into one single form covering the whole range of the concentration of the inert gas, even if the problem is of weakly nonlinear nature. The reason for this is that, as was pointed out in [4,5], when the concentration of inert gas becomes small in systems... 

Limitations to Relating the Polymer Response to Molecular and Oligomeric Properties, in many cases, the NLO response of the polymer can be derived entirely from the response of the molecular NLO chromophore units and their concentration in the polymer as a mediiun. The assiunption that the system is weakly interacting is justified if the nonlinearity is a linear function of molecular concentration (106,107). For large concentrations of molecules in a polymer matrix, the assumption of a weakly interacting system may break down, however, as dimerization or aggregation become important (108). 

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