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Appendix H
Response of Nonlinear Systems in the Linear Regime
In Blandford & McKee (1982), linear systems were analyzed in the linear regime. In this appendix a formalism is developed for analyzing nonlinear systems in the linear regime. Though the solution to this problem is a straightforward and probably necessary prerequisite for any comprehensive understanding of variability in AGN, it was not correctly obtained or applied in other works regarding AGN variability. We shall find that within the linear regime the analysis in Blandford & McKee (1982) is inadequate for general nonlinear systems.

Let us consider the generic system described in § F .3.1. Using the notation of § F .3.1 for

, and

, there is no reason that

should generally hold, and one may be forced to employ a fully nonlinear analysis method to accurately describe the system. However, let us assume here that the variations in

are sufficiently smaller than its mean, in which case, provided

is a smooth function, it can be approximated with

(H.1)

where the dimensionless, “asymptotic gain"

of changes in

for small and slow changes in

is defined by

(H.2)

where

is the error due to nonzero second order derivatives in

. The asymptotic gain has elsewhere been termed the “responsivity" (Krolik et al. 1991; Goad, O’Brien, & Gondhalekar 1993). Here it is an operator to distinguish between the various gains with the different “output" and “input" functions that will be required, though note that it is independent of the normalizations of these functions.

Let us extend the definition of gain by allowing a dependence upon the type of input signal. Consider a time-dependent local ionizing continuum flux or a Fourier component of it such as

. A dimensionless frequency-dependent gain or transfer function

with respect to

can then be defined as

(H.3)

Letting

denote the Fourier transform of

, etc., gives

(H.4)

and an analog of the Fourier transform of equation ( H.1 ) for a frequency-dependent gain,

(H.5)

With this notation,

is the dimensionless ratio of the amplitudes of variations of

, while -Im

is the delay in response.

Similarly, the asymptotic gain is Re

, while the “instantaneous component of the gain" is Re

Equation ( H.5 ) yields in the time domain

(H.6)

where we define the inverse Fourier transform of the gain of the output

with respect to the input

as the normalized linearized response function, which is

(H.7)

Here the lack of a caret denotes that the normalized response function has scaling other than that given to it by the inverse Fourier transform. Note that the integral of the linearized response function is just the asymptotic gain, which is only unity for actual linear systems.

Linearization is not always advantageous. In some cases, including those obeying equation ( F.1 ), the exact solution can easily be obtained from

(H.8)

where

(H.9)

is the response function of an “input-modified" system.

However, linearization can be quite useful, in some cases it allows complex systems to be accurately described by a single equivalent response function, which can drastically reduce the simulation time. Such is the case wherein one is interested in obtaining an observable quantity of a system with many clouds, wherein the convolution of

with a spatial linear response function must be evaluated. For instance, consider a hypothetical system wherein a physical cloud property

lags property

on a time scale

, while property

has a “direct" lag of

. The exact expression for

has two nested integrals over lags. However, upon linearization the transfer function of

for small variations in

(H.10)

or alternatively

(H.11)

which when applied (in eq. [ H.6 ]) requires only a single integral over lag. For future reference, note that when

, the two gain factors are “separable" from one another, i.e. for a restricted range of excitation frequencies one of the gain factors can be treated as a constant.

In this section it has been shown (eq. [ H.9 ]) that there is a “correction factor" of

in the expression for the “gain-corrected response function." Previous works (e.g., Blandford & McKee 1982) assumed that the systems themselves are linear, which is equivalent to assuming correction factors of unity. Some of the problems with making this assumption are pointed out in Goad, O’Brien, & Gondhalekar (1993) as well as § F .4 of this work. Note that the correction factors differ from unity in equation ( H.2 ) in nonlinear systems even if the perturbations are arbitrarily small and equation ( H.6 ) accurately describes the system.

Partially accounting for even higher order corrections due to nonlinearity is also possible within the linear regime and, in fact, is important for accurate interpretation of fits of linear models to nonlinear systems. If the variations are not infinitesimal, the above equations do not necessarily yield the “optimal" fit that would have obtained using real variability data. For instance, consider the case wherein

. The better-fitting optimal average for a sinusoidal-like input is the first Fourier coefficient of

(H.12)

Similarly, the observable asymptotic gain is approximately

(H.13)

where the inequalities can be removed only for the

case.

The sign in the Fourier transform used here, though different from that in several references (e.g., Blandford & McKee 1982), minimizes differences with the Laplace transform, which offers certain advantages in dealing with this type of problem.