Chapter II, this is applied to the general piecewise linear system, and a formula derived for the spectral density. For example, the spectral density can be obtained either directly or using the FP equation. Several special functions are used in Chapters III-V. The notations used are Abramowitz and Stegun [l ], except for the parabolic cylinder function, where the more familiar notation D)z) is used.
CHAPTER I
WHITE NOISE AND STOCHASTIC DIFFERENTIAL EQUATIONS
Before interpreting these conditions in terms of the behavior of sampling paths, some definitions will be given. It is thus seen that the distinguishing feature of a natural boundary of type (c) is that there is a positive probability that a path, starting from any point x, ends up in any given neighbor hood of the. Note that any of the irregular points shown can be omitted and P will still be independent of x.
CHAPTER II
SOLUTION OF THE FOKKER- PLANCK EQUATION
0 = xi, no problem is experienced; due to the continuity of p(x, s lx . 0) with respect to x. 0, this can be regarded as the limit of one of the two cases just discussed. 3, the Laplace transform of the autocorrelation can be explicitly formulated in terms of p and its derivatives with respect to x and x. • This can then be used to determine the.
THE LAPLACE-TRANSFORMED A UTOC ORRELA TION
The first term does not contain x explicitly and is linear and homogeneous in p; thus the operation (2. 60) simply replaces p with q. In fact the last term in (2.58,62) can always be omitted, and thus gk' ~ somewhat simplified. This formula (which also applies to multidimensional systems) was used by Caughey and Dienes [4, 8] to find the spectrum of.
THE GENERAL CASE--SOLUTION OF THE FOKKER-PLANCK EQUATION
For simplicity of presentation, it will be assumed throughout the general derivations of this and the next section that 1'.,. They will be used for the general derivations in Sections 3.1-2, but in the special cases treated in Sections 3.3-5, the results will be expressed in whichever formulation is greater. In this case, the solutions remain linearly independent and finite in the limit, and it is easily seen from the power series for M(cx., y, z) that.
In the next three sections, the methods developed in the previous section are used to obtain the Laplace transform. If the dimensionless transformed transition probability density (from depends on whether x < x ), then at the expense of reflection. The form of the restoring force f(x) in each of these cases is shown on the next page.
The graph of the spectral density is in this case given as one of the. Variation of this dimensionless parameter can be considered to represent variation of either slope t or intercept x. To find the limiting transformed transition density p(x, s lx0), it is necessary to find the constants K.
The inverse transformation of p can be found and corresponds to the transition density found by Smoluchowski,.
SOME TWO-INTERVAL CASES
Note that setting -t1 = -t2 in the above leads to the results obtained above for the linear system, while -t1 = -t, -t2 = oo leads to the rectifier discussed above. 0 with positive k) we approach the case with the noise rebound force, treated by Caughey and Dienes. See the discussion in the Introduction of the difference between two- and one-sided spectra.).
2 will not be assumed, as this does not provide any significant simplification in the results; those for continuous f(x) can be obtained by simply replacing k by (t. Such a trilinear characteristic can be used as a reasonable approximation to almost any continuous symmetric nonlinear characteristic with infinite endpoints~. 3.192.)--the formulas themselves for the symmetric trilinear case is not simple, and the increased accuracy of e.g. a five-interval approximation would probably not justify the trouble of working it out.
For the symmetric three-interval case, the set of simultaneous equations can be simplified somewhat by noting that. A check of the above formulas is obtained by noting that two of the cases considered in previous sections can be obtained by restricting processes to the symmetric three-interval case. Confluent hypergeometric functions and error functions can be evaluated using their rapidly convergent power series.
APPROXIMATION BY EQUIVALENT PIECEWISE LINEARIZATION
It would be interesting to find out whether the autocorrelation or spectrum of an arbitrary nonlinear system can be approximated arbitrarily closely (in a certain sense) by that of a piecewise linear system with a sufficient number of segments. Since the steady-state density corresponding to (3. 204) usually involves integrals that cannot be evaluated, the expectations in This results in small differences between the actual and equivalent linear mean and mean square.
If f(x) is continuous, the positive sign in (3. 2.15) implies the continuity of the equivalent restoring force tx + k; the negative sign means that e.g. To find out which sign represents a true minimum of I, it would be necessary to examine the second derivatives of I. For clarity from now on, the positive sign will be assumed, and continuous f(x) such that .
The method is to choose by eye a linear approximation of f(x), replace it in the right-hand side of to obtain improved values, and repeat until no further improvement is obtained. 8), page 124, the spectral density for the best three-part approximation is compared with that of the equivalent linear system.
2 o(tl -tz ), with the equivalent 3-pieces and equivalent linear approximations. a) EQUIVALENT 3-PART APPROXIMATION (b) EQUIVALENT LINEAR APPROXIMATION,.
CHAPTER IV
The boundary on the other side of the irregular point z will be regular or natural, depending on whether or not {.x +k-+O is present on this side as well. If it is regular, the correct boundary condition here is Q(z, t Ix) = 0, since Q is continuous at x = z (although aP need not be). From now on it will therefore be assumed that x1 and xn are the endpoints of such a regular interval.
-42) it is seen that the coefficients c~ are an order of magnitude smaller than the. is at least of order unity. The three examples treated below have the same restoring forces f(x) as three of the simpler cases treated in Chapter III. However, the possible appearance of irregular points in this chapter often allows a smaller interval (x . 1, xn) to be considered than in chapter III.
Then there is a trap at 0, and if the starting position x is positive, it is necessary to consider only the interval. 3 (the hard . limiter), since this produces a trap at x = 0 so that the process can be split into two independent subprocesses, at (-x. As in the previous case, these results can be verified by taking limits on sp and J sp dx.
There are four somewhat different cases in this example, depending on the relative signs of t and k.
CHAPTER V
PERFECT CORRELATION--GENERAL CASE
1 on both sides, so that the point is the exit boundary for the process on the right and the entry boundary for the one on the left, or s. 1 See Abramowitz and Stegun [l], Chapter 13, or Erdelyi et al. 5.28) and M, U denote Kummer's confluent hypergeometric functions of the first and second kind, respectively. 1 is infinite or finite and regular; and similarly at x •. is the output limit from above, .
Otherwise, both aP and Q must be. is an entry boundary from below, then since . dition is to be expected on the right side of a left shunt. 4, no boundary conditions are required for an exit boundary; however, the integrability of.p requires it. The above considerations lead to the following conditions. r eling of the coefficients on either side of the three types of left shunts that occur.
That is, the probability density on the left of a left shunt depends on that on the right, but that on the right does not depend on. All x to the right of the least irregular point lie in a regular interval bounded below by an output boundary or a regular absorbing boundary. According to the discussion in section 1.6, the stationary density P (x) is therefore zero except in the region to its left.
Zl} for x in the ith interval, which can be considered to consist entirely of left shunts.
PERFECT CORRELATION--EXAMPLES Only two cases will be worked out in detail
Note that according to Gray [22, 23] our system (being linear) should have the same mean, autocorrelation and spectrum as that. 1 has an additional left shunt at x = 0, consisting of an input. border below and a regular border above. 0 only one case will be worked out, the one where k = D 2 ) • 1 The restoring force f(x) for this case is shown on the next page.
ZERO CORRELATION CASE
Since irregular points cannot exist in these uncorrelated systems, each system can be treated in exactly the same way as the system with the same recovery function, as discussed in Chapter III. Fichera shows that the frontier can be divided into three regions: I:(l), where no boundary conditions can be imposed where the Dirichlet condition u = const. The output (x, y) of the second order system (A2-3) forms a two-dimensional Markov process, so the formula (0, 2) for the autocorrelation can be written.
This is not so, and in fact the above discussion of suitable boundary conditions shows that when solving in the region x > 0 one can specify v(o, y, t) for y < 0, but not for y: 2: O ; and vice-versa. The writer has no hope that useful results for second-order systems can be obtained by Robinson's method, or by any method for that matter. It is possible that an approximate solution can be obtained, but that should be the case.
Fichera, 110n a unified theory of boundary problems for elliptic-parabolic equations of second order11, Boundary Problems in Differential Equations, ed. Hille, 110n the integration problem for Fokker-Planck's equation in the theory of stochastic processes", 11 te. 1 of Random Vibration Volume 2, ed. Khas1minskii, "Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy prob- blade for parabolic equations", Theory of Prob.
Khazen, "Estimation of Probability Distribution Density for Random Processes in Systems with Piecewise Linear Type Nonlinearities," Theory of Prob.
