Nonlinear discontinuities

The nonlinearities discussed in Sections 10.1 and 10.1.1 are of a continuous (smooth) nature. That is, the nonlinear addition to spring force, or string tension, is a continuous function of displacement. A more extreme type of nonlinearity—discontinuous nonlinearity—tends to have a more profound effect on the system and spectrum. 

A network that is too large may require a large number of training patterns in order to avoid memorization and training time, while one that is too small may not train to an acceptable tolerance. Cybenko [30] has shown that one hidden layer with homogenous sigmoidal output functions is sufficient to form an arbitrary close approximation to any decisions boundaries for the outputs. They are also shown to be sufficient for any continuous nonlinear mappings. In practice, one hidden layer was found to be sufficient to solve most problems for the cases considered in this chapter. If discontinuities in the approximated functions are encountered, then more than one hidden layer is necessary. 

From the standpoint of the classical (analytical) theory with which we were concerned in this review, the situation is obviously absurd since each of these two equations is linear and of a dissipative type (since h > 0) trajectories of both of these equations are convergent spirals tending to approach a stable focus. However, if one carries out a simple analysis (see Reference 6, p. 608), one finds that change of equations for = 0, results in the change of the focus in a quasi-discontinuous manner, so that the trajectory can still be closed owing to the existence of two nonanalytic points on the -axis. If, however, the trajectory is closed, this means that there exists a stationary oscillation and in such a case the system (6-197) is nonlinear, although, from the standpoint of the differential equations, it is linear everywhere except at the two points at which the analyticity is lost. 

Nonlinear and discontinuous equations can be easily implemented, e.g., to simulate the effects of a temperature-limiting device or a digital voltmeter. ... 

S. HeikkilS and V. Lakshmikantham, Monotone iterative Techniques for Discontinuous Nonlinear Differential Equations (1994)... 

As mentioned earlier, nonlinear objective functions are sometimes nonsmooth due to the presence of functions like abs, min, max, or if-then-else statements, which can cause derivatives, or the function itself, to be discontinuous at some points. Unconstrained optimization methods that do not use derivatives are often able to solve nonsmooth NLP problems, whereas methods that use derivatives can fail. Methods employing derivatives can get stuck at a point of discontinuity, but the function-value-only methods are less affected. For smooth functions, however, methods that use derivatives are both more accurate and faster, and their advantage grows as the number of decision variables increases. Hence, we now turn our attention to unconstrained optimization methods that use only first partial derivatives of the objective function. 

Just as it is common to feel that if a little training is good then more must be better, some researchers have worked on the assumption that if a few neurons are good, many neurons must be better. This working assumption is often incorrect. Two hidden layers are all that is needed for an ANN to deal with discontinuous or nonlinear functions. More layers may be used, but they are not normally necessary except for specialized applications such as bottleneck networks,53 which are not covered in this chapter. The inclusion of extra... 

Contrary to the linear/nonlinear junetion, the transmittanee of a nonlinear step-like discontinuity (structure B, Fig.l) is less than unity for low-intensity light beams. Under the assumption of a unidireetional propagation, the transmittanee can be evaluated by matching the transverse eomponents of eleetrieal field of TE polarization at the plane of junetion ... 

The transmittance of the nonlinear step-like discontinuity in cylindrical waveguide has been evaluated under the assumption that profiles of low-intensity nonlinear modes can be approximated by profiles of linear modes. According to the results, nonlinear transmittance is less or greater than the linear one depending on waveguide parameters of the first and the second waveguides, Vi = kafafg 2= ka2(nf respectively. 

However, the difference between transverse profiles of linear and nonlinear modes can be significant for the considered range of powers (Fig.8). That is why numerical modeling of nonlinear mode propagation through the discontinuity is a reasonable way to study the spatial transformation of the mode field. 

Contrary to the linear/nonlinear junetion, the transmittance of the strueture B has a tendeney to grow with input power up to unity in the limit P(0) 1 (Fig. 17). The values of the nonlinear transmittances are different for light beams propagating in opposite directions through the discontinuity. 

Figure 16. Transmittance of nonlinear step-like discontinuity in comparison with linear transmittance (Fig. 15) for some values of initial power/ (3.3), fli = 3.0 pm,...

Moreover, the harmonic function h is not continuous across the discontinuities of N (this follows from (4.1.9), (4.1.7), and (4.1.3)). The magnitude of the appropriate jumps in h is a nonlinear function of the local values of ip and h themselves, so that h cannot be computed separately from

[Pg.111]

The discontinuity of the interface leads to two contributions to the second order nonlinear polarizability, the electric dipole effect due to the structural discontinuity and the quadrupole type contribution arising from the large electric field gradient at the surface. Under the electric dipole approximation, the nonlinear susceptibility of the centrosymmetric bulk medium 2 is zero. If the higher order magnetic dipole... 

The first two terms are electric quadrupole in character while the last term is magnetic dipolar. Under excitation by a single plane wave, the first term vanishes. In a homogeneous medium the second term vanishes by Gauss Law. The third term describes the induced polarization which is along the propagation direction. It can only radiate at the discontinuity of the surface. The full expression for the second-order nonlinear polarization in an isotropic medium is then written as the sum of the surface and bulk polarizations [78] ... 

When modelling the nonlinear response from a metal surface, two source currents are involved. The first and most difficult to describe is the source current from the surface which extends only a few angstroms into the metal. This surface current has components parallel and perpendicular to the surface. The latter is most sensitive to the details of the surface but is also the most difficult to calculate because of the discontinuity at the interface and the rapidly varying normal component of the electric field there. The bulk current is the second and is calculated to extend on the order of the optical skin depth into the metal. 

Nonlinear load — Electrical load that draws currents discontinuously or whose impedance varies during each cycle of the input AC voltage waveform. Figure 1.6 shows the waveform of a nonlinear current drawn by fluorescent lighting loads. 

---