This tutorial paper proposes a subclass of cellular neural networks (CNN) having no inputs (i.e., autonomous) as a universal active substrate or medium for modeling and generating many pattern formation and nonlinear wave phenomena from numerous disciplines, including biology, chemistry, ecology, engineering, and physics. Each CNN is defined mathematically by its cell dynamics (e.g., state equations) and synaptic law, which specifies each cell's interaction with its neighbors. We focus on reaction-diffusion CNNs having a linear synaptic law that approximates a spatial Laplacian operator. Such a synaptic law can be realized by one or more layers of linear resistor couplings. An autonomous CNN made of third-order universal cells and coupled to each other by only one layer of linear resistors provides a unified active medium for generating trigger (autowave) waves, target (concentric) waves, spiral waves, and scroll waves. When a second layer of linear resistors is added to couple a second capacitor voltage in each cell to its neighboring cells, the resulting CNN can be used to generate various turing patterns.< >

The qualitative nature of the time evolution in a piecewiselinear lossy resonant circuit driven by a sinusoidal voltage source is investigated by computer-aided analysis using exact analytical formulas. A surprising wealth of different nonlinear phenomena is discovered. They are: stable and unstable harmonics, subharmonics, and even apparently completely disordered aperiodic "chaotic" motions. In the latter case, the hyperbolicity, strange attractor, and broad-band frequency spectrum normally associated with chaotic motions have all been observed using nearly exact piecewise-linear solutions. These results represent the most reliable numerical confirmation to date of chaotic motions in a real physical circuit.

The canonic form of the three-port scattering matrix is deduced from the relations expressing the constraints of reciprocity and losslessness. One gives a particular construction of the denominator of the entries, which allows to solve the divisibility conditions resulting from the basic equations. The necessary and sufficient conditions are given which allow to synthesize the three-port as three two-ports connected in parallel with a shunt lossless impedance at the common port.

The transfer function obtained from the tuncated expansion of exp(p) in series of Bessel polynomials yield an approximation of the ideal filter. Thanks to a freely chosen parameter, it is possible to control the compromise between optimal amplitude and phase or between the rise time and the overshoot of the step response. Compared to the classical Thomson characteristic, which is linked to a single Bessel polynomial, the delay deviation in the passband is smaller.

Leapfrog RC active filters are known to be insensitive to the value of the elements thanks to their analogy to LC filters. It is shown that the class of LF filters amenable to such an analogy can be enlarged by considering nondissipative 2-ports between generalized dissipative terminations. There is no necessity for transmission zeroes at the origin or at infinity, which was the case in the previous approaches. The formulas linking the given transmittance to the LC 2-port description are derived and the optimal choice of the remaining degrees of freedom are discussed. The method presented here preserves the analogy between the LF design and its LC equivalent since this analogy guarantees the low sensitivity; yet it is totally free from all constraints originating from the LC implementation and irrelevant for the LF design.

The traditional method of realising attenuation poles by resonant circuits is questioned. It is shown that, under some conditions, it is possible to depart from the conventional solution.

Two equivalent linear circuits, which have neither capacitance loops nor inductance cut-sets, may be linked by a continuous equivalent transformation such that the individual sensitivities with respect to L's and C's are not invariant.

The choice of the extraction order of the attenuation poles in the classical synthesis of a ladder filter will influence the dispersion of the element values and the sensitivity of the transmittance s12 to the element values. The purpose of the letter is to investigate these two characteristics by an exhaustive study of a typical example.

The summed sensitivity invariants are derived by a 2-parameter dimensional analysis. This approach clarifies the limitation of these invariants to circuits with only two kinds of elements, and yields directly all generalisations to distributed or active networks.

The necessary and sufficient conditions are given for a function to be the group delay of a passive 2-port transmittance. In some important particular cases, more stringent conditions must be imposed. It appears that the realisability of the classical Thomson and Abele families in these cases is purely incidental, and that the last family is actually restricted to low ripples.

Two families of maximally flat approximations to the ideal filter can be deduced as particular cases of polynomials which are solutions of a hypergeometric equation. By studying the generalised hypergeometric function, one finds a continuous range of filter characteristics. This allows a control of the compromise to be made between the features of the transient response, the amplitude and the phas...

The necessary and sufficient conditions are established under which a multivariable transfer matrix can be synthetised as a cascade connection of lossless noncommensurate transmission lines. It is proved that, for this particular structure, the transmittance defines a unique reflectance.

The conditions for maximally flat attenuation are set up for a cascade connection of noncommensurate transmission lines. For structures with 2 and 3 lines, it is shown that arbitrary lengths cannot achieve a better maximally flat approximation than equal lengths.

A method is given for analysing the sensitivity of a 2-port transmission network independently of other effects inherent in the synthesis method. This method allows proof of the outstanding insensitivity of Cheby-shev filters; it also shows the importance of a minimum flat loss in predistorted filters.