This is the first to three expository articles, which aim to survey the chaos theory from an engineering point of view. This paper, in the first place, presents a brief introduction to chaos. Then, it is pointed out that the study of chaos from an engineering point of view is a challenging area in engineering science.

This is an expository article on chaotic phenomena in electrical and electronic circuits and systems. We briefly review the meaning of chaos, discovery of chaos, analytical method of chaos and future problems of chaos from electrical engineers' viewpoint. Electrical engineers have intently studied chaotic systems and are trying to find useful engineering applications of chaos.

A review is presented of the definitions of 'chaos' in the discrete system, the diagnosing methods of chaotic systems, and examples of engineering and/or biological chaos. First, enumerating physically intuitive pictures of one-dimensional chaos shows that there are many possible definitions of 'chaos' and that the 'observable chaos' is an important concept. Important roles of the Frobenius-Perron operator are discussed in theoretically studying statistical quantities of a completely chaotic orbit. In order to measure chaos, several quantities of a strange attractor are listed. Some of chaotic maps are shown to be applicable to a pseudorandom number generator. To examine biological chaos, macroscopic analyses and microscopic ones well be reviewed.

This paper discusses the behavior of a dynamical system described by the extended Liénard equation with an external force e(t)
f(x)g(x)e(t)
where f and g are not necessarily even or odd with respect to x, respectively. First, basic theorems on the existence of limit cycles and properties of singularities are proved in the case where e(t) is equal to a constant bias denoted by e(t)const.A cos ωτ, and effects of f and g on the portrait of trajectories of the systems are clarified. Then, the dynamical behavior of the system, where the external force is periodic, i.e., e(t)A cos ωt, is represented in relation to the singularity which varies periodically in time; an obtained result makes it clear and easy to understand the dynamical behavior. Further, some conditions which are necessary for the system mentioned above to generate a chaotic solution are presented. Finally, the results of the argument above are applied to the periodically forced van der Pol equation, and it is concluded that chaotic solutions hardly exist in this case.

An efficient algorithm is given for systematically calculating several statistics such as the invariant measure, the Kolmogorov-Sinai entropy, the autocorrelation function and the power spectrum of chaos in one-dimensional discrete dynamical systems defined by a map. The method is based on the Galerkin approximation to the Frobenius-Perron integral operator. Several numerical examples demonstrate that the proposed method can give approximations with high accuracy to the statistics of various one-dimensional chaos with the absolutely continuous invariant measure under the map.

A numerical method is proposed for identifying bifurcating solution of infinite dimensional nonlinear equations by making use of the infinite dimensional homotopy method. In this paper in the first place, in order to show the existence of bifurcating solutions for a certain class of the Fredholm operators, a mapping degree is defined which has the similar properties as in a finite dimensional space. Using this, under certain conditions a primary bifurcation point exists for a certain type of infinite dimensional nonlinear equations. Furthermore, in case of the Leray-Schauder operator, it is shown that a certain bifurcating solution of the Leray-Schauder operator equation can be identified by making use of the infinite dimensional homotopy method.

A hierarchical structure is observed in codimension 2, 1 and 0 homoclinic and heteroclinic bifurcations. Exact bifurcation equations make it possible to discern fine details of the bifurcation diagrams.

In a nonlinear dynamical circuit with sinusoidal external source, we frequently encounter various bifurcation phenomena of steady states such as jump and hysteresis phenomenon, frequency entrainment, etc. The steady state corresponds to a periodic solution of the circuit equations described by nonlinear ordinary differential equations. The generic bifurcations of the periodic solution are known as codimension one bifurcations: tangent bifurcation, period doubling bifurcation and the Hopf bifurcation. At a bifurcation value of parameters, if a periodic solution satisfies two bifurcation conditions, then the bifurcation refers as a codimension two bifurcation. This type of bifurcation may be observed in high dimensional systems with several parameters. In Ref.(1), we have classified codimension two bifurcations and proposed a numerical method for obtaining the bifurcation parameters. To illustrate the occurrences of some types of codimension two bifurcations, we analyzed a circuit described by 3-dimensional differential equation. For 3-dimensional system, however, two types of bifurcations never occur. In this paper, we shall treat 4-dimensional system as an illustrating example. In this example, we shall see all types of codimension two bifurcations defined in this paper. For a global property of bifurcation set of parameters, it is found that a type of codimension two bifurcation occurs successively together with the period doubling cascade and the Hopf bifurcations. This bifurcation sequence may cause a new route to the generation of chaotic oscillations.

This letter demonstrates an experimental result on chaos observed in a very practical electronic circuits called the phase-locked loops. Namely, we report that chaos can in fact occur in a practical FM demodulator circuit made of a phase-locked loop IC module (MC14046B) operating under widely employed high-damping case as well as very low-damping case.

The phase-locked loop (PLL) is a versatile functional device widely used in many electronic system. We found previously that the PLL can become chaotic under some operating conditions for wide range of system parameter values. The reason why the PLL can cause chaos is closely related to the homoclinic orbits of which existence are proved by Melnikov method. In this paper, we review the motivation and significance of chaos occurring in the PLL circuits. Then we confirm various chaotic characteristics of the phase-locked loop equation having the homoclinic orbits such as the fractal basin boundaries and sensitive dependence on initial conditions of a solution. At last, we investigate the route to chaos of the periodic solution of first type (PS1) by calculating the bifurcation diagram, and presents a new results that the PS1 can be chaotic via the period-doubling cascade.

This article discusses chaotic responses observed from a periodically forced neural-type oscillator introduced by Newcomb et al. Describing the circuit dynamics by an autonomous equation, we can derive a one-dimensional return map explicity and therefore can prove a sufficient condition of chaos generation in Lasota & Yorkes' sense. Also, our mapping procedure is valid for various other systems and seems to contribute for design of them.

Electroencephalographic (EEG) potentials are analysed by the Lyapunov spectrum in order to evaluate the orbital instability peculiar to deterministic chaos quantitatively. First, the Lyapunov spectra are estimated to confirm the existence of chaotic behavior in EEG data by the optimal approximation of Jacobian matrix in the reconstructed statespace. Second, the same method is applied to a neural network model with chaotic dynamics, the macroscopic average activity of which is analysed as a simple model of EEG data. The first analysis shows that the largest Lyapunov exponent is actually positive in the EEG data. On the other hand, the second analysis on the chaotic neural network shows that the positive largest Lyapunov exponent can be obtained by observing only the macroscopic average activity. Thus, these results indicate the possibility that one can know the existence of chaotic dynamics in the brain by analysing the Lyapunov spectrum of the macroscopic EEG data.

Canard is a new phenomenon of slow-fast systems, which was found by the numerical computations. Our primary purpose of this paper is to study the canard from the experimental viewpoint. The following results are obtained by the experimental observation of a nonlinear circuit: (1) Canard really appears in the actual circuit. But canard's life in the circuit is extremely short. (2) When a canard vanished, an irregular oscillations or a cycle with period two is still there, which usually does not occur in the two dimensional autonomous systems.

The elliptically-polarized nonlinear beam propagation in a two-dimensional optical guided-wave system containing Kerr media is solved numerically by using the finite-element method. Computed results for a nonlinear substrate exhibit novel transverse effects such as spatially modulational instabilities for solitons emitted from a film. Sensitiveness of the beam propagation on the initial state of polarization suggests a possibility for constructing new photonic devices.

The author shows experimentally how the nonlinear line transforms the noise spectrum and will discuss 1/f fluctuation phenomena which are widespread in nature. The line is a one-dimensional LC ladder type distributed line with loss, where the capacitor C changes with the line voltage. The experimental and computer-simulated results show a tendency that the white noise is transformed into the 1/f type one along the noise propagation.

A new method of test for goodness of fit of Laplace's distribution is being proposed in this paper. The new parameter H is defined by the two standard deviations which are estimated by the methods of moments and of maximum likelihood. If the number of data is large, the goodness of the Laplace's distribution is easily tested by the percentile of the normal distribution of the parameter H.

The basic selective-repeat ARQ scheme can achieve very high throughput. However, it encounters a resequencing problem. We analyze the average resequencing delay approximately under the assumption that the arrivals of NAK's are totally independent of the past history of the system. And, we examine its characteristics from numerical examples.

A design method of mode conversion type Y junction waveguides in lens-like media is proposed. The Y junction designed for reducing the mode conversion losses form the fundamental mode to higher order even modes can also reduce the mode conversion losses form the lowest odd mode to higher order odd modes.

The shielding effect of a new material composed of rubber with carbon and ferrite for microwave oven leakage is investigated. By adding an appropriate amount of carbon, the shielding effect is improved. A large attenuation constant, which is 6 dB/cm larger than that of a rubber ferrite, is relizable.

This letter describes the electrification properties of the human body by walking. A method is presented for measuring the electrified charge of the human body. The quantitative relationships between the electrified charge behaviors and the walking actions are also shown.

It was unknown whether there exists a language accepted by a two-way nondeterministic one counter automaton, but not accepted by any nondeterministic rebound automaton. This paper solves this problem, and shows that there exists such a language.

In the design of two-dimensional digital filters (2 DDF's), if the given 2 DDF design specifications can be decomposed into one-dimensional digital filter (1 DDF) specifications, the 2 DDF design problems can be reduced to 1 DDF ones. Thus the 2 DDF design problems can be made simpler. However, in the frequency domain design, the conventional decomposition methods can not avoid the problem that the 1 DDF magnitude specifications obtained from 2 DDF magnitude specification decomposition are not always nonnegative. Since negative values can not be regarded as magnitude specifications, design problem become intricate. Therefore, it is desirable in practice to develop a 2 DDF magnitude decomposition method which can guarantee the resulting 1 DDF magnitude specifications to be always nonnegative. Unfortunately, up to now, no such a method has ever been proposed. In this paper, we propose a new decomposition method called the Iterative Singular Value Decomposition (ISVD) for the decomposition of the given 2 DDF magnitude specification matrix. By the ISVD, the prescribed 2 DDF magnitude specification is decomposed into a pair of 1DDF ones, one of which is the magnitude response of a one-input/multi-output 1 DDF and the other is that of a multi-input/one-output 1 DDF. The ISVD guarantees that the resultant 1 DDF magnitude specifications are always nonnegative. The problem of designing a 2 DDF is then simplified through designing a pair of 1 DDF's with different delay elements. In our design method, 1 DDF's are designed by utilizing nonlinear optimization method to minimize the weighted magnitude square error functions. In the optimization process, a variable substitution method is proposed for transforming constrained optimization problems to unconstrained ones. As a result, no attentions should be paid to 1 DDF stability during the optimization process, and the stability of the resulting 1 DDF's is ensured. Three design examples are given to illustrate the effectiveness of the proposed method.

The relationship between a multirate quadrature mirror filters (QMF) system and an output sampling polyphase model of a periodically time varying digital filter is examined. It is proved that a polyphase model of a periodically time varying digital filter is equivalent to a specified multirate QMF system. Using the derived relationship, another type of polyphase model of a periodically time varying digital filter which is called an input sampling polyphase model is proposed and analyzed. Furthermore, the equivalence of two types of polyphase models, i.e., the conventional output sampling polyphase model and the proposed input sampling polyphase model, is discussed and conditions for their equivalence are obtained.

This paper proposes a new technique for designing 2-D diamond-shaped half-band FIR filters with maximally flat amplitude property. First, a 2-D half-band FIR filter is defined and several interesting properties of this filter are explained. Then a design method using these properties is explained and some design examples are shown. Finally, we discuss the advantages of our method by comparing with the previously proposed method.

Theoretical description with a potential is made for inhomogeneous structures of high field domain and current filament in semiconductors with a negative differential conductivity (NDC) appearing under voltage- and current-controlled conditions, respectively. The potential proposed here can describe systematically a route from homogeneous state to the patterned state through the instability of homogeneous state, whereas previously proposed potentials can describe only the patterned state. The potential is constructed from two internal variables: one is the variable dependent on the spatial coordinate which exhibits the spatial pattern in the NDC region, while another remains constant spatially but changes discontinuously its value when the patterned state bifurcates from a thermodynamic branch of the homogeneous state. The bifurcation to spatial pattern is examined in a similar way to the first-order phase transition in equilibrium systems. At the same time, the property of the resulting pattern is discussed from analogy with the phase separation.

The method of constructing a graph G with the maximum cardinality of minimum cut-sets, which is 2e/n, has been obtained by Harary in 1962, where n is the number of nodes and e is the number of edges. Afterward, the problem of finding a simple graph G which minimizes the number of minimum cut-sets with cardinality 2e/n subject to λ(G)2e/n was solved by Bauer, Boesch, Suffel and Tindell in 1985. Generalizing this, a necessary and sufficient condition for a simple graph with n nodes and e edges to minimize the number of cut-sets with cardinality z has been recently presented by Sun, Nagamochi and Kusunoki in 1989, where z is chosen such that 2e/nz22e/n3 for2e/n3, or z2, 3, for2e/n2. In this paper, we generalize the above results to multiple graphs, and give a necessary and sufficient condition for a multiple graph G with n nodes and e edges to minimize the number of minimum cut-sets whose cardinality is2e/n.

The growth of a primary root was studied for adzuki beans incubated under an external electric field. An electric current was applied to the solution around the root, which was laid horizontally. The electric potential and pH patterns were measured along the root. The change of the growth speed was observed when the electric field, corresponding to a current of 1-10 µA/cm², was applied for a period of 15 to 60 min. The application of the electric field in the direction from the base to the tip of a root activated the growth of the root, whereas the growth was inhibited by applying the electric field of the reverse polarity. The changes of the growth rates were both about 20% for the currents of 10 µA/cm². The root was generating an electric current pattern in the aqueous solution around its surface with the maximum current density of the order of 1 µA/cm². In the activation of the growth, the amplitude of the electric potential pattern and the acidification around the root were found to be increased. The opposite tendency was obtained in the growth inhibition. The degree of acidification corresponds to the change of the growth rate by a well-known acid growth mechanism.

The resequencing delay for a queueing system with multiple servers under a threshold-type scheduling is analyzed. We first derive the general expressions for the average total delay which is the sum of resequencing delay and queueing delay. The expressions are also applicable to the conventional M/M/n queueing system as the special case of zero-threshold policy. Next, the numerical calculation is carried out for the queueing system with 3 servers under a threshold-type scheduling to consider the impact of the multiple thresholds on the average resequencing delay. It is found that the resequencing delay as well as the queueing delay for a queueing system with multiple servers of different service rates can be recuced under a threshold-type scheduling by selecting the threshold values appropriately. The threshold-type scheduling is more effective to reduce the resequencing delay compared to the queueing delay especially in the region of modest traffic intensity and as the difference of the service rates becomes larger.

In ISDN and ATM, performance evaluation of packetized voice is an important problem. To analyse the superposed voice packets from a number of voice sources, the Markov modulated Poisson process (MMPP) has been used for approximating the superposed non-renewal process. In this approximation, a large computer time was needed for determining the MMPP parameters by the numerical inversion of Laplace transform. In this paper, introducing the index of skewness for counts as well as the index of dispersion, and using their conservation low, limiting property and differential coefficient at the origin, we propose a simplified method for determining the parameters.

The simultaneous measurement of five field components (three magnetic and two electric components) was carried out on board the ionospheric Aureol-3 satellite in a frequency range from 10 Hz to 1.5 kHz, and we have developed a systematic signal processing of direction finding for ionospheric and magnetospheric ELF emissions at eight selected frequencies mainly based on the likelihood concept. So, the purpose of this paper is to present the detailed description of those systematic direction finding measurements at frequencies above and below the proton gyrofrequency, but more emphasis is placed on the latter frequency range where two possible modes of propagation may coexist. This kind of systematic analyses will be of great use in the future wave analysis system on the spacecrafts.

An accurate numerical solution for the electromagnetic scattering from cascaded strip gratings is presented. The gratings are free-standing and must have common periodicity, but may be staggered. The propagation direction and the polarization of the incident plane wave are arbitrary. We derive a set of singular integral equations and solve it by the moment method, where the Chebyshev polynomials are chosen for the basis and the testing functions. By numerical calculations we examine the convergence of our solution and compare with other published data. Some numerical examples are presented to show the frequency selective characteristics of cascaded structures. This method is accurate and effective owing to the incorporation of the edge condition and the decomposition of the kernel into singular and regular parts.

A quasi-TEM wave characterization method is proposed for tight couplers which are constructed of coupled microstrip lines with conductor overlay. The characterization is carried out with consideration for strip conductor thickness by applying the rectangular boundary division method and a sequential optimization procedure. Estimated characteristics of a quarter-wavelength tight coupling section are given in the graphical form for the design of structural dimensions. A trial 3-dB hybrid coupler is designed with this method and its measured characteristics are compared with theoretical ones.

Contact contamination is verified by the measurement of static contact resistance so far. However, in case of the contamination of organic thin film, new evaluation method is needed in stead of static contact resistance. Then, quantitative analysis of the contact cleaning effect was conducted using contact activation characteristics, dynamic contact resistance pattern and so on when the ultrasonic cleaning, immersion cleaning, vapor cleaning and jet blow cleaning were performed for relay contacts. As a result ,it was clarified that characteristics of the dynamic contact resistance depends upon the nature of the organic film on the contact surface and the effect of each contact cleaning method was quantitatively compared.

Communication relays have been used for the control of low level circuits, and organic gases vaporized from relays or contamination of contact surface were studied as the factors lowering contact reliability so far. Conversely, contact materials of platinum group contacts like palladium and gold or silver alloy contacts were mainly evaluated as reliable contacts. Authors tried to clarify the contact failure mechanism of gold or silver alloy contacts at a extremely low electrical load which had minimal switching electrical energy. As a result, the oxides are detected from contact failure samples by the contact surface analysis, which suggests a small amount of impure substances in the gold alloy causes contact resistance increase. In addition, the influence of the ratio of oxygen and nitrogen on contact resistance was investigated. Then the contact failure has not occurred below 0.1% of oxygen concentration. As for some contact materials, the permissible contact force was also studied.

A reversible finite automaton (RFA) is a backward deterministic automaton, i.e., it can uniquely retrace its move sequence if the inverse sequence of its outputs is given. In this paper, we show a simple method to construct an RFA from Fredkin gates, which are reversible and bit-conserving logic gates, and unit wires (unit delays). The resulting circuit obtained by this method is garbage-less" in the sense that it has no inputs to which constants must be supplied nor outputs from which garbage signals are put out. We also show that a one-dimensional reversible partitioned cellular automaton, which are known to be computation universal, can be constructed from Fredkin gates and unit wires as a closed (thus garbage-less) infinite circuit.

After Diffie and Hellman's paper, there have been published many cryptosystems, where large composite numbers are used as the public keys, and the factorization of them are used as the secret keys. But, on the other hand, there also have been published many integer-factoring algorithms that factor composite numbers rapidly. So methodologies to construct primes that are strong for such algorithms are needed to guarantee the safety of such cryptosystems. Here we propose a randon polynomial time algorithm for constructing strong primes that uses a probabilistic primality testing algorithm.

In order to reconstruct the 3 D geometric model of a real object from multi-view line drawings some restrictions in a symmetric space of the original 3 D space have been set up. Based on these restriction a new algorithm of 3 D geometric model reconstruction is proposed. This method can work effectively under the more natural condition than other methods, and it is suitable for the situation of auxiliary views as well as the situation of perspective projection. Some practical computational results have been made which show that this algorithm is effective.