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.

The averaged equation for an arbitrary number of oscillators coupled by nonlinear coupling scheme invented by S. Nagano, is derived. This system is invented as a model of uni-cellular slime amoeba. By using the averaged equation, we investigate the synchronization characteristics of five coupled oscillators and a large number of coupled oscillators. In particular, we present the statistical property of coupled oscillators in terms of coupling factor γ. We also investigate the effect of linear and nonlinear coupling terms for achieving synchronization, and confirm that the nonlinear coupling term plays an important role for strong synchronization than linear coupling term does.

This paper describes the refined analysis of the angle-tracking performance of a coherent amplitude-comparison monopulse radar system in the presence of an interfering second target and, on the basis of the results, proposes a new improved system. The original system investigated by Kliger & Olenberger deteriorates the tracking performance for the wideband AGC in combination with narrowband PLL case. By appropriately combining systems with and without PLL, our new system is shown to have good tracking performance for all bandwidth" of the AGC and PLL relative to the Doppler separation.

This paper proposes a private communication system with chaos using fixed-point digital computation. When fixed-point computation is adopted, chaotic properties of the modulated signal should be checked carefully as well as calculation error problems (especially, overflow problems). In this paper, we propose a novel chaos modem system for private communications including a chaotic neuron type nonlinearity, an unstable digital filter and an overflow function. We demonstrate that the modulated signal reveals hyperchaotic property within 10,000 data point fixed-point computation, and evaluate the security of this system in view of the sensitivity of coefficients for demodulation.

A simple model of inductor-coupled bistable oscillators is shown to exhibit pulse wave propagation. We demonstrate numerically that there exists a pulse wave which propagates with a constant speed in comparatively wide parameter region. In particular, the propagating pulse wave can be observed in non-uniform lattice with noise. The propagating pulse wave can be observed for comparatively strong coupling case, and for weak coupling case no propagating pulse wave can be observed (propagation failure). We also demonstrate various interaction phenomena between two pulses.

A New carrier recovery subsystem for binary PSK systems using synchronized push-pull oscillator is described. A property that the push-pull oscillator has two stable states of which the phase difference is π, can be well applied for this purpose. This new subsystem does not include phase-locked loop (PLL), therefore pull-in time and pull-in range are not limited by the loop delay of PLL. The circuit of this push-pull oscillator is very simple. Yet, it is proven to achieve rather good performance, if a limiter which normalizes the input amplitude, when practically deformed on account of the band-limiting, is attached. Our experiments have had good agreement with the theory and computer simulation.

We investigate chaotic dynamics due to the homoclinic points observed from a widely used phase–locked loops operating as a frequency–modulated demodulator. Our purpose is to obtain parameter region of the homoclinic points using Melnikov method in a periodically-forced second–order nonlinear nonautonomous equation representing phase–locked loops. If the PLL equation has large damping (actually, this is the case of standard PLL), the unperturbed system becomes non–Hamiltonian. Therefore, one cannot obtain the saddle loop analytically in general, and hence it is very difficult to apply Melnikov method to such a system. Since the current PLL used in practice has a triangular phase detector (i.e., a periodic triangular shaped function) as its nonlinearity, we can use piecewise–linear method, and thus we are succeeded in deriving both the saddle loop and the Melnikov integral analytically even in the PLL equation with practical large damping. We have obtained many boundary curves for homoclinic tangency for a wide range of damping coefficients and modulation frequency. In particular, we treat the general case of β2ζ in this paper where β denotes the normalized natural frequency and ζ denotes the damping coefficient. We compare this results with our previous totally numerical results and have found that this method gives more accurate boundary curves than our previous method.

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.

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.

Secure communications via chaotic synchronization is experimentally demonstrated using 3-pieces of commercial integrated circuit phase-locked loops, MC14046. Different from the conventional chaotic synchronization secure communication systems where one channel is used, our system uses two channels to send one signal to be concealed. Namely, one channel is used to send a synchronizing chaotic signal. The other channel is used to send the informational signal superimposed on the chaotic masking signal at transmitter side. The synchronizing chaotic signal is applied as a common input to two identical PLL's located at both transmitter and receiver sides. It has been shown previously by us that the VCO inputs of almost identical two PLL's driven by a common chaotic signal become chaotic, and synchronized with each other. This synchronization is only possible for those who knows exact internal configuration and exact parameter values of the PLL at transmitter side. Therefore, we can use the synchronized VCO input signal as a masking signal which can be used as a key for secure communications. The advantage of this method compared to the previous one channel method is that informational signal frequency range does not affect the quality of recovered signal. Our experiments demonstrate good masking and recovery characteristics for sinusoidal, triangular, and square waves.

In this paper, we investigate the transitional dynamics and quasi-periodic solution appearing after the Saddle-Node (SN) bifurcation of a periodic solution in an inductor-coupled asymmetrical van der Pol oscillators with hard-type nonlinearity. In particular, we elucidate, by investigating global bifurcation of unstable manifold (UM) of saddles, that transitional dynamics and quasi-periodic solution after the SN bifurcation appear based on different structure of UM.