In coupled oscillator arrays, it is possible to control the inter-element phase shift up to 180 by free-running frequency distribution based on injection-locking phenomenon. In this paper, a new technique to control the inter-element phase shift electronically up to the maximum extent of 360 is reported. Oscillators are unilaterally coupled to the preceding oscillator through one of the two paths, which differ from each other 180 in electrical length and each includes an amplifier. Turning on the desired amplifier one can control the phase shift either -180 to 0 or 0 to 180. The technique was applied in a three-element oscillator array each coupled to a patch antenna via a round aperture. The radiation beam of the array could be scanned 47 in total.

We consider a system of coupled two oscillators with external force. At first we introduce the symmetrical property of the system. When the external force is not applied, the two oscillators are synchronized at the opposite phase. We obtain a bifurcation diagram of periodic solutions in the coupled system when the single oscillator has a stable anti-phase solution. We find that the synchronized oscillations eventually become in-phase when the amplitude of the external force is increased.

In this study, we show how changing a frequency in one of N chaotic circuits coupled by a resistor effects our system by means of both circuit experiment and computer calculation. In these N chaotic circuits, N-1 circuits are completely identical, and the remaining one has altered the value of the oscillation frequency. It is found that for the case of N = 3 when a value of a coupling resistor is gradually increased, only one circuit with different frequency exhibits bifurcation phenomena including inverse period-doubling bifurcation, and for larger value of coupling resistor, the chaotic circuit with different frequency suddenly stops oscillating and the remaining two chaotic circuits exhibit completely anti-phase synchronization.

When N oscillators are coupled by one resistor, we can see N-phase oscillation, because the system tends to minimize the current through the coupling resistor. Moreover, when the hard oscillators are coupled, we can see N, N - 1, , 3, 2-phase oscillation and get much more phase states. In this study, the two types of coupled oscillators networks with third and fifth-power nonlinear characteristics are proposed. One network has two-dimensional hexagonal structure and the other has two-dimensional lattice structure. In the hexagonal circuit, adjacent three oscillators are coupled by one coupling resistor. On the other hand, in the lattice circuit, four oscillators are coupled by one coupling resistor. In this paper we confirm the phenomena seen in the proposed networks by circuit experiments and numerical calculations. In the system with third-power nonlinear characteristics, we can see the phase patterns based on 3-phase oscillation in the hexagonal circuit, and based on anti-phase oscillation in lattice circuit. In the system with fifth-power nonlinear characteristics, we can see the phase patterns based on 3-phase and anti-phase oscillation in both hexagonal and lattice circuits. In particular, in these networks, we can see not only the synchronization based on 3-phase and anti-phase oscillation but the synchronization which is not based on 3-phase and anti-phase oscillation.

In this study, some oscillators with different oscillation frequencies, N - 1 oscillators have the same oscillation frequency and only the Nth oscillator has different frequency, coupled by a resistor are investigated. At first we consider nonresonance. By carrying out circuit experiments and computer calculations, we observe that oscillation of the Nth oscillator stops in some range of the frequency ratio and that others are synchronized as if the Nth oscillator does not exist. These phenomena are also analyzed theoretically by using the averaging method. Secondly, we investigate the resonance region where the fiequency ratio is nearly equal to 1. For this region we can observe interesting double-mode oscillation, that is, synchronization of envelopes of the double-mode oscillation and change of oscillation amplitude of the Nth oscillator.

In this study, multimode chaos observed from two coupled chaotic oscillators with hard nonlinearities is investigated. At first, a simple chaotic oscillator with hard nonlinearities is realized. It is confirmed that in this chaotic oscillator the origin is always asymptotically stable and that the solution, which is excited by giving relatively large initial conditions, undergoes period-doubling bifurcations and bifurcates to chaos. Next, the coexistence of four different modes of oscillations are observed from two coupled chaotic oscillators with hard nonlinearities by both of circuit experiments and computer calculations. One of the modes of oscillation is a nonresonant double-mode oscillation and this oscillation is stably generated even in the case that oscillation is chaotic. Namely, for this oscillation mode, chaotic oscillation and periodic oscillation can be simultaneously excited. This phenomenon has not been reported yet, and we name this phenomenon as double-mode chaos. Finally, the beat frequency of the double-mode chaos is confirmed to be changed by varying the value of the coupling capacitor.

In this study, we propose a system of N Wien-bridge oscillators with the same natural frequency coupled by one resistor, and investigate synchronization phenomena in the proposed system. Because the structure of the system is different from that of LC oscillators systems proposed in our previous works, this system cannot exhibit N-phase oscillations but 3-phase and in-phase oscillations. Also in this system, we can get an extremely large number of steady phase states by changing the initial states. In particular, when N is not so large, we can get more phase states in this system than that of the LC oscillators systems. Because this system does not include any inductors and is strong against phase error this system is much more suitable for applications on VLSI compared with coupled system of van der Pol type LC oscillators.

We propose an equivalent circuit model described by the Rössler equation. Then we can consider a coupled Rössler system with a physical meaning on the connection. We consider an oscillatory circuit such that two identical Rössler circuits are coupled by a resistor. We have studied three routes to entirely and almost synchronized chaotic attractors from phase-locked periodic oscillations. Moreover, to simplify understanding of synchronization phenomena in the coupled Rössler system, we investigate a mutually coupled map that shows analogous locking properties to the coupled Rössler System.

In this letter we propose a novel method of connection, called the hybrid connection, and find that a resistively coupled oscillator with hybrid connection has stable in-phase and anti-phase synchronized oscillations. Averaging method is used to investigate the stability of the synchronized oscillations. The theory is verified by the experimental results.

In this study, a ring of simple chaotic circuits coupled by inductors is investigated. An extremely simple three-dimensional autonomous circuit is considered as a chaotic subcircuit. By carrying out circuit experiments and computer calculations for two, three or four subcircuits case, various synchronization phenomena of chaos are confirmed to be stably generated. For the three subcircuits case, two different synchronization modes coexist, namely in-phase synchronization mode and three-phase synchronization mode. By investigating Poincar map, we can see that two types of synchronizations bifurcate to quasi-synchronized chaos via different bifurcation route, namely in-phase synchronization undergoes period-doubling route while three-phase synchronization undergoes torus breakdown. Further, we investigate the effect of the values of coupling inductors to bifurcation phenomena of two types of synchronizations.

In this study, we investigate the synchronization phenomena of coupled Wien bridge oscillators. The oscillator is characterized by a voltage controlled resistor with saturation. We use linear resistance to couple the oscillators. Two different kinds of coupling techniques, called current and voltage connections are proposed and they show completely opposite mode of synchronized oscillations. The dynamics of the two circuits are also derived to study the amplitude and phase dynamics of the synchronized state. The current connection has a simple resistive effect but stable phase mode is opposite to that of the voltage connection. The voltage connection has the coupling effect which is a combination of resistive and reactive couplings. Coupled three oscillators with current and voltage connection are also studied and stable tri-phase and in-phase synchronizations are observed, respectively. Averaging method is used to investigate the stability of synchronized mode of oscillations. Experimental results are also stated which agree well with the theory.

In this letter we propose a stabilizing method of phase control for resistively coupled oscillator networks. To demonstrate the effect of the control, we consider the coupled oscillator system containing only voltage type of connections. A state feedback technique to resistor sub-network is used to control the phase of synchronized oscillation. The technique is applied to two and three coupled oscillator cases. Finally we present experimental results, which agree well with the theory.

There have been many investigations of mutual synchronization of oscillators. In this article, N oscillators with the same natural frequencies mutually coupled by one resistor are analyzed. In this system, various synchronization phenomena can be observed because the system tends to minimize the current through the coupling resistor. When the nonlinear characteristics are third-power, we can observe N-phase oscillation, and this system can take (N 1)! phase states. When the nonlinear characteristics are fifth-power, we can observe (N 1),(N 2)3 and 2-phase oscillations as well as N-phase oscillations and we can get much more phase states from this system than that of the system with third-power nonlinear characteristics. Because of their coupling structure and huge number of steady states of the system, our system would be a structural element of cellular neural networks. In this study, it is confirmed that our systems can stably take huge number of phase states by theoretical analysis, computer calculations and circuit experiments.

Bifurcations of quasi–periodic responses in an oscillator described by conductively coupled van der Pol equations with a sinusoidal forcing term are investigated. According to the variation of three base frequencies, i.e., two natural frequencies of oscillators and the forcing frequency, various nonlinear phenomena such as harmonic or subharmonic synchronization, almost synchronization and complete desynchronization are ovserved. The most characteristic phenomenon observed in the four–dimensional nonautonomous system is the occurrence of a double Hopf bifurcation of periodic solutions. A quasi–periodic solution with three base spectra, which is generated by the double Hopf bifurcation, is studied through an investigation of properties of limit cycles observed in an averaged system for the original nonautonomous equations. The oscillatory circuit is particularly motivated by analysis of human circadian rhythms. The transition from an external desynchronization to a complete desynchronization in human rest–activity can be referred to a mechanism of the bifurcation of quasi–periodic solutions with two and three base spectra.