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.

Finding a nonnegative integer solution x for Ax = b (A Zmn, b Zm1) in Petri nets is NP-complete. Being NP-complete, even algorithms with theoretically bad worst case and with average complexity can be useful for a special class of problems, hence deserve investigation. Then a Grobner basis approach to integer programming problems was proposed in 1991 and some symbolic computation systems became to have useful tools for ideals, varieties, and algorithms for algebraic geometry. In this letter, Grobner basis approach is applied to three typical problems with respect to state equation in P/T Petri nets. In other words, after Grobner bases are derived by the tool Maple 7, we consider how to derive the T-invariants and particular solutions of the Petri nets by using them in this letter.

For fixed initial and destination states (i.e., markings), M0 and Md, there exist generally infinite firing count vectors in a Petri net. In this letter, it is shown that all fundamental particular solutions as well as all minimal T-invariants w.r.t. firing count vectors are needed to express an arbitrary firing count vector for the fixed M0 and Md. An algorithm for finding a special firing count vector which is expressed by using the only one specified fundamental particular solution is also given.

In this study, we investigate two oscillators which have the same natural frequency, mutually coupled by N-type piecewise-linear negative resistor. In this system, according to the negative range of the coupling negative resistor, the various inter-esting synchronization phenomena which are in-phase, opposite phase and doublemode-like oscillations are observed. Especially, we show doublemode-like oscillations that are not observed until now in mutually coupled van der Pol oscillators with the smooth cubic characteristics, although the ones with same natural frequencies are coupled. And we show the differences of the phenomena between two oscillators coupled by the smooth cubic negative resistor and the ones coupled by the piecewise-linear negative resistor.

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.

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.