This paper develops a design method and theoretical analysis for piecewise nonlinear oscillators that have desired circular limit cycles. Especially, the mathematical proof on existence, uniqueness, and stability of the limit cycle is shown for the piecewise nonlinear oscillator. In addition, the relationship between parameters in the oscillator and rotational directions and periods of the limit cycle trajectories is investigated. Then, some numerical simulations show that the piecewise nonlinear oscillator has a unique and stable limit cycle and the properties on rotational directions and periods hold.

In high-precision positioning systems, the limit cycles induced by friction effects result in a significant reduction in the positioning performance; particularly when the servo system utilizes a high gain controller. Accordingly, the current study presents a compensation scheme consisting of a dead-zone function and an integral term to limit the equivalent gain of unspecified controllers to the stable range. The proposed compensation scheme not only ensures that the feedback loop system remains stable, but also provides a simple and effective mechanism for preventing the users from inadvertently setting control gains which degrade the positioning performance of the system. The simulation results confirm the ability of the gain limit compensation scheme to suppress the effects of limit cycles and therefore demonstrate its feasibility for practical applications.

We show that a unit-grup, which represents a group of contiguous units with the same sign of output, is a dominant component for the dynamical behavior of a neural network with anti-symmetrical cyclie connections for the nearest neighbor connections and global connections. In transient state, it is shown that the unit-grup has the dynamics such that the amount n of units which belong to the unit-grup increases with time, and that the increasing rate of n decreases with increasing n. The dynamics cause the large difference of the number of limit-cycles between discrete and continuous time models. Additionally, the period of the limit-cycle depends on the size of the unit-grups. This dependency is obtained from computer simulations and two approximation methods. These approximations provide the lower and the upper bounds of the periods which depend on the gain of an activation function. Using these approximations, we also obtain detailed relations between a period and the other network parameters analytically.

This paper deals with a synthesis of a nonautonomous system with a stable limit cycle. We propose a synthesis method of a nonautonomous system whose transient trajectories converge to a prescribed limit cycle. We use receding horizon control to control a transient behavior of the nonautonomous system, and confirm its validity by simulation.

Limit cycle oscillations of rotor speed are substantially caused by inverter's dead time, when an induction motor (IM) drive operates in low frequency condition. In this paper, without any hardware modification, discontinuous PWM (DPWM3) modulate strategy possibly controls the unfavorable rotor speed limit cycle under no load operation condition. Simulated results are presented to demonstrate the effectiveness.

Evaluation of cyclic transitions in the discrete-time neural networks with antisymmetric and circular interconnection weights has been derived in an asymptotic mathematical form. The type and the number of limit cycles generated by circular networks, in which each neuron is connected only to its nearest neurons, have been investigated through analytical method. The results show that the estimated numbers of state vectors generating n- or 2n-periodic limit cycles are an exponential function of (1.6)n for a large number of neuron, n. The sufficient conditions for state vectors to generate limit cycles of period n or 2n are also given.

In this paper, we propose a synthesis method of hybrid systems with specified limit cycles. Several methods which sysnthesize a nonlinear system with prescribed limit cycles have been proposed. In these methods, the limit cycle is given by an algebraic equation, which will be called constraint equations, and its stability is guaranteed by a Lyapunov function derived from the constraint equation. In general, limit cycles of hybrid systems are nonsmooth due to the discontinuous vector fields. So the limit cycles are given by piecewise smooth constraint equations, we employ the piecewise smooth Lyapunov functions to construct desired nonsmooth limit cycles and guarantee their stability.

In this report, a design method of neural networks for limit cycle generator is described. First, the constraint conditions for the synaptic weights, which are given by the linear inequalities, are derived from the dynamics of neural networks. Next, the linear inequalities are solved by the linear programming method. The synaptic weights and other parameters are determined by the above solutions. Furthermore, neuro-based limit cycle generator is designed with analog electronic circuits and simulated by Spice. Finally, we confirm that our design method is efficient and practical for the design of neuro-based limit cycle generator.

In this paper, we propose the majority algorithm to choose the connection weights for the neural networks with quantized connection weights of 1 and 0. We also obtained the layered network to solve the parity problem with the input of arbitrary number N through an application of this algorithm. The network can be expected to have the same ability of generalization as the network trained with learning rules. This is because it is possible to decide the connection weights, regardless of the size of the training set. One can decide connection weights without learning according to our case study. Thus, we expect that the proposed algorithm may be applied for a real-time processing.

In order to investigate the dynamic behavior of quantized interconnection neural networks on neuro-chips, we have designed and fabricated hardware neural networks according to design rule of a 1.2 µm CMOS technology. To this end, we have developed programmable synaptic weights for the interconnection with three values of 1 and 0. We have tested the chip and verified the dynamic behavior of the networks in a circuit level. As a result of our study, we can provide the most straightforward application of networks for a dynamic pattern classifier. The proposed network is advantageous in that it does not need extra exemplar to classify shifted or reversed patterns.

This letter describes a new computational method to obtain the bifurcation parameter value of a limit cycle in nonlinear autonomous systems. The method can calculate a parameter value at which local bifurcations; tangent, period-doubling and Neimark-Sacker bifurcations are occurred by using properties of the characteristic equation for a fixed point of the Poincare mapping. Conventionally a period of the limit cycle is not used explicitly since the Poincare mapping needs only whether the orbit reaches a cross-section or not. In our method, the period is treated as an independent variable for Newton's method, so an accurate location of the fixed point, its period and the bifurcation parameter value can be calculated simultaneously. Although the number of variables increases, the Jacobian matrix becomes simple and the recurrence procedure converges rapidly compared with conventional methods.

A type of Lienard's equation +µf(x)+x=0, where f(x) is not an even function of x, is studied by Le Corbeiller as a model of various biological oscillations, such as breathing, and called two-stroke oscillators. A distinctive feature of this type of oscillators is that the parameter µ has the upper limit µ0 for the oscillator to have some stable limit cycle. This paper gives a numerical method for calculating this upper limit µ0.

In this paper, a simple method to investigate the dynamics of continuous-time neural networks based on the force (kinetic vector) derived from the equation of motion for neural networks instead of the energy function of the system has been described. The number of equilibrium points and limit cycles of one-dimensional neural networks with the asymmetric cyclic connection matrix has been investigated experimently by this method. Some types of equilibrium points and limit cycles have been theoretically analyzed. The relations between the properties of limit cycles and the number of connections also have been discussed.

We analyze dynamics of a simple hysteresis network (ab. SHN) which has only two parameters. We classify the periodic orbits and clarify the number of attractors and their domain of attraction. The SHN is a piecewise linear system, and therefore we can calculate the trajectory using exact solutions. We clarify the bifurcation sets on which equilibrium attractors bifurcate to the periodic orbits. We also give a sufficient condition for stability of the periodic orbits, and the stability is verified by laboratory experiment. The results of this paper may contribute to the development of an efficient multi functional artificial neural network.

As is well known, Linard's equation +µf (χ)+g(χ)=0 represents a wide class of oscillatory circuits as an extension of van der Pol's equation, and Linard's theorem guarantees the existence of a unique periodic solution which is orbitally stable. However, we sometimes meet such cases in engineering applications that the symmetry of the equation is violated, for instance, by a constant bias force. While, it has been known that asymmetric Linard's equation can have more than one periodic solution. The problem of finding the maximum number of such solutions, known as a special case of Hilbert's sixteenth problem, has recently been solved by T. Koga, one of the present authors. This paper first describes fundamental theorems due to T. Koga, and presents a solution to the synthesis problem of asymmetric Linard's systems, which generates an arbitrarily prescribed number of limit cycles, and which is considered to be important in relation to the stability of Linard's systems. Then, as application of this result, we give a method of determining parameters included in Linard's systems which may produce two limit cycles depending on the parameters. We also give a Linard's system which have three limit cycles. In addition, a new result on the parameter dependency of the number of limit cycles is presented.

This article discusses a synthesis procedure of a discrete-time asynchronous neural network whose information is a limit cycle. The synthesis procedure uses a novel connection matrix and can be reduced into a linear epuation. If all elements of desired limit cycles are independent at each transition step, the equation can be solved and all desired limit cycles can be stored. In some experiments, our procedure exhibits much better storing performance than previous ones.