Kazuo Horiuchi

A globally and quadratically convergent algorithm for solving nonlinear resistive networks

A globally convergent algorithm that is also quadratically convergent for solving bipolar transistor networks is proposed. The algorithm is based on the homotopy method using a rectangular subdivision. Since the algorithm uses rectangles, it is much more efficient than the conventional simplicial-type algorithms. It is shown that the algorithm is globally convergent for a general class of nonlinear resistive networks. Here, the term globally convergent means that a starting point which leads to the solution can be obtained easily. An efficient acceleration technique which improves the local convergence speed of the rectangular algorithm is proposed. By this technique, the sequence of the approximate solutions generated by the algorithm converges to the exact solution quadratically. Also, in this case the computational work involved in each iteration is almost identical to that of Newtons method. Therefore, the algorithm becomes as efficient as Newtons method when it arrives sufficiently close to the solution. It is also shown that sparse-matrix techniques can be introduced to the rectangular algorithm, and the partial linearity of the system of equations can be exploited to improve the computational efficiency. Some numerical examples are also given in order to demonstrate the effectiveness of the proposed algorithm. >

A Basic Theory for Available Operation of Extremely Complicated Large-Scale Network Systems

In this paper, we shall describe about a basic theory based on the concept of set-valued operators, suitable for available operation of extremely complicated large-scale network systems. Fundamental conditions for availability of system behaviors of such network systems are clarified in a form of fixed point theorem for system of set-valued operators. Here, the proof of this theorem is accomplished by the concept of Hausdorffs ball measure of non-compactness.

Geometric structure of mutually coupled phase-locked loops

Dynamical properties such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLLs) are problems of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLLs incorporating lag filters and triangular phase detectors. The fourth-order ordinary differential equation (ODE) governing the mutually coupled PLLs is reduced to the equivalent third-order ODE due to the symmetry, where the system is analyzed in the context of nonlinear dynamical system theory. An understanding as to how and when lock-in can be obtained or out-of-lock behavior persists, is provided by the geometric structure of the invariant manifolds generated in the vector field from the third-order ODE. In addition, a connection to the recently developed theory on chaos and bifurcations from degenerated homoclinic points is also found to exist. The two-parameter diagrams of the one-homoclinic orbit are obtained by graphical solution of a set of nonlinear (finite dimensional) equations. Their graphical results useful in determining whether the system undergoes lock-in or continues out-of-lock behavior, are verified by numerical simulations.

Fine Estimation Theory for Available Operation of Complicated Large-Scale Network Systems

In this paper, we shall construct mathematical theory based on the concept of set-valued mappings, suitable for available operation of extraordinarily complicated large-scale network systems by introducing some connected-block structures. A fine estimation technique for availability of system behaviors of such network systems are obtained finally in the form of fixed point theorem for a special system of fuzzy-set-valued mappings.

Risk analysis of fuzzy control systems with (n+1)-inputs and 1-output FLC

Abstract This paper proposes a fundamental method to analyze the uncertain fuzzy control systems by the methodology of functional analysis based on the fuzzy mapping concept. First of all, a completely continuous operator is defined, and some fuzzy mapping simultaneous equations are introduced by using the defined operator. In the next place, the existence conditions of the likely solutions of the fuzzy mapping simultaneous equations are discussed for the given membership grades. Last of all, it is shown that the simultaneous equations correspond to the input–output relation of the uncertain control systems with the ( n +1)-inputs and 1-output fuzzy logic controllers, and this mathematical discussion is applied to the risk analysis of the systems.

A Urabe type convergence theorem for a constructive simplified Newton method in infinite dimensional spaces

A constructive simplified Newton method is presented for calculating solutions of infinite dimensional nonlinear equations, which uses a projection scheme from an infinite dimensional space to finite dimensional subspaces. A convergence theorem of the method is shown based on Urabes theorem. For a class of successive approximation methods containing an infinite dimensional homotopy method, a stopping criterion is shown using the convergence theorem. The authors show that under a certain condition the stopping criterion is satisfied in finite cycles of iteration.<<ETX>>

A Fuzzy Estimation Theory for Available Operation of Extremely Complicated Large-Scale Network Systems

In this paper, we shall describe about a fuzzy estimation theory based on the concept of set-valued operators, suitable for available operation of extremely complicated large-scale network systems. Fundamental conditions for availability of system behaviors of such network systems are clarified in a form of β-level fixed point theorem for system of fuzzy-set-valued operators. Here, the proof of this theorem is accomplished in a weak topology introduced into the Banach space.

A PL homotopy continuation method with the use of an odd map for the artificial level

This note presents a new piecewise linear homotopy continuation method for solving a system of nonlinear equations. The important feature of the method is the use of an odd map for the artificial level of the homotopy. Some sufficient conditions for the global convergence of the method are given. They are different from the known conditions for the global convergence of the existing homotopy continuation methods. Specifically, they cover all the systems of nondegenerate linear equations.

Chaos from orbit-flip homoclinic orbits generated in real systems

A new class of chaotic systems is discovered that are generated in a practical, nonlinear, mutually coupled phase-locked loop (PLL) circuit. Presented theoretical results make it possible to understand experimental results of mutually coupled PLLs on the onset of chaos using the geometry of the invariant manifolds, while the resultant simple geometry and complex dynamics is expected to have applications in other areas, e.g., power systems or interacting bar magnets. Motivated by the numerical study of this system, the topological horseshoe is proven to be generated in the codimension 3 unfolding of a degenerated orbit-flip homoclinic point for this system. Qualitatively different type of bifurcation phenomena are also observed to appear depending on the phase detector (PD) characteristics.

Synchronization limit and chaos onset in mutually coupled phase-locked loops

Dynamical property such as lock-in or out-of-lock condition of mutually coupled phase-locked loops (PLLs) is a problem of practical interest. The present paper describes a study of such dynamical properties for mutually coupled PLLs incorporating lag filters and triangular phase detectors. The system is analysed in the context of nonlinear dynamical system theory. The symmetry of the mutually coupled PLLs system reduces the original 4th order ordinary differential equation (ODE) that governs the phase dynamics of the voltage-controlled oscillators (VCO) outputs to the 3rd order ODE, for which the geometric structure of the invariant manifolds provides an understanding as to how and when lock-in can be obtained or out-of-lock behavior persists. In addition, two-parameter diagrams of the one-homoclinic orbit are obtained by solving a set of nonlinear (finite dimensional) equations. This graphical results are confirmed to be useful in determining whether the system undergoes lock-in or continues out-of-lock behavior by numerical simulations. Presented theoretical results make it possible to understand experimental results of mutually coupled PLLs on the onset of chaos using the geometry of the invariant manifolds, where the resultant dynamical chaotic phenomena is postulated to represent an unfolding of the orbit-flip homoclinic point. Motivated by the numerical study of the system generated invariant manifolds, the topological horseshoe is proven to be generated even in the unfolding of a degenerated orbit-flip homoclinic point for the piecewise linear system under consideration.