Search

This paper considers the Alpazur oscillator, which is composed of a Rayleigh oscillator unit and a dc power supply controlled by a switch, and examines the bifurcation set by numerical calculation. In such a case, a bifurcation diagram is constructed in order to see the phase-space behavior of the system. In the conventional method, however, the calculation is possible only when the trajectory is continuous and differentiable. The method cannot directly be applied to the case where the trajectory is discontinuous or not smooth at a point (not differentiable) as in the case of on–off circuits. From such a viewpoint, this paper proposes a bifurcation trajectory tracking algorithm for the that nonlinear on–off circuit, containing a switch that is made on and off periodically by an external force. As the method of analysis, the composite Poincaré mapping is defined based on the periodic operation of the switch. The method of calculating the bifurcation parameter is described precisely. Using the proposed method, the Alpazur oscillator is analyzed. The phase-synchronized solutions that exist along the Neimark–Sacker bifurcation as well as the chaos are investigated. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 84(1): 75–83, 2001 [ABSTRACT FROM AUTHOR]

This paper derives a renormalization formula defined on the parameter space where mapping behavior is preserved, together with the equivalent potential function. In contrast to the universal function given by Feigenbaum, the behavior near the critical point is governed by the potential function. There are several interesting features which are shared by Feigenbaum's universal function, such as the representation of the critical point in terms of the unstable fixed point of the potential function, but the mapping differs from the scaling. The one-dimensional mapping is considered as an example, and the critical point and the scaling, which are major constants characterizing chaos from the potential function, are calculated precisely. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 81(8): 41–51, 1998 [ABSTRACT FROM AUTHOR]

This paper considers the case where hysteresis is given by the Preisach model and discusses the stability analysis for the periodic solution in the nonautonomous system containing hysteretic elements. The behavior of the system containing hysteretic elements is affected by the history of the system until it arrives at the steady state. In this paper, it is shown first that even periodic solutions with the same hysteretic loop may have different variational systems for the steady-state solution if the histories until arriving at the steady state are different, Next, two kinds of stability criteria are considered based on the difference of the history up to the steady state and the stability against the external disturbance is investigated. The dependence of the stability of the solution on the bifurcation of the periodic solution also is discussed. Finally, examples of numerical analysis are presented to verify the result of discussions. [ABSTRACT FROM AUTHOR]

A bifurcation phenomenon may occur in a nonlinear system where the number of equilibrium points or their properties change by a small change of parameter. The set of parameters that can produce the bifurcation phenomenon is called the bifurcation set. In general, it is difficult to determine the bifurcation set when the order of the system is increased. However, when the linear part of the system can be transformed into the controllable canonical form, the determination of the bifurcation set for the higher-order system can be reduced to that of the one-dimensional system. This paper derives the method to determine the bifurcation set for the compound discrete-time system based on that idea. First, a certain class of compound system composed of a series connection is considered and the kind of bifurcations that can be generated is analyzed. Then it is shown that the number of variables needed to determine the bifurcation set can be reduced to the number of decomposable subsystems in the case of a generally connected system. [ABSTRACT FROM AUTHOR]

This paper proposes a method for a numerical analysis of the bifurcation in the nonautonomous system containing hysteretic elements. Previously, the authors discussed the case where the characteristics of the hysteretic elements can be represented by Preisach's model, and proposed a method for steady-state analysis of the nonautonomous system containing hysteretic elements. First, a method is presented to investigate the change of the steady-state behavior with the variation of the system parameters by combining the proposed method and the continuation method. Then another method is presented in which the singular point on the characteristic curve (solution curve) is determined using the previously derived variational equations of the system. Finally, examples of analysis by the newly proposed method are presented, and the effects of the hysteresis loss and the ohmic loss on the bifurcation are compared. A remarkable difference between the two is indicated. [ABSTRACT FROM AUTHOR]

The forced Rayleigh equation is known to be the first differential equation for which the aperiodic oscillation is shown to exist. For this type of equation, however, it is not known whether chaos exists and neither has any detailed study been made on the bifurcation phenomenon. With this as background, the authors have repotted [1, 2] that a chaotic oscillation can exist in the forced Rayleigh equation. In [1] and [2], the authors accounted for the generation of chaos by regarding the nonlinear element as strongly dissipative (α→∞) and deriving the constrained equation so that the Poincaré map corresponds to the noninvertible circle map. In this paper, the forementioned constrained equation is used to analyze the bifurcation phenomenon in the forced Rayleigh equation. Interesting phenomena are verified, such as Arnold tongue, intermittency, period-doubling bifurcation and phase-locked chaos, which are accounted for by the forementioned mapping. It is shown especially that the transition to the torus and the intermittency can clearly be distinguished when the tangent bifurcation is generated. The Lyapunov exponent is calculated for the constrained equation and the case where α is finite, and it is verified numerically that the Lyapunov exponent converges to that of the constrained equation when α → ∞. [ABSTRACT FROM AUTHOR]

There are many reports on the bifurcation phenomena of a forced self-oscillatory system and a hard oscillator. One of them suggested the formation of a forced self-oscillatory system by the Duffing–van der Pol equation with linear restoring term. When the restoring force has a suitable negative linear term, the system has hard characteristics as well as soft oscillation. This paper deals with the bifurcation phenomena of subharmonic and higher-harmonic oscillations corresponding to the hard characteristics in the system. It is found that the frequencies at the unstable periodic points synchronize with the external force. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(9): 1–9, 1999 [ABSTRACT FROM AUTHOR]

It is critically important that the synchronize ion problem be studied as it pertains to oscillators with hard characteristics having two types of limit cycles where periodic external force is applied and oscillations are at a submultiple of the frequency of external force. This paper considers bifurcations of periodic states corresponding to harmonic synchronization and subharmonic synchronization of the order of one-half and one-third observed in a Duffing­Rayleigh equation with hard characteristics. As a result, it was discovered that the synchronization region corresponding to each limit cycle exists independently and, under certain conditions, the two regions have mutual coalescence. Moreover, in the region of subharmonic synchronization of the order of one-half it was observed that periodic doubling bifurcation intersects the Hopf bifurcation. As a result of analysis using Hopf bifurcations, it was also found that tangent bifurcations correlated with Arnold tongues and the associated resonance phenomena. [ABSTRACT FROM AUTHOR]

The forced Rayleigh equation is known as the differential equation for which the existence of the aperiodic solution is shown for the first time, From the viewpoint of the history of the research of chaos, it is very important to examine the bifurcation in this equation. The authors considered the forced Rayleigh oscillator containing a diode. By considering the limit α → ∞ for a parameter α contained in the system, the ideal diode constrained equation is defined. Then, the rigorous analysis of the bifurcation phenomenon is made for the forced Rayleigh oscillator. This paper considers the forced Rayleigh oscillator containing a large parameter α and shows that the bifurcation concerning the chaos generated in the oscillator can be accounted for by the ideal diode constrained equation by constructing and comparing the detailed bifurcation diagrams for the two systems. [ABSTRACT FROM AUTHOR]

This paper presents a global bifurcation scenario for chaotic dynamical systems that solve optimization problems. First, a chaotic dynamical system is constructed by a discretization of the gradient-descent dynamical system of the objective function. With an increase in the discretization parameter, local minimum solutions of the objective function bifurcate into chaotic attractors through a period-doubling bifurcation. The chaotic attractors are initially localized in the state space and eventually merge into a single global chaotic attractor via a series of crises. On the global chaotic attractor which visits a variety of local minima, “chaotic search” for a global minimum is realized. On the basis of the bifurcation scenario, we provide a guideline for tuning the bifurcation parameter value which gives rise to an efficient “chaotic search.” We also consider the efficiency of the “chaotic simulated annealing” algorithm in the light of its annealing schedule. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(2), 1–12, 1998 [ABSTRACT FROM AUTHOR]

This paper discusses bifurcation diagram reconstruction [1, 2] and shows by a numerical experiment that the correct bifurcation diagram of a dynamical system can be reconstructed from the signal of a chaotic dynamical system perturbed by a sufficiently large dynamical noise. It is also verified that the predictor derived using the proposed method not only can approximate the dynamical rules on the attractor, but also has an excellent global approximation ability on a sufficiently large open set containing the attractor. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(11): 30–45, 1997 [ABSTRACT FROM AUTHOR]

A learning algorithm is introduced for chaotic dynamical systems that solve nonlinear optimization problems. The algorithm controls the asymptotic measure of a chaotic dynamical system that searches for the optimum solution and improves the efficiency of chaotic search dynamics. Using several instances of 1- and 2-dimensional nonlinear optimization problems, the performance of the learning algorithm is demonstrated. We also show that the learning algorithm works as chaotic simulated annealing, which brings about gradual convergence of the chaotic search dynamics to a possible optimum solution. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(3): 10–21, 1999 [ABSTRACT FROM AUTHOR]