The transmission control protocol with a random early detection (TCP/RED) is an important algorithm for a TCP congestion control [1]. It has been expressed as a simple second-order discrete-time hybrid dynamical model, and shows unique and typical nonlinear phenomena, e.g., bifurcation phenomena or chaotic attractors [2], [3]. However, detailed behavior is unclear due to discontinuity that describes the switching of transmission phases in TCP/RED, but we have proposed its analysis method in previous study. This letter clarifies bifurcation structures with it.

We found a novel connecting orbit in the averaged Duffing-Rayleigh equation. The orbit starts from an unstable manifold of a saddle type equilibrium point and reaches to a stable manifold of a node type equilibrium. Although the connecting orbit is structurally stable in terms of the conventional definition of structural stability, it is structually unstable since a one-deimensional manifold into which the connecting orbit flows is unstable. We can consider the orbit is one of global bifurcations governing the differentiability of the closed orbit.

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.

Bifurcation Phenomena observed in a circuit containing two Josephson junctions coupled by a resistor are investigated. This circuit model has a mechanical analogue: Two damped pendula linked by a clutch exchanging kinetic energy of each pendulum. In this paper, firstly we study equilibria of the system. Bifurcations and topological properties of the equilibria are clarified. Secondly we analyze periodic solutions in the system by using suitable Poincare mapping and obtain a bifurcation diagram. There are two types of limit cycles distinguished by whether the motion is in S1R3 or T2R2, since at most two cyclic coordinates are included in the state space. There ia a typical structure of tangent bifurcation for 2-periodic solutions with a cusp point. We found chaotic orbits via the period-doubling cascade, and a long-period stepwise orbit.

In this article, we propose a square wave generator whose switching threshold values are switched by external inputs. This circuit is designed to simulate the synchronized luminescence of coupled fireflies. We investigate the behavior of the solutions in two coupled oscillators. The dynamics are demonstrated by a linear autonomous equation piecewisely, therefore, a one-dimensional return map is derived. We also prove the existence of stable in-phase synchronization in the coupled oscillator by using the return map, and we show the existence of regions of periodic solutions within a parameter space. Some theoretical results are confirmed by laboratory measurements.

We propose a stabilization method of unstable periodic orbits embedded in a chaotic attractor of continuous-time system by using discrete state feedback controller. The controller is designed systematically by the Poincar mapping and its derivatives. Although the output of the controller is applied periodically to system parameter as small perturbations discontinuously, the controlled orbit accomplishes C0. As the stability of a specific orbit is completely determined by the design of controller, we can also use the method to destabilize a stable periodic orbit. The destabilization method may be effectively applied to escape from a local minimum in various optimization problems. As an example of the stabilization and destabilization, some numerical results of Duffing's equation are illustrated.

The pendulum equation with a periodic impulsive force is investigated. This model described by a second order differential equation is also derived from dynamics of the stepping motor. In this paper, firstly, we analyze bifurcation phenomena of periodic solutions observed in a generalized pendulum equation with a periodic impulsive force. There exist two topologically different kinds of solution which can be chaotic by changing system parameters. We try to stabilize an unstable periodic orbit embedded in the chaotic attractor by small perturbations for the parameters. Secondly, we investigate the intermittent drive characteristics of two-phase hybrid stepping motor. We suggest that the unstable operations called pull-out are caused by bifurcations. Finally, we proposed a control method to avoid the pull-out by changing the repetitive frequency and stepping rate.

Some qualitative properties of an inductively coupled circuit containing two Josephson junction elements with a dc source are investigated. The system is described by a four–dimensional autonomous differential equation. However, the phase space can be regarded as S1×R3 because the system has a periodicity for the invariant transformation. In this paper, we study the properties of periodic solutions winding around S1 as a bifurcation problem. Firstly, we analyze equilibria in this system. The bifurcation diagram of equilibria and its topological classification are given. Secondly, the bifurcation diagram of the periodic solutions winding around S1 are calculated by using a suitable Poincar mapping, and some properties of periodic solutions are discussed. From these analyses, we clarify that a periodic solution so–called "caterpillar solution" is observed when the two Josephson junction circuits are weakly coupled.

In this paper, performance of chaos and burst noises injected to the Hopfield Neural Network for quadratic assignment problems is investigated. For the evaluation of the noises, two methods to appreciate finding a lot of nearly optimal solutions are proposed. By computer simulations, it is confirmed that the burst noise generated by the Gilbert model with a laminar part and a burst part achieved the good performance as the intermittency chaos noise near the three-periodic window.