Takuji Kousaka

Connecting border collision with saddle-node bifurcation in switched dynamical systems

Switched dynamical systems are known to exhibit border collision, in which a particular operation is terminated and a new operation is assumed as one or more parameters are varied. In this brief, we report a subtle relation between border collision and saddle-node bifurcation in such systems. Our main finding is that the border collision and the saddle-node bifurcation are actually linked together by unstable solutions which have been generated from the same saddle-node bifurcation. Since unstable solutions are not observable directly, such a subtle connection has not been known. This also explains why border collision manifests itself as a jump from an original stable operation to a new stable operation. Furthermore, as the saddle-node bifurcation and the border collision merge tangentially, the connection shortens and eventually vanishes, resulting in an apparently continuous transition at border collision in lieu of a jump. In this brief, we describe an effective method to track solutions regardless of their stability, allowing the subtle phenomenon to be uncovered.

BIFURCATION AND CHAOS IN COUPLED BVP OSCILLATORS

Bonhoffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

Bifurcation analysis of a piecewise smooth system with non‐linear characteristics

SUMMARY In previous works, there are no results about the bifurcation analysis for a piecewise smooth system with non-linear characteristics. The main purpose of this study is to calculate the bifurcation sets for a piecewise smooth system with non-linear characteristics. Werst propose a new method to track the bifurcation sets in the system. This method derives the composite discrete mapping, Poincare mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. As an illustrated example, we then apply this general methodology to the Rayleigh-type oscillator containing a state- period-dependent switch. In the circuit, we cannd many subharmonic bifurcation sets including global bifurcations. We also show the bifurcation sets for the border-collision bifurcations. Some theoretical results are veried by laboratory experiments. Copyright ? 2005 John Wiley & Sons, Ltd.

Analysis of border-collision bifurcation in a simple circuit

This paper considers a system interrupted by own state and a periodic interval. We know this system has prospects of occurrence of border-collision bifurcation. To analyze properties of the dynamics, we derive a one-dimensional map explicitly. We show some theorems and the existence of regions of periodic solution within two-parameter space. Some theoretical results are verified by laboratory experiments.

EXPERIMENTAL REALIZATION OF CONTROLLING CHAOS IN THE PERIODICALLY SWITCHED NONLINEAR CIRCUIT

This letter presents an experimental confirmation of controlling the chaotic behavior of a target unstable periodic orbit when the periodically switched nonlinear circuit has a chaotic attractor. The pole assignment for the corresponding discrete system derived from such a nonautonomous system via Poincare mapping works effectively, and the control unit is easily realized by the window comparator, sample-hold circuits, and so on.

General Consideration for Modeling and bifurcation Analysis of Switched Dynamical Systems

In much of the previous study of switched dynamical systems, it has been assumed that switching occurs at a common border between two regions in the same space as the system trajectory crosses the border. However, models arising from this consideration cannot cover systems whose trajectories do not actually cross the border. A typical example is the current-mode controlled boost converter whose trajectory is reflected at the border. In this paper, we propose a general method to model switched dynamical systems. Also, we suggest an analytical procedure to determine periodic solutions and their stability. The method is developed in terms of solution flows, and no solution has to be explicitly written. Most practical switched dynamical systems can be modeled and analyzed by this method.

Dynamical mechanism for interrupted circuit with switching delay

The unavoidable nonidealities with switching delay in current-mode-controlled buck converters have been reported in the literature. Investigations are carried out on the dynamical mechanism and its experimental validation on an interrupted circuit with switching delay. Switching delay is seen to influence a region of a two-valued function on a discrete map, and to induce the coexistence of a periodic orbit.

A subtle link in switched dynamical systems: saddle-node bifurcation meets border collision

Switched dynamical systems are known to exhibit border collision, in which a particular operation is terminated and a new operation is assumed as one or more parameters are varied. We report a subtle relation between border collision and saddle-node bifurcation in such systems. Our main finding is that the border collision and the saddle-node bifurcation are actually linked together by unstable solutions which have been generated from the same saddle-node bifurcation. Since unstable solutions are not observable directly, such a subtle relation has not been known. We describe an effective method to track solutions regardless of their stability, allowing the subtle phenomenon to be uncovered. Two typical DC-DC converters are observed to verify our finding.

Bifurcation analysis in hybrid nonlinear dynamical systems

In this paper, we investigate the bifurcation phenomena in the nonlinear dynamical system switched by a threshold of the state or a periodic interrupt. First, we propose a method to trace the bifurcation sets for above system. We derive the composite discrete mapping as Poincare mapping. As a result, it is possible to obtain the local bifurcation values in the parameter plane. We also propose an efficient analyzing method for border-collision bifurcations. As an illustrated example, we investigate the behavior of the Rayleigh-type oscillator switched by a threshold of the state or a periodic interrupt. In this system, we can find many subharmonic bifurcation sets including global bifurcations and border collision. Some theoretical results are verified by laboratory experiments.

Fundamental property of a nonlinear vibration system based on a rigid overhead wire-pantograph system

In an electric railway, contact loss occurs intermittently between overhead wire and a pantograph. The contact loss brings in some problems as development of wear. Although many researchers recently studied the phenomena, the problem was not sufficiently examined from the viewpoint of bifurcation theory. In this paper, we study the fundamental property of a nonlinear vibration system based on a rigid overhead wire-pantograph system [7]. First, we explain a behavior of the system. Then in order to show the existence domain of the periodic and non-periodic solutions in this system, we derive a two-parameter bifurcation diagram. Finally we try to explain the phenomena of this system with the help of an experimental apparatus.