Naohiko Inaba

Chaos via torus breakdown in a piecewise-linear forced van der Pol oscillator with a diode

Chaos via torus breakdown in the piecewise-linear forced van de Pol equation is studied rigorously by using the degenerate technique. The model is a negative resistance LC oscillator including a diode driven by a sinusoidal voltage source. The authors investigate an idealized case where the diode is assumed to operate as an ideal switch. In this case, the Poincare map is derived strictly as a one-dimensional return mapping of a circuit onto itself. This mapping clarifies the onset of chaos via torus breakdown observed in this circuit. The authors obtain the critical value of the bifurcation parameter analytically, which gives the boundary between the chaotic region and the torus region. This bridges the gaps between the abstract one-dimensional mapping and the real circuit. >

A singular bifurcation into instant chaos in a piecewise-linear circuit

Strange bifurcation route to chaos is found in a piecewise-linear second-order nonautonomous differential equation derived from a simple electronic circuit. When a limit cycle loses its stability, the attractor changes directly to chaos (instant chaos) without undergoing a period doubling bifurcation or an intermittency. The width of the attractors band is continuous at the bifurcation point, and the chaotic band grows larger continuously as the system parameter is varied. We call this bifurcation a singular bifurcation into instant chaos. The purpose of this paper is to show that the singular phenomenon arises from the piecewise-linearity of the system. To analyze this phenomenon in detail, the degenerate approach is applied. In this simplified case, the Poincare map is derived rigorously as a one-dimensional mapping. By analyzing it, we prove with computer assistance that the Liapunov exponent jumps discontinuously from minus to plus at the bifurcation point. >

Folded torus in the forced Rayleigh oscillator with a diode pair

It is known that the periodically forced Rayleigh equation is the first differential equation for which an aperiodic solution was ever discovered. However, it has not yet been clarified whether or not observable chaos exists in this equation. Chaotic oscillations observed in the forced Rayleigh oscillator are investigated in detail by using the piecewise-linear and degeneration technique. The model is a negative resistance LC oscillator with a pair of diodes driven by a sinusoidal source. The piecewise-linear constrained equation is derived from this circuit by idealizing the diode pair as a switch. The Poincare map of this equation is derived strictly as a one-dimensional return mapping on a circle (so-called circle map). This mapping becomes noninvertible when the amplitude of the forcing term is tuned larger. The folded torus observed in this oscillator is well explained by this mapping. >

Three-dimensional tori and Arnold tongues

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.

ANALYSIS OF TORUS BREAKDOWN INTO CHAOS IN A CONSTRAINT DUFFING VAN DER POL OSCILLATOR

The bifurcation structure of a constraint Duffing van der Pol oscillator with a diode is analyzed and an objective bifurcation diagram is illustrated in detail in this work. An idealized case, where the diode is assumed to operate as a switch, is considered. In this case, the Poincare map is constructed as a one-dimensional map: a circle map. The parameter boundary between a torus-generating region where the circle map is a diffeomorphism and a chaos-generating region where the circle map has extrema is derived explicitly, without solving the implicit equations, by adopting some novel ideas. On the bifurcation diagram, intermittency and a saddle-node bifurcation from the periodic state to the quasi-periodic state can be exactly distinguished. Laboratory experiment is also carried out and theoretical results are verified.

OPF chaos control in a circuit containing a feedback voltage pulse generator

In this paper, we discuss occasional proportional feedback (OPF) chaos control in a circuit containing a feedback voltage pulse generator. First, we show that the Poincare map of this oscillator without the chaos control is derived rigorously as a one-dimensional mapping. This mapping satisfies Li-Yorkes extended period three condition (1975). Hence, the mapping has an n-periodic point for any natural number n, and the appearance of chaos in Li-Yorkes sense is explained. Furthermore, OPF for piecewise-linear systems is applied to this oscillator. Each unstable orbit can be stabilized, and periodic orbits with one to ten periods are successfully stabilized in circuit experiments.

Chaotic phenomena in four circuits with an ideal diode due to the change of the oscillation frequency

Four simple chaos-generating circuits that include a diode are proposed for which both rigorous analyses and physical explanations of the generation of chaos are given. The idealized case in which the diode in the circuit operates as a simple on-off switching element is considered. In this case, the circuit equation is represented as an equation in which two linear second-order ordinary differential equations that have considerably different oscillation frequency are connected to each other by the switching operation of the diode. It is clarified that this effect causes a stretching and folding mechanism. Moreover, it is possible to derive a Poincare map strictly as a one-dimensional map that is similar to a logistic map, and it explains the generation of chaos.<<ETX>>

REVEALING THE TRICK OF TAMING CHAOS BY WEAK HARMONIC PERTURBATIONS

Taming chaos by weak harmonic perturbations has been a hot topic in recent years. In this paper, the authors investigate a scenario for the mechanism of taming chaos via bifurcation theory, and ass...

Experimental study of complex mixed-mode oscillations generated in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation.

Bifurcations of complex mixed-mode oscillations denoted as mixed-mode oscillation-incrementing bifurcations (MMOIBs) have frequently been observed in chemical experiments. In a previous study [K. Shimizu et al., Physica D 241, 1518 (2012)], we discovered an extremely simple dynamical circuit that exhibits MMOIBs. Our model was represented by a slow/fast Bonhoeffer-van der Pol circuit under weak periodic perturbation near a subcritical Andronov-Hopf bifurcation point. In this study, we experimentally and numerically verify that our dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and we observe MMOIBs and chaos with distinctive waveforms in real circuit experiments.

Synchronization of chaos in a pair of forced Rayleigh circuits with diodes

The use of a constrained equation is one of most powerful approaches in the analysis of chaos. In a previous paper (see ibid., vol. 38, no. 4, p. 398-409, 1991) we studied, in detail, chaos in a forced Rayleigh oscillator with a diode, by regarding the diode as a switch. The Poincare map was derived as a one-dimensional (1-D) mapping and, by this mapping, the generation of chaos was explained. In this paper, we study chaos synchronization of a pair of forced Rayleigh oscillators with diodes. By regarding the diodes as switches, chaos synchronization is accomplished very easily, without using a wide frequency band, and is realized in circuit experiments.