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Abstract: We analyse a piecewise-linear oscillator that consists of a three- resonant circuit with a hysteresis element. Three sets of two-dimensional linear equations, including a hysteresis function, represent the governing equations of the circuit, and all the Lyapunov exponents are calculated in a remarkably simple manner based on derived explicit solutions. Various dynamical phenomena such as two-torus, three-torus, and hyperchaos with four positive Lyapunov exponents are observed by Lyapunov analysis. We obtained a detailed bifurcation diagram in which novel bifurcation structure which we call a “two-torus Arnold tongue” is observed where two-torus generating regions exist in a three-torus generating region as if periodic states exist in a two-torus generating region. [Copyright &y& Elsevier]

Abstract: In this study, we propose a remarkably simple oscillator that exhibits extremely complicated behaviors. The second-order nonautonomous differential equation discussed in this Letter is considered to be one of the simplest dynamics that can produce mixed-mode oscillations (MMOs) and chaos. Our model uses a Bonhoeffer–van der Pol (BVP) oscillator under weak periodic perturbation. The parameter set of the BVP equation is chosen such that a focus and a relaxation oscillation coexist when no perturbation is applied. Under weak periodic perturbation, various types of MMOs and chaos with remarkably complicated waveforms are observed. [Copyright &y& Elsevier]

Abstract: This Letter investigates the period-doubling cascades of canards, generated in the extended Bonhoeffer–van der Pol oscillator. Canards appear by Andronov–Hopf bifurcations (AHBs) and it is confirmed that these AHBs are always supercritical in our system. The cascades of period-doubling bifurcation are followed by mixed-mode oscillations. The detailed two-parameter bifurcation diagrams are derived, and it is clarified that the period-doubling bifurcations arise from a narrow parameter value range at which the original canard in the non-extended equation is observed. [ABSTRACT FROM AUTHOR]

Abstract: The authors discuss the “embryology” of successive torus doubling via the bifurcation theory, and assert that the coupled map of a logistic map and a circle map has a structure capable of generating infinite number of torus doublings. [Copyright &y& Elsevier]

The response of a circuit which generates the duck solution is studied, where an extremely small periodic perturbation is applied. This perturbation breaks down the duck solution eventually, and then chaos occurs afterwards. To analyze this phenomenon in detail, the piecewise linear technique combined with the degenerate technique is employed. Through this technique, a one-dimensional Poincaré map is derived rigorously, hence chaos generation mechanism is well understood. The bifurcation diagrams are obtained for the circuit, and the parameter boundary between torus generating region and chaos generating region is derived exactly. From these bifurcation diagrams, periodic state breaks down when the forcing term is extremely small. [Copyright &y& Elsevier]

The bifurcation structure of fractional harmonic entrainments in the forced Rayleigh equation is analyzed in connection with the symmetry of the system. Due to the symmetry of the system, there are two types of fractional harmonic entrainment regions. One is the case in which an attractor symmetric in shape with respect to the origin is generated in the state space. Another is the case in which two attractors in an asymmetric shape appear in locations symmetric with respect to the origin. When the shapes of these bifurcation structures are studied on the parameter surface, it is found that the fractional harmonic entrainment regions of the two cases have different structures near the Neimark–Sacker bifurcation curve of the fundamental harmonic periodic solutions. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(3): 30–40, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10126 [ABSTRACT FROM AUTHOR]

We analyze the complex quasiperiodic oscillations and chaos generated by two coupled piecewise-constant hysteresis oscillators driven by a rectangular wave force. Oscillations generate Arnol’d resonance webs wherein lower dimensional resonance tongues extend such as that of a web in numerous directions. We provide the fundamental tools for bifurcation analysis of nonautonomous piecewise-constant oscillators. To optimize the outstanding simplicity of piecewise-constant circuits, we formulate a generalized procedure for calculating the Lyapunov exponents in nonautonomous piecewise-constant dynamics. The Lyapunov exponents in these dynamics can be calculated with a precision approximately similar to that of maps. We observe two-parameter Lyapunov diagrams near the fundamental resonance region called Chenciner bubbles, wherein the oscillation frequencies of the two oscillators and the force are synchronized with a ratio of 1:1:1. Inevitably, the hysteresis considerably distorts the Chenciner bubbles. This result suggests that the Chenciner bubbles do not necessarily occur due to simple phase-locking of two-dimensional tori that can be explained by homeomorphism on the circle. Furthermore, we observe the Farey sequence in the experimental measurements. [ABSTRACT FROM AUTHOR]

This report presents an extensive investigation of bifurcations of quasi-periodic oscillations based on an analysis of a coupled delayed logistic map. This map generates an invariant two-torus (IT 2 ) that corresponds to a three-torus in vector fields. We illustrate detailed Lyapunov diagrams and, by observing attractors, derive a quasi-periodic saddle–node (QSN) bifurcation boundary with a precision of 1 0 − 9 . We derive a stable invariant one-torus (IT 1 ) and a saddle IT 1 , which correspond to a stable two-torus and a saddle two-torus in vector fields, respectively. We confirmed that the QSN bifurcation boundary coincides with a saddle–node bifurcation point of a stable IT 1 and a saddle IT 1 . Our major concern in this study is whether the qualitative transition from an IT 1 to an IT 2 via QSN bifurcations includes phase-locking. We prove with a precision of 1 0 − 9 that there is no resonance at the bifurcation point. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we elucidate the extremely complicated bifurcation structure of a weakly driven relaxation oscillator by focusing on chaos, and notably, on complex mixed-mode oscillations (MMOs) generated in a simple dynamical model. Our model uses the Bonhoeffer–van der Pol (BVP) oscillator subjected to a weak periodic perturbation near a subcritical Andronov–Hopf bifurcation (AHB). The mechanisms underlying the chaotic dynamics can be explained using an approximate one-dimensional map. The MMOs that appear in this forced dynamical model may be more sophisticated than those appearing in three-variable slow–fast autonomous dynamics because the approximate one-dimensional mapping of the dynamics used herein is a circle map, whereas the one-dimensional first-return map that is derived from the three-variable slow–fast autonomous dynamics is usually a unimodal map. In this study, we generate novel bifurcations such as an MMO-incrementing bifurcation (MMOIB) and intermittently chaotic MMOs. MMOIBs trigger an MMO sequence that, upon varying a parameter, is followed by another type of MMO sequence. By constructing a two-parameter bifurcation diagram, we confirmed that MMOIBs occur successively. According to our numerical results, MMOIBs are often observed between two neighboring MMOs. Numerically, MMOIBs may occur as many times as desired. We also derive the universal constant of the associated successive MMOIBs. The existence of the universal constant suggests that MMOIBs could occur infinitely many times. Furthermore, intermittently chaotic MMOs appear in this dynamical circuit. The intermittently chaotic MMOs relate to a type of intermittent chaos that resembles MMOs at first glance, but includes rare bursts over a long time interval. Complex intermittently chaotic MMOs of various types are observed, and we clarify that the intermittently chaotic MMOs are generated by crisis-induced intermittency. [Copyright &y& Elsevier]

In this paper, we analyze the sudden change from chaos to oscillation death generated by the Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbation. The parameter values of the BVP oscillator are chosen such that a stable focus and a stable relaxation oscillation coexist if no perturbation is applied. In such a system, complicated bifurcation structure is expected to emerge when weak periodic perturbation is applied because the stable focus and the stable relaxation oscillation coexist in close proximity in the phase plane. We draw a bifurcation diagram of the fundamental harmonic entrainment. The bifurcation structure is complex because there coexist two bifurcation sets. One is the bifurcation set generated in the vicinity of the stable focus, and the other is that generated in the vicinity of the stable relaxation oscillation. By analyzing the bifurcation diagram in detail, we can explain the sudden change from chaos with complicated waveforms to oscillation death. We make it clear that this phenomenon is caused by a saddle-node bifurcation. [ABSTRACT FROM AUTHOR]