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ABSTRACT: We present herein an extensive analysis of the bifurcation structures of quasiperiodic oscillations generated by a threecoupled delayed logistic map. Oscillations generate an invariant threetorus, which corresponds to a fourdimensional torus in vector fields. We illustrate detailed twoparameter Lyapunov diagrams, which reveal a complex bifurcation structure called an Arnol'd resonance web. Our major concern in this study is to demonstrate that quasiperiodic saddlenode bifurcations from an invariant twotorus to an intermittent invariant threetorus occur because of a saddlenode bifurcation of a stable invariant twotorus and a saddle invariant twotorus. In addition, with some assumptions, we derive a bifurcation boundary between a stable invariant twotorus and a stable invariant threetorus due to a quasiperiodic Hopf bifurcation with a precision of . 
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ABSTRACT: Bifurcations of complex mixedmode oscillations denoted as mixedmode oscillationincrementing 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 Bonhoeffervan der Pol circuit under weak periodic perturbation near a subcritical AndronovHopf 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.Chaos (Woodbury, N.Y.) 02/2015; 25(2):023105. DOI:10.1063/1.4907741 · 1.76 Impact Factor 
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ABSTRACT: This report presents an extensive investigation of bifurcations of quasiperiodic oscillations based on an analysis of a coupled delayed logistic map. This map generates an invariant twotorus (IT22) that corresponds to a threetorus in vector fields. We illustrate detailed Lyapunov diagrams and, by observing attractors, derive a quasiperiodic saddlenode (QSN) bifurcation boundary with a precision of 10−910−9. We derive a stable invariant onetorus (IT11) and a saddle IT11, which correspond to a stable twotorus and a saddle twotorus in vector fields, respectively. We confirmed that the QSN bifurcation boundary coincides with a saddlenode bifurcation point of a stable IT11 and a saddle IT11. Our major concern in this study is whether the qualitative transition from an IT11 to an IT22 via QSN bifurcations includes phaselocking. We prove with a precision of 10−910−9 that there is no resonance at the bifurcation point.Physica D Nonlinear Phenomena 09/2014; DOI:10.1016/j.physd.2014.09.001 · 1.83 Impact Factor 
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ABSTRACT: This study elucidates the bifurcation structure causing chaos disappearance in a foursegment piecewise linear Bonhoeffer–van der Pol oscillator with a diode under a weak periodic perturbation. The parameter values of this oscillator are chosen such that stable focus and stable relaxation oscillation can coexist in close proximity in the phase plane if no perturbation is applied. Chaos disappearance occurs through a previously unreported novel and unconventional bifurcation mechanism. To rigorously analyze these phenomena, the diode in this oscillator is assumed to operate as a switch. In this case, the governing equation is represented as a constraint equation, and the Poincaré map is constructed as an onedimensional map. By analyzing the Poincaré map, we clearly demonstrate why the stable relaxation oscillation that exists when no perturbation is applied disappears via chaotic oscillation when an extremely weak perturbation is applied.Nonlinear Dynamics 05/2014; 76(3):17111723. DOI:10.1007/s1107101412405 · 2.42 Impact Factor 
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ABSTRACT: This study analyzes an Arnold resonance web, which includes complicated quasiperiodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a twodimensional invariant torus (IT), which corresponds to a threedimensional torus in vector fields. Numerous onedimensional invariant closed curves (ICCs), which correspond to twodimensional tori in vector fields, exist in a very complicated but reasonable manner inside an ITgenerating region. Periodic solutions emerge at the intersections of two different thin ICCgenerating regions, which we call ICCArnold tongues, because all three independentfrequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICCArnold tongues through a NeimarkSacker bifurcation in the neighborhood of a quasiperiodic Hopf bifurcation (or a quasiperiodic NeimarkSacker bifurcation) boundary.Chaos (Woodbury, N.Y.) 03/2014; 24(1):013137. DOI:10.1063/1.4869303 · 1.76 Impact Factor 
01/2014; 1:887890. DOI:10.15248/proc.1.887

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ABSTRACT: In this paper, we discuss the bifurcation of a limit cycle to a threetorus in a piecewise linear thirdorder forced oscillator. A threetorus cannot be generated in thirdorder autonomous oscillators; our dynamical model exhibits a threetorus of minimal dimension. We adopt a thirdorder piecewise linear oscillator that exhibits a twotorus and apply a periodic perturbation to this oscillator. First, appropriate parameter values are selected to induce a limit cycle in the oscillator. In addition, this limit cycle is synchronized to the periodic perturbation. When the angular frequency of the periodic perturbation decreases, the oscillator is desynchronized, and a twotorus appears via a saddlenode bifurcation. This was verified by tracking the fixed point corresponding to the limit cycle on the Poincaré map and calculating the eigenvalues of the fixed point. Furthermore, the variation of a bifurcation parameter results in the generation of a threetorus via a quasiperiodic NeimarkSacker bifurcation. This bifurcation is identified as a quasiperiodic NeimarkSacker bifurcation from the observation of the second and third degenerate negative Lyapunov exponents. It was confirmed that all of the three Lyapunov exponents become zero at the quasiperiodic NeimarkSacker bifurcation point.09/2013; 2013(9). DOI:10.1093/ptep/ptt070 
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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 mixedmode 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 onedimensional map. The MMOs that appear in this forced dynamical model may be more sophisticated than those appearing in threevariable slow–fast autonomous dynamics because the approximate onedimensional mapping of the dynamics used herein is a circle map, whereas the onedimensional firstreturn map that is derived from the threevariable slow–fast autonomous dynamics is usually a unimodal map. In this study, we generate novel bifurcations such as an MMOincrementing 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 twoparameter 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 crisisinduced intermittency.Physica D Nonlinear Phenomena 09/2012; 241(18):1518–1526. DOI:10.1016/j.physd.2012.05.014 · 1.83 Impact Factor 
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ABSTRACT: We analyse a piecewiselinear oscillator that consists of a threeLCLC resonant circuit with a hysteresis element. Three sets of twodimensional 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 twotorus, threetorus, 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 “twotorus Arnold tongue” is observed where twotorus generating regions exist in a threetorus generating region as if periodic states exist in a twotorus generating region.Physica D Nonlinear Phenomena 07/2012; 241(14):1169–1178. DOI:10.1016/j.physd.2012.03.011 · 1.83 Impact Factor 
Conference Paper: Computational sensitivity of a high dimensional dynamical oscillator
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ABSTRACT: The difficulty arises when we carry out Lyapunov analysis for a high dimensional oscillator. Our model is an eightdimensional oscillator with a hysteresis element. This oscillator is piecewiselinear, and therefore, the explicit solution in each branch are obtained explicitly. We define the return map rigorously by using these explicit solutions. Numerical results show that we cannot often obtain a stationary solution even if we remove the transient 100,000 iterations of the return map. Furthermore, we encounter the following case: Lyapunov exponents are calculated by averaging 1,000,000 iterations of the Jacobian matrix of the return map to calculate the Lyapunov exponents. However, we cannot simply estimate and classify the solutions from the value of Lyapunov exponents in some cases even if the objective attractor is not chaotic, because the structure of oscillators with high dimensions are extremely complex.Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012; 01/2012 
Conference Paper: Chaos and Oscillation Death in a Weakly Driven Bonhoeffervan der Pol Oscillator near a Subcritical AndronovHopf Bifurcation
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ABSTRACT: We carry out bifurcation analysis for a piecewiselinear Bonhoeffervan der Pol oscillator under weak periodic perturbation. The parameter values are chosen such that a stable focus and a stable relaxation oscillation coexist when no perturbation is applied. When we apply weak periodic perturbation, complicated phenomena such as sudden change from chaos to oscillation death emerge. We analyze these phenomena by applying a piecewiselinear technique combined with a degenerate technique.Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012; 01/2012 
01/2012; 3(4):508520. DOI:10.1587/nolta.3.508

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ABSTRACT: In this paper, we analyze the sudden change from chaos to oscillation death generated by the Bonhoeffervan 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 saddlenode bifurcation.Physical Review E 11/2011; 84(5 Pt 2):056209. DOI:10.1103/PhysRevE.84.056209 · 2.33 Impact Factor 
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ABSTRACT: By using a remarkably simple and natural degenerate technique, the mechanism of chaos via torus breakdown observed in a simple fourdimensional autonomous circuit including two diodes is investigated rigorously. This degenerate technique is uniquely comparable to the wellknown slow–fast singular perturbation method. The idealized case where the diodes are assumed to operate as switches is considered. In this case, the governing equation is represented by a constrained equation, and the Poincaré mapping is derived rigorously as the circle map. The torus breakdown in the fourdimensional autonomous circuit is well explained by the Poincaré mapping. The theoretical results are verified by laboratory experiment.Physica D Nonlinear Phenomena 05/2011; 240(11):903912. DOI:10.1016/j.physd.2011.01.005 · 1.83 Impact Factor 
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ABSTRACT: In this study, we propose a remarkably simple oscillator that exhibits extremely complicated behaviors. The secondorder nonautonomous differential equation discussed in this Letter is considered to be one of the simplest dynamics that can produce mixedmode 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.Physics Letters A 04/2011; 375(14):15661569. DOI:10.1016/j.physleta.2011.02.053 · 1.63 Impact Factor 
Conference Paper: Collapse of mixedmode oscillations and chaos in the extended Bonhoeffervan Der pol oscillator under weak periodic perturbation.
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ABSTRACT: Mixedmode oscillations in a slowfast dynamical system under weak perturbation are studied numerically. First, we make a bandlimited extremely weak Gaussian noise, and apply this noise to this oscillator. Then, we observe random phenomenon from numerical study even if the noise is extremely weak. The mixedmode oscillations are submerged by chaos due to extremely weak noise. We imagine that mixedmode oscillations in a slowfast systems are delicate to the noise. In order to make clear the mechanism of generation of chaos, we assume that weak perturbation is periodic. From this assumption, we can calculate Lyapunov exponent, and draw a bifurcation diagram. In this bifurcation diagram, perioddoubling bifurcations take place when the amplitude of the periodic perturbation is extremely small. We suspect the observability of the mixedmode oscillation of the slowfast dynamical system by experiment from this numerical result.20th European Conference on Circuit Theory and Design, ECCTD 2011, Linkoping, Sweden, Aug. 2931, 2011; 01/2011 
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ABSTRACT: This Letter investigates the perioddoubling 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 perioddoubling bifurcation are followed by mixedmode oscillations. The detailed twoparameter bifurcation diagrams are derived, and it is clarified that the perioddoubling bifurcations arise from a narrow parameter value range at which the original canard in the nonextended equation is observed.Physics Letters A 08/2010; DOI:10.1016/j.physleta.2010.07.033 · 1.63 Impact Factor 
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ABSTRACT: 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 Poincaré map is constructed as a onedimensional map: a circle map. The parameter boundary between a torusgenerating region where the circle map is a diffeomorphism and a chaosgenerating 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 saddlenode bifurcation from the periodic state to the quasiperiodic state can be exactly distinguished. Laboratory experiment is also carried out and theoretical results are verified.International Journal of Bifurcation and Chaos 04/2008; 18(4). DOI:10.1142/S0218127408020835 · 1.02 Impact Factor 
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ABSTRACT: Two coexisting stable canards are discussed in a periodically driven singularly perturbed van der Pol equation, where the amplitude of the driving force is extremely small. Canard breakdown into chaos via perioddoubling bifurcations is also observed.Physics Letters A 04/2007; 363(56):404410. DOI:10.1016/j.physleta.2006.11.039 · 1.63 Impact Factor 
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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.Physics Letters A 01/2006; 348(3):187194. DOI:10.1016/j.physleta.2005.08.089 · 1.63 Impact Factor

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