Naohiko Inaba

Mathematical Physics, Theoretical Physics, Computational Physics

24.72

Publications

  • Munehisa SEKIKAWA · Naohiko INABA
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    ABSTRACT: This study investigates quasiperiodic bifurcations generated in a coupled delayed logistic map. Since a delayed logistic map generates an invariant closed curve (ICC), a coupled delayed logistic map exhibits an invariant torus (IT). In a parameter region generating IT, ICCgenerating regions extend in many directions like a web. This bifurcation structure is called an Arnol'd resonance web. In this study, we investigate the bifurcation structure of Chenciner bubbles, which are complete synchronization regions in the parameter space, and illustrate a complete bifurcation set for one of Chenciner bubbles. The bifurcation boundary of the Chenciner bubbles is surrounded by saddle-node bifurcation curves and Neimark-Sacker bifurcation curves. Copyright © 2015 The Institute of Electronics, Information and Communication Engineers.
    No preview · Article · Dec 2015 · IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
  • Munehisa Sekikawa · Naohiko Inaba
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    ABSTRACT: We discuss a complicated bifurcation structure involving several quasiperiodic bifurcations generated in a three-coupled delayed logistic map where a doubly twisted Neimark-Sacker bifurcation causes a transition from two coexisting periodic attractors to two coexisting invariant closed circles (ICCs) corresponding to two two-dimensional tori in a vector field. Such bifurcation structures are observed in Arnol'd tongues. Lyapunov and bifurcation analyses suggest that the two coexisting ICCs and the two coexisting periodic solutions almost overlap in the two-parameter bifurcation diagram.
    No preview · Article · Oct 2015 · Physics Letters A
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    ABSTRACT: An extensive bifurcation analysis of partial and complete synchronizations of three-frequency quasi-periodic oscillations generated in an electric circuit is presented. Our model uses two-coupled hysteresis oscillators and a rectangular wave forcing term. The governing equation of the circuit is represented by a piecewise-constant dynamics generating a three-dimensional torus. The Lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation. By analyzing this extremely simple circuit, we clearly demonstrate that it generates an extremely complex bifurcation structure called Arnol'd resonance web. Inevitably, chaos is observed in the neighborhood of Chenciner bubbles around which regions generating three-dimensional tori emanate. Furthermore, the numerical results are experimentally verified.
    No preview · Article · Aug 2015 · Physica D Nonlinear Phenomena
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    ABSTRACT: We present herein an extensive analysis of the bifurcation structures of quasi-periodic oscillations generated by a three-coupled delayed logistic map. Oscillations generate an invariant three-torus, which corresponds to a four-dimensional torus in vector fields. We illustrate detailed two-parameter Lyapunov diagrams, which reveal a complex bifurcation structure called an Arnol'd resonance web. Our major concern in this study is to demonstrate that quasi-periodic saddle-node bifurcations from an invariant two-torus to an intermittent invariant three-torus occur because of a saddle-node bifurcation of a stable invariant two-torus and a saddle invariant two-torus. In addition, with some assumptions, we derive a bifurcation boundary between a stable invariant two-torus and a stable invariant three-torus due to a quasi-periodic Hopf bifurcation with a precision of .
    No preview · Article · Mar 2015 · Physics Letters A
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    S. Hidaka · K. Kamiyama · T. Endo · N. Inaba · M. Sekikawa
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    ABSTRACT: In this study, we investigate an invariant three-torus (IT3) and the related bifurcations generated in a three-coupled delayed logistic map. Here, IT3 in this map corresponds to a four-torus in vector fields. First, we reveal that, to observe a clear Lyapunov diagrams for quasi-periodic oscillations, it is necessary and important to remove a large number of transient iterations, for example, 10,000,000 transient iteration count as well as 10,000,000 stationary iteration count to evaluate Lyapunov exponents in this higher dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by observing the graph of Lyapunov exponents, the global transition from an invariant two-torus (IT2) to an IT3 is a Neimark-Sacker type bifurcation that should be called a one-dimensional higher quasi-periodic Hopf bifurcation of an IT2. In addition, we confirm that another bifurcation route from an IT2 to an IT3 would be caused by a saddle-node type bifurcation that should be called a one dimensional higher quasi-periodic saddle-node bifurcation of an IT2.
    Full-text · Article · Feb 2015
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    Kuniyasu Shimizu · Munehisa Sekikawa · Naohiko Inaba
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    ABSTRACT: 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.
    Full-text · Article · Feb 2015 · Chaos (Woodbury, N.Y.)
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    ABSTRACT: This study investigates an invariant three-torus (IT3) and related quasi-periodic bifurcations generated in a three-coupled delayed logistic map. The IT3 generated in this map corresponds to a four-dimensional torus in vector fields. First, to numerically calculate a clear Lyapunov diagram for quasi-periodic oscillations, we demonstrate that it is necessary that the number of iterations deleted as a transient state is large and similar to the number of iterations to be averaged as a stationary state. For example, it is necessary to delete 10,000,000 transient iterations to appropriately evaluate the Lyapunov exponents if they are averaged for 10,000,000 stationary state iterations in this higher-dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by analyzing a graph of the Lyapunov exponents, we show that the local bifurcation transition from an invariant two-torus(IT2) to an IT3 is caused by a Neimark-Sacker bifurcation, which can be a one-dimension-higher quasi-periodic Hopf (QH) bifurcation. In addition, we confirm that another bifurcation route from an IT2 to an IT3 can be generated by a saddle-node bifurcation, which can be denoted as a one-dimension-higher quasi-periodic saddle-node bifurcation. Furthermore, we find a codimension-two bifurcation point at which the curves of a QH bifurcation and a quasi-periodic saddle-node bifurcation intersect.
    Full-text · Article · Jan 2015
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    ABSTRACT: 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 (IT22) 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 10−910−9. We derive a stable invariant one-torus (IT11) and a saddle IT11, 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 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 phase-locking. We prove with a precision of 10−910−9 that there is no resonance at the bifurcation point.
    No preview · Article · Sep 2014 · Physica D Nonlinear Phenomena
  • Naohiko Inaba · Munehisa Sekikawa
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    ABSTRACT: This study elucidates the bifurcation structure causing chaos disappearance in a four-segment 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 one-dimensional 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.
    No preview · Article · May 2014 · Nonlinear Dynamics
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    ABSTRACT: 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.
    Full-text · Article · Mar 2014 · Chaos (Woodbury, N.Y.)
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    ABSTRACT: Bifurcation transitions between a 1D invariant closed curve (ICC), corresponding to a 2D torus in vector fields, and a 2D invariant torus (IT), corresponding to a 3D torus in vector fields, have been the subjects of intensive research in recent years. An existing hypothesis involves the bifurcation boundary between a region generating an ICC and a region generating an IT. It asserts that an IT would be generated from a stable fixed point as a consequence of two Hopf (or two Neimark–Sacker) bifurcations. We assume that this hypothesis may puzzle many researchers because it is difficult to assess its validity, although it seems to be a reasonable bifurcation scenario at first glance. To verify this hypothesis, we conduct a detailed Lyapunov analysis for a coupled delayed logistic map that can generate an IT, and indicate that this hypothesis does not hold according to numerical results. Furthermore, we show that a saddle-node bifurcation of unstable periodic points does not coincide with the bifurcation b
    No preview · Article · Feb 2014 · Progress of Theoretical and Experimental Physics

  • No preview · Article · Jan 2014
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    Kaoru Itoh · Naohiko Inaba · Munehisa Sekikawa · Tetsuro Endo
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    ABSTRACT: In this paper, we discuss the bifurcation of a limit cycle to a three-torus in a piecewise linear third-order forced oscillator. A three-torus cannot be generated in third-order autonomous oscillators; our dynamical model exhibits a three-torus of minimal dimension. We adopt a third-order piecewise linear oscillator that exhibits a two-torus 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 two-torus appears via a saddle-node 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 three-torus via a quasi-periodic Neimark-Sacker bifurcation. This bifurcation is identified as a quasi-periodic Neimark-Sacker 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 quasi-periodic Neimark-Sacker bifurcation point.
    Full-text · Article · Sep 2013 · Progress of Theoretical and Experimental Physics
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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 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.
    Full-text · Article · Sep 2012 · Physica D Nonlinear Phenomena
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    ABSTRACT: We analyse a piecewise-linear oscillator that consists of a three-LCLC 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.
    Full-text · Article · Jul 2012 · Physica D Nonlinear Phenomena
  • Naohiko Inaba · Munehisa Sekikawa · Tetsuro Endo

    No preview · Article · Jan 2012
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    ABSTRACT: The difficulty arises when we carry out Lyapunov analysis for a high dimensional oscillator. Our model is an eight-dimensional oscillator with a hysteresis element. This oscillator is piecewise-linear, 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.
    Full-text · Conference Paper · Jan 2012
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    ABSTRACT: We carry out bifurcation analysis for a piecewise-linear Bonhoeffer-van 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 piecewise-linear technique combined with a degenerate technique.
    Full-text · Conference Paper · Jan 2012
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    ABSTRACT: 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.
    Full-text · Article · Nov 2011 · Physical Review E
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    Naohiko Inaba · Yoshifumi Nishio · Tetsuro Endo
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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 four-dimensional autonomous circuit including two diodes is investigated rigorously. This degenerate technique is uniquely comparable to the well-known 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 four-dimensional autonomous circuit is well explained by the Poincaré mapping. The theoretical results are verified by laboratory experiment.
    Full-text · Article · May 2011 · Physica D Nonlinear Phenomena

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