Naohiko Inaba Meiji University, Tokyo
Meiji University, Tokyo
Mathematical Physics, Theoretical Physics, Computational Physics
Publications
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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 saddlenode bifurcation curves and NeimarkSacker bifurcation curves. Copyright © 2015 The Institute of Electronics, Information and Communication Engineers.  [Show abstract] [Hide abstract]
ABSTRACT: We discuss a complicated bifurcation structure involving several quasiperiodic bifurcations generated in a threecoupled delayed logistic map where a doubly twisted NeimarkSacker bifurcation causes a transition from two coexisting periodic attractors to two coexisting invariant closed circles (ICCs) corresponding to two twodimensional 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 twoparameter bifurcation diagram.  [Show abstract] [Hide abstract]
ABSTRACT: An extensive bifurcation analysis of partial and complete synchronizations of threefrequency quasiperiodic oscillations generated in an electric circuit is presented. Our model uses twocoupled hysteresis oscillators and a rectangular wave forcing term. The governing equation of the circuit is represented by a piecewiseconstant dynamics generating a threedimensional 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 threedimensional tori emanate. Furthermore, the numerical results are experimentally verified.  [Show abstract] [Hide abstract]
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 .  [Show abstract] [Hide abstract]
ABSTRACT: In this study, we investigate an invariant threetorus (IT3) and the related bifurcations generated in a threecoupled delayed logistic map. Here, IT3 in this map corresponds to a fourtorus in vector fields. First, we reveal that, to observe a clear Lyapunov diagrams for quasiperiodic 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 discretetime 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 twotorus (IT2) to an IT3 is a NeimarkSacker type bifurcation that should be called a onedimensional higher quasiperiodic Hopf bifurcation of an IT2. In addition, we confirm that another bifurcation route from an IT2 to an IT3 would be caused by a saddlenode type bifurcation that should be called a one dimensional higher quasiperiodic saddlenode bifurcation of an IT2.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
ABSTRACT: This study investigates an invariant threetorus (IT3) and related quasiperiodic bifurcations generated in a threecoupled delayed logistic map. The IT3 generated in this map corresponds to a fourdimensional torus in vector fields. First, to numerically calculate a clear Lyapunov diagram for quasiperiodic 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 higherdimensional discretetime 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 twotorus(IT2) to an IT3 is caused by a NeimarkSacker bifurcation, which can be a onedimensionhigher quasiperiodic Hopf (QH) bifurcation. In addition, we confirm that another bifurcation route from an IT2 to an IT3 can be generated by a saddlenode bifurcation, which can be denoted as a onedimensionhigher quasiperiodic saddlenode bifurcation. Furthermore, we find a codimensiontwo bifurcation point at which the curves of a QH bifurcation and a quasiperiodic saddlenode bifurcation intersect.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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 saddlenode bifurcation of unstable periodic points does not coincide with the bifurcation b 
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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.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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. 

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. 
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.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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.
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