# Naohiko Inaba Meiji University, Tokyo

Meiji University, Tokyo

## Mathematical Physics, Theoretical Physics, Computational Physics

## Publications

- [Show abstract] [Hide abstract]

**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 .Physics Letters A 03/2015; 379(7). DOI:10.1016/j.physleta.2014.12.022 · 1.63 Impact Factor - [Show abstract] [Hide abstract]

**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.Chaos (Woodbury, N.Y.) 02/2015; 25(2):023105. DOI:10.1063/1.4907741 · 1.76 Impact Factor - 01/2015; 6(3):433-442. DOI:10.1587/nolta.6.433
- [Show abstract] [Hide abstract]

**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.Physica D Nonlinear Phenomena 09/2014; 289. DOI:10.1016/j.physd.2014.09.001 · 1.83 Impact Factor - [Show abstract] [Hide abstract]

**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.Nonlinear Dynamics 05/2014; 76(3):1711-1723. DOI:10.1007/s11071-014-1240-5 · 2.85 Impact Factor - [Show abstract] [Hide abstract]

**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.Chaos (Woodbury, N.Y.) 03/2014; 24(1):013137. DOI:10.1063/1.4869303 · 1.76 Impact Factor - 01/2014; 1:887-890. DOI:10.15248/proc.1.887
- [Show abstract] [Hide abstract]

**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.Progress of Theoretical and Experimental Physics 09/2013; 2013(9). DOI:10.1093/ptep/ptt070 - [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 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.Physica D Nonlinear Phenomena 09/2012; 241(18):1518–1526. DOI:10.1016/j.physd.2012.05.014 · 1.83 Impact Factor - [Show abstract] [Hide abstract]

**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.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

[Show abstract] [Hide abstract]

**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.Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012; 01/2012 -
##### Conference Paper: Chaos and Oscillation Death in a Weakly Driven Bonhoeffer-van der Pol Oscillator near a Subcritical Andronov-Hopf Bifurcation

[Show abstract] [Hide abstract]

**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.Nonlinear Dynamics of Electronic Systems, Proceedings of NDES 2012; 01/2012 - 01/2012; 3(4):508-520. DOI:10.1587/nolta.3.508
- [Show abstract] [Hide abstract]

**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.Physical Review E 11/2011; 84(5 Pt 2):056209. DOI:10.1103/PhysRevE.84.056209 · 2.33 Impact Factor - [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 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.Physica D Nonlinear Phenomena 05/2011; 240(11):903-912. DOI:10.1016/j.physd.2011.01.005 · 1.83 Impact Factor - [Show abstract] [Hide abstract]

**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.Physics Letters A 04/2011; 375(14):1566-1569. DOI:10.1016/j.physleta.2011.02.053 · 1.63 Impact Factor -
##### Conference Paper: Collapse of mixed-mode oscillations and chaos in the extended Bonhoeffer-van Der pol oscillator under weak periodic perturbation.

[Show abstract] [Hide abstract]

**ABSTRACT:**Mixed-mode oscillations in a slow-fast dynamical system under weak perturbation are studied numerically. First, we make a band-limited 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 mixed-mode oscillations are submerged by chaos due to extremely weak noise. We imagine that mixed-mode oscillations in a slow-fast 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, period-doubling bifurcations take place when the amplitude of the periodic perturbation is extremely small. We suspect the observability of the mixed-mode oscillation of the slow-fast dynamical system by experiment from this numerical result.20th European Conference on Circuit Theory and Design, ECCTD 2011, Linkoping, Sweden, Aug. 29-31, 2011; 01/2011 - [Show abstract] [Hide abstract]

**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.Physics Letters A 08/2010; 374(36). DOI:10.1016/j.physleta.2010.07.033 · 1.63 Impact Factor - [Show abstract] [Hide abstract]

**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 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.International Journal of Bifurcation and Chaos 04/2008; 18(4). DOI:10.1142/S0218127408020835 · 1.02 Impact Factor - [Show abstract] [Hide abstract]

**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 period-doubling bifurcations is also observed.Physics Letters A 04/2007; 363(5-6):404-410. DOI:10.1016/j.physleta.2006.11.039 · 1.63 Impact Factor

- Questions & Answers
- Open Reviews
- Research Feedback