A method to suppress local minima for symmetrical DOPO networks

Coherent Ising machine (CIM) implemented by degenerate optical parametric oscillator(DOPO) networks can solve some combinatorial optimization problems. However, whenthe network structure has a certain type of symmetry, optimal solutions caused by pitchforkbifurcations are not always detected since local minima grasp the search process. In addition,a uniform pump rate for DOPOs in the conventional control operation cannot overcome thisproblem. In this paper proposes a method to avoid trapping of the local minima by applying acontrol input in a pump rate of an appropriate node. This control may break the symmetricalproperty and the bifurcation structure temporarily, then it guides the trajectory into the globalminima. We show several simulation results.

Nonlinear Theory and Its Applications, IEICE


Bending angles of a broken line causing bifurcations and chaos

We replace the cubic characteristics in the Duffing equation by two line segments connected at a point and investigate how an angle of that broken line conducts bifurcations to periodic orbits. Firstly we discuss differences in periodic orbits between the Duffing equation and a forced planar system including the broken line. In the latter system, a grazing bifurcation split the parameter space into the linear and nonlinear response domains. Also, we show that bifurcations of non-resonant periodic orbits appeared in the former system are suppressed in the latter system. Secondly, we obtain bifurcation diagrams by changing a slant parameter of the broken line. We also nd the parameter set that a homoclinic bifurcation arises and the corresponding horseshoe map. It is clari ed that a grazing bifurcation and tangent bifurcations form boundaries between linear and nonlinear responses. Finally, we explore the piecewise linear functions that show the minimum bending angles exhibiting bifurcation and chaos.


Influence of bifurcation structures revealed by refinement of a nonlinear conductance in Josephson junction element

We conduct a bifurcation analysis of a single-junction superconducting quantum interferometer with an external flux. We approximate the current-voltage characteristics of the conductance in the equivalent circuit of the JJ by using two types of function: a linear function and a piecewise linear (PWL) function. We describe a method to compute the local stability of the solution orbit and to solve the bifurcation problem. As the result, we reveal the bifurcation structure of the systems in a two-dimensional parameter plane. By making a comparison between the linear and PWL cases, we find a difference in the shapes of their bifurcation sets in the two-dimensional parameter plane even though there are no differences in the one-dimensional bifurcation diagrams or the trajectories. As for the influence of piecewise linearization, we discovered that grazing bifurcations terminate the calculation of the local bifurcations, because they drastically change the stability of the periodic orbit.

外部磁束によってドライブされるSQUID における分岐現象を検討した．コンダクタンス成分を線形，ならびに区分線形で近似表現した場合のジョセフソン接合素子の特性を検討した．分岐問題の解法について記述した． 結果として，パラメータ平面における分岐構造を明らかにできた．線形および区分線形コンダクタンスのそれぞれの結果を比較し，一次元分岐図や相図では見出すことができなかった相違を発見することができた．また，grazing 分岐が区分線形系では発生し，他の分岐を終端させていることも判った．


A general method to stabilize unstable periodic orbits for switched dynamical systems with a periodically moving threshold

In the previous study, a method to control chaos for switched dynamical systems with constant threshold value has been proposed. In this paper, we extend this method to the systems including a periodically moving threshold. The main control scheme is based on the pole placement, then a small control perturbation added to the moving threshold value can stabilize an unstable periodic orbit embedded within a chaotic attractor. For an arbitrary periodic function of the threshold movement, we mathematically derive the variational equations, the state feedback parameters, and a control gain by composing a suitable Poincaré map. As examples, we illustrate control implementations for systems with thresholds whose movement waveforms are sinusoidal and sawtooth-shape, and unstable one and two periodic orbits in each circuit are stabilized in numerical and circuit experiments. In these experiments, we confirm enough convergence of the control input.

先行研究では，固定したしきい値をもつ断続力学系におけるカオス制御を検討した．本論文では，それらのアイディアを拡張し，周期的に摂動するしきい値に対応した．極配置法を用いて合理的に小さい制御入力が，しきい値の周期摂動に上乗せされることにより，カオスに埋め込まれた不安定周期軌道が安定化される． 任意の周期関数をしきい値摂動に選んだとしても，ポアンカレ写像によって適切に変分方程式および制御器を導出できる．例としてしきい値が正弦波で摂動される場合ならびにのこぎり波で摂動される場合についての回路実装に対し，モデルに対する数値実験および回路実験を実施し，提案手法が有効であることを示す．

International Journal of Circuit Theory and Applications


Bifurcation analysis of eight coupled degenerate optical parametric oscillators

A degenerate optical parametric oscillator (DOPO) network realized as a coherent Ising machine can be used to solve combinatorial optimization problems. Both theoretical and experimental investigations into the performance of DOPO networks have been presented previously. However a problem remains, namely that the dynamics of the DOPO network itself can lower the search success rates of globally optimal solutions for Ising problems. This paper shows that the problem is caused by pitchfork bifurcations due to the symmetry structure of coupled DOPOs. Some two-parameter bifurcation diagrams of equilibrium points express the performance deterioration. It is shown that the emergence of non-ground states regarding local minima hampers the system from reaching the ground states corresponding to the global minimum. We then describe a parametric strategy for leading a system to the ground state by actively utilizing the bifurcation phenomena. By adjusting the parameters to break particular symmetry, we find appropriate parameter sets that allow the coherent Ising machine to obtain the globally optimal solution alone.

コヒーレントイジングマシンとしての縮退光パラメトリック発振器(DOPO)ネットワークは，組み合わせ最適化問題を解くことができる．先行研究ではその性能に関する理論的・実験的側面検討が行われた．しかし，DOPOネットワークはイジング問題としての最適解求解率からすると性能が低いことが示されている．本論文ではこの低い求解率はDOPOネットワーク自身のもつ対称性構造を原因としたピッチフォーク分岐が原因であることを指摘する．いくつかの2パラメータ分岐図は，性能劣化を如実に示す．この対称性からくる分岐構造によって，最適解に対応するground stateに系が到達する前に，局所解としての非ground state に状態がトラップされる． 本論文ではさらに，この局所解の発生を回避する戦略(パラメータ摂動方法)も示す．すなわち，対称性を能動的に破壊することにより，最適解へと状態を導くアイディアを示す．


Clock pulse modulation for ripple reduction in buck-converter circuits

DC-DC switching converters which are frequently treated as hybrid dynamical systems exhibit complex behavior due to nonlinear and interrupt characteristics. For synchronous buck-converters, we propose a method to control chaotic behavior by pulse frequency modulation. An input voltage, a duty ratio of PWMs, and so on, affect to the regulation characteristics of converters directly, but a frequency of PWMs is determined by the frequency characteristics of the converter and is set as a fixed value. The proposed chaos-control method suppresses chaotic responses by slightly perturbing the pulse frequency alone, therefore our method can stabilize unstable periodic orbits without influence on the voltage regulation scheme. To simplify the feedback controller, the condition of dimension reduction for the controlling gain vector is derived. The proposed controller acheves the stabilization without a current sensor. Numerical simulation and circuit implementation demonstrate the validity of this method.

しばしばハイブリッドシステムの例として取り上げられるDC-DCコンバータは，非線形断続特性によって複雑な振る舞いを呈する．同期型バックコンバータに対し，われわれはパルス周波数変調によるカオス制御手法を提案する．入力電圧やPWMのデューティ比などはコンバータの整流特性に直接影響するため，固定値とされる．本手法では，パルス周波数のみ摂動をかけ，カオス的応答を抑制し，結果として電圧整流には影響を出さずに不安定周期解を安定化させることができる．フィードバック制御器を単純化させるため，制御ベクトルの低次元化条件も導出した．制御器は電流センサを用いてないことは特筆できる．数値計算結果ならびに回路実験は，本提案の有効性を裏付ける．


Nonlinear resonance and devil's staircase in a forced planer system containing a piecewise linear hysteresis

The Duffing equation describes a periodically forced oscillator model with a nonlinear elasticity. In its circuitry, a saturable-iron core often exhibits a hysteresis. However, a few studies about the Duffing equation has discussed the effects of the hysteresis because of difficulties in their mathematical treatment. In this paper, we investigatea forced planer system demonstrating the Duffing equation containing the hysteresis. For simplicity, we approximate the hysteresis to a piecewise linear function. Since the solutions are expressed by combinations of some dynamical systems and switching conditions, a finite-state machine is derived from the hybrid system approach, and then bifurcation theory can be applied to it. We topologically classify periodic solutions and compute local and grazing bifurcation sets accurately. In comparison with the Duffing equation, we discuss the effects caused by the hysteresis, such as the devil's staircase in resonant solutions.

ダフィング方程式は非線形弾性をもつ強制振動系である．その回路実装においては非線形特性は飽和鉄心によって与えられ，しばしばそれはヒステリシス特性を示す．しかし, ダフィング方程式におけるヒステリシスの影響を検討した研究は殆ど存在せず，それはヒステリシスがもたらす数学的取り扱いの難しさからであると考えられる．この論文では，ヒステリシスを含むダフィング方程式を扱う．ただし，ヒステリシスは区分線形特性で与えることする．このとき系は可微分な有限オートマトンで記述されるハイブリッドシステムとなり，よって分岐理論が適用できる．発生する周期解の位相的分類を行い，それらの局所分岐および grazing 分岐をシューティング法で求めた．ダフィング方程式の結果と比較し，ヒステリシスの影響として，悪魔の階段が共振解に対して出現することを示す．


A Method to Detect Rogue Access Points in a Campus without Decrypting WLAN Frames

キャンパス内の不正アクセスポイントを無線フレームの復号なく検出する手法

Security problems of rogue APs exist because there are many wireless access points (APs) in a campus. In many cases, there are not only university-wide network administrators but also subnet administrators and the former ones usually do not know details of each subnet. We propose a method for the university-wide network administrators to detect rogue APs without decrypting WLAN frames. HTTP packets of captive portal detection are used in this method. As a result of our evaluation, we confirmed that it is possible to associate an IP address as seen from wired upstream network and a frame from WLAN.

大学に無線LAN アクセスポイント(以下，AP と呼ぶ)を導入するにあたり，許可を得ず接続されたAP や正規のAP を装ったAP 等の不正AP に関するセキュリティ上の課題が存在する．大学では，全学としてのネットワーク管理者(以下，全学ネットワーク管理者と呼ぶ)だけでなく，各サブネットワーク毎に管理者を指定していることが多く，ネットワーク全体を把握している管理者が存在しない可能性がある．本稿では，全学ネットワーク管理者の視点で，サブネットワーク管理者との連携を追加で要することなく，検査対象AP のキャンパスネットワークへの接続の有無を判断することにより不正AP を検出する手法について提案する．提案手法では，Windows やAndroid 等のOS に導入されている，Captive Portal Detection (以下，CPD と呼ぶ)を利用する．キャンパスネットワークの上流と無線LAN 通信区間の二箇所におけるCPD 用HTTP 通信の時間差から，検査対象AP のキャンパスネットワークへの接続を確認する．なお，本手法では無線LAN での通信において，WPA2 等の復号を行わず推定している．評価実験では，提案手法によりネットワーク上流から見た各サブネットワークのIP アドレスと無線LAN クライアントが接続したAP の紐付けが可能であることを確認した．

The Transactions of the Institute of Electronics, Information and Communication Engineers B

電子情報通信学会論文誌(B)


Multivalued discrete tomography using dynamical system that describes competition

Multivalued discrete tomography involves reconstructing images composed of three or more gray levels from projections. We propose a method based on the continuous-time optimization approach with a nonlinear dynamical system that effectively utilizes competition dynamics to solve the problem of multivalued discrete tomography. We perform theoretical analysis to understand how the system obtains the desired multivalued reconstructed image. Numerical experiments illustrate that the proposed method also works well when the number of pixels is comparatively high even if the exact labels are unknown.

離散値トモグラフィは3つ以上の複数レベルの濃度値を持つ画像を，投影から再構成する問題である．我々はこれら離散値トモグラフィを解く，競争原理を効果的に導入した非線形力学系に基づく連続時間最適化手法を提案する．この系がいかに画像再構成を達成するかに関する理論解析を与えた．また，数値実験では，ラベル数を与えていない高解像度画像であっても適切な結果が得られることを示している．


Bifurcation Analysis of the Nagumo-Sato Model and its Coupled Systems

The Nagumo-Sato model is a simple mathematical expression of a single neuron, and it is catego- rized as a discrete-time hybrid dynamical system. To compute bifurcation sets in such a discrete- time hybrid dynamical system accurately, conditions for periodic solutions and bifurcations are formulated herewith as a boundary value problem, and Newton's method is implemented to solve that problem. As the results of the analysis, the following properties are obtained: border- collision bifurcations play a dominant role in dynamical behavior of the model; chaotic regions are distinguished by tangent bifurcations; and multi-stable attractors are observed in its coupled system. We demonstrate several bifurcation diagrams and corresponding topological properties of periodic solutions.

佐藤南雲モデルは1つの神経細胞のダイナミクスを記述するもっとも簡素な数学モデルである．このような離散力学系ハイブリッドシステムにおける周期点の分岐集合を精度良く求めるために，境界値問題への定式化およびそのニュートン法によるシューティングを適用した．この系では border-collision 分岐がドミナントに発生し，接線分岐で区切られたカオス領域，複数のアトラクタによる多重安定領域が存在することが判明した．対応する分岐図と位相的分類に関する結果を示している．

International Journal of Bifurcation and Chaos


Calculation Method of Local Bifurcation Point in Piecewise Nonlinear Discrete-Time Dynamical Systems

This paper proposes a method for calculation of localbifurcation points in discrete-time dynamical systems withpiecewise nonlinear characteristics (PNDDS). First, ann-dimensional PNDDS, which has two piecewise nonlinearmaps, is shown and its variational equation is derived. Next,a calculation method for the local bifurcation points thatutilizes the conditional equation for the periodic solutionand the characteristic equation is proposed. It is essential tocalculate the derivatives of the map with an initial value andwith a bifurcation parameter to obtain the bifurcation pointscontinuously in the parameter space. The above calculationprocess is a key component of the proposed method, and isexplained in detail. Finally, we apply the proposed methodto a two-dimensional PNDDS and calculate the local bifurcationpoints in order to confirm the validity of the proposedmethod.


Can a Pseudo Periodic Orbit Avoid a Catastrophic Transition?

We propose a resilient control scheme to avoid catastrophic transitions associated with saddlenode bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark-Sacker bifurcations whose periodic orbits are kept existing through bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic obit can restore a stable limit cycle which disappeared after the saddle-node bifurcation.

力学系における周期解の接線分岐に関係した，カタストロフィックな過渡現象を回避するレジリエント制御の一手法を提案している．従来のフィードバック制御をもとにしたカオス制御手法では，分岐前後でターゲットとなる周期解は，安定性が変化するものの存在し続けることを利用している．レジリエント制御の立場からは，従って，パラメータ変化に対し，周期倍分岐，ネイマルクサッカー分岐を抑制することができる．ところが接線分岐は分岐後にターゲットとなる数学的集合が消滅するため，従来のカオス制御手法を適用できない．そこでわれわれは，疑似周期軌道を定義する．この軌道は存在し得ないが，接線分岐直後のベクトル空間では過渡的にはアトラクタ的性質を有している．この疑似周期軌道に，External force control を適用し安定化を測れば，パラメータ変化に対して接線分岐を抑制する制御が行える．手法と数値計算的エビデンスを示したあと，実際の電子回路において実現性や性能を実証した．


A computation method for non-autonomous systems with discontinuous characteristics

We propose a computation method to obtain bifurcation sets of periodic solutions in non-autonomous systems with discontinuous properties. If the system has discontinuity for the states and/or the vector field, conventional methods cannot be applied. We have developed a method for autonomous systems with discontinuity by taking the Poincare mapping on the switching point in the preceded study, however, this idea does not work well for some non-autonomous systems with discontinuity. We overcome this difficulty by extending the system to an autonomous system. As a result, bifurcation sets of periodic solutions are solved accurately with a shooting method. We show two numerical examples and demonstrate the corresponding laboratory experiment.

断続特性を持つ非自律系にみられる周期解の分岐集合を計算する手法を提案している．ベクトル空間や状態に断続性が含まれる場合，古典的な解析手法は適用できない．われわれは先行研究で断続自律系の分岐問題を解くアルゴリズムを考案しているが，それを非自律断続系に置き換えてるだけでは収束しない．そこで非自律系を自律系化し，改めて断続特性をポアンカレ写像として評価する．アルゴリズムは安定となり，ある例題で分岐集合を求めることができた．ハードウェア実験との整合性も確認している．


Controlling chaos of hybrid systems by variable threshold values

We try to stabilize unstable periodic orbits embedded in the given chaotic hybrid dynamical system by perturbation of a threshold value. In the conventional chaos control methods, a control input is designed by state-feedback value which is proportional to the difference between the target orbit and the current state, and it is applied into the specific system parameter as a small perturbation. During transition state, the control system consumes a certain control energy given by integration of such perturbations. In our method, we changes the threshold value dynamically to control the chaotic orbit. Unlike OGY method or the delayed feedback control, no actual control input is added into the system. The state-feedback is utilized only to determine the dynamic threshold value, thus the orbit starting from the current threshold value reaches the next controlled threshold value without any control energy. We obtain the variation of the threshold value from the composite Poincare map, and the controller is designed by the linear feedback theory with this variation. We demonstrate this method in simple hybrid chaotic systems and show performances with evaluating basins of attractions.

しきい値の変動を制御量として対象のカオス力学系に埋め込まれた不安定周期軌道を安定化する．この手法では過渡応答における制御エネルギーが発生しない．しきい値の変分についても解析的に求める手段を応用しており，制御フィードバックゲインも指定極に合わせて正確に算出可能である．2つのカオス的ハイブリッドシステムへの適用例を挙げ，そのうち1つには回路実装も与える．


区分非線形離散時間力学系における局所的分岐点の計算法

This paper proposes a calculation method of local bifurcation points for discrete-time dynamical systems with piecewise nonlinear characteristic (PNDDS). First, an n-dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is defined. Next, the calculation method of the local bifurcation points is proposed using the conditional equation of the periodic solution and the characteristic equation. Here, we have to calculate derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key of the proposed method, and we explain it in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points for confirming its validity.

IEEJ Transactions on Electronics, Information and Systems

電気学会論文誌C (電子，情報，システム部門誌)


Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit: analytic extension and numerical validation

In the research field of nonlinear dynamical system theory, it is well known that a homoclinic/heteroclinic point leads to unpredictable motions, such as chaos. Melnikov's method enables us to judge whether the system has a homoclinic/heteroclinic orbit. Therefore, in order to assess a vessel's safety with respect to capsizing, Melnikov's method has been applied for investigations of the chaos that appears in beam sea rolling. This is because chaos is closely related to capsizing incidents. In a previous paper (Maki et al. in J Mar Sci Technol 15:102 106 2010), a formula to predict the capsizing boundary by applying Melnikov's method to analytically obtain the non-Hamiltonian heteroclinic orbit was proposed. However, in that paper, only limited numerical investigation was carried out. Therefore, further comparative research between the analytical and numerical results is conducted, with the result being that the formula is validated.

船の横揺れ現象に現れるカオス現象は，船舶の転覆と密接な関係があるため，古くから研究の対象となってきた，本研究では，これまでに著者らが提案した転覆限界を見積もるための理論計算手法と数値実験結果を比較した．その結果，理論的に求めた転覆限界を示すパラメータ値において，初期値平面のフラクタル的浸食が生じることを確認し，手法の妥当性を確認した．

Journal of Marine Science and Technology


Analytical methods to predict the surf-riding threshold and the wave-blocking threshold in astern seas

For the safe design and operation of high-speed craft, it is important to predict the behavior of these craft in following and astern quartering seas as it is well known that serious problems can occur when a vessel is forced by the waves to travel at wave speed. The surf-riding threshold is the lower speed limit above which the vessel will be forced to travel at wave speed (usually on the face of the wave) which is generally accepted to be a prerequisite for broaching. While a vessel travelling below this speed will experience significant changes in its longitudinal speed in the wave, it will not be forced to travel at wave speed. For high-speed craft, the wave-blocking threshold also becomes important. This is the upper speed, below which the vessel will also be forced to travel at wave speed (usually on the back of the wave) and is related to the possibility of bow diving. By the application of a polynomial approximation to the waveinduced surge force, including the nonlinear surge equation, an analytical formula to predict both the surf-riding and the wave-blocking thresholds is proposed. Comparative results of the surf-riding threshold and wave-blocking threshold predicted utilizing the proposed formula and the thresholds predicted using numerical bifurcation analysis indicate fairly good agreement. In addition, previously proposed analytical formulae are examined.

追波中の波乗り現象の発生及び消失の境界を求めるため，理論的・実験的検討を行った．その中で，前後揺れ方程式の非線形項を多項式近似することにより，新たな波乗り限界推定公式を導き，実験との比較を行った．その結果，今回提案した手法は，実験結果をよく説明することが分かった．


Bifurcation Analysis of a Simple 3D Oscillator and Chaos Synchronization of its Coupled Systems

Tamasevicius et al. proposed a simple 3d chaotic oscillator for educational purpose. In fact the oscillator can be implemented very easily and it shows typical bifurcation scenario so that it is a suitable training object for introductory education for students. However, as far as we know, no concrete studies on bifurcations or applications on this oscillator have been investigated. In this paper, we make a thorough investigation on local bifurcations of periodic solutions in this oscillator by using a shooting method. Based on results of the analysis, we study chaos synchronization phenomena in diffusively coupled oscillators. Both bifurcation sets of periodic solutions and parameter regions of in-phase synchronized solutions are revealed. An experimental laboratory of chaos synchronization is also demonstrated.

Tamasevicius らはダイオードとインバータからなる簡素な構成の3階自律系カオス発振器を提案している．これは周期解の典型的な局所分岐シナリオを呈し，学生実験などで適切な素材であると言われている．しかしながらその分岐構造や応用はまだ報告されていない．そこでわれわれは局所分岐を大域的なパラメータ平面について求め，分岐構造を明らかにした．また，それらの結合系におけるカオス同期化現象について，その存在パラメータ範囲などを分岐図を通じて明らかにした．


Bifurcation analysis with threshold values for interrupt autonomous systems

断続自律系方程式における分岐のしきい値解析

スイッチや状態変数の跳躍で表される断続力学系の研究が盛んに行われている．複数の異なるベク トル場が接合して力学系を構成する場合，しばしばしきい値が接合面(断続面)の位置を決める条件となる．系 に現れる周期軌道は，しきい値をパラメタとしたとき，その値の変化に対して様々な分岐現象を生じる．状態変 数やパラメタが断続面において微分不可能となる断続系では，従来の解析手法の適用に困難をもたらすが，断続 面を局所断面とし局所座標系を構成するPoincare 写像を定義すれば，シューティング法による分岐解析が行な えることが示されている．しかしながら，しきい値の定義式は一般にモデル方程式に陽に含まれないため，分岐 パラメタ値としてのしきい値に対するシューティング解法は知られていない．本論文では，断続系の分岐理論を 拡張し，しきい値をパラメタとする変分方程式を導出し，求積する手法について述べる．さらに，具体例として 結合Izhikevich ニューロンモデルを取り上げ，分岐構造の詳細を説明する

The Transactions of the Institute of Electronics, Information and Communication Engineers A

電子情報通信学会論文誌(A)


Analytical formulae for predicting the surf-riding threshold for a ship in following seas

Making use of Melnikov's method, a generalized formula for predicting the surf-riding threshold is developed as an extension to the applications of Kan and Spyrou. A new analytical formula for calculating the surfriding threshold of a ship in following seas is also proposed in light of nonlinear dynamical system theory. By applying a continuous piecewise linear approximation to the waveinduced surge force, a heteroclinic bifurcation point is obtained analytically with an uncoupled surge equation. Results calculated using these formulae are presented, and they show good agreement with those obtained utilizing numerical bifurcation analysis. Further, it was confirmed that the surf-riding threshold obtained using the proposed formulae agrees reasonably well with that obtained experimentally for an unconventional vessel.

追波中での波乗り現象の発生を予測する手法を得るため，非線形力学系理論の観点から検討を実施した．その中で，前後揺れ方程式中の非線形項を区分的に線形化することにより，波乗りの限界を理論的に見積もる手法を提案し，実験により検証を行い，その有効性を確認した．


Bifurcation Analysis of Izhikevich Neuron Model

An autonomous differential equation with jumps has been proposed by Izhikevich for a demonstration of various spiking waves by changing its parameters. Some well-known firing patterns of this model are shown by numerical simulations, however bifurcation analysis of the model has not been investigated in detail. In this paper, we apply our numerical computation algorithm to this model. Although it is a challenge to manage jumps with the Poincar´e mapping and we obtain bifurcation parameter values by a shooting method. As a result, we conclude that some spiking modes are caused by local bifurcation of limit cycles.

状態にジャンプを伴うイジケビッチが提案した自律系神経モデルは，系のパラメータを変化させると，自然界で観測される様々なスパイク・バースト振動モードを再現する．われわれはこのモデルに関する分岐問題を解く．幾つかのスパイクの変遷は分岐現象によって説明できることを示す．

Dynamics of Continuous, Discrete and Impulsive Systems. Series A: Mathematical Analysis


Bifurcations observed in BVP oscillator with periodically switched RC circuit

Many works have been devoted to the study of the behavior of the Bonhoe er-van der Pol (BVP) oscillator in physiological and biochemical modeling studies. In this paper, bifurcation and chaos in BVP oscillator, which is intermittently connected to a parallel RC load, is investigated. By using the periodicity of switching action we introduce a composite discrete mapping as the Poincar´e mapping that is useful for the analysis of piecewise nonlinear systems. Adjusting the switching period close to the integral multiple of the period of self-oscillation, we find various synchronizations. In particular the fundamental synchronization is discussed. Under the appropriate switching ratios, we illustrate that the synchronized state bifurcates to chaotic states by the period doubling cascades or the torus breaking down.

RCサブ回路に断続的に接続するBVP発振器の分岐問題について検討する．スイッチの周期に合わせたポアンカレ写像を考えることにより，可微分な合成写像を導くことができ，解析が進められる．自然周期に近いスイッチ周波数を選べば，様々な同期現象が認められる．周期解ロック領域，準周期解領域，カオス領域等を分岐図で説明する．


Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit

The chaos that appears in the ship roll equation in beam seas known as the escape equation has been intensively investigated because it is closely related to capsizing incidents. In particular, many applications of the Melnikov integral formula have been reported in the literature; however, in all the analytical works concerning the escape equation, the Melnikov integral is formulated utilizing a separatrix for the Hamiltonian part or a numerically obtained heteroclinic orbit for the non-Hamiltonian part of the original escape equation. To overcome such limitations, this article attempts to utilise an analytical expression for the non-Hamiltonian part. As a result, an analytical procedure is provided that makes use of a heteroclinic orbit of the non-Hamiltonian part within the framework of the Melnikov integral formula.

船の横揺れ現象に現れるカオス現象は，船舶の転覆と密接な関係があるため，古くから研究の対象となってきた，本研究では，非線形横揺れ運動方程式(転覆方程式)のノンハミルトニアン部の解を解析的に求め，それについてメルニコフの手法を適用することで転覆の限界を得た．

Journal of Marine Science and Technology


Chaos Control Implementation by Interrupted Electric Circuit with Switching Delay

We have demonstrated that the chaotic circuit with a switching delay is modeled by a return map, and a controller for the suppression of chaos is proposed. A circuit representing a controller stabilizing a period-1 unstable periodic orbit in an interrupted electric circuit with a certain switching delay is also discussed.

スイッチの遅れを持つカオス回路においてリターンマップを定義し，カオスを抑制する制御器を提案した．具体的な該当する系において周期1の不安定周期軌道を安定化した．

International Journal of Bifurcation and Chaos


Numerical Bifurcation Analysis Framework for Autonomous Piecewise-Smooth Dynamical Systems

In this paper, we consider a numerical method for the bifurcation analysis method of nonlinear piecewise-smooth systems. While linear piecewise-smooth systems can be analyzed rigorously, nonlinear piecewise-smooth systems cannot lead to an analytical solution. Derived from both continuous and discrete system analysis approaches, our method uses a Poincar´e map to transform the results of partial analysis over the continuous components into those issued from a discrete mapping. We then apply conventional methods in order to find critical parameter values and obtain bifurcation diagrams in the parameter space. The numerical procedure is fully described and illustrated by the analysis results of various versions of the Alpazur oscillator.

区分的に滑らかな(区分非線形な)自律系対する，統一的・数値的な分岐解析の手法を提案する．これらの系は解析解が求まらず，また断続点において微分が不可となる，解析が困難な系であるが，われわれは問題を，可微分な写像に変換し，その導関数や微分値も数値的に得ることができるため，統一的な手順でアトラクタの解析が行える．例としてAlpazur 発振器回路の特性を変更した様々なバージョンの回路について解析例を挙げる．


Bifurcation Analysis of Iterative Image Reconstruction Method for Computed Tomography

Of the iterative image reconstruction algorithms for computed tomography (CT), the power multiplicative algebraic reconstruction technique (PMART) is known to have good properties for speeding convergence and maximizing entropy. We analyze here bifurcations of fixed and periodic points that correspond to reconstructed images observed using PMART with an image made of multiple pixels and we investigate an extended PMART, which is a dynamical class for accelerating convergence. The convergence process for the state in the neighborhood of the true reconstructed image can be reduced to the property of a fixed point observed in the dynamical system. To investigate the speed of convergence, we present a computational method of obtaining parameter sets in which the given real or absolute values of the characteristic multiplier are equal. The advantage of the extended PMART is verified by comparing it with the standard multiplicative algebraic reconstruction technique (MART) using numerical experiments.

コンピュータトモグラフィーにおける反復画像再構成アルゴリズムとしてPMARTが広く知られている．本論文ではPMARTの反復過程を分岐問題と捉え，観測され固定点や周期点について解析する．また，加速アルゴリズムを付与したPMARTについても検討する．固定点問題を解き，分岐図を示すとともに，収束速度などの性能について言及する．MARTとの比較を行い，数値結果とともにPMARTの優位性を述べる．


Bifurcation analysis of current coupled BVP oscillators

The Bonhöffer van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

BVP発振器はニューロンを模擬するシンプルなモデルである．BVP発振器はその独立変数に応じたポートが存在し，複数のBVP発振器をこれらのポートを通じて結合させると様々な力学的特性を表現できる．本論文は2つのBVP発振器の電流ポート同士を抵抗で結合した系の分岐問題を検討する．特徴的な結果として，周期解の逆相同期が安定に存在することが分岐図より示されることが挙げられる．


Stable torus and its bifurcation phenomena in a Three-Dimensional Autonomous Circuit

It is well known that a stable torus is observed as a result after the system meets the super-critical Neimark Sacker bifurcation for a limit cycle. Although tori are easily observed in two-dimensional and periodically forced dynamical systems, there is a few papers about stable tori in three-dimensional autonomous systems. Besides, as physical circuit implementations, such circuits contain very special active elements, or have difficulty in realizing. In this paper, we show a very simple circuit of three-dimensional autonomous system, an extended Bonhöffer van der Pol (BVP) oscillator, which is demonstrating a stable torus. Firstly we explain a discovery of the torus in a computer simulation of the model equation. To implement it as a circuitry, we design a new nonlinear resistor. Although this contains an FET and an op-amp, it is simpler than any other nonlinear resistors proposed in previous papers. We confirm that this BVP oscillator can generate a stable torus in a real circuitry. We thoroughly investigate the bifurcation phenomena of various limit cycles and tori in this circuit, i.e., a super-critical Neimark Sacker and tangent bifurcations of limit cycles are concretely obtained, furthermore, phase locking and chaos regions are clarified in a bifurcation diagram.

安定なトーラスは通常，リミットサイクルの超臨界なNeimark-Sacker分岐を経て得られることが知られている．2次元非自律系では容易に観測されるトーラスではあるが，3次元自律系で安定なトーラスが観測される報告は多くなく，観測されたとしても極めて技巧的な能動素子を含んでいる．本論文では極めてシンプルなBVP発振器をもとに構成された3次元自律系回路で安定なトーラスが生じることを示す．またこのトーラスに関わる様々な局所分岐を計算し，分岐図として表した．


Control of chaos in a piecewise smooth nonlinear system

This paper shows the stabilization of the unstable periodic orbit of any given piecewise smooth system with linear and/or nonlinear characteristics. By utilizing the periodicity of the switching action, we construct the Poincare´ mapping including all information of the original system. This mapping offers a first step toward extending a novel technique for controlling chaos based on the appropriate state feedback in piecewise smooth nonlinear systems. We also apply this approach to Rayleigh type oscillator described by the piecewise smooth nonlinear systems.

本論文では区分的に滑らかな非線形力学系におけるカオス中の不安定周期解を安定化する一般論を示す．スイッチ等の動作における周期性を利用し，ポアンカレ写像を構成し，これをカオス制御のフィードバックゲイン設計に用いる．レイリー発振器の区分非線形系にこの技術を適用し，制御可能であることを数値的に示す．


A design method of bursting using 2-parameter bifurcation diagrams in FitzHugh-Nagumo model

Spiking and bursting observed in nerve membranes seem to be important when we investigate information representation model in the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose a design method to generate bursting responses in FitzHugh Nagumo model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the two-parameter bifurcation diagram.

脳における情報処理表現を検討する上で神経膜の興奮性に基づくスパイクやバースト発振の出現は重要である．これらの発振の定性的性質の分類はなされているが，われわれはFitzHugh-Nagumoの単独モデルに周期外力を印加するモデルを提案し，さまざまなバースト発振を設計できることを示す．


Performance of Chaos and Burst Noises Injected to the Hopfield NN for Quadratic Assignment Problems

In this paper, performance of chaos and burst noises injected to the Hopfield Neural Network for quadratic assignment problems is investigated. For the evaluation of the noises, two methods to appreciate finding a lot of nearly optimal solutions are proposed. By computer simulations, it is confirmed that the burst noise generated by the Gilbert model with a laminar part and a burst part achieved the good performance as the intermittency chaos noise near the three-periodic window.

本論文では，二次割り当て問題を解く Hopfield Neural Network にカオスノイズとバーストノイズを注入した際の性能についての調査を行っている．さらに最適解に近い解をいかに多く見つけるかを判断すること ができる評価方法を提案している．コンピュータシミュレーションの結果，インターミッテンシー·カオスのラミナー部とバースト部の存在をモデリングしたギルバートモデルによって得られるバーストノイズが3周期窓付近のインターミッテンシー·カオスと同様のよい性能であることが確認されている．

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


Bifurcation and chaos in coupled BVP oscillators

Bonh¨offer van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

抵抗で電圧結合された BVP 発振器系における平衡点および周期解の分岐について検討している．FETをインバータで特性反転させたものを非線形素子とすると，それはシグモイド関数をよく近似しており，単一のBVP回路においては硬い発振が観測される．十分大きい値の抵抗によりこれらを結合すると周期倍分岐，ピッチフォーク分岐を経て，カオスが生じる．回路パラメータからの数理モデルの導出とその数値計算結果を示し，結合系内部の抵抗値による2パラメータ平面における分岐集合を求めた．これらによりカオスの存在領域や，カオスの発生メカニズムが明らかとなった．また，数値計算結果は極めてよい精度で回路実験結果と合致することも確認された．

International Journal of Bifurcation and Chaos


Experimental Investigation of Controlling Chaos using Pole Assignment in the Forced BVP Oscillator

周期外力を印加したBVP発振器における極配置法に基づくカオス制御の実験的検討

In previous work(1), we proposed a control method of the chaotic attractor in the non-autonomous systems and stabilized the unstable periodic orbits observed in the Duffing's equation. Our method can be used as a general technique for controlling chaos, however, the control function was not demonstrated in the laboratory experiments. With this in mind, in this paper, we consider the experimental control of the chaotic behavior in the circuit model, called the forced BVP oscillator. The pole assignment for the corresponding discrete system derived from such a non-autonomous system via Poincaré mapping works effectively, and the control unit is easily realized by the window comparator, sample-hold circuits, and so on. In particular, we try to stabilize the unstable 1 and 3-periodic orbits in the chaotic attractor in the numerical and experimental simulations.

強制BVP発振器におけるカオスについて，カオス制御器の実回路実装について検討する．ウィンドウコンパレータ，サンプルホールド器を使用し，極配置法に基づく制御器が実現される．幾つかの不安定周期軌道を本手法により安定化した．

IEEJ Transactions on Electronics, Information and Systems

電気学会論文誌C (電子，情報，システム部門誌)


Bifurcation of Burst Response in an Amari-Hopfield Neuron Pair with a Periodic External Force


Controlling chaos in a state-dependent nonlinear system

In this paper, we propose a general method for controlling chaos in a nonlinear dynamical system containing a state-dependent switch. The pole assignment for the corresponding discrete system derived from such a nonsmooth system via Poincar e mapping works effectively. As an illustrative example, we consider controlling the chaos in the Rayleigh-type oscillator with a state-dependent switch, which is changed by the hysteresis comparator. The unstable one- and two-periodic orbits in the chaotic attractor are stabilized in both numerical and experimental simulations.

状態に依存し切り替わるスイッチを有する非線形力学系に関するカオス制御の一般的手法について提案する．ポアンカレ写像を巧妙に定義すれば，それから導かれる離散写像やそのヤコビは適切に評価でき，制御系設計に活かされる．例としてレイリータイプ発振器に状態依存型スイッチが組込まれた系について，そのカオス中に含まれる不安定周期軌道を安定化する．


Chaos and Control of Periodically Switched Nonlinear Systems

This paper discusses chaos and its control in periodically switched nonlinear systems. Since the trajectory of the system is not differentiable at break points, the Poincar sections should be naturally attached at these points and then the Poincar mapping is derived as a composite of local mappings defined among these sections. This map enable us to calculate the location and its stabilities of the periodic points in the chaotic attractor. We also consider chaos control problem based on the Poincar map. The controlled system is fully described as the differentiable map, there fore the controller can be designed by using conventional pole assignment technique. As an illustrated example, we consider a Rayleigh type oscillator containing a periodic switch. We firstly discuss bifurcation sets of the system. The dynamic properties of periodic or bits and chaotic attractors e clarified. We also investigate bifurcation structure of phase locked regions along a Neimark-Sacker bifurcation set. Some theoretical results are con firmed by laboratory experiments. Secondly we try to stabilize one and two-period unstable periodic orbits by using the proposed method. The controller is calculated easily because all information can be obtained from the Poincar map. The results of numerical simulations are demonstrated.

本論文では周期スイッチを有する非線形系におけるカオスとその制御について議論している．そのような系では，軌道がブレークポイントで微分が不可能となり，従来の手法をそのまま適用はできない そこでポアンカレ写像をそのブレークポイントに取付け，周期解に対応する固定点方程式を可微分になるよう構成した．これにより各種局所分岐が計算できる．系にカオスが生じる場合，それに埋め込まれた不安定周期軌道を安定化する制御手法を検討した．

The Latin American Applied Research Journal


Controlling bifurcations of discrete maps

Bifurcation control has attracted increasing attention in recent years. A simple and unfied state-feedback methodology is developed in this paper for various bifurcation controls for discrete-time systems. The control task can be either shifting an existing bifurcation or creating a new one, for transcritical, pitchfork, saddle-node, and period- doubling bifurcations. Some computer simulations are included to illustrate the methodology and verify the theoretical results.

離散写像における分岐制御の手法を提案する．ピッチフォーク分岐，接線分岐，周期倍分岐などの分岐現象を生成する制御タスクについて述べ，理論を裏付ける数値シミュレーションについて示す．


Anticontrol of chaos by making a nonchaotic system chaotic has led to the discovery of some new chaotic systems, particularly the continuous-time three-dimensional autonomous Chen's equation with only two quadratic terms. This paper further investigates some basic dynamical properties and various bifurcations of Chen's equation, thereby revealing its different features from some other chaotic models such as its origin, the Lorenz system.

非カオス的な力学系に不安定化制御を適用する研究のなかで，新たなカオス力学系が発見された．このチェン方程式と呼ばれる三次元自律系について，平衡点や周期解の分岐について詳細に解析を行った．チェン方程式とローレンツ方程式と定性的に比較し，位相的差異について議論する．


Bifurcation of Switched Nonlinear Dynamical Systems

This paper proposes a method to trace bifurcation sets for a piecewise-defined differential equation. In this system, the trajectory is continuous, but it is not differentiable at break points of the characteristics. We define the Poincare mapping by suitable local sections and local mappings, and thereby it is possible to calculate bifurcation parameter values. As an illustrated example, we analyze the behavior of a two-dimensional nonlinear autonomous system whose state space is constrained on two half planes concerned with state dependent switching characteristics. From investigation of bifurcation diagrams, we conclude that the tangent and global bifurcations play an important role for generating various periodic solutions and chaos. Some theoretical results are confirmed by laboratory experiments

状態の条件によってスイッチが動作し，従う力学系が切り替えられる機構をもつハイブリッドシステムの分岐問題の計算一般理論を記述している．本論文では，スイッチが切り替わるタイミングでポアンカレ写像を定義し，その時点における軌道の変分量を数値積分し，その値を写像によって誘導される差分方程式系のヤコビ行列とすることを提案し，さらに固定点·周期点，および分岐パラメータの計算アルゴリズムを述べている．従来取扱いが困難であった軌道の不可微分点の扱いが可能となり，数値的に厳密な計算が実現される．例題としてある自律系ハイブリッドシステムを取り上げ，分岐集合の計算や，カオスとの関連を述べている．本論文は IEEE/ISCAS 会議にて投稿を推薦されたものである．

IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing


Synchronization of Electric Fireflies by using Square Wave Generators

In this article, we propose a square wave generator whose switching threshold values are switched by external inputs. This circuit is designed to simulate the synchronized luminescence of coupled fireflies. We investigate the behavior of the solutions in two coupled oscillators. The dynamics are demonstrated by a linear autonomous equation piecewisely, therefore, a one-dimensional return map is derived. We also prove the existence of stable in-phase synchronization in the coupled oscillator by using the return map, and we show the existence of regions of periodic solutions within a parameter space. Some theoretical results are confirmed by laboratory measurements.

方形波発振を行うマルチバイブレータを，フォトカップラとフォトダイオードによって結合した場合に生じる同期現象について議論する．このモデルは，水を半分入れたペットボトルをパイプでつないだシステムを模擬している．初期値とパラメータによって準周期を基本とした周期解の同期が豊富に生じる．本論文ではモデル方程式の分岐解析を行い，これらパラメータ平面における同期化領域を詳細に調べた．系は区分線形系であり，分岐条件は適切なポアンカレ写像を設けることにより厳密に解析が行えた．

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


Bifurcation phenomena in the Josephson junction circuit coupled by a resistor

Bifurcation phenomena observed in a circuit containing two Josephson junctions coupled by a resistor are investigated. This circuit model has a mechanical analogue; Two damped pendular linked by a clutch exchanging kinetic energy of each pendulum. In this paper, firstly we study equilibria of the system. Bifurcations and topological properties of the equilibria are clarified. Secondly we analyse periodic solutions in the system by using suitable Poincare mapping and obtain a bifurcation diagram. There are two types of limit cycles distinguished by whether the motion is in S1 * R3 or T2 * R2, since at most two cyclic coordinates are included in the state space. There is a typical structure of tangent bifurcation for 2-periodic solutions with a cusp point. We found chaotic orbits via the period-doubling cascade, and a long-period stepwise orbit.

抵抗で結合されるジョセフソン接合素子の数理モデルについて検討した．状態に関する正弦関数で記述されるこのモデルは，そのモジュロや対象性を大きく反映した特異な応答を示す．本論文ではこの系にみられる分岐現象およびカオスについて議論する．モデルの力学的アナログである，クラッチで接続される2つの振り子モデルを考えた場合，合理的な結果を示していることがわかった．クラッチの摩擦係数と入力とのパラメータ平面における分岐集合を求め，クラッチを介して伝達する2つめの振り子へのエネルギーの変化によって，解の位相的分類が行えることを示す．


A Study of the Pendulum Equation with a Periodic Impulsive Force --- Bifurcation and Control

The pendulum equation with a periodic impulsive force is investigated. This model described by a second order differential equation is also derived from dynamics of the stepping motor. In this paper, firstly, we analyze bifurcation phenomena of periodic solutions observed in a generalized pendulum equation with a periodic impulsive force. There exist two topologically different kinds of solution which can be chaotic by changing system parameters. We try to stabilize an unstable periodic orbit embedded in the chaotic attractor by small perturbations for the parameters. Secondly, we investigate the intermittent drive characteristics of two-phase hybrid stepping motor. We suggest that the unstable operations called pull-out are caused by bifurcations. Finally, we proposed a control method to avoid the pull-out by changing the repetitive frequency and stepping rate.

ステッピングモータはスピンドルの一回転に多くのステップがあり，対称性と断続特性を用いて原点周りのシリンダー相平面に運動を集約することができる．本論文では，このステッピングモータ集約モデルにおいて，外力に対する応答について，その周期解の分岐現象を解析した．ステッピングモータの脱調は，この相平面における周期解の周期倍分岐連鎖後のカオスであることを示す．そして同時にカオス状態において埋め込まれている不安定周期軌道を線形制御理論を用いて安定化を図った．


Bifurcation of an inductively coupled Josephson junction circuit

Some qualitative properties of an inductively coupled circuit containing two Josephson junction elements with a dc source are investigated. The system is described by a four-dimensional autonomous differential equation. However, the phase space can be regarded as S^1×R^3 because the system has a periodicity for the invariant transformation. In this paper, we study the properties of periodic solutions winding around S^1 as a bifurcation problem. Firstly, we analyze equilibria in this system. The bifurcation diagram of equilibria and its topological classification are given. Secondly, the bifurcation diagram of the periodic solutions winding around S^1 are calculated by using a suitable Poincare mapping, and some properties of periodic solutions are discussed. From these analyses, we clarify that a periodic solution so-called "caterpillar solution" [1] is observed when the two Josephson junction circuits are weakly coupled.

2つのジョセフソン接合素子の等価回路を抵抗結合し，直流電源を 接続した回路における分岐現象について検討した． おのおのの等価回路は，摩擦のある振り子と等価であり，抵抗結合によって， 個々の系の角速度の情報がお互いに影響を及ぼすようになる． そのこの系はトーラス空間上の運動となる． 力学的には，入力トルクにより振り子1を駆動し，そのエネルギーの一部を クラッチによって振り子2に伝達しているシステムとなっている． この報告では，結合係数と入力をパラメータに取り，平衡点についての 定性的性質を考察し，さらに周期解の分岐について述べる．M. Levi が発見しているうなり振動モードについて，その分岐と発生メカニズムを解析し，このモードは，周期解の周期倍分岐による二周期解であること，およびそのパラメータ平面における存在領域を示した．


Bifurcation of heteroclinic orbits in a circuit containing a Josephson junction element

The circuit equation of the circuit containing a Josephson junction and the equation of motion of a pendulum in which an elastic restoring force is applied in the direction of rotation are described in terms of a second-order ordinary differential equation containing the sum of a trigonometric term on the state and a linear term. In such a dynamic system, the number of saddles and sinks changes according to the parameter values and no periodicity exists with respect to the location of the saddles and sinks, thus analysis is difficult. In this paper, a circuit containing a Josephson junction is chosen as an example of such a dynamic system. Classification by the tangent bifurcation of the saddles and sinks and the results of the analysis based on the heteroclinic orbits on the phase plane are presented. The heteroclinic orbit presents a global bifurcation on the phase plane due to its structural instability. The parameters generated by this orbit are used as the values for the bifurcation, and the bifurcation set was derived on the parameter plane. As a result, the unique linking number is defined for a particular parameter region surrounded by the bifurcation curve. This value can be used as a guideline for rotation of the orbit on the phase plane. Further, by combining it with the bifurcation ensemble, a classification of the stable sinks in the basin is made possible.

Electronics and Communications in Japan, Part 3 : Fundamental Electronic Science


Vibration suppressing control of flexible rotary crane using tip position sensor

For vibration suppressing control of flexible rotary cranes, two control methods by the use of a tip position sensor have been proposed. One is a control system that suppresses transient vibrations generated by the impact of switching of driving speeds when Bang-Bang control, open-loop anti-swing control, is completed. The other is a servo system to which vibration suppressing feedback is added. Experiments have confirmed that either system is remarkably effective in suppressing vibration.

柔軟なブームをもつ旋回クレーンについて，その吊り荷揺れメカニズムのモデリングと制御を検討した．チップ位置センサを用いてブームの揺れを計測し，PID制御を施す．実際に実機を試作し，精密にモデルのパラメータを導出した．センサにはレーザコリメータとカメラを用い，データ処理および制御量の決定はパーソナルコンピュータを用いている．計算機シミュレーションと実機による計測により，提案手法の有効性を示している．


Efficient Dynamic Simulation Method of Cranes with a Spherical Pendulum

球面振子を有するクレーンのシミュレーション法

旋回型クレーンにおいて，クレーン本体と吊り荷を一つの開ループリンク機構としてモデリングを行うことを提案している．受動関節と仮想リンクという新しい概念を導入し，従来使用されている開ループリンク機構ロボット用に開発された高速シミュレーション方法が適用可能となった．本文ではリンクの構成問題の詳述と，適用例，それらの数値計算結果を示している．適用例では，実験用に製作したクレーンの実パラメータを用いてシミュレーションを行い，実機とシミュレーションとの差違についても検討している．また，結果の3次元ソリッドモデルによるグラフィクス表示も試みている．

Transactions of The Japan Society of Mechanical Engineers, Series C

日本機械学会論文集(C編)


A Questionnaire System with An Authentication Framework Considering Online Activities of Users

It is difficult to distribute online questionnaire adequately to those subjects who satisfy requirements of investigations. In addition, low questionnaire response rates, because of factors such as overlooking request emails and noncooperation to answer the questionnaires, sometimes annoy the investigators. Now that today's organizational systems adopt a unified authentication framework of which logging function stores authentication records of users in federated systems, the possible solution for these problems arises by utilizing the log data and adding the questionnaire process to the framework. The present paper proposes a flow controlling mechanism arranged in a major unified authentication framework and describes how to utilize log data on the framework to select target persons for questionnaires and increase the response rates.

学内の各種アンケートはアンケート対象者へ適切に依頼することが困難である．加えて，依頼メールの見落とし，アンケートに答えない，などの要因でアンケート回収率は低くなり，依頼者を困らせる．昨今組織で運用される情報システムでは，連携するシステムにユーザの認証情報を格納する統一認証フレームワークを採用している．そこで，このログデータを活用し，アンケートシステムを認証プロセス内に紛れ込ませることにより，アンケートの回収率を向上させる機構を提案する． 主要認証フレームワークにフロー制御機構を採り入れ，ログデータよりアンケート対象者を選択させる．

IEEJ Transactions on Electrical and Electronic Engineering (TEEE)


Steady state analysis of the TCP network with RED algorithm

The transmission control protocol with a random early detection (TCP/RED) is an important algorithm for a TCP congestion control. It has been expressed as a simple second-order discrete-time hybrid dynamical model, and shows unique and typical nonlinear phenomena, e.g., bifurcation phenomena or chaotic attractors. However, detailed behavior is unclear due to discontinuity that describes the switching of transmission phases in TCP/RED, but we have proposed its analysis method in previous study. This letter clarifies bifurcation structures with it.

TCP通信におけるREDは重要な技術である．そのダイナミクスは簡素な2次元離散ハイブリッドシステムで表わされ，分岐現象やカオスなど，典型的な非線形現象を呈する．しかし，TCP/RED の送信フェーズにおけるスイッチングを決める不連続性により，詳細な振る舞いはまだ明確にされていない．前報にて解析方法を提案したので，本論文では分岐構造を明らかにする．

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


Bifurcations and Phase Differences of Nonlinear Coupled Dictyostelium Systems

Nagano proposed a mathematical model of coupled Dictyostelium systems and demonstrated the strong tendency of in-phase synchronization. In this study, we investigate the mechanisms of this phenomenon by obtaining bifurcation sets and analyzing the phase differences of responses. From the analysis results, the model has a wide parameter range giving a stable 1-periodic solution. Thus, the one-to-one synchronization is maintained regardless of changes in the environment parameters. For example, even if the frequency ratio of two oscillators is extremely unbalanced, the model with an appropriate coupling factor shows a 1-periodic solution.

Naganoらは結合粘菌発振系において強く同相同期がかかることを示唆している．本論文では，この系に対する分岐構造計算および位相差の計算によって同期現象の機序を検討する．結果として，系は広く1-周期解のパラメータレンジを持ち，パラメータ変動には大変鈍感に1対1同相同期を維持することがわかった．2つの発振器が相当アンバランスなパラメータであっても，適当な結合度合いで容易に同相同期が達成される．


Computational Method of Border-collision Bifurcation Point for Piecewise Nonlinear Discrete-Time Dynamical Systems

区分非線形離散時間力学系に生じるBorder-collision 分岐の分岐パラメータ計算法

IEEJ Transactions on Electronics, Information and Systems

電気学会論文誌C (電子，情報，システム部門誌)


Bifurcations in two-dimensional Hindmarsh-Rose type model

We analyze a two-dimensional Hindmarsh Rose type model exhibiting properties of both Class 1 and Class 2 neurons. Although the system is two-dimensional and contains only four parameters, the obtained bifurcation diagrams show that the bifurcation structure satisfies conditions for emergence of both features with constant stimuli.

クラス1およびクラス2のニューロ応答を行うHindmarsh Rose型2次元非自律系を解析した．直流バイアスを印加する環境下でありながら，4つのパラメータによりクラス1および2の特徴を示すことが出来る最も簡単な2次元系であることが示される．

International Journal of Bifurcation and Chaos


In this letter, we investigate the cross-coupled BVP oscillators. A single BVP oscillator has two terminals which can extract an independent state variable, so in the preceding works, several coupling systems have been studied. Synchronization modes and chaos in these systems are classified as results of bifurcation problems. We revisit a coupled oscillator, and clarify new results which have not been reported before, i.e. stable tori and its breakdown, and chaotic motions. Also classification of synchronized periodic solutions is done by a bifurcation diagram.

交差結合されたBVP発振器を解析している．単体のBVP発振器は2つの独立変数にかかるポートを持っており，今までにも様々なBVP結合系が検討されてきている．本論文では，分岐問題を解き，同期モードをそれらの分類により説明している．さらに他の結合系に比較し，まだ報告のない応答について述べる．安定なトーラスや，その崩壊，カオスの出現などを示す．


Experimental Realization of Controlling Chaos in the Periodically Switched Nonlinear Circuit

This letter presents an experimental confirmation of controlling the chaotic behavior of a target unstable periodic orbit when the periodically switched nonlinear circuit has a chaotic attractor. The pole assignment for the corresponding discrete system derived from such a nonautonomous system via Poincar´e mapping works effectively, and the control unit is easily realized by the window comparator, sample-hold circuits, and so on.


Bifurcation in asymmetrically coupled BVP oscillators

BVP oscillator is the simplest mathematical model describing dynamical behavior of the neural activity. The large scale neural network can often be described naturally by coupled systems of BVP oscillators. However, even if two BVP oscillators are merely coupled by a linear element, the whole system exhibits a complicated behavior. In this paper, we analyze coupled BVP oscillators with asymmetrical coupling structure, with each oscillator having a different internal resistance. We present a complete four-dimensional system, which shows a rich variety of bifurcation phenomena, and strange attractors. We calculate bifurcation diagrams and confirm relaxation phenomena in the laboratory experiments. We also briefly report a conspicuous strange attractor.

BVP発振器は神経回路の動特性を模擬した最も単純な回路であり，大規模な神経回路網はBVP発振器の結合系で表現される．しかしながらたった2つのBVP発振器を，線形受動素子で結合させた場合であっても極めて複雑な応答が生じる．この論文では非対称に結合されたBVP発振器を解析するが，1つの4次元自律系と扱うべき，多様な応答が生じている．解の位相的性質は，分岐問題を解くことにより説明される．実際の回路実装における応答と，解析結果の対照についても述べる．


On Synchronization and Control of Coupled Wilson-Cowan Neural Oscillators

This paper investigates the complex dynamics, synchronization and control of chaos in a system of strongly connected Wilson Cowan neural oscillators. Some typical synchronized periodic solutions are analyzed by using the Poincar e mapping method, for which bifurcation diagrams are obtained. It is shown that topological change of the synchronization mode is mainly caused and carried out by the Neimark Sacker bifurcation. Finally, a simple feedback control method is presented for stabilizing an in-phase synchronizing periodic solution embedded in the chaotic attractor of a higher-dimensional model of such coupled neural oscillators.

本論文では，結合強度の大きいWilson-Cowan型神経発信回路の結合系の複雑な挙動，同期やカオスの制御について検討している．結果として，当該力学系では同期モードの変化は主にNeimark-Sacker分岐によるものであることを，分岐図等をもとに示す．また，同相同期などを保持し続けるための制御則を提案し，シミュレーションにより有用性を示している．


Chaos in a Simple Hybrid System and its Control

A simple hybrid system exhibiting chaos is developed and a chaos control method realised. It is shown that, since the dynamics can be expressed by a return map, the system has an observable chaos. The control method, which is based on the pole assignment technique, can stabilise any unstable periodic orbit in the chaos. The theoretical result is verified by laboratory experiments.

抵抗とキャパシタ，比較器とフリップフロップのみからなるカオス回路を提案し，解析している．一階の常微分方程式がしきい値とクロックパルスによって切り替えられるこの系は，連続状態と離散イベントから構成されるハイブリッドシステムとなっている．系の振る舞いは厳密解を用いて，skew tent map と呼ばれる区分線形写像に変換することができる．回路実験とシミュレーション結果を示した．


On Impulsive Control of a Periodically Forced Chaotic Pendulum System

We consider impulsive control of a periodically forced pendulum system which has rich chaos and bifurcation phenomena. A new impulsive control method for chaos suppression of this pendulum system is developed. Some simple sufficient conditions for driving the chaotic state to its zero equilibrium are presented, and some criteria for eventually, exponentially asymptotical stability are established. This work provides a rigorous theoretical analysis to support some early experimental observations on controlling chaos in the periodically forced pendulum system.

周期外力を印加されたカオス的振り子は多様な分岐現象を示す．この系の平衡点に関するインパルス制御を考える．解析的に安定な制御則に関する十分条件を示し，シミュレーションを行った．制御則は指数漸近的であることが理論的に示されている．


A Simple Approach to Calculation and Control of Unstable Periodic Orbits in Chaotic Piecewise Linear Systems

This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor, and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given.

この論文は区分線形な自律系のカオス中に埋め込まれた不安定周期軌道を安定化する単純な手法を提案している．当該アルゴリズムは所望のUPOを検索することに使え，また，それらは簡素な状態フィードバックによって安定化可能である．制御系の安定性について，簡単な解析も行っている．


This Letter reports the finding of a new chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.

ローレンツ方程式の安定平衡点について不安定化制御を試みる過程で発見した自律系のカオスについて報告している．不安定な平衡点付近まで十分軌道が巻き込まれる，特徴のある形状のカオスアトラクタを示す．すでに有名なカオス力学系として知られるローレンツ方程式，レスラー方程式の双方と比較して，発見した方程式は，位相的に異なる力学系となっていることを確認した．この論文自体は以上の点についての速報であるが，出版後，他の論文から多くの引用がなされるようになった．


A computation of bifurcation parameter values for limit cycles

This letter describes a new computational method to obtain the bifurcation parameter value of a limit cycle in nonlinear autonomous systems. The method can calculate a parameter value at which local bifurcations; tangent, period-doubling and Neimark-Sacker bifurcations are occurred by using properties of the characteristic equation for a fixed point of the Poincare mapping. Conventionally a period of the limit cycle is not used explicitly since the Poincare mapping needs only whether the orbit reaches a cross-section or not. In our method, the period is treated as an independent variable for Newton's method, so an accurate location of the fixed point, its period and the bifurcation parameter value can be calculated simultaneously. Although the number of variables increases, the Jacobian matrix becomes simple and the recurrence procedure converges rapidly compared with conventional methods.

本論文では，非線形自律系としてモデル化された力学系について，リミットサイクルの分岐パラメータ値を計算する新たな計算アルゴリズムを提案している．リミットサイクルの周期は，パラメータ値に依存して変化する値であるため本来変数と扱うべきである．従来法ではポアンカレ写像を取るための数値積分時間を得るため，定数と扱っていた．しかし，この扱いにより，むしろ状態やパラメータに関する方程式は周期に対する依存度が高くなり，ヤコビは複雑となり，分岐パラメータ計算アルゴリズムにおいては収束性に影響を与えていた．本アルゴリズムでは周期を変数とし，方程式の独立性も高くなって，収束性能も高くなったことを示している．

IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences


On Unstable Saddle-Node Connecting Orbit in a Planer Autonomous System

We found a novel connecting orbit in the averaged Duffing-Rayleigh equation. The orbit starts from an unstable manifold of a saddle type equilibrium point and reaches to a stable manifold of a node type equilibrium. Although the connecting orbit is structurally stable in terms of the conventional definition of structural stability, it is structually unstable since a one-dimensional manifold into which the connecting orbit flows is unstable. We can consider the orbit is one of global bifurcations governing the differentiability of the closed orbit.

われわれはこれまでに報告のない新しいタイプのコネクティングオービットを，ダフィングレイリー方程式において発見した．これはサドル型平衡点の不安定多様体から，安定結節点の安定多様体をつなぐものであり，構造安定性に関する新たな議論を喚起する．この軌道を含む分岐図を精密に計算し，解の位相的性質を分類した