Calculation method for unstable periodic points in two-to-one maps using symbolic dynamical system

In this study, we have focused on the two-to-one maps and developed the numerical method to calculate the unstable periodic points (UPPs), based on the theory of the symbolic dynamical system. The core technique of the method is the definition of a non-deterministic map G. From the experimental result of three typical maps: logistic map, tent map, and Bernoulli map, we have confirmed the proposed method works very well within the defined errors. Our method has the following advantages: the method converges rapidly as the period of the target UPP is larger; we can choose the target UPP regardless of its cause (any bifurcation is not a matter); we can find the UPPs that are always unstable in the given parameter range. The convergence of the method is guaranteed by two standpoints: the corresponding symbolic dynamical system, and the asymptotic stability of UPP of G. Hereby, the error of the convergence is scalable according to the numeric precision of the software

Homoclinic bifurcation analysis for logistic map

In this study, we have developed the method to obtain the homoclinic bifurcation parameter of an arbitrary targeted fixed point in the logistic map Tr. We have considered the geometrical structure of Tr around x =0.5 and derived the core condition of the bifurcation occurrence. As the result of numerical experiment, we have calculated the exact bifurcation parameter of the fixed point of Trℓ with ℓ≤256. We have also discussed the Feigenbaum constants found in the bifurcation parameter and the fixed point coordinate sequences. This fact implies the local stability of the fixed point and global structure around it are in association via the constants

Bifurcation and Controlling Chaos in Nonlinear Dynamical System

本論文は，非線形常微分方程式で記述された力学系の分岐現象の解析，およびカ オス制御の問題と応用について述べている. 強制外力を加えた振り子結合系，ジョセフソン接合素子の結合系(超電導量子干 渉計)などの力学系は，常微分方程式に三角関数を含んでいる.そのため，状態空 間とパラメータ空間との積空間において周期性を有し，観測される現象もその周期 性を反映して複雑になっている.これら力学系にみられる平衡点，周期解の分岐現 象を解析し，解の振舞いの定性的性質を検討する. 第2章では，回転方向に弾性復元力をもつ振り子を取り上げ，パラメータの変化 に伴い発生するヘテロクリニック軌道の分岐構造を解明した.また， 2個の振り子 を，弾性復元力のある梁で接続した場合と，クラッチを有する剛体で接続した場合 について，定トルクを加えたとき発生する平衡点，周期解の分岐の詳細を検討した. これらの解析により，従来知られていなかった，高次元回転系の大域的な解の振舞 いを定性的に説明することができている. 連続系においてカオス応答がみられるとき，そのカオスアトラクタ中に埋め込ま れている不安定な周期軌道を安定化する問題をカオス制御という.第3章では，カ オス制御理論についての新手法，およびその応用を述べている.従来のカオス制御 系の手法の多くは，離散系に対してのみ制御器を構成していたが，本手法では，ポア ンカレ写像から導かれる離散系の周期点を安定化する制御器を設計し，状態フィー ドバックを極配置法で構成する.その制御入力を元の微分方程式系のパラメータへ 摂動として加える.つまり，差分方程式系と微分方程式との組合せによる合成力学 系での制御系設計を提案している.応用例としてステッピングモータにみられる脱 調現象の制御問題を取り上げた.この脱調現象は周期倍分岐連鎖によって引き起こ されるカオスであり実用上望ましくない運転状態である.まずステッピングモー タが周期インパルス列で駆動される振り子と等価であることを導出し，第2章の結 果を踏まえ，ポアンカレ写像の構成と分岐の解析を行なった.また，分岐を抑制す る制御器の設計とその制御系の数値シミュレーションを行ない，提案した制御方法 の有用性を示した

A Consideration of an ICT Tool for Chalk and Talk Lectures

学生の数理的思考を促し，演繹・帰納を通じて一定の技術を体得させるには，板書に併せた講述が効果的と考えられる。一方でプロジェクタなどの視覚装置を用いて画像・図面などを提示する教育効果も極めて大きい。本稿ではこれらの折衷案として，ICT ツールを用いて手書きと視聴覚コンテンツ両方の提示を実現する，安価な一つの方法を提案する。遠隔講義や大人数講義での実使用を通して明らかになってきた可能性と問題点について述べる。Many teachers notice that it is important to represent abstract images, development of equations, and summaries on a blackboard in lectures for promotion of students’ understanding. On the other hand, visual aids are also effective since figures and pictures encourage intuitive comprehension. We report new possibilities and problems with this tool through experience of operations in several lectures including remote lectures and massive classrooms

粗悪雑誌と業績評価

本コラムでは著者の経験や主観に基づく，粗悪雑誌が選択される仕組み，粗悪雑誌を回避する方法について述べる．以下では粗悪雑誌についての定義は了解されているものと仮定した上で話を進める．研究分野によって実情と合わない点も多いかと思うが，ご容赦願いたい

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, when the network structure has a certain type of symmetry, optimal solutions are not always detected since the search process may be trapped by local minima. In addition, a uniform pump rate for DOPOs in the conventional operation cannot overcome this problem. In this paper proposes a method to avoid trapping of the local minima by applying a control input in a pump rate of an appropriate node. This controller breaks the symmetrical property and causes to change the bifurcation structure temporarily, then it guides transient responses into the global minima. We show several numerical simulation results

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 find the parameter set that a homoclinic bifurcation arises and the corresponding horseshoe map. It is clarified 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

Computation of bifurcations : Automatic provisioning of variational equations

In the conventional implementations for solving bifurcation problems, Jacobian matrix and its partial derivatives regarding the given problem should be provided manually. This process is not so easy, thus it often induces human errors like computation failures, typing error, especially if the system is higher order. In this paper, we develop a preprocessor that gives Jacobian matrix and partial derivatives symbolically by using SymPy packages on the Python platform. Possibilities about the inclusion of errors are minimized by symbolic derivations and reducing loop structures. It imposes a user only on putting an expression of the equation into a JSON format file. We demonstrate bifurcation calculations for discrete neuron dynamical systems. The system includes an exponential function, which makes the calculation of derivatives complicated, but we show that it can be implemented simply by using symbolic differentiation

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

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 investigate a forced planer system obtained by replacing a cubic term in the Duffing equation with a hysteresis function. 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