Kyohei Kamiyama

Three-dimensional tori and Arnold tongues

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.

Classification of Bifurcations of Quasi-Periodic Solutions Using Lyapunov Bundles

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincare section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability ...

BIFURCATION OF QUASI-PERIODIC OSCILLATIONS IN MUTUALLY COUPLED HARD-TYPE OSCILLATORS: DEMONSTRATION OF UNSTABLE QUASI-PERIODIC ORBITS

In this paper, we obtain bifurcations of quasi-periodic orbits occuring in mutually coupled hard-type oscillators by using our recently developed computer algorithm to directly determine the unstab...

Quasi-Periodic Bifurcations of Higher-Dimensional Tori

We classify the local bifurcations of quasi-periodic d-dimensional tori in maps (abbr. MTd) and in flows (abbr. FTd) for d ≥ 1. It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MTd can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark–Sacker bifurcations. Furthermore, normal bifurcations of FTd can be classified into three classes: saddle-node, double covering, and Neimark–Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MTd and FTd: namely, saddle-node cycle and heteroclinic cycle bifurcations of the (d − 1)-dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST1) and zero-dimensional tori in sections (abbr. ST0). The bifurcations of ST1 can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark–Sacker bifurcations. The bifurcations of ST0 can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark–Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST1/ST0 and the bifurcations of MTd/FTd. We present examples of all of these bifurcations.

BIFURCATION ANALYSIS OF THE PROPAGATING WAVE AND THE SWITCHING SOLUTIONS IN A RING OF SIX-COUPLED BISTABLE OSCILLATORS — BIFURCATION STARTING FROM TYPE 2 STANDING WAVE SOLUTION

In this paper, we analyze the bifurcation of Type 2 periodic solution in a ring of six-coupled bistable oscillators. We show that pitchfork and heteroclinic bifurcations, which induce chaos, cause a change from periodic (standing wave) solution to quasi-periodic (propagating wave) solution or the inverse when the coupling strength is varied. We also explain the existence of the switching solution, and presume that the birth and death of this switching solution are due to a pitchfork bifurcation of a quasi-periodic solution.

Study on the Artificial Synthesis of Human Voice Using Radial Basis Function Networks

In this study, we introduce the method of reconstructing more natural synthetic voice by using radial basis function network (RBF) that is one of neural network that is suitable for function approximation problems and following and synthesizing vocal fluctuations. In the synthetic simulation of RBF, we have set the Gaussian function based on parameters and tried to reconstruct the vocal fluctuations. With respect to parameter estimation, we have adopted to nonlinear least-squares method for making much account of the nonlinearity of human voice. When we have reproduced the synthesized speech, we have tried to reconstruct the nonlinear fluctuations of amplitude by adding normal random number. We have made a comparison the real voice and the synthetic voice obtained from simulation. As a consequence, we have found that it was possible to synthesize the vocal fluctuations for a short time.

SUPERCRITICAL PITCHFORK BIFURCATION OF THE QUASI-PERIODIC SWITCHING SOLUTION IN A RING OF SIX-COUPLED, BISTABLE OSCILLATORS

In our preceding study, we investigated the appearance and disappearance scenario of (quasi-periodic) switching solution in a ring of six-coupled, bistable oscillators. This system was expressed as twelve-dimensional phase space, which was divided into linear subspace H and its complimentary linear subspace H⊥. We investigated the stability of switching solution in H by calculating the conditional Lyapunov exponent to the direction of H⊥. As a result, we found a critical coupling factor αcrt at which switching solution lost its stability in the direction of H⊥. However, we did not know what type of bifurcation occurred at this point. In this study, we elucidate that switching solution causes a supercritical pitchfork bifurcation at αcrt.

Homoclinic Cycle Bifurcations in Planar Maps

In this study, bifurcations of an invariant closed curve (ICC) generated from a homoclinic connection of a saddle fixed point are analyzed in a planar map. Such bifurcations are called homoclinic cycle (HCC) bifurcations of the saddle fixed point. We examine the HCC bifurcation structure and the properties of the generated ICC. A planar map that can accurately control the stable and unstable manifolds of the saddle fixed point is designed for this analysis and the results indicate that the HCC bifurcation depends upon a product of two eigenvalues of the saddle fixed point, and the generated ICC is a chaotic attractor with a positive Lyapunov exponent.

Electronic Circuit Experiments and SPICE Simulation of Double Covering Bifurcation of 2-Torus Quasi-Periodic Flow in Phase-Locked Loop Circuit

Double covering (DC) bifurcation of a 2-torus quasi-periodic flow in a phase-locked loop circuit was experimentally investigated using an electronic circuit and via SPICE simulation; in the circuit...

Computational sensitivity in the analysis of Torus and its bifurcations

In this study, we investigate an invariant three-torus (IT3) and the related bifurcations generated in a three-coupled delayed logistic map. Here, IT3 in this map corresponds to a four-torus in vector fields. First, we reveal that, to observe a clear Lyapunov diagrams for quasi-periodic oscillations, it is necessary and important to remove a large number of transient iterations, for example, 10,000,000 transient iteration count as well as 10,000,000 stationary iteration count to evaluate Lyapunov exponents in this higher dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by observing the graph of Lyapunov exponents, the global transition from an invariant two-torus (IT2) to an IT3 is a Neimark-Sacker type bifurcation that should be called a one-dimensional higher quasi-periodic Hopf bifurcation of an IT2. In addition, we confirm that another bifurcation route from an IT2 to an IT3 would be caused by a saddle-node type bifurcation that should be called a one dimensional higher quasi-periodic saddle-node bifurcation of an IT2.