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Magnified view of the bifurcation diagram of Fig. 4 near the boundary between 0 D and 2 D with ε 1 = 0.025 and ε 2 = 0.035.
Source publication
Bifurcation transitions between a 1D invariant closed curve (ICC), corresponding to a 2D torus in vector fields, and a 2D invariant torus (IT), corresponding to a 3D torus in vector fields, have been the subjects of intensive research in recent years. An existing hypothesis involves the bifurcation boundary between a region generating an ICC and a...
Contexts in source publication
Context 1
... we discuss the NS bifurcation of the fixed point denoted by the red line. Figure 6 shows a mag- nified view of Fig. 4 near the NS bifurcation of 0 D. Figure 6 shows that the NS bifurcation boundary between 0 D and 2 D denoted by the red line completely coincides with the boundary between the fixed-point-generating region and an ICC-generating region. The red line is derived using a shooting algorithm defined in Ref. [21]. ...
Context 2
... we discuss the NS bifurcation of the fixed point denoted by the red line. Figure 6 shows a mag- nified view of Fig. 4 near the NS bifurcation of 0 D. Figure 6 shows that the NS bifurcation boundary between 0 D and 2 D denoted by the red line completely coincides with the boundary between the fixed-point-generating region and an ICC-generating region. The red line is derived using a shooting algorithm defined in Ref. [21]. ...
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Citations
... The development of computing power has led to a new surge of interest in systems with multi-frequency quasiperiodic dynamics. [12][13][14][15][16][17][18][19] Bifurcations of tori were classified as the saddle-node, Hopf, and torus doubling types. 12,13 Various bifurcation mechanisms of the transition from two-frequency to three-frequency quasiperiodic dynamics were investigated. ...
... 12,13 Various bifurcation mechanisms of the transition from two-frequency to three-frequency quasiperiodic dynamics were investigated. [14][15][16] The synchronous regimes in the periodically forced resonant limit cycle on a torus, 17 in coupled multimode oscillators with time-delayed feedback, 18 and coupled autonomous quasiperiodic generators 19 were examined. Analysis of torus doubling bifurcations for three and four dimension maps, which began many years ago by Kaneko,20,21 was recently performed for the simplest 4D model of a real radio-physical generator. ...
The dynamics of three three-dimensional repressilators globally coupled by a quorum sensing mechanism was numerically studied. This number (three) of coupled repressilators is sufficient to obtain such a set of self-consistent oscillation frequencies of signal molecules in the mean field that results in the appearance of self-organized quasiperiodicity and its complex evolution over wide areas of model parameters. Numerically analyzing the invariant curves as a function of coupling strength, we observed torus doubling, three torus arising via quasiperiodic Hopf bifurcation, the emergence of resonant cycles, and secondary Neimark–Sacker bifurcation. A gradual increase in the oscillation amplitude leads to chaotizations of the tori and to the birth of weak, but multidimensional chaos.
... In contrast, higher-dimensional tori generated in higher-dimensional dynamics have also been the subjects of intensive research in recent years [7][8][9][10][11][12][13][14][15][16][17][18]. In our previous study, serious difficulties were encountered during the numerical analysis of a high-dimensional oscillator [18]. ...
This study investigates an invariant three-torus (IT3) and related quasi-periodic bifurcations generated in a three-coupled delayed logistic map. The IT3 generated in this map corresponds to a four-dimensional torus in vector fields. First, to numerically calculate a clear Lyapunov diagram for quasi-periodic oscillations, we demonstrate that it is necessary that the number of iterations deleted as a transient state is large and similar to the number of iterations to be averaged as a stationary state. For example, it is necessary to delete 10,000,000 transient iterations to appropriately evaluate the Lyapunov exponents if they are averaged for 10,000,000 stationary state iterations in this higher-dimensional discrete-time dynamical system even if the parameter values are not chosen near the bifurcation boundaries. Second, by analyzing a graph of the Lyapunov exponents, we show that the local bifurcation transition from an invariant two-torus(IT2) to an IT3 is caused by a Neimark-Sacker bifurcation, which can be a one-dimension-higher quasi-periodic Hopf (QH) bifurcation. In addition, we confirm that another bifurcation route from an IT2 to an IT3 can be generated by a saddle-node bifurcation, which can be denoted as a one-dimension-higher quasi-periodic saddle-node bifurcation. Furthermore, we find a codimension-two bifurcation point at which the curves of a QH bifurcation and a quasi-periodic saddle-node bifurcation intersect.
... In contrast, higher-dimensional tori generated in higher-dimensional dynamics have also been the subjects of intensive research in recent years [7][8][9][10][11][12][13][14][15][16][17][18]. In our previous study, serious difficulties were encountered during the numerical analysis of a high-dimensional oscillator [18]. ...
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.
... Recently, much attention has been paid to partial-entrainment regions where two of the three independent frequency components of a three-dimensional torus become synchronized. 1,[8][9][10] Such entrainment regions could be caused by a saddlenode (SN) bifurcation of a stable two-dimensional torus and a saddle two-dimensional torus, which generates an extremely complex quasi-periodic entrainment region called an Arnold resonance web. In this study, we numerically investigate the quasi-periodic bifurcation structure generated by a coupled delayed logistic map that can exhibit an IT that corresponds to a three-dimensional torus in vector fields. ...
... Such phenomena include hyperchaos 7,8 and high-dimensional tori. 1,[8][9][10][11][12][13][14][15][16][17][18][19][20] In particular, partial and complete synchronization phenomena of the threefrequency oscillations of a three-dimensional torus have recently been the subject of intensive research. 1,[8][9][10] The partial synchronization of a three-dimensional torus indicates a state where two of the three independent frequency components become synchronized. ...
... 1,[8][9][10][11][12][13][14][15][16][17][18][19][20] In particular, partial and complete synchronization phenomena of the threefrequency oscillations of a three-dimensional torus have recently been the subject of intensive research. 1,[8][9][10] The partial synchronization of a three-dimensional torus indicates a state where two of the three independent frequency components become synchronized. Vitolo, Broer, and Sim o have mathematically investigated bifurcations of quasi-periodic oscillations. ...
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.
An extensive bifurcation analysis of partial and complete synchronizations of three-frequency quasi-periodic oscillations generated in an electric circuit is presented. Our model uses two-coupled hysteresis oscillators and a rectangular wave forcing term. The governing equation of the circuit is represented by a piecewise-constant dynamics generating a three-dimensional torus. The Lyapunov exponents are precisely calculated using explicit solutions without numerically solving any implicit equation. By analyzing this extremely simple circuit, we clearly demonstrate that it generates an extremely complex bifurcation structure called Arnol'd resonance web. Inevitably, chaos is observed in the neighborhood of Chenciner bubbles around which regions generating three-dimensional tori emanate. Furthermore, the numerical results are experimentally verified.
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 .
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.
