Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
ISSN-L : 2185-4106
Regular Section
Bifurcation structure of an invariant three-torus and its computational sensitivity generated in a three-coupled delayed logistic map
Shuya HidakaNaohiko InabaKyohei KamiyamaMunehisa SekikawaTetsuro Endo
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2015 Volume 6 Issue 3 Pages 433-442


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.

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© 2015 The Institute of Electronics, Information and Communication Engineers
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