Phase reduction of weakly perturbed limit cycle oscillations in time-delay systems

Novičenko, Viktor; Pyragas, K.

doi:10.1016/j.physd.2012.03.001

Cited by 49 publications

(85 citation statements)

References 24 publications

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“…In Fig. 2, we compare the estimated rotation number with that of the direct calculations of the coupled map lattice (14). In Eq.…”

Section: Phase Reduction Of Quasi-periodic Attractorsmentioning

confidence: 99%

“…3, we show an example of orbit stabilization by the DFC for the coupled map lattice (14). To obtain the unstable QPO, we choose the parameters (a, b, d) ¼ (0.79, 0.02, 0.06) such that jl 1 j; jl 2;3 j > 1.…”

Section: Stabilization Of Unstable Qpo By Dfcmentioning

confidence: 99%

“…14,15 In the continuous-time case, the time-delay systems have infinite-dimensional phase space and hence the standard adjoint method cannot be directly applied. In the discrete-time case, since the system is finite-dimensional even if there is time delay, we can directly apply the adjoint method to the DFC system.…”

Section: Phase Reduction Of Dfcmentioning

confidence: 99%

“…In the discrete-time case, since the system is finite-dimensional even if there is time delay, we can directly apply the adjoint method to the DFC system. However, the results in the literature 14,15 suggest that the adjoint method can be reduced to a smaller problem in the discrete-time case.…”

Section: Phase Reduction Of Dfcmentioning

confidence: 99%

“…However, the amplitude of the PRC should be normalized again for the delayd. 14 We obtain the PRC of the DFĈ Z n with the delayd from that of the unstable QPO Z n…”

Section: Evaluation Of Dfc For Time-delay Mismatchmentioning

confidence: 99%

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Delayed feedback control and phase reduction of unstable quasi-periodic orbits

Ichinose

Komuro

2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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The delayed feedback control (DFC) is applied to stabilize unstable quasi-periodic orbits (QPOs) in discrete-time systems. The feedback input is given by the difference between the current state and a time-delayed state in the DFC. However, there is an inevitable time-delay mismatch in QPOs. To evaluate the influence of the time-delay mismatch on the DFC, we propose a phase reduction method for QPOs and construct a phase response curve (PRC) from unstable QPOs directly. Using the PRC, we estimate the rotation number of QPO stabilized by the DFC. We show that the orbit of the DFC is consistent with the unstable QPO perturbed by a small state difference resulting from the time-delay mismatch, implying that the DFC can certainly stabilize the unstable QPO.

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“…In Fig. 2, we compare the estimated rotation number with that of the direct calculations of the coupled map lattice (14). In Eq.…”

Section: Phase Reduction Of Quasi-periodic Attractorsmentioning

confidence: 99%

Section: Stabilization Of Unstable Qpo By Dfcmentioning

confidence: 99%

Section: Phase Reduction Of Dfcmentioning

confidence: 99%

Section: Phase Reduction Of Dfcmentioning

confidence: 99%

“…However, the amplitude of the PRC should be normalized again for the delayd. 14 We obtain the PRC of the DFĈ Z n with the delayd from that of the unstable QPO Z n…”

Section: Evaluation Of Dfc For Time-delay Mismatchmentioning

confidence: 99%

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Delayed feedback control and phase reduction of unstable quasi-periodic orbits

Ichinose

Komuro

2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Evaluation of heuristic reductions of a model for the segmentation clock in zebrafish

Shimono

Yamaguchi

Ishimatsu

et al. 2017

IEEJ Transactions Elec Engng

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Time-delayed interactions between DNA, mRNA, and proteins play an important role in the periodic somite segmentation of vertebrates. A mathematical model of the segmentation clock in zebrafish that illustrates the dynamics between proteins and mRNA has recently been proposed; however, the complexity of the model makes solving it analytically unrealistic. In this study, we derived a first-order delay-differential equation (DDE) model by following two heuristic reduction approaches: simplifying protein-mRNA interactions and protein dimerization. Then, we calculated the eigenvalues of the model and found that the maximum eigenvalue and the oscillatory dynamics are qualitatively equivalent when some parameters are varied. This result enabled us to illustrate the effectiveness of using eigenvalues to assess the dynamics of the reduced model. Further, we extended the reduction methods to a complete model of somite segmentation considering the interaction of four genes, and derived a reduced model with four variables and four delay terms. Although the oscillation period of our reduced model decreased by ∼10% compared to the original model, the amplitude differed by less than 10%. The dynamic features for genetic conditions are also qualitatively reproduced by the reduced model. Our model demonstrates that it is possible to identify key parameters and to reveal the interactions among the parameters.

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