Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators

Le, Luan Ba; Konishi, Keiji; Hara, Naoyuki

doi:10.1007/s11071-011-0077-4

“…Many natural systems are mathematically modeled by nonlinear delay differential equations which contain one or more time delays. Successful examples include blood production in patients with leukemia [5], dynamics of optical systems [6,7], population dynamics [8], physiological model [9], El Niño/southern oscillation [10], the Lorentz force with Liénard-Weichert potentials [11], neural network with three neurons [12], delay feedback control and synchronization [13,14], etc. The presence of a delay in a system makes the system infinite-dimensional, and may lead to an unstable and oscillatory response.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of a first time-delay chaotic system

Li

¹

,

Wang

²

,

Zhou

³

2019

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This paper deals with the dynamic behavior of the chaotic nonlinear time delay systems of general formẋ(t) = g(x(t), x(t-τ)). We carry out stability analysis to identify the parameter zone for which the system shows a stable equilibrium response. Through the bifurcation analysis, we establish that the system shows a stable limit cycle through supercritical Hopf bifurcation beyond certain values of delay and parameters. Next, a numerical simulation of the prototype system is used to show that the system has different behaviors: stability, periodicity and chaos with the variation of delay and other parameters, which demonstrates the validity of our method. We give the single-and two-parameter bifurcation diagrams which are employed to explore the dynamics of the system over the whole parameter space.

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“…One Oscillator Namajūnas et al (1995), Hövel and Scholl (2005) Ahlborn and Parlitz (2004), Le et al (2012b) Two oscillators Aronson et al (1990) Osipov et al (1997) Besides the number of supply chain partners mapped, the kind of interaction between these supply chain partners can differ. Depending on the specific supply chain property described by oscillators (e.g.…”

Section: Static Connection Delay Connection Multiple Delay Connectionmentioning

confidence: 99%

Modelling Oscillations in the Supply Chain: The Case of a Just-in-Sequence Supply Process from the Automotive Industry

Klug

¹

2021

Preprint

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Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.

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“…One Oscillator Namajūnas et al (1995), Hövel and Schöll (2005) Ahlborn and Parlitz ( 2004), Le et al (2012b) Two oscillators Aronson et al (1990), Woafo and Kraenkel (2002), Le et al (2010) Lu et al (1997, Reddy et al (1998), Konishi et al ( 2008) Konishi et al (2008), Le et al (2012c) Three oscillators Yamaguchi and Shimizu (1984), Le et al (2012a) Reddy et al (1998, Le et al ( 2013) Konishi et al (2010), Bera et al (2017) Network oscillators Osipov et al (1997) Burbidge (1984) about the problem of multi-cycle ordering, whereby each item has its own ordering cycle and is considered independently of any other required item. He summarised this effect by the "ordering cycle law" and states that "If the various components made in a factory are ordered and made to different time cycles, they will generate high amplitude and unpredictable variations in both stocks and load as the many contributing component stock cycles drift in and out of phase".…”

Section: Multiple Delay Connectionmentioning

confidence: 99%

Modelling oscillations in the supply chain: the case of a just-in-sequence supply process from the automotive industry

Klug

¹

2021

J Bus Econ

3

0

View full text Add to dashboard Cite

Pervasive and ubiquitous oscillations, mapping the repetitive variation in time of a specific state, are well known as abundant phenomena in research and practice. Motivated by the success of oscillators in the modelling, analysis and control of dynamical systems, we developed a related approach for the dynamic description of supply chains. This paper aims to introduce a generic oscillator model for supply chains by the original application of oscillator equations. Therefore an established oscillator model for deductive modelling of supply chain echelons is used. With the help of coupled van der Pol oscillators, the dynamical interaction of an inventory system is described and applied to a real-life supply process in automotive industry. According to its reductionist approach only two differential equations are used to analyse a Just-in-Sequence supply process in car industry. Based on the fact that any oscillatory state can be reduced to the phase of the oscillation (phase reduction), a phase space map is generated. This compact visual reference of the supply process can act as the quantitative basis for an adaptive control mechanism during its operation. By delaying or accelerating the inventory oscillations of the supplier stock a detuned coupled supply process can be re-synchronised without changing the amplitude. An additional analysis of Hilbert transform is applied to determine the boundaries of phase-locking between the inventory oscillation phases, where the instantaneous phases are bounded. Furthermore parameters of the synchronisation threshold and the transient phases between synchronous and non-synchronous regimes have been investigated, supported by an Arnold tongue representation. The investigations show that with the help of a generic oscillatory model it is possible to measure and quantify phenomena of inventory dynamics in supply chains. Especially the analysis of synchronisation phenomena with the help of phase space and Arnold tongue representations foster developments of performance measurement in supply chain management.

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Design and experimental verification of multiple delay feedback control for time-delay nonlinear oscillators

Cited by 24 publications

References 55 publications

Bifurcation analysis of a first time-delay chaotic system

Bifurcation analysis of a first time-delay chaotic system

Modelling Oscillations in the Supply Chain: The Case of a Just-in-Sequence Supply Process from the Automotive Industry

Modelling oscillations in the supply chain: the case of a just-in-sequence supply process from the automotive industry

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