Nonlinear Dynamics Of Delay Systems Research Articles

Nonlinear dynamics of delay systems: an overview

Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue 'Nonlinear dynamics of delay systems', which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Delayed dynamical systems: networks, chimeras and reservoir computing.

We present a systematic approach to reveal the correspondence between time delay dynamics and networks of coupled oscillators. After early demonstrations of the usefulness of spatio-temporal representations of time-delay system dynamics, extensive research on optoelectronic feedback loops has revealed their immense potential for realizing complex system dynamics such as chimeras in rings of coupled oscillators and applications to reservoir computing. Delayed dynamical systems have been enriched in recent years through the application of digital signal processing techniques. Very recently, we have showed that one can significantly extend the capabilities and implement networks with arbitrary topologies through the use of field programmable gate arrays. This architecture allows the design of appropriate filters and multiple time delays, and greatly extends the possibilities for exploring synchronization patterns in arbitrary network topologies. This has enabled us to explore complex dynamics on networks with nodes that can be perfectly identical, introduce parameter heterogeneities and multiple time delays, as well as change network topologies to control the formation and evolution of patterns of synchrony. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Class-C semiconductor lasers with time-delayed optical feedback.

We perform a linear stability analysis and numerical bifurcation diagrams of a class-C laser with time-delayed optical feedback. We employ a rate equation system based on the Maxwell-Bloch equations, and study the influence of the dephasing time on the laser dynamics. We find a stabilizing effect of an intermediate dephasing time, i.e. when moving from a class-B to a class-C laser. At long dephasing times, a destabilization of the laser solution occurs by a feedback-induced unlocking of Rabi oscillations at the second laser threshold. We predict an optimum resistance to time-delayed optical feedback for dephasing times close to the photon cavity lifetime. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Transmission time delays organize the brain network synchronization.

The timing of activity across brain regions can be described by its phases for oscillatory processes, and is of crucial importance for brain functioning. The structure of the brain constrains its dynamics through the delays due to propagation and the strengths of the white matter tracts. We use self-sustained delay-coupled, non-isochronous, nonlinearly damped and chaotic oscillators to study how spatio-temporal organization of the brain governs phase lags between the coherent activity of its regions. In silico results for the brain network model demonstrate a robust switching from in- to anti-phase synchronization by increasing the frequency, with a consistent lagging of the stronger connected regions. Relative phases are well predicted by an earlier analysis of Kuramoto oscillators, confirming the spatial heterogeneity of time delays as a crucial mechanism in shaping the functional brain architecture. Increased frequency and coupling are also shown to distort the oscillators by decreasing their amplitude, and stronger regions have lower, but more synchronized activity. These results indicate specific features in the phase relationships within the brain that need to hold for a wide range of local oscillatory dynamics, given that the time delays of the connectome are proportional to the lengths of the structural pathways.This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

Complex partial synchronization patterns in networks of delay-coupled neurons.

We study the spatio-temporal dynamics of a multiplex network of delay-coupled FitzHugh-Nagumo oscillators with non-local and fractal connectivities. Apart from chimera states, a new regime of coexistence of slow and fast oscillations is found. An analytical explanation for the emergence of such coexisting partial synchronization patterns is given. Furthermore, we propose a control scheme for the number of fast and slow neurons in each layer. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

The effect of state dependence in a delay differential equation model for the El Niño Southern Oscillation.

Delay differential equations (DDEs) have been used successfully in the past to model climate systems at a conceptual level. An important aspect of these models is the existence of feedback loops that feature a delay time, usually associated with the time required to transport energy through the atmosphere and/or oceans across the globe. So far, such delays are generally assumed to be constant. Recent studies have demonstrated that even simple DDEs with non-constant delay times, which change depending on the state of the system, can produce surprisingly rich dynamical behaviour. Here, we present arguments for the state dependence of the delay in a DDE model for the El Niño Southern Oscillation phenomenon in the climate system. We then conduct a bifurcation analysis by means of continuation software to investigate the effect of state dependence in the delay on the observed dynamics of the system. More specifically, we show that the underlying delay-induced structure of resonance regions may change considerably in the presence of state dependence. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Resonant Doppler effect in systems with variable delay.

We demonstrate that a time-varying delay in nonlinear systems leads to a rich variety of dynamical behaviour, which cannot be observed in systems with constant delay. We show that the effect of the delay variation is similar to the Doppler effect with self-feedback. We distinguish between the non-resonant and the resonant Doppler effect corresponding to the dichotomy between conservative delays and dissipative delays. The non-resonant Doppler effect leads to a quasi-periodic frequency modulation of the signal, but the qualitative properties of the solution are the same as for constant delays. By contrast, the resonant Doppler effect leads to fundamentally different solutions characterized by low- and high-frequency phases with a clear separation between them. This is equivalent to time-multiplexed dynamics and can be used to design systems with well-defined multistable solutions or temporal switching between different chaotic and periodic dynamics. We systematically study chaotic dynamics in systems with large dissipative delay, which we call generalized laminar chaos. We derive a criterion for the occurrence of different orders of generalized laminar chaos, where the order is related to the dimension of the chaotic attractor. The recently found laminar chaos with constant plateaus in the low-frequency phases is the zeroth-order case with a very low dimension compared to the known high dimension of turbulent chaos in systems with conservative delay. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Fokker-Planck representations of non-Markov Langevin equations: application to delayed systems.

Noise and time delays, or history-dependent processes, play an integral part in many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n = 2 by converting the time non-local Langevin equation to a time-local one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Travelling fronts in time-delayed reaction-diffusion systems.

We review a series of key travelling front problems in reaction-diffusion systems with a time-delayed feedback, appearing in ecology, nonlinear optics and neurobiology. For each problem, we determine asymptotic approximations for the wave shape and its speed. Particular attention is devoted to their validity and all analytical solutions are compared to solutions obtained numerically. We also extend the work by Erneux et al. (Erneux et al. 2010 Phil. Trans. R. Soc. A 368, 483-493 (doi:10.1098/rsta.2009.0228)) by considering the case of a slowly propagating front subject to a weak delayed feedback. The delay may either speed up the front in the same direction or reverse its direction. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Delay-induced switched states in a slow-fast system.

We consider the two-component delay system εx'(t) = - x(t) - y(t) + f(x(t - 1)), y'(t) = ηx(t) with small para- meters ε, η and positive feedback function f. Previously, such systems have been reported to model switching in optoelectronic experiments, where each switching induces another one after approximately one delay time, related to one round trip of the signal. In this paper, we study these delay-induced switched states. We provide conditions for their existence and show how the formal limits ε → 0 and/or η → 0 facilitate our understanding of this phenomenon. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Experimental observations on unsafe zones in milling processes.

The unsafe zone in machining is a region of the parameter space where steady-state cutting operations may switch to regenerative chatter for certain perturbations, and vice versa. In the case of milling processes, this phenomenon is related to the existence of an unstable quasi-periodic oscillation, the in-sets of which limit the basin of attraction of the stable periodic motion that corresponds to the chatter-free cutting process. The mathematical model is a system of time-periodic nonlinear delay differential equations. It is studied by means of a nonlinear extension of the semidiscretization method, which enables the estimation of the parameter ranges where the unsafe (also called bistable) zones appear. The theoretical results are checked with thorough experimental work: first, step-by-step parameter variations are adapted to identify hysteresis loops, then harmonic burst excitations are used to estimate the extents of the unsafe zones. The hysteresis loops are accurately distinguished from the dynamic bifurcation phenomenon that is related to the dynamic effect of slowly varying parameters. The experimental results confirm the existence of the bistable parameter regions. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Correction to: 'Growth model for large branched three-dimensional hydraulic crack system in gas or oil shale'.

You have accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Chau Viet T., Bažant Zdeněk P. and Su Yewang 2019Correction to: ‘Growth model for large branched three-dimensional hydraulic crack system in gas or oil shale’Phil. Trans. R. Soc. A.3772019022020190220http://doi.org/10.1098/rsta.2019.0220SectionYou have accessCorrectionsCorrection to: ‘Growth model for large branched three-dimensional hydraulic crack system in gas or oil shale’ Viet T. Chau Viet T. Chau Google Scholar Find this author on PubMed Search for more papers by this author , Zdeněk P. Bažant Zdeněk P. Bažant Google Scholar Find this author on PubMed Search for more papers by this author and Yewang Su Yewang Su Google Scholar Find this author on PubMed Search for more papers by this author Viet T. Chau Viet T. Chau Google Scholar Find this author on PubMed Search for more papers by this author , Zdeněk P. Bažant Zdeněk P. Bažant Google Scholar Find this author on PubMed Search for more papers by this author and Yewang Su Yewang Su Google Scholar Find this author on PubMed Search for more papers by this author Published:22 July 2019https://doi.org/10.1098/rsta.2019.0220This article corrects the followingResearch ArticleGrowth model for large branched three-dimensional hydraulic crack system in gas or oil shalehttps://doi.org/10.1098/rsta.2015.0418 Viet T. Chau, Zdeněk P. Bažant and Yewang Su volume 374issue 2078Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences13 October 2016Phil. Trans. R. Soc. A374, 20150418 (Published online 13 October 2016) (doi:10.1098/rsta.2015.0418)Recent reanalysis revealed that the branching indicated by the computer program in this article was, in fact, triggered by a programming glitch which caused an unintended abrupt change of Biot coefficient for transverse water pressure on the crack. This change suddenly increased the pressure on the solid phase and triggered dynamic response. Such a sudden trigger is probably unrealistic and requires correction. The correction nevertheless reveals a useful fact, namely that producing fluid pressure shocks within the cracks would locally stimulate branching. A partly revised model, which is based mostly on the mathematical formulation from this paper leads to crack branching similar to what is seen in the figures of this paper, has been published by Rahimi-Aghdam et al. [1].Footnotes© 2019 The Author(s)Published by the Royal Society. All rights reserved.ReferencesReference1. Rahimi-Aghdam Set al.2019Branching of hydraulic cracks enabling permeability of gas or oil shale with closed natural fractures. Proc. Natl Acad. Sci. USA 116, 1532–1537. (doi:10.1073/pnas.1818529116) Crossref, PubMed, ISI, Google Scholar Previous Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsRelated articlesGrowth model for large branched three-dimensional hydraulic crack system in gas or oil shale13 October 2016Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences This Issue09 September 2019Volume 377Issue 2153Theme issue ‘Nonlinear dynamics of delay systems’ organised and edited by Andreas Otto, Wolfram Just and Günter Radons Article InformationDOI:https://doi.org/10.1098/rsta.2019.0220PubMed:31329068Published by:Royal SocietyPrint ISSN:1364-503XOnline ISSN:1471-2962History: Published online22/07/2019Published in print09/09/2019 License:© 2019 The Author(s)Published by the Royal Society. All rights reserved. Citations and impact Subjectscivil engineeringfluid mechanicsmechanical engineering

Geometric analysis of synchronization in neuronal networks with global inhibition and coupling delays.

We study synaptically coupled neuronal networks to identify the role of coupling delays in network synchronized behaviour. We consider a network of excitable, relaxation oscillator neurons where two distinct populations, one excitatory and one inhibitory, are coupled with time-delayed synapses. The excitatory population is uncoupled, while the inhibitory population is tightly coupled without time delay. A geometric singular perturbation analysis yields existence and stability conditions for periodic solutions where the excitatory cells are synchronized and different phase relationships between the excitatory and inhibitory populations can occur, along with formulae for the periods of such solutions. In particular, we show that if there are no delays in the coupling, oscillations where the excitatory population is synchronized cannot occur. Numerical simulations are conducted to supplement and validate the analytical results. The analysis helps to explain how coupling delays in either excitatory or inhibitory synapses contribute to producing synchronized rhythms. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Fast consensus in a large-scale multi-agent system with directed graphs using time-delayed measurements.

This article is on fast-consensus reaching in a class of multi-agent systems (MAS). We present an analytical approach to tune controllers for the agents based on the premise that delayed measurements in the controller can be preferable to standard controllers relying only on current measurements. Controller tuning in this setting is however challenging due to the presence of delays. To tackle this problem, we propose an analytic geometry approach. The key contribution is that the tuning can be implemented for complex eigenvalues of the arising graph Laplacian of the network, complementing the current state of the art, which is limited to real eigenvalues. Results, therefore, extend our knowledge beyond symmetric graphs and enable the study of the MAS under directed graphs. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Acting together, destabilizing influences can stabilize human balance

The causes of falling in the elderly are multi-factorial. Three factors that influence balance stability are the time delay, a sensory dead zone and the maximum ankle torque that can be generated by muscular contraction. Here, the effects of these contributions are evaluated in the context of a model of an inverted pendulum stabilized by time-delayed proportional-derivative (PD) feedback. The effect of the sensory dead zone is to produce a hybrid type of control in which the PD feedback is switched ON or OFF depending on whether or not the controlled variable is larger or smaller than the detection threshold, Π. It is shown that, as Π increases, the region in the plane of control parameters where the balance time (BT) is greater than 60 s is increased slightly. However, when maximum ankle torque is also limited, there is a dramatic increase in the parameter region associated with BTs greater than 60 s. This increase is due to the effects of a torque limitation on over-control associated with bang-bang type switching controllers. These observations show that acting together influences, which are typically thought to destabilize balance, can actually stabilize balance. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.