Solution multistability in first-order nonlinear differential delay equations

Losson, Jérôme; Mackey, Michael C.; Longtin, André

doi:10.1063/1.165982

Cited by 94 publications

(47 citation statements)

References 24 publications

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“…The existence of such time delays is absolutely essential to ensure that the solution is always uniquely determined (Milton et al 1989;). Multistability and other complex dynamics readily arise in these examples (Losson et al 1993;Foss et al 1997). Although the effects of noise have received comparatively little attention Eurich & Milton 1996), it is obvious that switching between coexisting attractors will add further complexity.…”

Section: Discussionmentioning

confidence: 99%

Balancing with positive feedback: the case for discontinuous control

Milton

Townsend

King

et al. 2009

Phil. Trans. R. Soc. A.

103

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Experimental observations indicate that positive feedback plays an important role for maintaining human balance in the upright position. This observation is used to motivate an investigation of a simple switch-like controller for postural sway in which corrective movements are made only when the vertical displacement angle exceeds a certain threshold. This mechanism is shown to be consistent with the experimentally observed variations in the two-point correlation for human postural sway. Analysis of first-passage times for this model suggests that this control strategy may slow escape by taking advantage of two intrinsic properties of a stochastic unstable first-order delay differential equation: (i) time delay and (ii) the possibility that the dynamics can be 'temporarily confined' near the origin.

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Section: Discussionmentioning

confidence: 99%

Balancing with positive feedback: the case for discontinuous control

Milton

Townsend

King

et al. 2009

Phil. Trans. R. Soc. A.

103

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“…It is even possible to construct the solution using a compass and a ruler [3]! Moreover, experimental paradigms using piecewise constant types of feedback can be developed and used to directly compare prediction with observation [51,52,58,72,64]. (20) observed as a function of C and τ (see [28] for details).…”

Section: Stabilizing the Upright Positionmentioning

confidence: 99%

Human Balance Control: Dead Zones, Intermittency, and Micro-chaos

Milton

Insperger

Stépàn

2015

Mathematical Approaches to Biological Systems

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Summary. The development of strategies to minimize the risk of falling in the elderly represents a major challenge for aging, in industrialized societies. The corrective movements made by humans to maintain balance are small amplitude, intermittent and ballistic. Small amplitude, complex oscillations (micro-chaos) frequently arise in industrial settings when a time-delayed digital processor attempts to stabilize an unstable equilibrium. Taken together these observations motivate considerations of the effects of a sensory threshold on the stabilization of an inverted pendulum by time-delayed feedback. In the resulting switching-type delay differential equations, the sensory threshold is a strong small-scale nonlinearity which has no effect on large-scale stabilization, but may produce complex, small amplitude dynamics including limit cycle oscillations and micro-chaos. A close mathematical relationship exists between a scalar model for balance control and the micro-chaotic map that arises in some models of digitally controlled machines. Surprisingly, transient, timedependent, bounded solutions (transient stabilization) can arise even for parameter ranges where the equilibrium is asymptotically unstable. In other words the combination of a sensory threshold with a time-delayed sampled feedback can increase the range of parameter values for which balance can be maintained, at least transiently. Neuro-biological observations suggest that sensory thresholds can be manipulated either passively by changing posture or actively using efferent feedback. Thus it may be possible to minimize the risk of falling by means of strategies that manipulate sensory thresholds by using physiotherapy and appropriate exercises.

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“…5,35,36 The dependence of the solutions on the initial function in the MG model and in other first order DDEs was investigated by Losson et al 5 They showed, by means of numerical and experimental methods, that the basins of attraction of piecewise constant feedback DDE can possess a complex structure at all scales measurable numerically. Specifically in relation with the MG model, Losson et al focus on a family of sinusoidal IFs, which parameter values give rise to limit cycles either with strictly positive or negative values of the variable.…”

Section: The Mg Modelmentioning

confidence: 99%

“…30 Several groups have proposed experimental implementations of the MG model via electronic circuits. 5,19,29,[31][32][33] An electronic implementation based on an analog delay line was considered in Refs. 19,28, and 31 to address the problem of synchronization and control of high dimensional chaos.…”

Section: The Mg Modelmentioning

confidence: 99%

“…Time-delayed systems often display coexisting, high-dimensional attractors. [4][5][6][7][8][9] Time-delays occur in a wide range of real-world systems, either due to couplings or to feedback loops (for recent reviews, see Refs. 10 and 11), and many practical applications have been demonstrated, for example, it has been shown that delay-dynamical systems, even in their simplest manifestation, can perform efficient information processing, 12 and the high dimensionality of the chaotic dynamics that they generate can be exploited for implementing ultra-fast random number generators.…”

Section: Introductionmentioning

confidence: 99%

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Organization and identification of solutions in the time-delayed Mackey-Glass model

Amil

Cabeza

Masoller

et al. 2015

Chaos

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Copyright 2015 AIP Publishing. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP PublishingMultistability in the long term dynamics of the Mackey-Glass (MG) delayed model is analyzed by using an electronic circuit capable of controlling the initial conditions. The system's phase-space is explored by varying the parameter values of two families of initial functions. The evolution equation of the electronic circuit is derived and it is shown that, in the continuous limit, it exactly corresponds to the MG model. In practice, when using a finite set of capacitors, an excellent agreement between the experimental observations and the numerical simulations is manifested. As the delay is increased, different periodic or aperiodic solutions appear. We observe abundant periodic solutions that have the same period but a different alternation of peaks of dissimilar amplitudes and propose a novel symbolic method to classify these solutions.Peer ReviewedPostprint (published version

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Solution multistability in first-order nonlinear differential delay equations

Cited by 94 publications

References 24 publications

Balancing with positive feedback: the case for discontinuous control

Balancing with positive feedback: the case for discontinuous control

Human Balance Control: Dead Zones, Intermittency, and Micro-chaos

Organization and identification of solutions in the time-delayed Mackey-Glass model

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