The Simulation of Variable Delay

Seddon, James; Johnson, Richard A.

doi:10.1109/tc.1968.5008879

Cited by 12 publications

(6 citation statements)

References 9 publications

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“…On the other hand, a nonlinear time-scale transformation θ = Φ(t) can be used to transform a transport with an arbitrary time-varying velocity to a transport with constant velocity, where the distance in the new time scale may change but the delay class remains the same. The known transformation of time-varying delays which are implicitly defined by equation (2.7) to constant delays is applied in [44][45][46][47]. This type of delays is also called variable transport delay [48] in engineering applications, where the intrinsic constant delay τ c is related to the constant transport distance d c .…”

Section: Time-varying Delay and Doppler Effectmentioning

confidence: 99%

Resonant Doppler effect in systems with variable delay

Müller-Bender

Otto

Radons

2019

Phil. Trans. R. Soc. A.

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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’.

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Section: Time-varying Delay and Doppler Effectmentioning

confidence: 99%

Resonant Doppler effect in systems with variable delay

Müller-Bender

Otto

Radons

2019

Phil. Trans. R. Soc. A.

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“…(1) and its inverse t ¼ Φ −1 ðφÞ. The system in the new time scale φ with the new state variable zðφÞ ¼ y½Φ −1 ðφÞ can be written as [5][6][7][8][9][10][11] The function ΦðtÞ is assumed to be bijective and differentiable almost everywhere to ensure a one-to-one mapping between Eqs. (1) and (2).…”

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confidence: 99%

“…For example, if φ ¼ ΦðtÞ specifies a distance for a transport over time t, the delay τðtÞ in Eq. (4) specifies the traveling time for a transport with variable velocity Φ 0 ðtÞ over the constant distance c. In the literature, these delays are called variable transport delays [11,28], pipe delays [6], or threshold-type delays [7,16,29]. On the other hand, there are variable delays, which are not defined by Eq.…”

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confidence: 99%

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Universal Dichotomy for Dynamical Systems with Variable Delay

2017

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We show that the dynamics of systems with a time-dependent delay is fundamentally affected by the functional form of the retarded argument. Associating with the latter an iterated map, the access map, and a corresponding Koopman operator, we identify two universality classes. Members in the first are equivalent to systems with a constant delay. The new, second class is characterized by the mode-locking behavior of their access maps and by an asymptotically linear, instead of a logarithmic, scaling of the Lyapunov spectrum. The membership depends in a fractal manner only on the parameters of the delay.

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“…For some systems with time-varying delay it is known that they can be transformed to systems with constant delay because the delay is defined by an intrinsic constant delay and a known timescale transformation [13]. This type of systems arises, for example, in biological models with threshold delays, where the intrinsic constant delay represents the evolutionary steps, which have to be passed to reach the state of adulthood and the corresponding timescale represents the grade of evolution [3,14].…”

Section: Introductionmentioning

confidence: 99%

From dynamical systems with time-varying delay to circle maps and Koopman operators

2017

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In the present paper we investigate the influence of the retarded access by a time-varying delay on the dynamics of delay systems. We show that there are two universality classes of delays, which lead to fundamental differences in dynamical quantities such as the Lyapunov spectrum. Therefore we introduce an operator theoretic framework, where the solution operator of the delay system is decomposed into the Koopman operator describing the delay access and an operator similar to the solution operator known from systems with constant delay. The Koopman operator corresponds to an iterated map, called access map, which is defined by the iteration of the delayed argument of the delay equation. The dynamics of this one-dimensional iterated map determines the universality classes of the infinite-dimensional state dynamics governed by the delay differential equation. In this way, we connect the theory of time-delay systems with the theory of circle maps and the framework of the Koopman operator. In the present paper we extend our previous work [Otto, Müller, and Radons, Phys. Rev. Lett. 118, 044104 (2017)], by elaborating the mathematical details and presenting further results also on the Lyapunov vectors.

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The Simulation of Variable Delay

Cited by 12 publications

References 9 publications

Resonant Doppler effect in systems with variable delay

Resonant Doppler effect in systems with variable delay

Universal Dichotomy for Dynamical Systems with Variable Delay

From dynamical systems with time-varying delay to circle maps and Koopman operators

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