Resonant Doppler effect in systems with variable delay

Abstract

<jats:p>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.</jats:p> <jats:p>This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.</jats:p>

Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

“…Our results support the statements in Ref. [29,51], where we have conjectured that the intensity levels of the laminar phases can be used to encode information for computational or cryptographic purposes. For example, due to the robustness of the mapping between the intensity levels of the laminar phases, a chaos based logic based on the ideas behind Ref.…”

“…This is because the laminar phases inside the state interval are pairwise independent for p > 1 as pointed out in Ref. [29], and for p > q the state interval is longer than the delay period. As a result, the first part vanishes.…”

“…This means, roughly speaking, that the map defined by Eq. (4a) and the time-varying delay τ (t) has no influence on the dynamics of the limit map except for a quasiperiodic frequency modulation [29]. Due to the stretching and folding by the map z = F (z) = µ f (z), the limit map dynamics is characterized by oscillations, whereby the characteristic frequency of a solution segment z n (t) grows with n, i.e, it grows with each iteration of the limit map.…”

“…Since the repulsive periodic points t * k separate p basins of attraction of periodic points inside each state interval I n , each of the latter contains p laminar phases (see Ref. [29,51] for details). Since the access map is invertible, the intensity levels of the laminar phases inside the state interval I n are successively mapped to the intensity levels in I n+1 by z = µ f (z).…”

“…Due to environmental fluctuations, time-delays are typically not constant, but rather time-varying. If a system involves transport processes with time-varying velocity or time-varying transport distance, the delay is timevarying [28,29]. Although models with time-varying delay are often more realistic, there are only a few papers on such systems with and without noise.…”