Resonant Doppler effect in systems with variable delay

Abstract

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’.

1. Introduction

In nature, time delays are induced due to transport processes or evolution processes and often cannot be neglected. Dynamical systems with time delay can show a rich variety of complex dynamical behaviour. An overview is given in the theme issue which is introduced by Just et al. [1].

Coupled optical systems involve delays of different orders of magnitude and thus naturally show a rich variety of dynamics of different complexity. For instance, an extensive review of systems consisting of delay-coupled semiconductor lasers is given in [2], and the dependence of the complexity on the delay length for coupled fibre laser oscillators is analysed in [3]. From the latter, it becomes clear that increasing the delay typically leads to an increase of the complexity of the dynamics and in [4] typical features of the dynamics related to the large delay limit are demonstrated. The increase of the complexity can appear in various ways. On the one hand, for periodic dynamics, the complexity increases by the amount of information which is stored in the state of the system as in the case of multistable periodic solutions [5,6]. And on the other hand, for chaotic dynamics, the amount of information which is produced by the system increases because the entropy per delay interval increases [3]. As a consequence for chaotic systems with a large constant delay, the attractor dimension scales typically linear with the delay, and thus can increase arbitrarily [7–9]. Similar to multistable periodic solutions, multiple chaotic attractors [8] can be observed. Systems with large delay show spatio-temporal phenomena [9], such as spatio-temporal intermittency and defects [10], convective instabilities [11] and Eckhaus instabilities [12].

In nature, the delay is typically not constant. In general, the delays are state-dependent. If, however, the delay generating mechanism is not influenced by the delay itself, the delay can be considered to be time-varying. For instance, if one considers the evolution of biological species under fluctuating environmental conditions, the delay given by the time needed to reach the ability to reproduce is time-varying. Although time-varying delays are often more realistic than constant delays, there exist only few results for the influence of a time-varying delay on the dynamics of a system. A variable delay induces specific types of synchronization in coupled systems [13] and enriches the dynamics of maps [14], time continuous systems [15–17] and spatially extended systems [18]. The effect of a time-varying delay on the structure of chaotic attractors, on synchronization and possible applications to cryptography are discussed in [19–22]. A time-varying delay can stabilize systems [23], which can be used in engineering applications such as turning and milling [24–26]. If the delay varies very fast, it can be approximated by a constant but distributed delay [27]. This approach was used, for example, to stabilize unstable steady states (amplitude death) [28–32] and for the stability analysis of a laser model with time-varying delay [33]. Especially, in the limit of a large delay, some general aspects of systems with time-varying delay can be derived from the theory of singularly perturbed systems with state-dependent delay, which are analysed in [34–36].

In our previous work [37,38], we found that there are two types of time-varying delay and demonstrated that they are related to fundamental differences in the dynamics of the involved systems. Systems with a so-called conservative delay are equivalent to systems with a constant delay, where equivalent means that the system with a time-varying delay can be transformed into a system with a constant delay by a suitable time-scale transformation, which preserves dynamical quantities such as the Lyapunov exponents. Hence, the characteristic dynamics of systems with a conservative delay equals the characteristic dynamics of systems with a constant delay, which are extensively analysed in the literature. Contrarily, systems with a so-called dissipative delay do not show this equivalence and their characteristic dynamics differs fundamentally from the dynamics of systems with constant delay. The differences are found, for instance, in the tangent space dynamics with respect to the asymptotic scaling behaviour of the Lyapunov spectrum and the localization properties of the covariant Lyapunov vectors [38]. In our recent work [39] we demonstrated that a large time-varying delay can lead to a new type of chaos, which is called laminar chaos and is fundamentally different from the chaotic dynamics which can be observed in systems with constant delay. In this paper, we provide additional results on the influence of the delay classes on the dynamics of delay systems. We derive general mechanisms behind the feedback by a variable delay, which are generalizations of the well-known Doppler effect. Using these results, we show that introducing a time-varying delay can lead to a variety of new types of chaotic dynamics, which are characterized by different degrees of complexity. Furthermore, we exploit these mechanisms to design so-called time-multiplexed dynamics, which is characterized by a periodic switching between different dynamical behaviours. The latter can be chosen arbitrarily from a certain pool of chaotic and periodic dynamics. This opens new possibilities for the processing and the storage of data.

2. Time-varying delay and Doppler effect

In the following, we provide an intuition for the origin and the consequences of variable delays in dynamical systems. The physical mechanism behind time delays can always be understood as a transport process with a finite velocity over a finite distance, where the time delay is the travelling time of a signal from the source to a sensor. A variable time delay τ(t) can be generated either by varying the velocity v or by varying the distance d of the signal propagation or both. Since a variable velocity can be transformed into a variable distance via a nonlinear time-scale transformation (see below), without loss of generality, we consider a transport with constant velocity v over a time-varying distance d = d(t). This is illustrated in figure 1 via an ideal propagation of a sound signal, where the distance between the speaker and the microphone varies periodically with period T. More precisely, the microphone records a sound signal at time t, which was the output signal of the speaker at the retarded time t′ = R(t) < t. The retarded time is defined by

$$t^{\prime }=R(t)=t-\tau (t),$$ 2.1

which maps the current time t to the retarded time t′. The time delay τ(t) is proportional to the distance and is given by

$$\tau (t)=\frac{d(t)}{v}=\frac{1}{v}(\overline{d}+\mathrm{\Delta }d(t)),$$ 2.2

where Δd(t) specifies the variation around the mean distance $\overline{d}$. Using this set-up, many features of time-varying delays can be understood in terms of the Doppler effect, which comes immediately to mind if one thinks about a passing ambulance. The speaker is driven by a periodic signal and thus, it emits plane waves with constant frequency and wavelength, which is indicated by the equidistant black lines in figure 1a. Due to the periodic movement of the microphone, the frequency of the measured signal changes periodically, whereby the measured frequency is higher or lower than the emitted frequency at the times where the distance d(t) decreases or increases, respectively. In a time-delay system, the recorded signal at the sensor affects the signal at the source. This is illustrated in figure 1b by the feedback line between the microphone and the speaker. For simplicity, the feedback is characterized by a function f that acts on the recorded signal at the microphone instantaneously and modifies the source, i.e. the output of the speaker. Since the measured signal at the microphone is a retarded version of the speaker output, the signal z(t), which is generated at the speaker, is given by

$$z(t)=f(z(R(t))).$$ 2.3

The above equation is a functional equation describing one of the simplest time-delay systems with variable delay.

Figure 1.

(a) Classical Doppler effect versus (b) Doppler effect with self-feedback. The output (c) of the Doppler effect with self-feedback shows two types of dynamics depending on the parameters of the time-varying microphone position $\overline{d}+\mathrm{\Delta }d(t)$. For a sinusoidal variation Δd(t) = A_dsin(2πt)/(2π), the parameters of the microphone position $\overline{d}+\mathrm{\Delta }d(t)$ for these two types of dynamics are visualized in (d). The white regions form a (fat) Cantor set leading to a conservative delay corresponding to the non-resonant Doppler effect, whereas the black regions are called Arnold tongues leading to a dissipative delay corresponding to the resonant Doppler effect. (Online version in colour.)

The question is now how the Doppler effect influences the output z(t) of the speaker-microphone system with self-feedback, or in general, how a time-varying delay influences the solution of equation (2.3). Note that our analysis is not restricted to this simple equation. The concept is quite general and can be applied also to delay differential equations (DDE), as for instance equation (3.1), which will be analysed in §3. Equation (2.3) can be solved iteratively: If we divide the solution z(t) into suitable segments z_n(t) with t∈(t_n−1, t_n] and t_n−1 = R(t_n) the whole signal can be constructed from the given initial signal z₀(t) by the recurrence relation

$$z_n(t)=f(z_{n-1}(R(t))),$$ 2.4

which is solved by

$$z_n(t)=f^n(z_0(R^n(t))).$$ 2.5

For reasons which will become clear in §3 the mapping of the function z_n−1(t) to z_n(t) in equation (2.4) is called limit map. We call an interval (t_n−1, t_n] a state interval, since z_n(t), with t inside this interval, characterizes the whole memory of the system at time t_n, which is given by the sound wave between the speaker and the microphone. This can be seen as a kind of space–time representation, which is similar to the space–time representation for systems with constant delay in [40], with the difference that the size of the ‘space’, i.e. the length of the intervals (t_n−1, t_n] varies with the discrete ‘time’ n. From equation (2.5), it becomes clear that the two iterated maps z′ = f(z) and t′ = R(t) influence the output of the system. Since f is specific to the system and we are interested in the dynamical properties induced by the time-varying delay, we first focus our analysis on the map t′ = R(t). We call this map access map, because it describes fundamental properties of the repeated access of the delay system to its history at the retarded time. Now let us restrict our analysis to a periodic variation of the microphone position, which causes a periodic variation of the delay. We assume that the delay period equals one, i.e. τ(t + 1) = τ(t), which can always be achieved by a linear time-scale transformation. This means that we measure time in units of the period of the delay variation. We further assume that R(t) is invertible, which means that the microphone does not recede from the speaker faster than the sound velocity. Then the dynamics of the access map can be analysed by its reduced version t′ = R(t) mod 1 which is a circle map, cf. [41,42].

In [37,38,43], we have introduced a classification of time-varying delays by the dynamics of the corresponding (reduced) access map, which is based on known results on the dynamics of invertible circle maps, cf. [41,42]. One finds that the reduced access map can show two types of dynamical behaviour. For conservative delay, the reduced access map is characterized by marginally stable quasi-periodic motion. In this case, the mean distance between two arbitrary orbits is preserved and thus the Lyapunov exponent of the access map is given by λ[R] = 0. By contrast, for dissipative delay, the reduced access map is characterized by mode-locking and has stable periodic orbits or fixed points with λ[R] < 0. For a sinusoidal variation of the distance d(t) with mean distance $\overline{d}$ and amplitude A_d the dependence of the delay class on the parameters is illustrated in figure 1d. The black regions, which are called Arnold tongues [42], correspond to dissipative delays, and the white regions, which form a fat fractal [42] for each fixed A_d, correspond to conservative delays.

From equation (2.3), it is clear that the delay type will have a significant effect on the resulting waveform. In figure 1c exemplary signals for conservative and dissipative delays are shown, where for both delays the initial signal was a sinusoidal signal and f(z) = z. For conservative delays, the signal is modulated quasi-periodically. This can be understood by using the above-mentioned theory for circle maps. In particular, for conservative delays, the reduced access map is topological conjugate to the pure rotation θ′ = θ − τ_c mod 1. Hence, there is an invertible function Φ such that

$$\mathit{\Phi }(R({\mathit{\Phi }}^{-1}(t)))=\theta -{\tau }_{\mathrm{c}}\iff \mathit{\Phi }(t)-\mathit{\Phi }(R(t))={\tau }_{\mathrm{c}}.$$ 2.6

It follows that the system is equivalent to a system with a constant delay τ_c and can be transformed to the latter by introducing the new time scale θ = Φ(t) and the new variable y(θ) = z(Φ⁻¹(θ))) [37,38,43]. By equation (2.6), our exemplary process, where the delay τ(t) is caused by the transport of a signal over a time-varying distance d(t), can be mapped to a process, where the same delay τ(t) is caused by the transport of the signal over the constant distance $d_{\mathrm{c}}=\overline{v}{\tau }_{\mathrm{c}}$ with the time-varying velocity v(t), where $\overline{v}$ is the time-average of v(t). This becomes clear by interpreting the quantity $\overline{v}\hspace{0.17em}\mathit{\Phi }(t)$ as the distance which was covered by the signal with the periodically time-varying velocity $v(t)=\overline{v}\hspace{0.17em}\dot{\mathit{\Phi }}(t)$ and rewriting equation (2.6) as

$${\int }_{t-\tau (t)}^{t}v(t^{\prime })\hspace{0.17em}\mathrm{d}{t}^{\prime }=d_{\mathrm{c}}.$$ 2.7

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 [44–47]. 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. In population dynamics, it is also called threshold delay [49,50], where d_c is typically interpreted as a threshold which must be reached by passing different evolutionary steps to become able to reproduce. This interpretation leads to a simple explanation of the signal in figure 1c. The signal is modulated quasi-periodically because of the interplay of two generally incommensurate periods, which are the characteristic transmission time ${\tau }_{\mathrm{c}}=d_{\mathrm{c}}/\overline{v}$ and the period of the time-varying velocity v(t).

Contrarily, the signal which was generated with a dissipative delay, is characterized by a periodic switching between low- and high-frequency phases. After each iteration of equation (2.3), the frequencies in the low- (high) frequency phases decrease (increase) and the phases get wider (narrower). This is a consequence of the mode-locking behaviour of the access map. From the Doppler effect, it follows that the frequency of the signal measured by the microphone increases or decreases, if the distance d(t), i.e. the delay decreases or increases, respectively. For a mode-locking access map corresponding to dissipative delay, on average the distance d(t) decreases (increases) at times where the frequency of the signal is already high (low), because the distance d(t) decreased (increased) on average already at these times in the previous round trips of the feedback loop. The term ‘on average’ means that the statements hold for every qth iteration of equation (2.4), i.e. z_n+q(t) = f^q(z_n(R^q(t))), where q denotes the period of the periodic orbits of the reduced access map. If the reduced access map has q-periodic orbits, there is a natural number p such that there are infinitely many points t^(*)_l fulfilling R^q(t^(*)_l) + p = t^(*)_l, which we simply call periodic points of the access map t′ = R(t). If t^(*)_l is a stable (unstable) periodic point of the access map, (R^q)′(t^(*)_l) < 1 ((R^q)′(t^(*)_l) > 1), the frequency of z_n+q(t) in the neighbourhood of t^(*)_l is smaller (larger) than the frequency of z_n(t) in the neighbourhood of t^(*)_l − p. Due to the interaction between the Doppler effect and the resonance from the mode-locking behaviour of the access map, we call the effect behind dissipative delays resonant Doppler effect. By contrast, we call the effect behind conservative delays non-resonant Doppler effect. These mechanisms behind dissipative and conservative delays are fundamental and present in all time-delay systems.

3. Resonant Doppler effect in nonlinear delay differential equations

In §2, we have described the fundamentals of the resonant and non-resonant Doppler effect for the simple time delay system equation (2.3) by using the trivial choice f(z) = z. Here, we describe the effect of time-varying delays in nonlinear systems and differential equations with delay (DDEs). Since the influence of the non-resonant Doppler effect can be mapped to the influence of a constant delay, we focus our analysis to the resonant Doppler effect. We consider DDEs of the form

$$\frac{1}{T}\dot{z}(t)=-z(t)+f(z(R(t))),\mathrm{with}\hspace{0.17em}R(t)=t-\tau (t).$$ 3.1

Systems with the structure of DDE (3.1) occur in several applications. For instance, the Ikeda equation, with f(z) = μ sin(z), was used for describing chaos in an optical cavity with a nonlinear optical medium [51,52]. A further example is the Mackey–Glass equation, with f(z) = μ z/(1 + z¹⁰), which models oscillations in blood production [53]. The DDE with the quadratic nonlinearity, f(z) = μ z(1 − z) is used in [54] to derive general properties of solutions of DDE (3.1). Since systems described by DDE (3.1) can be easily realized in optoelectronic set-ups, they are used for fast realizations of chaos communication [55–58] and reservoir computing [59,60]. In the past, much attention has been paid to the singular limit of a large parameter T, which can be interpreted as the limit of a large delay as one can easily verify by a linear time-scale transformation of DDE (3.1). The so-called slowly oscillating periodic solutions, which are characterized by slow oscillations within a period of the order of the delay, and the connected transition layer phenomena where intensively investigated for constant delay [54,61,62] and for state-dependent delay [34]. For constant delay, the multistability of these solutions was also investigated [63–65]. The analysis in [39] goes beyond those periodic solutions and establishes a theory of the dynamics of a slowly oscillating chaotic solution, which is called laminar chaos and is found only in systems with dissipative delay. These solutions are fundamentally different from the slowly oscillating solutions, which are known for constant delay systems. In fact, they are a result of the Doppler effect with self-feedback due to a time-varying dissipative delay. In the following, we present a systematic analysis of the dynamics due to dissipative delays for the nonlinear DDE (3.1). It opens a rich variety of periodic and chaotic dynamics, which may be interesting for applications.

For T → ∞, the derivative term on the left-hand side of DDE (3.1) vanishes and equation (3.1) becomes equation (2.3), which can be solved via the recurrence relation (2.4). In this context, equation (2.4) is called limit map as it was done for the analogous relation for constant delay [64]. Thus, the main effect of the interaction of a nonlinearity with a time-varying delay can be understood by studying the limit map equation (2.4) with nonlinear f. For finite T, there is an additional effect due to the time derivative in equation (3.1). In this case, the DDE (3.1) can be solved by the method of steps, which makes use of the variation of constants approach to solve the DDE (3.1) in one solution segment [44]

$$z_{n+1}(t)=z_n(t_n){\mathrm{e}}^{-T(t-t_n)}+{\int }_{t_n}^t\hspace{0.17em}\mathrm{d}{t}^{\prime }\hspace{0.17em}T{\mathrm{e}}^{-T(t-t^{\prime })}f(z_n(R(t^{\prime }))).$$ 3.2

Following the analysis in [39], the solution operator defined by equation (3.2) can be decomposed into the application of the limit map given by equation (2.4) and the subsequent application of a integral operator with the kernel Te^{−T(t−t′)}. Since for large T, the integral kernel can be seen as an approximation of the delta distribution [8], the integral operator can be interpreted as a smoothing operator, which damps high frequencies and approaches the identity for T → ∞. Consequently, for large T, the analysis can be reasonably done in terms of the limit map, keeping in mind the influence of the smoothing operator.

(a). Intrinsic time-multiplexed dynamics

In time-multiplexed dynamics, the corresponding trajectories appear as a sequence of segments of trajectories from independent dynamical systems, which can exhibit stable equilibria, periodic dynamics and chaotic dynamics. Time-multiplexing is relevant for many applications such as the transmission of multiple signals over a shared signal path [66] and for the electro-optical realization of reservoir computing [59,60]. In [67], the concept of time-multiplexing is used for the electro-optical implementation of arbitrary networks of coupled maps. In the following, we show how the resonant Doppler effect leads to an intrinsic time-multiplexed dynamics with a periodic switching between different solution states. This is illustrated in figure 2.

Figure 2.

Structure of the solutions which is induced by the resonant Doppler effect. (a) The exemplary solution z(t) shows laminar chaos, which is caused by a dissipative delay, i.e. the resonant Doppler effect and is explained in §§3b. The solution segment z_n(t) = z(t) with t inside the state interval (t_n−1, t_n] can be divided into the solution segments z_n,l = z(t), with t∈U_n,l, where the U_n,l are the basins of attraction of the stable periodic points of the access map. The boundaries of the U_n,l are given by the repulsive periodic points of the access map, whose locations are marked by the thin grey lines. The periodic points are the solutions t* of R^q(t*_n,l) + p = t*_n,l for some $p\in \mathbb{N}$ (see inset), where q denotes the period of the corresponding periodic orbits (here p = 3 and q = 2). The natural number p determines the number of U_n,l inside each state interval. As indicated by the arrows for the exemplary solution segment inside U_n,1, the solution segment inside U_n,l is mapped by the limit map to the solution segment U_n+1,l. This means that, for large T, the solution segments z_n,l inside one state interval (fixed n) are almost pairwise independent and are strictly pairwise independent for T → ∞. (b) Two stable periodic solutions of equation (3.1) with large T, where the map z′ = f(z) has two stable fixed points z*₁ = 0 and z*₂ = 1 and R(t) is the same as in (a). The red vertical lines represent the boundaries of the U_n,l. The solutions were generated with different initial conditions z₀(t). For the top and the bottom trajectory, we chose z₀(t) = 0 if t∈U_0,1 and z₀(t) = 0 if t∈U_0,1∪U_0,2, respectively, and z₀(t) = 1 otherwise. (Online version in colour.)

For dissipative delays, there exist infinitely many periodic points t^(*)_l. The unstable ones may define boundaries between different basins of attraction of the access map equation (2.1), and therefore, it is possible that the dynamics of equation (3.1) on the left- and right-hand side of an unstable periodic point t^(*)_l may be rather independent. More precisely, if we set the initial time t₀ to an unstable periodic point of the access map, the boundaries of the state intervals (t_n−1, t_n] are also unstable periodic points of the access map, since t_n−1 = R(t_n). Then, according to [38], each state interval (t_n−1, t_n] can be divided into p subintervals U_n,l, $l=1,2\dots ,p$, which are the basins of attraction of the attractive periodic points of the access map, i.e. they fulfil R^q(U_n,l) + p = U_n,l and if t, s∈U_n,l one has |Rⁿ(t) − Rⁿ(s)| → 0 in the limit n → ∞. As shown in [39], for T → ∞ the limit map maps the solution segment z_n,l(t) = z(t) with t∈U_n,l to the solution segment z_n+1,l(t) with t∈U_n+1,l (figure 2a). For large but finite T the integral operator in equation (3.2) leads to a smooth transition between two neighbouring solution segments, where the relaxation time decreases with increasing T. Thus, for dissipative delay and T → ∞ (large T), the dynamical system divides into at least p independent (almost independent) systems and the solution switches between them periodically in time, which can be viewed as time-multiplexing.

This property can be used to construct multistable periodic solutions. Figure 2b shows two stable periodic solutions of equation (3.1), where f(z) has two fixed points z*₁ = 0 and z*₂ = 1. This means that the solution in one segment becomes constant z_n,l(t) ≈ z*_k and the solution in the corresponding segment of the following state intervals is the same, i.e. z_n′,l(t) ≈ z*_k for n′ > n and k = 1 or k = 2. A temporal switching between these two states can be obtained by using different initial conditions for different solution segments. In figure 2b, there is a sinusoidal dissipative delay with p = 3, which means that every third solution segment (divided by the red vertical lines) contains the same state. The p-periodic solutions in figure 2b coexist with the steady states z*₁ = 0 and z*₂ = 1, which are obtained if the initial functions are chosen appropriately. Further multi-stable solutions can be obtained if the map f exhibits periodic dynamics. Note that in contrast to the slowly oscillating periodic solutions in systems with constant delay, the period of the solutions originated from the resonant Doppler effect is locked and does not change with the parameter T as known for the former [54,61,62]. In §§3c, the intrinsic time-multiplexing induced by the resonant Doppler effect is exploited to design more complex time-multiplexed dynamics, which consists of different chaotic and periodic dynamics.

(b). Resonant Doppler effect and chaos

Another characteristic of the resonant Doppler effect is given by the contracting property of the access map t′ = R(t), which leads to the low-frequencies phases inside the U_n,l and the high-frequency regions at the boundaries (cf. figure 1). The dynamics of the limit map given by equation (2.4) is characterized by the access map and the map z′ = f(z), respectively. If the latter exhibits chaotic dynamics, the competition between the contraction by the access map R and the expansion by the map f leads to different kinds of chaotic dynamics. In figure 3a–c, three exemplary chaotic trajectories of DDE (3.1) with large T are presented, where we have chosen the nonlinearity f(z) = 4 z(1 − z) corresponding to chaotic dynamics and different sinusoidally time-varying delays. The trajectory in figure 3a corresponds to a conservative delay and shows a dynamics which we call turbulent chaos, in accordance with the term ‘optical turbulence’ introduced in [52]. Turbulent chaos was originally observed in systems with constant delay and is characterized by fast fluctuations on a scale much smaller than the delay length. The mechanism behind turbulent chaos can be explained via the limit map. The consecutive iteration of the map f increases the characteristic frequency due to the repeated stretching and folding of the function values. The access map R leads to a quasi-periodic modulation of the signal but does not change the asymptotic behaviour. For large but finite T, the characteristic frequency of the solution is bounded because the additional smoothing operator in equation (3.2) acts as a low pass filter, where the cut-off frequency is proportional to T. The trajectories in figure 3b,c correspond to two different dissipative delays, and thus demonstrate the influence of the resonant Doppler effect. We call this dynamics generalized laminar chaos. It is characterized by low-frequency phases, the laminar phases, which are periodically interrupted by short irregular bursts around the repulsive periodic points of the access map. In figure 3c, the low-frequency phases degenerate to constant laminar phases. This special case is simply called laminar chaos and was introduced in [39].

Figure 3.

Chaotic trajectories of DDE (3.1) with f(z) = 4 z(1 − z), T = 200, and sinusoidal delay τ(t) = τ₀ + A/(2π)sin(2π t) for (a) a conservative delay (A = 0.90, τ₀ = 1.54) leading to turbulent chaos known from systems with constant delay [52], (b) a dissipative delay (A = 0.86, τ₀ = 1.50) leading to generalized laminar chaos as indicated by the periodic switching between phases of high and low frequency, and for (c) a dissipative delay (A = 0.90, τ₀ = 1.50) leading to laminar chaos characterized by constant laminar phases and burst-like transitions between them. For generalized laminar chaos, the burst-like transitions between the laminar phases are located at the repulsive periodic points of the access map. The criterion (3.3) for laminar chaos with constant plateaus is visualized by the Lyapunov chart (d) where the contours are regions of equal Lyapunov exponents of the access map R(t). The dashed red contour corresponds to λ[R] = − λ[f] = − log2. (Online version in colour.)

In the following, we describe the properties of generalized laminar chaos in the limit T → ∞. In extension to the known criterion for laminar chaos [39], we derive necessary conditions for the existence of different orders of this type of chaos. According to [39] laminar chaos with constant plateaus can be observed only if λ[f] > 0 and

$$\mathrm{\lambda }[f]+\mathrm{\lambda }[R]<0.$$ 3.3

In particular, if equation (3.3) holds, it can be shown via the limit map equation (2.4) that the time derivative vanishes, ${\dot{z}}_n(t)\to 0$ for n → ∞ at almost all t. This results in the constant plateaus. Only at the unstable periodic points of the access map ${\dot{z}}_n(t)$ does not vanish and irregular bursts occur. For our exemplary system, the criterion is illustrated in figure 3d, which is a Lyapunov chart of the access map as function of the delay amplitude A and the mean delay τ₀. The dashed red contour corresponds to λ[f] + λ[R] = 0, and thus the criterion for laminar chaos is fulfilled by all parameters above this line. If the criterion for laminar chaos is not fulfilled, the absolute value $|{\dot{z}}_n(t)|$ of the derivative of the state under the evolution of the limit map increases exponentially for n → ∞ at almost all t∈(t_n−1, t_n]. Hence, the characteristic frequency increases by each iteration as in the case of turbulent chaos, but there are still fundamental differences to turbulent chaos, which we explain in the following.

We consider the evolution of perturbations of the initial state z₀(t) under iterations of the limit map equation (2.4). As described in §§3a, the solution inside one state interval can be divided into segments z_n,l = z(t), t∈U_n,l, with one stable periodic point t*_l of the access map inside each subinterval. Since for T → ∞ the solution segments inside one state interval are pairwise independent, the perturbations can be analysed separately inside each of them. We define δz₀ as the initial perturbation in one segment U_0,l of the initial state z₀(t), which is given by the power series

$$\delta z_0=\delta z_0^{[N]}(t):=\sum _{k=N}^{\mathrm{\infty }}{\alpha }_k\hspace{0.17em}{(t-t^{\ast })}^k.$$ 3.4

In the above equation, we dropped the index l for brevity and t* is the attractive periodic point of the access map in U_0,l. We call δz^[N]₀(t)∈O(t^N) perturbation of order N, since lim_t → t*δz^[N]₀(t) (t − t*)^−N < ∞. For N = 0, the initial perturbation δz^[0]₀(t) is an arbitrary analytic function in U_0,l. Starting with the perturbed initial state ${\tilde{z}}_0(t)$, the perturbed state after n-iterations of the limit map is given by

$${\tilde{z}}_n(t)=z_n(t)+\delta z_n^{[N]}(t).$$ 3.5

With equation (2.4) we obtain for ${\tilde{z}}_n(t)$

$${\tilde{z}}_n(t)=f^n[z_0(R^n(t))+\delta z_0^{[N]}(R^n(t))]\approx f^n(z_0(R^n(t)))+{(f^n)}^{\prime }(z_0(R^n(t)))\hspace{0.17em}\delta z_0^{[N]}(R^n(t)),$$ 3.6

which leads together with equation (3.4) to the following approximation:

$$\delta z_n^{[N]}(t)\approx \sum _{k=N}^{\mathrm{\infty }}{\alpha }_k\hspace{0.17em}{(f^n)}^{\prime }(z_0(R^n(t))){(R^n(t)-t^{\ast })}^k.$$ 3.7

For large n, we have Rⁿ(t) → t*, |(fⁿ)′(z₀(Rⁿ(t)))|∼e^n λ[f] and |(Rⁿ(t) − t*)^k|∼e^{n k λ[R](t*)}, where t∈U_n,l. This means that if K(t*) is the smallest natural number such that

$$\mathrm{\lambda }[f]+K(t^{\ast })\hspace{0.17em}\mathrm{\lambda }[R](t^{\ast })<0,$$ 3.8

perturbations of the order K(t*) vanish, i.e. lim_{n → ∞}δz^[K(t*)]_n(t) = 0. In other words, the monomials of order N < K(t*) characterize the unstable directions and the monomials with larger order N≥K(t*) characterize the stable directions inside the subintervals U_n,l. If there are m subintervals U_n,l, which correspond to the stable periodic points t*_l inside each state interval (t_n−1, t_n], there are

$$N_u=\sum _{l=1}^{m}K(t_l^{\ast })$$ 3.9

unstable directions. If the criterion equation (3.8) holds, it can be shown that perturbations of the order N≥K(t*) need not to be small to vanish. Thus, it follows directly for the solution segments z_n,l(t) for n → ∞

$$z_{n,l}(t)\to f^n(\sum _{k=0}^{K(t_l^{\ast })-1}\frac{z_{0,l}^{(k)}(t_l^{\ast })}{k!}\hspace{0.17em}{(R^n(t)-t_l^{\ast })}^k),\mathrm{where}\hspace{0.17em}t\in U_{n,l}$$ 3.10

and z_0,l(t) is expressed by its Taylor series at t*_l. Thus, N_u components determine the position on the attractor, if one neglects the influence of the transitions at the unstable periodic points, which are the boundaries of the U_n,l. For sawtooth-shaped delay, this approximation is exact and the limit map dynamics is characterized by N_u positive Lyapunov exponents. We call the generalized laminar chaos, which fulfils equation (3.8), (K(t*) − 1)-order laminar chaos according to the upper limit of the sum in equation (3.10).

In figure 4, these results are visualized by the time evolution of exemplary trajectories and perturbed trajectories under the iterations of the limit map, equation (2.4), by using the sinusoidal and sawtooth-shaped delays which are illustrated in figure 5. To obtain comparable results, the delays where chosen in such a way that the related access maps have one attractive fixed point t* inside each state interval with the same position and Lyapunov exponent. The location of t* inside the initial state interval is indicated by the dashed red line in figure 4. The unstable fixed points and the jumps of the access maps for sinusoidal and sawtooth-shaped delay, respectively, are located at the integer numbers. In the left column of figure 4, the initial states z₀(t) are perturbed by perturbations δz₀(t) of order N = K(t*). As a result, the perturbations vanish inside the basins of attraction of the attractive periodic points t* + j, $j\in \mathbb{N}$, of the access map. This means that for sawtooth-shaped delay the perturbations vanish everywhere and for sinusoidal delay the perturbation grows only at the unstable fixed points of the access map, where the irregular busts are located. According to the analysis above, the perturbations vanish only because of the contraction property of the access map. In the vicinity of the repulsive periodic points, the access map is not contracting, and thus the perturbations grow, which causes the burst-like transitions between the low-frequency or laminar phases. In the right column of figure 4, the initial states z₀(t) are perturbed by perturbations δz^N₀(t) with N < K(t*). Consequently, the perturbations grow everywhere, that is, the intensity levels of the laminar phases in figure 4g,j diverge and the oscillating states in (h) and (k) diverge. The latter becomes evident by the magnifications (i) and (l) of (h) and (k), respectively.

Figure 4.

Time evolution of original state and perturbed state under iterations of the limit map, equation (2.4), with f(z) = 4 z(1 − z) for sinusoidal (top) and sawtooth-shaped (bottom) delays which are illustrated in figure 5 (A = 0.9 for (a,d,g,j) and A = 0.3 for (b,e,h,k). The unperturbed solutions, which are obtained from a sinusoidal initial function z₀(t), and the perturbed solutions with initial function ${\tilde{z}}_0(t)=z_0(t)+\delta z_0^{[N]}(t)$ are represented by the thin blue and thick orange lines, respectively. The perturbations δz^[N]₀(t) are defined by (3.4). The smallest natural number K = K(t*), such that the criterion equation (3.8) holds, is K = 1 for (a,d,g,j) and K = 2 (b,e,h,k), leading to laminar chaos and first-order laminar chaos, respectively. (c,f ,i,l) are magnifications of (b,e,h,k), respectively. In (a–f ), the order N of the initial perturbation δz^[N]₀(t) is larger than or equal to K, which means that the perturbation vanishes almost everywhere, for t → ∞. For sinusoidal delay (a–c), the perturbation grows in a region near the repulsive periodic points of the access map, which shrinks with each iteration of the limit map, whereas for sawtooth-shaped delay (d–f ) the perturbation vanishes everywhere. In (g–l) N = K − 1 < K, the perturbations grow and the trajectories diverge from each other. (Online version in colour.)

Figure 5.

(a) Sinusoidal (thick) and sawtooth-shaped (thin) delay variation ν(t) and (b) reduced access maps R(t) mod 1 = t − τ(t) mod 1 which correspond to the delay τ(t) = 1 + ν(t). For both of the delays, the Lyapunov exponent of the attractive fixed point of the reduced access map is given by λ[R] = log(1 − A). (Online version in colour.)

For T → ∞, we have seen that generalized laminar chaos is characterized by a finite number N_u of unstable directions if one neglects the burst-like transitions between the low-frequency phases. For large but finite T the burst-like transitions at the unstable periodic points of the access map are smoothed by the integral operator in equation (3.2) and a dynamics similar to figure 3b appears. By contrast, for T → ∞, turbulent chaos is characterized by an infinite number of unstable directions, which for the latter are simply all function values of the initial state z₀(t). In a DDE with finite T, the number of these unstable directions is bounded but increases with increasing T, which is known from systems with constant delay [7–9]. This leads to the conclusion that, even for finite T, generalized laminar chaos is a very low-dimensional phenomenon compared to turbulent chaos, as already indicated in [39].

(c). Time-multiplexing with chaotic signals

With the results from §§3b, we are able to construct more complex time-multiplexed dynamics and especially time-multiplexing with chaotic signals or two different types of chaos. As we have demonstrated already in §§3a, for dissipative delay, the solution z_n(t) in the nth state interval can be divided into subintervals U_n,l, which are the basins of attraction of the attractive periodic points of the access map. Based on this concept, there are two mechanisms, which can be exploited for creating the time-multiplexing between different types of dynamics including chaos.

The idea of the first mechanism is to find a delay, where the Lyapunov exponents λ[R](t*) of the access map are different in different pairwise-independent solution segments U_n,l. Depending on the nonlinearity f in the DDE (3.1), this can lead to a periodic switching between different orders of mode-locked chaos or in the extreme case a periodic switching between laminar chaos and turbulent chaos. The dynamics of the reduced access map inside the segments U_n,l can be characterized either by the attraction to different stable periodic orbits with different Lyapunov exponent or by marginally stable dynamics.

In figure 6, the reduced access map R(t) mod 1 and a resulting trajectory of an exemplary system with a periodic switching between laminar chaos and turbulent chaos are visualized. The system is defined by DDE (3.1) with f(z) = 4 z(1 − z). The delay varies periodically with period 1 and is defined by

$$\tau (t)=1-\{\begin{array}{ll}\frac{0.9}{4\pi }\sin (4\pi \hspace{0.17em}t),& \mathrm{if}\hspace{0.17em}\{t\}\le 0.5\\ 0.9\hspace{0.17em}(\{t\}-0.5)\hspace{0.17em}{e}^{(\{t\}-0.5)/(\{t\}-0.6)}& \mathrm{if}\hspace{0.17em}0.5<\{t\}<0.6\\ 0,& \mathrm{else}\end{array},\mathrm{where}\hspace{0.17em}\{t\}=t-\lfloor t\rfloor .$$ 3.11

The domain of the reduced access map can be divided into three invariant sets U₁ = [0, 0.5], U₂ = [0.5, 0.6] and U₃ = [0.6, 1]. On U₁ the access map has a stable fixed point with λ[R] = log[0.1] < 0 and thus the delay inside [n, n + 0.5], $n\in \mathbb{Z}$, is dissipative. Together with λ[f] = log[2], which implies that the criterion (3.3) is fulfilled, it is clear that we observe laminar chaos in this time intervals. The delay on U₃ is conservative because it is constant. Thus, we observe turbulent chaos inside the time intervals [n + 0.6, n + 1]. Inside U₂, the reduced access map has a marginally stable fixed point at 0.6, which represents a limiting chase of a dissipative delay with λ[R] = 0. Being exact, according to equation (3.8), the dynamics has to be characterized as infinite-order laminar chaos, which shares the property with turbulent chaos, that the limit map dynamics is characterized by an infinite number of unstable directions. Roughly speaking, we can characterize the dynamics of the delay system inside the time intervals [n + 0.5, n + 0.6] as turbulent chaos and merge them with the time intervals which correspond to U₃. To observe a periodic switching between laminar and turbulent chaos in a DDE, the separation between the conservative region and the dissipative region of the reduced access map must be structurally stable, since for finite T, the limit map dynamics is perturbed by the smoothing operator. The smoothing operator induces a temporal drift δ of structures in the space–time representation of z(t) [5], such that, for large T, z_n+1(t) ≈ f(z_n(R(t − δ))) = f[z_n(R(t) − Δ(t))], where Δ(t) > 0. In our example, the stability against this perturbation is achieved by the structural stability of the unstable fixed point at 0.5, which requires the existence of the regions related to U₂ between the regions of dissipative and conservative delay. If the reduced access map inside U₂ would be substituted by the bisectrix as in U₃, the unstable fixed point would not be structurally stable. Hence, there would be arbitrary small perturbations, which lead to pure laminar chaos, since all points inside [0, 1] are attracted by the stable fixed point of the perturbed reduced access map.

Figure 6.

Exemplary dynamical system showing time-multiplexing of laminar and turbulent chaos. The system defined by DDE (3.1), with f(z) = 4 z(1 − z), and the delay was chosen, such that the reduced access map (a) has a dissipative and a conservative region. They are separated by an unstable fixed point, which is structurally stable, i.e. it survives small perturbations of the shape of the map. A magnification of the unstable fixed point is shown in (b) and an exemplary trajectory, which illustrates the periodic switching between laminar chaos and turbulent chaos, is shown in (c). (Online version in colour.)

The idea behind the second mechanism for time-multiplexing between different subsystems is to modify the map f. More precisely, the time-multiplexed dynamics is achieved by switching the nonlinearity of the DDE (3.1) between the intervals U_n,l. Hence, we define our system by

$$\frac{1}{T}\dot{z}(t)=-z(t)+f(t,z(R(t))),\mathrm{with}\hspace{0.17em}f(t,z)=f_l(z)\mathrm{for}\hspace{0.17em}t\in U_{n,l}.$$ 3.12

In figure 7b, the trajectory of an exemplary time-multiplexed system is visualized, where the dynamics switches between laminar chaos and the relaxation to the equilibrium z = 0.5. The delay is chosen such that there are two basins of attraction U_n,l inside each state interval. For t∈U_n,1 and t∈U_n,2, the map z′ = f(t, z) is chaotic or has a stable fixed point, respectively, leading to the time-multiplexing between laminar chaos and steady state dynamics.

Figure 7.

Temporal switching between different nonlinearities. The systems switches periodically with period 2 between two quadratic nonlinearities f_k = μ_kz(1 − z) (a), where in the first half period f(t, z) = f₁(z) and in the second half period f(t, z) = f₂(z). The two nonlinearities f_k correspond to the logistic map with the bifurcation parameters μ₁ = 4.0 and μ₂ = 2.0, where the map z′ = f(z) exhibits chaotic dynamics or has a stable fixed point, respectively. The delay τ(t) = 2 − 0.9sin(2πt)/(2π) was chosen such that the access map has two fixed points inside each state interval. The two U_n,l coincide with the first and the second half period, which leads to the result that the trajectory (b) periodically switches between laminar chaos and the relaxation to an equilibrium. (Online version in colour.)

4. Conclusion

We have investigated the influence of a time-varying delay τ(t) on the dynamics of time-delay systems. We have shown that the central effect due to a time-varying delay is equivalent to the Doppler effect, where a varying distance between a signal source and a sensor leads to a frequency shift of the measured signal. In addition, in time delay systems, there is a feedback from the sensor signal to the source, and therefore, time delay systems can be interpreted as a Doppler effect with self-feedback. From the previous literature, it is known that there are two types of time-varying delays, which are called conservative and dissipative delay. These types are related to the specific dynamics of the so-called access map formed by the retarded argument R(t) = t − τ(t). Conservative delays lead to marginally stable (periodic or quasi-periodic) access map dynamics, whereas dissipative delays lead to mode-locking. Corresponding to the two types of time-varying delay, there are two types of the Doppler effect with self-feedback. The so-called non-resonant Doppler effect is caused by conservative delays and leads to a dynamics of the delay system, which is equivalent to the dynamics of a system with a constant delay because conservative delays can be transformed to constant delays. Contrarily, the resonant Doppler effect is caused by dissipative delays and the dynamical behaviour is qualitatively different from the dynamics in systems with constant delay.

In this paper, we have extended the results in [39] for a class of systems with large time-varying delay, where the dynamics can change drastically with the delay class. In the latter reference, it was shown that under certain conditions, dissipative delays can lead to low-dimensional laminar chaos, whereas a conservative delay leads to high-dimensional turbulent chaos. In this paper, we have demonstrated that there are infinitely many different types of laminar chaotic solutions, which we call generalized laminar chaos. In all these cases, the solution is characterized by low-frequency phases, which are periodically interrupted by bursts with high frequencies. For the original laminar chaos or zeroth-order laminar chaos, the low-frequency phases are constant. For generalized laminar chaos, the shape of the laminar phases depends on the order of the laminar chaos. Roughly speaking, the order of the laminar chaos determines the number of unstable directions (or positive Lyapunov exponents), which saturates for increasing delay. As a result, for large delay, the dimension of generalized laminar chaos is potentially much lower than the dimension of turbulent chaos, which for constant delay is proportional to the delay [7–9]. We further have shown that the resonant Doppler effect leads to time-multiplexed dynamical systems. For large delay, the trajectories of the resulting system are characterized by a periodic switching between different dynamics, which can be useful for applications in data storage and transmission applications.

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Authors' contributions

D.M.-B. developed the theory, performed the numerical analysis and drafted the manuscript. A.O. and G.R. contributed to the theory. G.R. supervised the work. All authors carefully read and approved the manuscript.

Competing interests

The authors declare that they have no competing interests.

Funding

We acknowledge partial support from the German Research Foundation (DFG) under the Grant No. 321138034.