Amplitude chimeras and bump states with and without frequency entanglement: a toy model

Starting from the toy model discussed in the introduction and in [9, 50], we now consider the case where the frequency ω is constant, but the amplitude r is time dependent, r(t), and is governed by bistable dynamics, as follows:

$$\begin{align} \frac{\textrm{d}x}{\textrm{d}t} & = -ax+\frac{ar}{\sqrt{x^2+y^2}}x-\boxed{\omega} \,y \end{align} \tag{ 3a }$$

$$\begin{align} \frac{\textrm{d}y}{\textrm{d}t} & = -ay+\frac{ar}{\sqrt{x^2+y^2}}y+ \boxed{\omega} \,x \end{align} \tag{ 3b }$$

$$\begin{align} \frac{\textrm{d} r}{\textrm{d}t} & = c_r \left(r -r_l\right)\left(r -r_c\right)\left(r -r_h\right) . \end{align} \tag{ 3c }$$

The variable ω is boxed in equations (3a ) and (3b ) for highlighting purposes and for comparison with equations (6a ) and (6b ), later on. The time dependent amplitude is here represented by small r, instead of capital R used in the Introduction and in [9, 50], where it refers to constant amplitude oscillations. Similarly, the constant c_r in equation (3c ) governing the amplitude dynamics takes the subscript r to stress that it is related to the amplitude r modulations. The variant model, equation (3), can be also considered as an 'intensity-dependent self-interaction' model, because now the nonlinear term $a r/{\sqrt{x^2+y^2}}$ not only includes the inverse intensity term $(\sqrt{x^2+y^2})^{-1}$ which multiplies the x or y-variables, but also involves the overall variable r which asymptotically tends to either r_l or r_h , depending on the initial conditions.

When $c_r = 0$, the amplitude takes a constant value $r(t) = R$ = constant and the model reduces to

$$\begin{align} \frac{\textrm{d}x}{\textrm{d}t} & = -ax+\frac{aR}{\sqrt{x^2+y^2}}x-\omega y \end{align} \tag{ 4a }$$

$$\begin{align} \frac{\textrm{d}y}{\textrm{d}t} & = -ay+\frac{aR}{\sqrt{x^2+y^2}}y+ \omega x . \end{align} \tag{ 4b }$$

The solution is analytically obtained as equations (5), see also [9].

$$\begin{align} x\left(t\right) & = R\left(1-Ae^{-at}\right) \cos \left(\omega t\right) \end{align} \tag{ 5a }$$

$$\begin{align} y\left(t\right) & = R\left(1-Ae^{-at}\right) \sin \left(\omega t\right). \end{align} \tag{ 5b }$$

Starting from any initial state determined as $x^2 +y^2 = R^2(1-A)^2$ with $0\unicode{x2A7D} A \lt1$, the trajectory relaxes exponentially to a circle of constant radius R and constant phase velocity ω. The relaxation exponent is denoted by a.

When $c_r \gt 0$, then $r = r (t)$ becomes time dependent. Equation (3c ) has three fixed points, $r = r_l , r_c$ and r_h . The subscripts l, c and h denote the low, center and high amplitude fixed points, respectively. From the point of view of stability, r_c is an attractor and r_l and r_h are repellers. Starting with an arbitrary initial $r(t = 0)$-value the system can only end-up with final amplitude $r = r_c$.

If $c_r \lt 0$ the amplitude becomes also time dependent, $r = r(t)$, but, stability-wise, the central fixed point r_c is a repeller and r_l and r_h are attractors. The system now may exhibit amplitude bistability. Depending on the initial amplitude $r(t = 0)$ the system can end-up in one of the two stable fixed points, r_l or r_h .

As an example, in figure 1(a) we present the temporal evolution of equation (3) starting from two different initial conditions, called I (black lines) and II (red lines). Other parameters are exactly the same: a = 1.0, ω = 1.0, $r_l = 2.0$, $r_c = 3.0$ and $r_h = 4.0$. In panel a) we note that the two trajectories have equal frequencies but different amplitudes. The equality of the frequencies is numerically verified and shown in plot figure 1(b).

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This last option, $c_r \lt 0$, will be used in the next section, section 3, when many two-amplitude oscillators will be coupled in a ring. We will numerically demonstrate that when the different coupled oscillators start with different, random, initial amplitudes there will be a split of the network in domains of alternating low r_l and high r_h amplitudes.

Before introducing the coupled system, we present in section 2.2 an extension of the amplitude bistable toy model where amplitude r and frequency ω are entangled and, as a result, they influence each other and complexify the overall motion.

For the amplitude-frequency entanglement we consider the following extension of the amplitude modulation toy model:

$$\begin{align} \frac{\textrm{d}x}{\textrm{d}t} & = -ax+\frac{ar}{\sqrt{x^2+y^2}}x-\boxed{r \omega} \,y \end{align} \tag{ 6a }$$

$$\begin{align} \frac{\textrm{d}y}{\textrm{d}dt} & = -ay+\frac{ar}{\sqrt{x^2+y^2}}y+ \boxed{r \omega } \,x \end{align} \tag{ 6b }$$

$$\begin{align} \frac{\textrm{d} r}{\textrm{d}t} & = c_r \left(r -r_l\right)\left(r -r_c\right)\left(r -r_h\right) . \end{align} \tag{ 6c }$$

Namely, we introduce a factor of r multiplying the constant phase velocity ω in the last terms of equations (6a ) and (6b ) (see boxed expressions and compare with the case without entanglement presented in equations (3a ) and (3b )). This is the only differences between systems equations (3) and (6). The solutions for the variables x and y can be given analytically in terms of r and t

$$\begin{align} x\left(t\right) & = r\left(1-Ae^{-at}\right) \cos \left(r \omega t\right) \end{align} \tag{ 7a }$$

$$\begin{align} y\left(t\right) & = r\left(1-Ae^{-at}\right) \sin \left(r \omega t\right), \end{align} \tag{ 7b }$$

while the evolution of r is governed by the last, independent equation (6c ). As previously described, if $c_r \lt0$, the last equation (6c ) has two attracting values for the amplitude r, namely, r_l and r_h . Depending on the initial conditions r may reach any of the two fixed points and then the state variables x and y will perform oscillations with variable phase velocity, equal to $r\omega$. This way the bifurcation of the r-variable in equation (6c ) influences not only the amplitude of the oscillations but also modifies its frequency.

We would like to point out that the nonlinear terms in equation (6) induce non-isochronisity during the temporal evolution because of the frequency variations (frequency dependence on amplitude). Only at the asymptotic state the uncoupled elements reach the steady state limit cycle and then the dynamics becomes isochronous, since the ranges r (and, consequently, the corresponding frequencies) attain their fixed values. Namely, at the asymptotic limit the amplitude dependent frequencies attain asymptotic values 'ωr_l ' or 'ωr_h ', depending on the initial r-values. After entering the asymptotics, the (uncoupled) equation (6) attain the limiting dynamics and behave as a perfect chronometer, also known by the term isochrone. For stable periodic oscillations isochrones coincide with the isophases (lines or regions on the phase space where a particular phase is maintained), while for complex, stochastic and chaotic oscillations the idea of isochrones becomes ambiguous and we talk about non-isochronous oscillations [51]. In this case, it is sometimes possible to introduce optimal isophases as Poincare surfaces having phases as constant as possible [51–53], but this falls beyond the scope of the present study.

To compare the entangled dynamics, equation (6), with the case without entanglement, equation (3), in figure 1(c) we present the evolution of the $x-$variable starting from precisely the same initial conditions and parameters as in figure 1(a), but with the entanglement as described in equation (6). We can now notice that not only the amplitude of the oscillations but also the frequencies change as a result of the entanglement. The frequency change in the two trajectories is further verified by numerically computing the corresponding, asymptotic, mean phase velocities, see figure 1(d).

The amplitude-frequency entanglement may be a source of false interpretation when observing different frequencies in a network, as will be discussed in the next section. In such cases the frequency, by mistake, may be considered as a variable in the system, while in reality variability comes from the entanglement with the amplitude.

Consider a ring network containing N nodes. We denote by x_i and y_i the state variables on the ith node and by r_i its amplitude. We use the simplest nonlocal coupling scheme where each oscillator is linearly coupled to S closest neighbors on its left and S neighbors on its right. Therefore, each element is coupled to 2S closest neighbors, in total. Furthermore, let us denote by small letter $s = 2S/N$, the ratio of coupled elements 2S over the total number of elements N in the network. The parameter s expresses the extent of nonlocal connectivity in the network. The coupling strength is denoted by σ and is common for the x_i and y_i variables and maybe different, denoted by σ_r , in the r_i variables. The phase velocity ω, relaxation exponent a and $c_r \lt0$ are constant and common to all oscillators. The coupled system dynamics reads:

$$\begin{align} \frac{\textrm{d}x_i}{\textrm{d}t} & = -ax_i+\frac{ar_i}{\sqrt{x_i^2+y_i^2}}x_i-\boxed{\omega} \,y_i + \frac{\sigma}{2S}\sum_{j = i-S}^{i+S}\left[ x_j-x_i\right] \end{align} \tag{ 8a }$$

$$\begin{align} \frac{\textrm{d}y_i}{\textrm{d}t} & = -ay_i+\frac{ar_i}{\sqrt{x_i^2+y_i^2}}y_i+\boxed{\omega} \,x_i + \frac{\sigma}{2S}\sum_{j = i-S}^{i+S}\left[ y_j-y_i\right] \end{align} \tag{ 8b }$$

$$\begin{align} \frac{\textrm{d}r_i}{\textrm{d}t} & = c_r \left(r_i -r_l\right)\left(r_i -r_c\right)\left(r_i -r_h\right)+\frac{\sigma_r}{2S}\sum_{j = i-S}^{i+S}\left[ r_j-r_i\right] . \end{align} \tag{ 8c }$$

In equation (8) all indices are taken$\mod{N}$. In the simulations, random initial conditions are considered in all 3N variables $(x_i, y_i, r_i)$, $i = 1, \ldots , N$. Since the frequency ω is common to all elements and there is no explicit frequency bifurcation in the system, we do not expect to observe chimera states with domains where different frequencies develop. On the other hand, equation (8c ) allows for the choice of different amplitudes in different domains of the network, depending on the local choice of initial amplitudes. This way amplitude chimeras may develop, where domains with different amplitudes alternate in the network. This scenario may be on the basis of the amplitude chimeras observed in nonlocally coupled chaotic systems [12–15], among other interesting synchronization phenomena.

As an example, we present in figure 2 the amplitude chimera for the working parameter set with specific parameter values: S = 40, σ = 0.3 and $\sigma_{r} = 0.9$. In panel (a) we present the temporal evolution of the $x-$variables for oscillators 100 and 200, in panel (b) the ω-variable profile and in (c) the spacetime plot of the $x-$variable for all elements. The amplitude chimera has two domains of high amplitude oscillations as shown in figure 2(c) with color label covering the range [−3,3] (black-purple-red-yellow regions) and two domains of lower amplitudes with color label ranging between [−1,1] (red-purple regions). All elements have the same frequency, ω = 1 (see figure 2(b)), and the spacetime plot demonstrates that there is a continuation in the spatial profiles, although the amplitude of the oscillations are different in the different domains. This finding is typical for pure amplitude chimeras, where the different network regions are characterized by their proper amplitudes and not by different frequencies as in the frequency chimeras. Figure 2 will be used for comparison with figure 3 in the next section to demonstrate the effects of entanglement.

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Consider now the case of a ring network of oscillators where each element performs under frequency-amplitude entanglement. Namely, each oscillator is linked to S-elements to its left and right and its frequency ω is modulated by its amplitude r_i . The system of equations describing the network evolution is:

$$\begin{align} \frac{\textrm{d}x_i}{\textrm{d}t} & = -ax_i+\frac{ar_i}{\sqrt{x_i^2+y_i^2}}x_i-\boxed{\omega r_i} \,y_i + \frac{\sigma}{2S}\sum_{j = i-S}^{i+S}\left[ x_j-x_i\right] \end{align} \tag{ 9a }$$

$$\begin{align} \frac{\textrm{d}y_i}{\textrm{d}t} & = -ay_i+\frac{ar_i}{\sqrt{x_i^2+y_i^2}}y_i+\boxed{\omega r_i} \,x_i + \frac{\sigma}{2S}\sum_{j = i-S}^{i+S}\left[ y_j-y_i\right] \end{align} \tag{ 9b }$$

$$\begin{align} \frac{\textrm{d}r_i}{\textrm{d}t} & = c_r \left(r_i -r_l\right)\left(r_i -r_c\right)\left(r_i -r_h\right)+\frac{\sigma_r}{2S}\sum_{j = i-S}^{i+S}\left[ r_j-r_i\right] . \end{align} \tag{ 9c }$$

Note the boxed expressions $\boxed{\omega r_i}$ in equations (9a ) and (9b ), which account for the entanglement of the oscillator identical internal frequencies ω with the individual amplitudes $r_i(t)$. All other terms in equation (9) are the same as in equation (8). Typical simulation results are depicted in figure 3. Initial conditions are identical to the ones used in figure 2.

In panel 3(a), we present the temporal evolution of oscillators Nos. 100 and 200. We note that they have different amplitude and different frequency. Regarding frequency, in 100 Time Units (TUs) oscillator No. 100 has performed approx. ∼8.5 cycles, while No. 200 has performed ∼25 cycles. This is verified in panel 3(b), where the mean phase velocity $\omega_{200} = 3\cdot\omega_{100}$. We recognize here the effect of entanglement, which modifies the mean phase velocity in certain regions of the network, creating a symbiosis of amplitude-and-frequency chimeras. This is further verified in the spacetime plot, panel 3(c), where the oscillations in regions between ∼[200–300] and ∼[600–750] have high frequencies and amplitudes, while in regions ∼[1–200], ∼[300–600] and ∼[750 1000] they have low amplitude and frequency. In between the regions of constant amplitude (or frequency) the elements have variable amplitude (or frequency) to bridge the gap between these different values.

As discussed in the previous section, the frequency variations in the network come as a result of the entanglement between a variable amplitude r and a constant ω characterizing the uncoupled elements. The random initial conditions drive parts of the system to select one or the other possible amplitude and, as a consequence, they also create domains of different frequencies.

In this section we introduce the frequency and amplitude deviations as measurable indicators that quantify the level of network synchronization in both entangled and non-entangled scenarios.

To compute the limiting frequency (or mean phase velocity) ω_i of an individual oscillator i in the network we count the number of full cycles $q_i(\Delta T )$ that the oscillator i has performed within a relatively large time interval ΔT. Then, the mean phase velocity is defined as the number of complete cycles per unit time, namely

$$\begin{equation} \omega_i = \frac{2\pi q_i\left(\Delta T\right)}{\Delta T}. \end{equation} \tag{ 10 }$$

To avoid transient phenomena, the time period ΔT used for the recording of the limiting (asymptotic) ω-values needs to be set after the system (network) has attained its steady state (i.e. after the transient period).

The above frequency definition concerns individual oscillators in the network. To quantify collective effects, comparative indices such as maximum and minimum mean phase velocities need to be introduced in the system. For any given network, let us denote by ω_max and ω_min the maximum and minimum mean phase velocity recorded among all elements, respectively. Note that ω_max and ω_min need to be recorded after the transient period, when the network has reached the steady state. Given that $\omega_\textrm{max} \unicode{x2A7E} \omega_\textrm{min} \unicode{x2A7E} 0$, let us denote $\Delta\omega = \omega_\textrm{max} - \omega_\textrm{min} \unicode{x2A7E} 0$. If $\Delta \omega = 0$ (within a tolerable error ε > 0) then all oscillator in the system have the same frequency and then frequency chimera states are not developed. Alternatively, if $\Delta \omega \gt \epsilon$ the network has the possibility to support frequency chimera states, since the characteristic of the typical (frequency) chimeras is the non-negligible frequency difference between coherent and incoherent domains.

Similarly to the definition of the frequency deviation in the typical chimera states, in the amplitude chimeras we compute the maximum and minimum amplitudes recorded in the network. If r_i denotes the limiting value of the amplitude attained by oscillator i after a long time (i.e. after the transient), let us denote by $r_\textrm{max} = \max \{r_1, r_2,\ldots r_N \}\unicode{x2A7E} 0$ and by $r_\textrm{min} = \min \{r_1, r_2,\ldots r_N \} \unicode{x2A7E} 0$. By their definition $r_\textrm{max} \unicode{x2A7E} r_\textrm{min} \unicode{x2A7E} 0$. The amplitude divergence of the network may be quantified by the difference $\Delta r = r_\textrm{max} - r_\textrm{min} \unicode{x2A7E} 0$. Similarly to the case of frequencies, if $\Delta r = 0$ then all network elements are characterized by the same amplitude and amplitude chimeras are not developed. Alternatively, if $\Delta r \unicode{x2A7E} 0$ (with an interval of toleration) then amplitude chimeras are possible. In the most complex case where both $\Delta r$ and $\Delta \omega$ are non-negligible, then chaotic chimeras emerge with both amplitude and frequency variations.

Apart the quantitative indices $\Delta r$ and $\Delta \omega$ introduced above, a number of other indices have been proposed in the literature, such as the Kuramoto order parameter, Fourier spectra and various correlation indices [1, 54] mainly for the study of frequency chimeras. In the present study, primarily $\Delta \omega$ and $\Delta r$ will be used for discrimination between frequency and amplitude chimeras.

In the next sections we explore the effects of various parameters in the formation of domains of alternating amplitude/frequency values. Without loss of generality, in the rest of the manuscript we use the following working parameter set: $c_r = -1$ (to ensure of the existence of one repulsive fixed point, r_c , surrounded by two attractive ones r_l and r_h ), $r_l = 1$, $r_c = 2$, $r_h = 3$, ω = 1 (constant frequency of all oscillators), S = 40 (unless otherwise stated), a = 1.0 and σ = 0.3. The special case $r_l = 0$ related to the presence of bump states is discussed in section 5.

To start,we here present color coded 2D maps of the maximum, minimum and difference of r values formed in the system under the working parameter set defined at the end of section 3.3, namely: $c_r = -1$, $r_l = 1$, $r_c = 2$, $r_h = 3$, R = 1, a = 1.0 and σ = 0.3. The scanning parameters, σ_r and $s = 2S/N$, are in the ranges: $0 \unicode{x2A7D} \sigma_r \unicode{x2A7D} 1$ and $0 \unicode{x2A7D} s \unicode{x2A7D} 0.2$.

Starting with the minimum r-values, in figure 4(b) we note that the minimum values are mostly around or slightly higher than the value r_l (set to 1 in the simulations). Thus, the coupling in the system does not significantly affect the low fixed point of the amplitudes. On the other hand, $r_\textrm{max}$ changes drastically as shown in figure 4(a): for large values of σ_r and s, the $r_h ( = 3)$ amplitude fixed point is absorbed by the low amplitude fixed point $r_l = 1$. For this reason, in figure 4(c) the difference $\Delta r = 0$ is recorded, where the black color dominates. In these cases all elements have the same amplitude and therefore no amplitude chimeras are possible. For large values of σ_r and low values of s, the $r_h ( = 3)$ amplitude fixed point remains close to its original values, while the lowest amplitudes (min values) remain close to $r_l = 1$ and in this case multi-amplitude chimeras are possible. In the corresponding regions of figure 4(c) the difference Δr takes maximum values (close to 2) and the yellow color dominates. These regions correspond to well developed chimeras. The parameters of the example presented in figure 2 are denoted by the letter A in the three panels of figure 4. One may recognize the difference in the amplitude in panels 2(a) and (c). Finally, we note also intermediate values of the difference Δr for intermediate values of σ_r .

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As a second indicative example, we present in figure 5 the case of parameters s = 0.02 and $\sigma_r = 0.5$, which are denoted by the letter B in figure 4. The reduction in the values of the coupling range s causes the split of the network in many regions of alternating high and low amplitude dynamics. This inverse relation between the coupling range and the number of distinct domains is also known to take place in other dynamical systems coupled in networks [55, 56]. To be more precise, in figure 5(a) we present a typical snapshot of the system's state variables x_i at 200 TU (black dots) together with the system amplitudes r_i (red crosses). It is easily understood that the system splits into domains of low amplitude (close to $r_l = 1$) and high amplitudes $r_h = 3$. In fact, figure 5(b) demonstrates that the oscillators reach their final amplitudes as early as 30TUs and maintain these values afterwards. For shorter times (t < 30 TU) many regions of different amplitudes are present, as imposed by the random initial conditions. As time advances, and with the influence of the coupling terms σ_r in the system, adjacent regions merge and at the final/steady state six regions of high (and six of low) amplitudes prevail. A discontinuous change of the variable r_i and x_i can be observed in the borders between the twelve different domains and this is confirmed by the spacetime plots in figure 5(c). In this respect, and in view of figure 5(b) these states can be considered as amplitude death states.

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To get a better comparison of the measures $r_\textrm{max}$, $r_\textrm{min}$ and Δr we plot in the same graph the three measures as a function of σ_r for constant s = 0.16. The results are shown in figure 6. In the left part of the graph, where $\sigma_r \lt 0.45$, the maximum amplitude in the system stays close to $r_h = 3$ with a tendency to decrease as the coupling range increases. Indeed, as σ_r increases the interactions in the system lower the amplitude to approach values closer to r_l . Similarly, the minimum amplitude in the system stays close to $r_l = 1$ with a tendency to increase as the coupling strength increases. As σ_r increases the interactions in the system enhance the lower amplitudes to approach the maximum values, close to r_h . Around a critical value $\sigma_r\sim 0.5$, the amplitudes collapse to the lowest value $r_\textrm{max}\sim r_\textrm{min} \sim 1$ and $\Delta r = 0$. This behavior persists in the interval $0.5 \unicode{x2A7D} \sigma_r \unicode{x2A7D} 0.7$ where all amplitudes are equal in the system and amplitude chimeras are not possible. Increasing further the σ_r values, in the interval $0.7 \unicode{x2A7D} \sigma_r \unicode{x2A7D} 0.85$, a divergence of amplitudes develops in the network and the maximum amplitudes reach values close (but lower) than r_h , while the minimum values stay near r_l indicating a region where amplitude chimeras are present. Finally, for $\sigma_r \gt 0.85$ $r_\textrm{max} = r_\textrm{min}$ and amplitude chimeras disappear.

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To show that in this case (absence of entanglement) we have no significant difference in the frequencies (mean phase velocities) of the elements, we present the color coded maps of ω_max and Δω. The map for ω_min is not shown because it is very similar to ω_max since Δω's are mostly 0 in the system.

From figure 7(a) we note a change in the colors of the color map representing the ω_max values, but the change is negligible as can be seen from the ω-scale shown on the right of the panel. The same conclusions are drawn from panel (b), representing the ω-differences in the system. Therefore, frequency chimeras are not supported in this system without entanglement and only amplitude chimeras are observed in the parameter regions discussed above.

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We now turn to the case where entanglement is introduced in the system as discussed in section 3.2 and this induces frequency chimeras in addition to amplitude ones. The parameters are the same as in figure 3 and in this section we explore the region of parameters were frequency chimeras occur. The scanning parameters are again in the range $0 \unicode{x2A7D} \sigma_r \unicode{x2A7D} 1$ and $0 \unicode{x2A7D} s = 2S/N \unicode{x2A7D} 0.2$.

In figure 8, we use color coded maps to present only the variations ω_max, ω_min and $\Delta \omega = \omega_\textrm{max}-\omega_\textrm{min}$. We do not need to show the maps for the $r_\textrm{max}$, $r_\textrm{min}$ and Δr because they are identical with the ones in figure 4. This is because the amplitudes are formed only via equation (9c ) which evolves independently of ω and is identical to equation (8c ). These last two equations develop independently of equations (9a ) and (9b ) (or equations (8a ) and (8b ) for the non-entangled case). In the simulations, while the values of r are formed via equation (9c ) (or equation (8c ) for the non-entangled case) they are fed in equations (9a ) and (9b ) and they contribute to the formation of the ω-profiles. Note, that the mechanisms of r-feeding in the entangled and non-entangled case are different as the boxed terms in equations (8a ) and (9a ) demonstrate, and that is why in the first case the ω-profiles are vaguely modified, while in the second case they vary considerably following closely the amplitude variations, see figure 8. Note in panel (8c ) the Δω variations which are in the range $0 \unicode{x2A7D} \Delta \omega \unicode{x2A7D} 2.5$ and compare with the non-entangled case, figure 7(b), where the variations are negligible, $0 \unicode{x2A7D} \Delta \omega \unicode{x2A7D} 0.02$.

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In parallel with the discussions in the previous section we also investigate the synchronization properties at parameter position denoted by letter B on maps figure 8, in the case of entanglement. In figure 9(a) we show a typical profile of the state variable x_i . Many coherent regions are formed separated by small incoherent ones as may be observed in this figure. We recall that the amplitude (r_i ) profile is identical to the one in figure 5(b) since they are governed by the same equations (see equations (8c ) and (9c )). On the other hand the mean phase velocity profiles are different as can be seen in figure 9(b), because it is governed by the entanglement terms (see equations (9a ) and (9b )) and as a result a number of successive domains with binary frequencies are observed. The presence of these domains becomes clear in the spacetime plots, see figure 9(c). In contrast to the case of no entanglement, at the borders between coherent regions of different frequencies one may observed small incoherent domains. The effect here (figures 9(b) and (c)) is vaguely observable due to the small size (large number) of the coherent regions and is not as clear as in figures 3(b) and (c), where only a few (four) coherent regions are observed with larger sizes.

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To have a visual idea of the frequency variations in the system, we also present the values of ω_max, ω_min and Δω for constant s = 0.14. The results are shown in plot figure 10. The form of the three curves follow closely the ones in figure 6, as the r-variations induce the ω ones. Namely, for low σ_r values the maximum frequency in the system stays close to the product $r_h \cdot \omega = 3\cdot 1 = 3$ with a tendency to decrease as the coupling strength increases. In the middle range of parameter σ_r there is an abrupt drop and the maximum and minimum frequencies take close values, while in the large σ_r range the maximum frequency increases again close to the product $r_h \cdot \omega = 3$. The minimum frequency in the system stays always close to the product $r_l \cdot \omega = 1\cdot 1 = 1$ and varies slightly above it. As a result, the frequency divergence Δω remains close to 0 in the middle of σ_r range and frequency chimeras are not supported in this area, while for σ_r values outside this range frequency chimeras are possible, as demonstrated in figure 3. We also note a tendency for the maximum and minimum frequency to approach each other as $\sigma_r\to 1$, similarly to the tendency of $r_\textrm{min}$ and $r_\textrm{max}$ in the case without entanglement, see figure 6. (Note that figures 6 and 10 have been recorded for different s values, 0.16 and 0.14 respectively).

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As a final remark related to effects of entanglement, we would like to recall that in all case (with or without entanglement) the parameter ω, which primarily governs the oscillators frequency, is kept constant. Only in the case of entanglement the frequencies of the different oscillators are carried along with the amplitude r variations. In the case of no entanglement the frequency of all oscillators remains close to the constant ω value. Amplitude variations leading to amplitude chimeras take place in both cases in the present model.

We would like to add that it is possible to form a more complex extension of these models with four equations, to explicitly incorporate both amplitude and frequency variations. From the point of view of the dynamical system, this means to add also equation (1c ) in equation (3) and we end up with a system of four equations and four variables: $x, y, r, \omega$. In this case one may combine different frequencies and amplitudes in the system which, together with the coupling terms, will induce even more complex synchronization patterns. Furthermore, the four variable model ($x, y, r, \omega$) may be considered with and without entanglement. These extensions fall outside the scope of the present investigation and are left for future studies.

Starting with the case of no entanglement and $r_l = 0$, we present in figure 11(a) a snapshot of the x_i -variables of the ring nodes in black color. All nodes follow each other in synchrony, however, they do not attain a common maximum amplitude as the red curve in figure 11(a) indicates. This red curve presents the maximum amplitude that each element reaches. One clearly understands that there are two quiescent regions Q$_1 \sim$ [0–100, 800–1000] and Q$_2 \sim$ [400-500] where the maximum amplitude is 0 (note that due to the ring connectivity with periodic boundary conditions the elements 0–100 and 800–1000 belong to the same quiescent region, Q₁). This means that the elements in regions Q₁ and Q₂ do not oscillate in reality but they remain inactive fluctuating around the $r_l = 0$ fixed point. The elements in the intermediate regions [100–400] and [500–800] perform synchronous oscillations of variable continuous amplitudes as the red curve in figure 11(a) shows. The synchrony in the system is demonstrated in figure 11(b), where all frequencies recorded are basically the same, $\omega_i = 1$ for $i = 1,\ldots 1000$.

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The spacetime plot in figure 11(c) gives a clearer picture of the temporal evolution in the system. In fact, the regions Q₁ and Q₂ do not change in space and time and they remain constantly near $x_i = 0$. It is interesting to compare figures 2(c) and 11(c) : in the former case regions Q₁ and Q₂ perform oscillations with lower amplitudes than the intermediate regions ([100–400] and [500–800]), while in the latter case regions Q₁ and Q₂ are quiescent. The two intermediate regions in figure 11(c), [100–400] and [500–800], present oscillations in time, with their state variables x_i taking antipodal values. This can be better observed in 11(a), where the elements of region [100–400] are in the negative values at the same time that the ones in region [500–800] are on the positive axis.

We stress here, that these are the amplitude bump states. Namely, the quiescent regions oscillate with null amplitudes while the active regions coexist with the quiescent ones and they present a variety of amplitudes, as shown in figure 11(a) (red points). The amplitudes of the active regions may vary in the interval $(r_l = 0,r_h]$. The interval is open on the left side because the 0-amplitude characterizes the quiescent regions, while in the active regions all elements have amplitudes above 0 and they may potentially reach values as large as r_h .

In the case of amplitude-frequency entanglement the system evolution changes drastically. The elements loose synchrony because the amplitude variations influence the node frequencies. That is the reason why some nearby elements move incoherently, as can be seen in figure 12(a), where the black dots indicate the positions of the system's state variables in a typical snapshot.

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To be more precise, in figure 12(a) we observe a typical system snapshot at t = 195TU (black dots) together with the maximum amplitude r_i attained by each node. As previously, regions Q₁ and Q₂ are quiescent and the state variables x_i stay near 0. At the intermediate regions there are a number of elements that are incoherent (at the borders with regions Q₁ and Q₂) while in the middle of these regions the elements become coherent again. The reason for this behavior is understood if we inspect the mean phase velocities ω_i , in figure 12(b). Due to the amplitude-frequency entanglement the intermediate regions acquire a variation of frequencies ranging from a minimum $r_l \cdot \omega = 0$ (in regions Q₁ and Q₂) to a maximum $r_h \cdot \omega = 3\cdot 1 = 3$ (in the intermediate regions). We note, that due to the nonlinearity of the system equations together with the coupling, the maximum ω_i values never reach as high as 3. To bridge the gap between the quiescent elements and the ones oscillating with frequencies close to ω_max, the mediating elements (regions [120–180], [270–325], [510–560] and [700–750]) have variable frequencies and therefore their x_i state variables are incoherent. The different regions can be also observed in figure 12(c), where the quiescent regions are colored red, the regions of constant frequency close to ω_max show the oscillations while the regions between oscillatory and quiescent elements have mixed colors indicating incoherence. As a last remark, we stress the comparison between images figures 3(c) and 12(c). Both cases present similar oscillations in the high (close to ω_max) frequencies. However, in regions Q₁ and Q₂ the vanishing of the fixed point amplitude $r_l = 0$ together with the amplitude-frequency entanglement causes the cease of oscillations in figure 12(c), while low frequency oscillations are maintained in figure 3(c).

In the case of entanglement frequency-and-amplitude bump states are produced. This is because the amplitude drives the frequency and the active regions present variability both in amplitude and in frequency. To the contrary, the elements that belong to quiescent regions are inactive and their amplitudes as well as frequencies register nearly null.