# Emergent Spatial–Temporal Patterns in a Ring of Locally Coupled Population Oscillators

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Logistic Equation as a Map of a Nonlinear Operator

_{1}, the following condition is satisfied:

_{2}is an arbitrary number.

_{i}. Obviously, along with (3), the conditions for all points with period P are satisfied:

#### 2.2. Algorithm for Finding Solutions for a Set of Initial Conditions

_{1}) “blind” iterations, where we care not about the intermediate results but only about matrix $B\left({N}_{1}\right)={\widehat{F}}^{{N}_{1}}B\left(0\right)$, and then some number (N

_{2}) of “measuring” iterations, where the set of all intermediate matrices is preserved: $B\left({N}_{1}+1\right),B\left({N}_{1}+2\right),\dots ,B\left({N}_{1}+{N}_{2}\right)$.

_{s}with a given accuracy $\u03f5$ for a point in the parametric space r, D for random initial conditions:

#### 2.3. Spectral Entropy of an Arbitrary Valid Signal

#### 2.4. Relative Spectral Entropy of Residuals (Temporal Entropy)

#### 2.5. Relative Spectral Entropy of Solutions (Spatial Entropy)

#### 2.6. Mean Value of Generalized Solutions and Entropy of the Mean Solution

## 3. Results

_{T}(see Model), which, under given initial conditions, characterizes oscillatory processes in a chain of oscillators considered as an integral system. Figure 3 shows the dependence of the probability of low-entropy dynamics, E

_{T}< 0.3, on the model parameters under initial conditions that were set randomly. It can be seen that for small values of the chain length (Figure 4, L = 2 and L = 5), the behavior of the model qualitatively resembles that of the pointwise (at D = 0) logistic map. In particular, this behavior depends significantly on the value of the parameter r. For example, for r values in the range of 3–3.5, the dynamics of the model is regular, i.e., low-entropy, with E

_{T}< 0.3 at any values of the parameter D. With increasing r, the dynamics becomes irregular, highly entropic, and the probability of low numerical values of E

_{T}is close to zero (Figure 4). The narrow vertical regions of low-entropy dynamics at L = 2 and L = 5 (Figure 4) resemble the windows of regularity in the pointwise logistic map (cf. Figure 1). A significant difference from the pointwise map is the presence of regions with regular dynamics in a wide range of high values of r (Figure 4, regions indicated in yellow at r > 3.6). The presence of such regions indicates that for any r in Models (1)–(2) at relatively small values of the parameter L, there will be such a value of the parameter D at which the dynamics of the model is low-entropy. In the space of parameters (r, D), note also the enlargement of such regions that correspond to intermediate (between 0 and 1) values of the probability of observing low-entropy regimes as the length of the oscillator chain increases (in Figure 4, these regions are marked with colors other than yellow and dark blue). At L = 2 and L = 5, such regions are hardly noticeable, but with increasing L, their area increases distinctively. This means that the dynamics of the oscillator chain at r > 3.6 depends more and more noticeably on the initial conditions with increasing chain length. At the same time, at r ≤ 3.5, the increase in L does not affect the regime of the dynamics of the oscillator chain; the dynamics remain low-entropy (Figure 4).

_{x}(see model) that was established by the time the numerical experiment ended. The dependence of E

_{x}on the parameters of Models (1)–(2) is shown in Figure 5.

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Bifurcation diagram of the logistic map x$\left(t+1\right)=rx\left(t\right)\left(1-x\left(t\right)\right)$.

**Figure 3.**Examples of spatial and temporal structures for different parameter values of Models (1)–(2) for L = 100. The left column demonstrates the values of Models (1)–(2) in each oscillator at the end of the numerical experiment (see model); the middle column shows the last 150 values in each oscillator, and the right column shows the last 20 values in the first (I = 1) oscillator. (

**a**,

**b**) r = 3.141, D = 0.16 (for different initial conditions); (

**c**) r = 3.545, D = 0.16; (

**d**) r = 3.808, D = 0.32; (

**e**) r = 3.909, D = 0.02.

**Figure 4.**Probability of observing low temporal entropy (E

_{t}< 0.3; Equation (4)) for different chain lengths L of Models (1)–(2) under different initial conditions (100 numerical experiments for each pair of parameters r and D).

**Figure 5.**Values of the averaged spatial entropy for different lengths (L = 50, 100, 200) of the oscillator chain (1) under different initial conditions (100 numerical experiments for each pair of the parameters r and D). The white color indicates the regions in which the values of x in all oscillators are the same at the end of the computations.

**Figure 6.**Averaged values of $\overline{x}$ in each oscillator at time t (see model) for different initial conditions depending on the value of the parameter D; r = 3.808. (

**a**) L = 20, (

**b**) L = 100, (

**c**) L = 200.

**Figure 7.**The values of $\mathrm{x}$ in each oscillator of the chain for two different initial conditions (left and middle columns) and the average values of $\overline{x}$ for each oscillator in the chain for 100 different initial values (right column); r = 3.8; D = 0.32. (

**a**) L = 90, (

**b**) L = 93, (

**c**) L = 95.

**Figure 8.**Examples of spatial structures for different initial conditions (first and second columns) and for average values of $\overline{x}$ over 100 random initial conditions (third column); L = 100. (

**a**) r = 3.65, D = 0.01; (

**b**) r = 3.65, D = 0.05; (

**c**) r = 3.4, D = 0.01; (

**d**) r = 3.4, D = 0.05; (

**e**) r = 3.808, D = 0.33.

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**MDPI and ACS Style**

Rusakov, A.V.; Tikhonov, D.A.; Nurieva, N.I.; Medvinsky, A.B.
Emergent Spatial–Temporal Patterns in a Ring of Locally Coupled Population Oscillators. *Mathematics* **2023**, *11*, 4970.
https://doi.org/10.3390/math11244970

**AMA Style**

Rusakov AV, Tikhonov DA, Nurieva NI, Medvinsky AB.
Emergent Spatial–Temporal Patterns in a Ring of Locally Coupled Population Oscillators. *Mathematics*. 2023; 11(24):4970.
https://doi.org/10.3390/math11244970

**Chicago/Turabian Style**

Rusakov, Alexey V., Dmitry A. Tikhonov, Nailya I. Nurieva, and Alexander B. Medvinsky.
2023. "Emergent Spatial–Temporal Patterns in a Ring of Locally Coupled Population Oscillators" *Mathematics* 11, no. 24: 4970.
https://doi.org/10.3390/math11244970