# Stability of Bounded Dynamical Networks with Symmetry

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Formulation

- Incidence conditions: A curve i has two ends at $s=0$ and at $s={L}_{i}\left(t\right)$.
- (a)
- If at the point $s=0$, curve i intersects with two other curves, namely curves j and q, at their starting point for example, then:$${G}_{i}(0,t)={G}_{j}(0,t)={G}_{q}(0,t).$$Then, at the other end of curve i, at point $s={L}_{i}\left(t\right)$, curve i intersects:
- with two other curves, namely curves p and r at their ending points:$${G}_{i}({L}_{i}\left(t\right),t)={G}_{p}({L}_{p}\left(t\right),t)={G}_{r}({L}_{r}\left(t\right),t),$$
- at the boundary $\partial \Omega $:$$b\left({G}_{i}({L}_{i}\left(t\right),t)\right)=0.$$

- (b)
- If at the point $s=0$, curve i intersects with the boundary instead of two other curves, then:$$b\left({G}_{i}(0,t)\right)=0.$$In this case, at the other end of curve i, at point $s={L}_{i}\left(t\right)$, curve i intersects with two other curves, namely curves p and r, at their ending point:$${G}_{i}({L}_{i}\left(t\right),t)={G}_{p}({L}_{p}\left(t\right),t)={G}_{r}({L}_{r}\left(t\right),t).$$

- Angle conditions:
- (a)
- At the point at which the curves intersect: If curves $i,j,q$ intersect at their starting points, for example, then:$$\begin{array}{c}{G}_{is}(0,t)\xb7{G}_{js}(0,t)=cos\frac{2\pi}{3},\\ {G}_{qs}(0,t)\xb7{G}_{is}(0,t)=cos\frac{2\pi}{3};\end{array}$$
- (b)
- At $\partial \Omega $:$$<{G}_{is}({L}_{i}\left(t\right),t),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\nabla b\left({G}_{i}\right)>=0.$$

**Remark**

**1.**

**Remark**

**2.**

#### 1.2. Changing the Domain

- Incidence conditions: A curve i has two ends at $x=0$ and at $x={l}_{i}$.
- (a)
- If at the point $x=0$, curve i intersects with two other curves, namely curves j and q, at their starting points for example, then:$${\Gamma}_{i}(0,t)={\Gamma}_{j}(0,t)={\Gamma}_{q}(0,t).$$Then, at the other end of curve i, at point $x={l}_{i}$, curve i intersects:
- with two other curves, namely curves p and r, at their ending points for example:$${\Gamma}_{i}({l}_{i},t)={\Gamma}_{p}({l}_{p},t)={\Gamma}_{r}({l}_{r},t),$$
- at the boundary $\partial \Omega $:$$b\left({\Gamma}_{i}({l}_{i},t)\right)=0.$$

- (b)
- If at the point $x=0$, curve i intersects with the boundary instead of two other curves then:$$b\left({\Gamma}_{i}(0,t)\right)=0.$$In this case, at the other end of curve i, at point $x={l}_{i}$, curve i intersects with two other curves, namely curves p and r, at their ending points for example:$${\Gamma}_{i}({l}_{i},t)={\Gamma}_{p}({l}_{p},t)={\Gamma}_{r}({l}_{r},t).$$

- Angle conditions:
- (a)
- At the point at which the curves intersect: If curves $i,j,q$ intersect at their starting point for example, then:$$\begin{array}{c}\frac{{\Gamma}_{is}(0,t)}{|{\Gamma}_{is}(0,t)|}\xb7\frac{{\Gamma}_{js}(0,t)}{|{\Gamma}_{js}(0,t)|}=cos\frac{2\pi}{3},\\ \frac{{\Gamma}_{qs}(0,t)}{|{\Gamma}_{qs}(0,t)|}\xb7\frac{{\Gamma}_{is}(0,t)}{|{\Gamma}_{is}(0,t)|}=cos\frac{2\pi}{3};\end{array}$$
- (b)
- at $\partial \Omega $:$$<{\Gamma}_{is}({l}_{i},t),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\nabla b\left({\Gamma}_{i}\right)>=0.$$

#### 1.3. Stability

- Incidence conditions: A curve i has two ends at $x=0$ and at $x={l}_{i}$.
- (a)
- If at the point $x=0$, curve i intersects with two other curves, namely curves j and q, at their starting points for example, then:$$\begin{array}{c}{h}_{i}^{N}\left(0\right)+{h}_{j}^{N}\left(0\right)+{h}_{q}^{N}\left(0\right)=0,\\ {h}_{i}^{T}\left(0\right)+{h}_{j}^{T}\left(0\right)+{h}_{q}^{T}\left(0\right)=0;\end{array}$$Then, at the other end of curve i, at point $x={l}_{i}$ curve i intersects:
- with two other curves, namely curve p and r, at their ending points for example:$$\begin{array}{c}{h}_{i}^{N}\left({l}_{i}\right)+{h}_{p}^{N}\left({l}_{p}\right)+{h}_{r}^{N}\left({l}_{r}\right)=0,\\ {h}_{i}^{T}\left({l}_{i}\right)+{h}_{p}^{T}\left({l}_{p}\right)+{h}_{r}^{T}\left({l}_{r}\right)=0;\end{array}$$
- at the boundary $\partial \Omega $:$${h}_{i}^{T}\left({l}_{i}\right)=0.$$

- (b)
- If at the point $x=0$, curve i intersects with the boundary instead of two other curves, then:$${h}_{i}^{T}\left(0\right)=0.$$In this case, at the other end of curve i, at point $x={l}_{i}$, curve i intersects with two other curves, namely curves p and r, at their ending points for example:$$\begin{array}{c}{h}_{i}^{N}\left({l}_{i}\right)+{h}_{p}^{N}\left({l}_{p}\right)+{h}_{r}^{N}\left({l}_{r}\right)=0,\\ {h}_{i}^{T}\left({l}_{i}\right)+{h}_{p}^{T}\left({l}_{p}\right)+{h}_{r}^{T}\left({l}_{r}\right)=0;\end{array}$$

- Angle conditions:
- (a)
- At the point at which the curves intersect: If curves $i,j,q$ intersect at their starting points for example, then:$$\begin{array}{c}{h}_{i}^{\prime N}\left(0\right)+{h}_{i}^{T}\left(0\right){k}_{i}={h}_{j}^{\prime N}\left(0\right)+{h}_{j}^{T}\left(0\right){k}_{j}={h}_{q}^{\prime N}\left(0\right)+{h}_{q}^{T}\left(0\right){k}_{q};\end{array}$$
- (b)
- at $\partial \Omega $:$${K}_{i}^{\partial \Omega}{h}_{i}^{N}\left({l}_{i}\right)={h}_{i}^{\prime N}\left({l}_{i}\right).$$

**Lemma**

**1.**

## 2. A Bounded Network with Two Inner Triple Junctions

**Definition**

**1.**

- Incidence at the junctions:$$\begin{array}{c}{\Gamma}_{1}(0,t)={\Gamma}_{2}(0,t)={\Gamma}_{3}(0,t);\\ {\Gamma}_{3}({l}_{3},t)={\Gamma}_{4}({l}_{4},t)={\Gamma}_{5}({l}_{5},t);\\ {\Gamma}_{5}(0,t)={\Gamma}_{6}(0,t)={\Gamma}_{7}(0,t);\\ {\Gamma}_{8}({l}_{8},t)={\Gamma}_{9}({l}_{9},t)={\Gamma}_{10}({l}_{10},t);\\ {\Gamma}_{10}(0,t)={\Gamma}_{4}(0,t)={\Gamma}_{11}(0,t);\\ {\Gamma}_{11}({l}_{11},t)={\Gamma}_{12}({l}_{12},t)={\Gamma}_{13}({l}_{13},t).\end{array}$$
- Incidence at the boundary $\partial \Omega $:$$\begin{array}{c}b\left({\Gamma}_{i}({l}_{i},t)\right)=0,\phantom{\rule{1.em}{0ex}}i=1,2,6,7;\\ b\left({\Gamma}_{i}(0,t)\right)=0\phantom{\rule{1.em}{0ex}}i=8,9,12,13.\end{array}$$
- Angle conditions at the junctions:$$\begin{array}{c}\frac{{\Gamma}_{ix}(0,t)}{|{\Gamma}_{ix}(0,t)|}\xb7\frac{{\Gamma}_{(i+1)x}(0,t)}{|{\Gamma}_{(i+1)x}(0,t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=1,2,5,6;\\ \frac{{\Gamma}_{ix}({l}_{i},t)}{|{\Gamma}_{ix}({l}_{i},t)|}\xb7\frac{{\Gamma}_{(i+1)x}({l}_{i+1},t)}{|{\Gamma}_{(i+1)x}({l}_{i+1},t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=3,4,8,9,11,12;\\ \frac{{\Gamma}_{10x}(0,t)}{|{\Gamma}_{10x}(0,t)|}\xb7\frac{{\Gamma}_{4x}(0,t)}{|{\Gamma}_{4x}(0,t)|}=cos\frac{2\pi}{3};\\ \frac{{\Gamma}_{4x}(0,t)}{|{\Gamma}_{4x}(0,t)|}\xb7\frac{{\Gamma}_{11x}(0,t)}{|{\Gamma}_{11x}(0,t)|}=cos\frac{2\pi}{3}.\end{array}$$
- Angle conditions at $\partial \Omega $:$$\begin{array}{c}<{\Gamma}_{is}({l}_{i},t),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\nabla b\left({\Gamma}_{i}({l}_{i},t)\right)>=0,\phantom{\rule{1.em}{0ex}}i=1,2,6,7;\\ <{\Gamma}_{is}(0,t),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\nabla b\left({\Gamma}_{i}(0,t)\right)>=0,\phantom{\rule{1.em}{0ex}}i=8,9,12,13.\end{array}$$

- Incidence at the junctions:$$\begin{array}{c}{h}_{i}^{N}\left(0\right)+{h}_{i+1}^{N}\left(0\right)+{h}_{i+2}^{N}\left(0\right)=0,\phantom{\rule{1.em}{0ex}}i=1,5;\\ {h}_{i}^{T}\left(0\right)+{h}_{i+1}^{T}\left(0\right)+{h}_{i+2}^{T}\left(0\right)=0,\phantom{\rule{1.em}{0ex}}i=1,5;\\ {h}_{i}^{N}\left({l}_{i}\right)+{h}_{i+1}^{N}\left({l}_{i+1}\right)+{h}_{i+2}^{N}\left({l}_{i+2}\right)=0,\phantom{\rule{1.em}{0ex}}i=3,8,11;\\ {h}_{i}^{T}\left({l}_{i}\right)+{h}_{i+1}^{T}\left({l}_{i+1}\right)+{h}_{i+2}^{T}\left({l}_{i+2}\right)=0,\phantom{\rule{1.em}{0ex}}i=3,8,11;\\ {h}_{10}^{N}\left(0\right)+{h}_{4}^{N}\left(0\right)+{h}_{11}^{N}\left(0\right)=0,\\ {h}_{10}^{T}\left(0\right)+{h}_{4}^{T}\left(0\right)+{h}_{11}^{T}\left(0\right)=0.\end{array}$$
- Incidence at the boundary $\partial \Omega $:$$\begin{array}{c}{h}_{i}^{T}\left({l}_{i}\right)=0,\phantom{\rule{1.em}{0ex}}i=1,2,6,7;\\ {h}_{i}^{T}\left(0\right)=0,\phantom{\rule{1.em}{0ex}}i=8,9,12,13.\end{array}$$
- Angle conditions at the junctions:$$\begin{array}{c}{h}_{1}^{\prime N}\left(0\right)+{h}_{1}^{T}\left(0\right){k}_{1}={h}_{2}^{\prime N}\left(0\right)+{h}_{2}^{T}\left(0\right){k}_{2}={h}_{3}^{\prime N}\left(0\right)+{h}_{3}^{T}\left(0\right){k}_{3};\\ {h}_{3}^{\prime N}\left({l}_{3}\right)+{h}_{3}^{T}\left({l}_{3}\right){k}_{3}={h}_{4}^{\prime N}\left({l}_{4}\right)+{h}_{4}^{T}\left({l}_{4}\right){k}_{4}={h}_{5}^{\prime N}\left({l}_{5}\right)+{h}_{5}^{T}\left({l}_{5}\right){k}_{5};\\ {h}_{5}^{\prime N}\left(0\right)+{h}_{5}^{T}\left(0\right){k}_{5}={h}_{6}^{\prime N}\left(0\right)+{h}_{6}^{T}\left(0\right){k}_{6}={h}_{7}^{\prime N}\left(0\right)+{h}_{7}^{T}\left(0\right){k}_{7};\\ {h}_{8}^{\prime N}\left({l}_{8}\right)+{h}_{8}^{T}\left({l}_{8}\right){k}_{8}={h}_{9}^{\prime N}\left({l}_{9}\right)+{h}_{9}^{T}\left({l}_{9}\right){k}_{9}={h}_{10}^{\prime N}\left({l}_{10}\right)+{h}_{10}^{T}\left({l}_{10}\right){k}_{10};\\ {h}_{10}^{\prime N}\left(0\right)+{h}_{10}^{T}\left(0\right){k}_{10}={h}_{4}^{\prime N}\left(0\right)+{h}_{4}^{T}\left(0\right){k}_{4}={h}_{11}^{\prime N}\left(0\right)+{h}_{11}^{T}\left(0\right){k}_{11};\\ {h}_{11}^{\prime N}\left({l}_{11}\right)+{h}_{11}^{T}\left({l}_{11}\right){k}_{11}={h}_{12}^{\prime N}\left({l}_{12}\right)+{h}_{12}^{T}\left({l}_{12}\right){k}_{12}={h}_{13}^{\prime N}\left({l}_{13}\right)+{h}_{13}^{T}\left({l}_{13}\right){k}_{13}.\end{array}$$
- Angle conditions at $\partial \Omega $:$$\begin{array}{c}{K}_{i}^{\partial \Omega}{h}_{i}^{N}\left({l}_{i}\right)={h}_{i}^{\prime N}\left({l}_{i}\right),\phantom{\rule{1.em}{0ex}}i=1,2,6,7;\\ {K}_{i}^{\partial \Omega}{h}_{i}^{N}\left(0\right)={h}_{i}^{\prime N}\left(0\right),\phantom{\rule{1.em}{0ex}}i=8,9,12,13.\end{array}$$

**Definition**

**2.**

**Theorem**

**1.**

- (a)
- If the domain Ω is convex (an ellipse for example) at the points at which the steady state of the network meets the boundary, then the steady state is unstable.
- (b)
- If the domain Ω is non-degenerate concave at the points at which the steady state of the network meets the boundary, then the steady state is stable.
- (c)
- If the domain Ω is flat at the points at which the steady state of the network meets the boundary, then the steady state is neutrally stable.

**Proof.**

## 3. Hexagonal Network with an Internal Hexagon

**Definition**

**3.**

**2**, 3

**4**, 5

**8**, 9

**10**, 11

**14**, 15

**16**, 17

**20**, 21

**22**, 23

**26**, 27

**28**, 29

**32**, 33

**34**, 35

- Incidence at the junctions:$$\begin{array}{c}{\Gamma}_{i}(0,t)={\Gamma}_{i+1}(0,t)={\Gamma}_{i+2}(0,t),\phantom{\rule{1.em}{0ex}}i=1,5,9,13,17,21,25,29,33,36;\\ {\Gamma}_{i}({l}_{i},t)={\Gamma}_{i+1}({l}_{i+1},t)={\Gamma}_{i+2}({l}_{i+2},t),\phantom{\rule{1.em}{0ex}}i=3,7,11,15,19,23,27,31;\\ {\Gamma}_{35}({l}_{35},t)={\Gamma}_{1}({l}_{1},t)={\Gamma}_{36}({l}_{36},t);\\ {\Gamma}_{38}({l}_{38},t)={\Gamma}_{6}({l}_{6},t)={\Gamma}_{39}({l}_{39},t);\\ {\Gamma}_{39}(0,t)={\Gamma}_{12}(0,t)={\Gamma}_{40}(0,t);\\ {\Gamma}_{40}({l}_{40},t)={\Gamma}_{18}({l}_{18},t)={\Gamma}_{41}({l}_{41},t);\\ {\Gamma}_{41}(0,t)={\Gamma}_{24}(0,t)={\Gamma}_{42}(0,t);\\ {\Gamma}_{42}({l}_{42},t)={\Gamma}_{30}({l}_{30},t)={\Gamma}_{37}({l}_{37},t).\end{array}$$
- Incidence at the boundary $\partial \Omega $:$$\begin{array}{c}b\left({\Gamma}_{i}({l}_{i},t)\right)=0,\phantom{\rule{1.em}{0ex}}i=2,10,14,22,26,34;\\ b\left({\Gamma}_{i}(0,t)\right)=0\phantom{\rule{1.em}{0ex}}i=4,8,16,20,28,32.\end{array}$$
- Angle conditions at the junctions:$$\begin{array}{c}\frac{{\Gamma}_{ix}(0,t)}{|{\Gamma}_{ix}(0,t)|}\xb7\frac{{\Gamma}_{(i+1)x}(0,t)}{|{\Gamma}_{(i+1)x}(0,t)|}=cos\frac{2\pi}{3},\\ \mathrm{for}\phantom{\rule{1.em}{0ex}}i=1,2,5,6,9,10,13,14,17,18,21,22,25,26,29,30,33,34,36,37;\\ \frac{{\Gamma}_{ix}({l}_{i},t)}{|{\Gamma}_{ix}({l}_{i},t)|}\xb7\frac{{\Gamma}_{(i+1)x}({l}_{i+1},t)}{|{\Gamma}_{(i+1)x}({l}_{i+1},t)|}=cos\frac{2\pi}{3},\\ \mathrm{for}\phantom{\rule{1.em}{0ex}}i=3,4,7,8,11,12,15,16,19,20,23,24,27,28,31,32;\\ \frac{{\Gamma}_{ix}({l}_{i},t)}{|{\Gamma}_{ix}({l}_{i},t)|}\xb7\frac{{\Gamma}_{1x}({l}_{1},t)}{|{\Gamma}_{1x}({l}_{1},t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=35,36;\\ \frac{{\Gamma}_{ix}({l}_{i},t)}{|{\Gamma}_{ix}({l}_{i},t)|}\xb7\frac{{\Gamma}_{6x}({l}_{6},t)}{|{\Gamma}_{6x}({l}_{6},t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=38,39;\\ \frac{{\Gamma}_{ix}({l}_{i},t)}{|{\Gamma}_{ix}({l}_{i},t)|}\xb7\frac{{\Gamma}_{18x}({l}_{18},t)}{|{\Gamma}_{18x}({l}_{18},t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=40,41;\\ \frac{{\Gamma}_{ix}({l}_{i},t)}{|{\Gamma}_{ix}({l}_{i},t)|}\xb7\frac{{\Gamma}_{30x}({l}_{30},t)}{|{\Gamma}_{30x}({l}_{30},t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=37,42;\\ \frac{{\Gamma}_{ix}(0,t)}{|{\Gamma}_{ix}(0,t)|}\xb7\frac{{\Gamma}_{12x}(0,t)}{|{\Gamma}_{12x}(0,t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=39,40;\\ \frac{{\Gamma}_{ix}(0,t)}{|{\Gamma}_{ix}(0,t)|}\xb7\frac{{\Gamma}_{24x}(0,t)}{|{\Gamma}_{24x}(0,t)|}=cos\frac{2\pi}{3},\phantom{\rule{1.em}{0ex}}i=41,42.\end{array}$$
- Angle conditions at $\partial \Omega $:$$\begin{array}{c}<{\Gamma}_{is}({l}_{i},t),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\nabla b\left({\Gamma}_{i}({l}_{i},t)\right)>=0,\phantom{\rule{1.em}{0ex}}i=2,10,14,22,26,34;\\ <{\Gamma}_{is}(0,t),\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\nabla b\left({\Gamma}_{i}(0,t)\right)>=0,\phantom{\rule{1.em}{0ex}}i=4,8,16,20,28,32.\end{array}$$

- Incidence at the junctions:$$\begin{array}{c}{h}_{i}^{N}\left(0\right)+{h}_{i+1}^{N}\left(0\right)+{h}_{i+2}^{N}\left(0\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{i}^{T}\left(0\right)+{h}_{i+1}^{T}\left(0\right)+{h}_{i+2}^{T}\left(0\right)=0,\\ \mathrm{for}\phantom{\rule{1.em}{0ex}}i=1,5,9,13,17,21,25,29,33,36;\\ {h}_{i}^{N}\left({l}_{i}\right)+{h}_{i+1}^{N}\left({l}_{i+1}\right)+{h}_{i+2}^{N}\left({l}_{i+2}\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{i}^{T}\left({l}_{i}\right)+{h}_{i+1}^{T}\left({l}_{i+1}\right)+{h}_{i+2}^{T}\left({l}_{i+2}\right)=0,\\ \mathrm{for}\phantom{\rule{1.em}{0ex}}i=3,7,11,15,19,23,27,31;\\ {h}_{35}^{N}\left({l}_{35}\right)+{h}_{36}^{N}\left({l}_{36}\right)+{h}_{1}^{N}\left({l}_{1}\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{35}^{T}\left({l}_{35}\right)+{h}_{36}^{T}\left({l}_{36}\right)+{h}_{1}^{T}\left({l}_{1}\right)=0;\\ {h}_{38}^{N}\left({l}_{38}\right)+{h}_{39}^{N}\left({l}_{39}\right)+{h}_{6}^{N}\left({l}_{6}\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{38}^{T}\left({l}_{38}\right)+{h}_{39}^{T}\left({l}_{39}\right)+{h}_{6}^{T}\left({l}_{6}\right)=0;\\ {h}_{40}^{N}\left({l}_{40}\right)+{h}_{41}^{N}\left({l}_{41}\right)+{h}_{18}^{N}\left({l}_{18}\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{40}^{T}\left({l}_{40}\right)+{h}_{41}^{T}\left({l}_{41}\right)+{h}_{18}^{T}\left({l}_{18}\right)=0;\\ {h}_{42}^{N}\left({l}_{42}\right)+{h}_{30}^{N}\left({l}_{30}\right)+{h}_{37}^{N}\left({l}_{37}\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{42}^{T}\left({l}_{42}\right)+{h}_{30}^{T}\left({l}_{30}\right)+{h}_{37}^{T}\left({l}_{37}\right)=0;\\ {h}_{39}^{N}\left(0\right)+{h}_{40}^{N}\left(0\right)+{h}_{12}^{N}\left(0\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{39}^{T}\left(0\right)+{h}_{40}^{T}\left(0\right)+{h}_{12}^{T}\left(0\right)=0;\\ {h}_{41}^{N}\left(0\right)+{h}_{42}^{N}\left(0\right)+{h}_{24}^{N}\left(0\right)=0\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{41}^{T}\left(0\right)+{h}_{42}^{T}\left(0\right)+{h}_{24}^{T}\left(0\right)=0.\end{array}$$
- Incidence at the boundary $\partial \Omega $:$$\begin{array}{c}{h}_{i}^{T}\left({l}_{i}\right)=0,\phantom{\rule{1.em}{0ex}}i=2,10,14,22,26,34\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{h}_{i}^{T}\left(0\right)=0,\phantom{\rule{1.em}{0ex}}i=4,8,16,20,28,32.\end{array}$$
- Angle conditions at the junctions:$$\begin{array}{c}{h}_{i}^{\prime N}\left(0\right)+{h}_{i}^{T}\left(0\right){k}_{i}={h}_{i+1}^{\prime N}\left(0\right)+{h}_{i+1}^{T}\left(0\right){k}_{i+1}={h}_{i+2}^{\prime N}\left(0\right)+{h}_{i+2}^{T}\left(0\right){k}_{i+2},\\ \mathrm{for}\phantom{\rule{1.em}{0ex}}i=1,5,9,13,17,21,25,29,33,36;\\ {h}_{i}^{\prime N}\left({l}_{i}\right)+{h}_{i}^{T}\left({l}_{i}\right){k}_{i}={h}_{i+1}^{\prime N}\left({l}_{i+1}\right)+{h}_{i+1}^{T}\left({l}_{i+1}\right){k}_{i+1}={h}_{i+2}^{\prime N}\left({l}_{i+2}\right)+{h}_{i+2}^{T}\left({l}_{i+2}\right){k}_{i+2},\\ \mathrm{for}\phantom{\rule{1.em}{0ex}}i=3,7,11,15,19,23,27,31;\\ {h}_{35}^{\prime N}\left({l}_{35}\right)+{h}_{35}^{T}\left({l}_{35}\right){k}_{35}={h}_{36}^{\prime N}\left({l}_{36}\right)+{h}_{36}^{T}\left({l}_{36}\right){k}_{36}={h}_{1}^{\prime N}\left({l}_{1}\right)+{h}_{1}^{T}\left({l}_{1}\right){k}_{1};\\ {h}_{38}^{\prime N}\left({l}_{38}\right)+{h}_{38}^{T}\left({l}_{38}\right){k}_{38}={h}_{39}^{\prime N}\left({l}_{39}\right)+{h}_{39}^{T}\left({l}_{39}\right){k}_{39}={h}_{6}^{\prime N}\left({l}_{6}\right)+{h}_{6}^{T}\left({l}_{6}\right){k}_{6};\\ {h}_{40}^{\prime N}\left({l}_{40}\right)+{h}_{40}^{T}\left({l}_{40}\right){k}_{40}={h}_{41}^{\prime N}\left({l}_{41}\right)+{h}_{41}^{T}\left({l}_{41}\right){k}_{41}={h}_{18}^{\prime N}\left({l}_{18}\right)+{h}_{18}^{T}\left({l}_{18}\right){k}_{18};\\ {h}_{37}^{\prime N}\left({l}_{37}\right)+{h}_{37}^{T}\left({l}_{37}\right){k}_{37}={h}_{42}^{\prime N}\left({l}_{42}\right)+{h}_{42}^{T}\left({l}_{42}\right){k}_{42}={h}_{30}^{\prime N}\left({l}_{30}\right)+{h}_{30}^{T}\left({l}_{30}\right){k}_{30};\\ {h}_{39}^{\prime N}\left(0\right)+{h}_{39}^{T}\left(0\right){k}_{39}={h}_{40}^{\prime N}\left(0\right)+{h}_{40}^{T}\left(0\right){k}_{40}={h}_{12}^{\prime N}\left(0\right)+{h}_{12}^{T}\left(0\right){k}_{12};\\ {h}_{41}^{\prime N}\left(0\right)+{h}_{41}^{T}\left(0\right){k}_{41}={h}_{42}^{\prime N}\left(0\right)+{h}_{42}^{T}\left(0\right){k}_{42}={h}_{24}^{\prime N}\left(0\right)+{h}_{24}^{T}\left(0\right){k}_{24}.\end{array}$$
- Angle conditions at $\partial \Omega $:$$\begin{array}{c}{K}_{i}^{\partial \Omega}{h}_{i}^{N}\left({l}_{i}\right)={h}_{i}^{\prime N}\left({l}_{i}\right),\phantom{\rule{1.em}{0ex}}i=2,10,14,22,26,34;\\ {K}_{i}^{\partial \Omega}{h}_{i}^{N}\left(0\right)={h}_{i}^{\prime N}\left(0\right),\phantom{\rule{1.em}{0ex}}i=4,8,16,20,28,32.\end{array}$$

**Theorem**

**2.**

- 1.
- If the domain Ω is convex (an ellipse for example) at the points where the steady state of the network meets the boundary, then the steady state is unstable.
- 2.
- If the domain Ω is non-degenerate concave at the points where the steady state of the network meets the boundary, then the steady state is stable.
- 3.
- If the domain Ω is flat at the points where the steady state of the network meets the boundary, then the steady state is neutrally stable.

**Proof.**

## 4. Conclusions

- Grain growth problems: the increase in the size of grains (crystallites) in a material at high temperatures;
- Neural networks that can be applied on the decomposition system (3) for ${k}_{i}=0$;
- Continuum modelling of electromechanical dynamics in large-scale power systems;
- Electrical networks of power systems that are not static and change their shape in time. This type of network can include synchronization problems of nonlinear circuits in dynamic electrical networks with general topologies and power system cascading risk assessment based on complex network theory.

## Acknowledgments

## Conflicts of Interest

## References and Note

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Dassios, I.K.
Stability of Bounded Dynamical Networks with Symmetry. *Symmetry* **2018**, *10*, 121.
https://doi.org/10.3390/sym10040121

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Dassios IK.
Stability of Bounded Dynamical Networks with Symmetry. *Symmetry*. 2018; 10(4):121.
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**Chicago/Turabian Style**

Dassios, Ioannis K.
2018. "Stability of Bounded Dynamical Networks with Symmetry" *Symmetry* 10, no. 4: 121.
https://doi.org/10.3390/sym10040121