A Turing–Hopf Bifurcation Scenario for Pattern Formation on Growing Domains

Abstract

In this paper, we study the emergence of different patterns that are formed on both static and growing domains and their bifurcation structure. One of these is the so-called Turing–Hopf morphogenetic mechanism. The reactive part we consider is of FitzHugh–Nagumo type. The analysis was carried out on a flat square by considering both fixed and growing domain. In both scenarios, sufficient conditions on the parameter values are given for the formation of specific space–time structures or patterns. A series of numerical solutions of the corresponding initial and boundary value problems are obtained, and a comparison between the resulting patterns on the fixed domain and those arising when the domain grows is established. We emphasize the role of growth of the domain in the selection of patterns. The paper ends by listing some open problems in this area.

Appendices

Appendix 1: The Parameter Spaces in the Not Growth Case

The goal of these two first final appendixes is to present the detailed analysis with the aim of showing the sets where the parameters appearing in the different ODE or PDE systems through this paper, satisfy the appropriate conditions, are not empty. The parameters appearing in our systems are: a,  b,  c and d. Thus, we are going to investigate the required conditions on different parameter planes in two separate cases: 1. with no growing domain and 2. in the isotropic growing domain case.

Along this appendix, we will assume the conditions $0<b<1$ and $\left( -0.6+\frac{a}{b}\right) >0$ fulfill. These are to ensure the existence of a unique positive equilibrium of the kinetic part of our system.

1.1 The Set Where $0<\bar{u}<1$, with $0<b<1$ and Conditions for Turing Bifurcation

Just to refresh ourselves, we have that whenever the following conditions

$$\begin{aligned} \mathbf{I.} \left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}\right]< & {} 0,\quad \mathbf{II.} \left[ 1-b\left( 1-\bar{u}^2\right) \right]>0,\nonumber \\ \mathbf{III.} \left[ dc\left( 1{-}\bar{u}^2\right) {-}\frac{b}{c}\right]>&0,\quad \mathbf{IV.} \left[ dc\left( 1-\bar{u}^2\right) -\frac{b}{c}\right] ^2-4d\left[ 1{-}b\left( 1{-}\bar{u}^2\right) \right] >0,\nonumber \\ \end{aligned}$$

(64)

hold, the reaction–diffusion system

$$\begin{aligned} \begin{aligned} u_t&= \Delta u +c\left( u-\frac{u^3}{3}+v-0.6\right) \\ v_t&= d \Delta v -\frac{u-a+bv}{c}, \end{aligned} \end{aligned}$$

(65)

exhibits a Turing bifurcation.

Fig. 11

Blue shady region where $0<b<1$ and $0<\bar{u}<1$ (Color figure online)

Note that, for $0<b<1$ whenever the parameter c satisfies the inequality $0<c<c^*$, where

$$\begin{aligned} c^*=\sqrt{\frac{b}{1-\bar{u}^2}}, \end{aligned}$$

the condition I. holds. The critical value, $c^*$, of c is well defined whenever $0<\bar{u}<1$. The explicit form of $\bar{u}$, which depends on the parameters a and b in a massive way given in (4). We claim that, restricting b as $0<b<1$, the set of values of a for which $0<\bar{u}<1$ is a nonempty set. Since the parameters a and b must be positive, we consider the positive quadrant of the plane ab. Then by using: the explicit form of $\bar{u}$ and the Maple software, the set of the parameters on the plane ab where $0<\bar{u}<1$ is determined. This is all the set of points situated at the left of the line $l_2$ appearing in Fig. 11. Note the line $l_2$ appears in the first quadrant of the cd plane. The justification of this is as follows. Once we select the parameters in such a way the condition I is fulfilled for all $c\in (0,c^*)$, no matter which value of d is. Hence, for each $c\in (0,c^*)$ in such a plane, we have a vertical straight line.

Now, let see the condition II. Since the set of parameters for which $0<\bar{u}^2<1$ with $0<b<1$ is a nonempty set, it is straightforward the verify $0<(1-b)<\left[ 1-b\left( 1-\bar{u}^2\right) \right]$.

For the condition III, given we require $d>1$, for fixed $b\in (0,1)$ we can select $\bar{u}\in (0,1)$ in such a way, varying c that condition holds. In fact, it is enough to choose $d>1$ such that

$$\begin{aligned} d>\frac{b}{c^2\left( 1-\bar{u}^2\right) }. \end{aligned}$$

Actually, in the first quadrant of the cd plane the right-hand side of the previous inequality (for $0<b<1$) it represents the positive branch of a hyperbola with asymptotic behavior toward the coordinate axis. Therefore, for all the points (c, d)—with $d>1$—above such a hyperbola, the condition III, is satisfied.

Once we put in geometrical terms the conditions I–III for the occurrence of the Turing bifurcation, let us see the fourth condition

$$\begin{aligned} \left\{ \left[ dc\left( 1-\bar{u}^2\right) -\frac{b}{c}\right] ^2-4d\left[ 1-b\left( 1-\bar{u}^2\right) \right] \right\} >0. \end{aligned}$$

In order to define the boundary of the region where the previous conditions holds, let see where the equality

$$\begin{aligned} \left\{ \left[ dc\left( 1-\bar{u}^2\right) -\frac{b}{c}\right] ^2-4d\left[ 1-b\left( 1-\bar{u}^2\right) \right] \right\} =0, \end{aligned}$$

is satisfied. Hence, d takes the following values

$$\begin{aligned} d_1,d_2=\frac{b(\bar{u}^2-1)+2\pm 2\sqrt{1-b\left( 1-\bar{u}^2\right) }}{c^2\left( 1-\bar{u}^2\right) ^2}. \end{aligned}$$

Therefore, in addition to being $d>1$ whenever d belongs the complement of the interval

$$\begin{aligned} \left[ \frac{b(\bar{u}^2-1)+2-2\sqrt{1-b\left( 1-\bar{u}^2\right) }}{c^2\left( 1-\bar{u}^2\right) ^2},\frac{b(\bar{u}^2-1)+2+2\sqrt{1-b\left( 1-\bar{u}^2\right) }}{c^2\left( 1-\bar{u}^2\right) ^2}\right] , \end{aligned}$$

the condition IV fulfills. Thus, by choosing $b\in (0,1)$ and $0<\bar{u}<1$ as previously we showed, and with the aim of having a geometrical image of the conditions, we note that the extreme values, $d_1$ and $d_2$, of the above interval define—for $c\ne 0$—two symmetrical hyperbolas in the cd plane. Given $c>0$, only one of these is of interest. Therefore, all the couples (c, d) of the plane cd with $d>1$ and allocated above the mentioned hyperbola satisfy the condition IV.

Fig. 12

On the shady region all the conditions for triggering the Turing morphogenetic mechanism in the reaction–diffusion system (65) are satisfied

Once we have determined the set of parameters where each single condition holds for the occurrence of the Turing bifurcation, we now proceed to put all of them together. We do it in the cd plane by taking into account that in addition to the previous conditions, c and d are restricted as follows: $c>0$ and $d>1$. Therefore, by choosing the parameters values in the shady region appearing in Fig. 12, we guarantee the Turing bifurcation occurs.

1.2 Conditions for the Hopf Bifurcation

Here we are going to determine the conditions for the emergence of a limit cycle in the kinetic part of the system (3) by means of a Hopf bifurcation. Just remind, the characteristic polynomial of the Jacobian matrix at the equilibrium $(\bar{u},\bar{v})$ is

$$\begin{aligned} \lambda ^2-\lambda \left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}\right] +1-b\left( 1-\bar{u}^2\right) , \end{aligned}$$

(66)

whose roots are

$$\begin{aligned} \lambda _1, \lambda _2=\frac{c\left( 1-\bar{u}^2\right) -\frac{b}{c}\pm \sqrt{\left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}\right] ^2-4\left[ 1-b\left( 1-\bar{u}^2\right) \right] }}{2}. \end{aligned}$$

(67)

First, we require $\lambda _1$ and $\lambda _2$ must be complex, i.e., they should have the form

$$\begin{aligned} \lambda _1 (c)=\alpha (c)+ i\,\beta (c)\quad \mathrm{and } \quad \lambda _2 (c)=\bar{\lambda }_1(c) \end{aligned}$$

with $\beta (c)\ne 0$. This happens whenever the inequality

$$\begin{aligned} \left\{ \left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}\right] ^2-4\left[ 1-b\left( 1-\bar{u}^2\right) \right] \right\} <0, \end{aligned}$$

(68)

holds. In the plane cd,  for fixed b and $\bar{u}$ the above inequality represents the blue region shown in Fig. 13.

Fig. 13

Region where the conditions to obtain a Hopf bifurcation are fulfilled

In addition, we require the existence of a critical value, $c^*,$ of c such that the real part, $\alpha$, of $\lambda _1$ and $\lambda _2$ vanishes, i.e., $\alpha (c^*)=0$. This critical value is given by

$$\begin{aligned} c^*=\sqrt{\frac{b}{1-\bar{u}^2}}. \end{aligned}$$

(69)

Note $c^*$ is defined whenever we are able of choosing the parameter values such that $0<\bar{u}<1$. In fact, this is feasible as it was already shown in previous section. See Fig. 11. Therefore, $c^*$ is the bifurcation value of c for which the real part of the eigenvalues changes of sign.

On the other side, one of the conditions at the very beginning of this appendix implies $b<\frac{a}{0.6}=\frac{5}{3}a$ which, in the ab plane (Fig. 11) it corresponds to the region situated at the right of the straight line $l_1$; meanwhile, in the same ab plane, from the explicit form of $\bar{u}$, taking into account $0<\bar{u}<1$, the region where this happens is that at the left of the line $l_2.$ Therefore, the intersection where this and the previous condition hold is that blue shady region in Fig. 11.

In the main part of this paper, we carried out the needed additional analysis in order to see the stability of the limit cycle which, according to the Hopf bifurcation theorem, emerges.

1.3 For the Turing–Hopf Bifurcation

In order to determine the region where both Turing and Hopf bifurcations coexist, we find the intersection of the shaded regions in Figs. 12 and 13. From this, we obtain the region shown in Fig. 12.

Appendix 2: The Parameter Spaces in the Growing Case

Here, in order to determinate the set of parameters where the different bifurcations occur, we are going to proceed in a similar fashion as we did it in the previous appendix. The new ingredient here is the inclusion of the isotropic growth of the domain, by means of an exponential growth function.

1.1 For the Turing Bifurcation

Proceeding in a similar way as we did in the case with no growth, we determine the regions where the system exhibits a Turing bifurcation. The FitzHugh–Nagumo equations with exponential isotropic growth become:

$$\begin{aligned} \begin{aligned} u_t&= \frac{1}{\rho ^2(t)} \left( u_{\zeta \zeta } + u_{\eta \eta } \right) + c\left( u-\frac{u^3}{3}+v-0.6\right) -2ku,\\ v_t&= \frac{d}{\rho ^2(t)} \left( v_{\zeta \zeta } + v_{\eta \eta } \right) + -\frac{u-a+bv}{c}-2kv. \end{aligned} \end{aligned}$$

(70)

The conditions for the existence of a Turing bifurcation with $h_*=2k$ are

$$\begin{aligned}&(f_u+g_v)-2h_*<0,\, \end{aligned}$$

(71)

$$\begin{aligned}&(f_ug_v-f_vg_u)-h_* (f_u+g_v)+h^2>0,\, \end{aligned}$$

(72)

$$\begin{aligned}&(df_u+g_v)-h_*(1+d)>0,\, \end{aligned}$$

(73)

$$\begin{aligned}&\left[ (df_u{+}g_v)-h_*(1+d)\right] ^2-4 d \left[ (f_ug_v{-}f_vg_u){-}h_* (f_u{+}g_v)+h^2 \right] >0, \end{aligned}$$

(74)

which for the system we consider are

$$\begin{aligned}&\left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}-4k\right] <0, \end{aligned}$$

(75)

$$\begin{aligned}&\left[ 1-\left( 1-\bar{u}^2\right) (b+2kc)+\frac{2bk}{c}+4k^2\right] >0, \end{aligned}$$

(76)

$$\begin{aligned}&\left\{ \left( dc\left( 1-\bar{u}^2\right) -\frac{b}{c}\right) -2k(1+d)\right\} >0, \end{aligned}$$

(77)

$$\begin{aligned}&\left[ dc\left( 1-\bar{u}^2\right) -\frac{b}{c}-2k(1+d)\right] ^2\nonumber \\&\quad -\,4d \left[ 1-\left( 1-\bar{u}^2\right) (b+2kc)+\frac{2bk}{c} +4k^2\right] >0, \end{aligned}$$

(78)

By condition (75)

$$\begin{aligned} 0<c<\frac{2k+\sqrt{4k^2+\left( 1-\bar{u}^2\right) b}}{1-\bar{u}^2}=c^*, \end{aligned}$$

which in Fig. 14 is represented by the region between the d axis and the line $l_1.$ Also for the region $c<-c^*,$ which is not of biological interest.

From the condition (76) we obtain c must satisfy

$$\begin{aligned} 0<c<\frac{1-b\left( 1-\bar{u}^2\right) +\sqrt{[1-\left( 1-\bar{u}^2\right) b]^2+16bk^2\left( 1-\bar{u}^2\right) }}{4k\left( 1-\bar{u}^2\right) }, \end{aligned}$$

From condition (77)

$$\begin{aligned} d>\frac{b+2kc}{c^2\left( 1-\bar{u}^2\right) -2k}, \end{aligned}$$

which determines the region to the right of the hyperbola $l_2.$

Now, from condition (77), we have the following regions: a) between the d axis and the line $l_{3a};$ b), lying to the right of $l_{3b}.$

Therefore, the region where we have a Turing bifurcation is represented by the blue shaded region in Fig. 14.

Fig. 14

Region where are fulfilled all conditions to obtain a Turing bifurcation

1.2 For the Hopf Bifurcation

We now determine the region in parameter space where the conditions for a Hopf bifurcation are met. Given that the characteristic polynomial associated with the Jacobian matrix of the linearized system is

$$\begin{aligned} \lambda ^2-\lambda \left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}-4k\right] +1-\left( 1-\bar{u}^2\right) (b+2kc)+\frac{2bk}{c}+4k^2. \end{aligned}$$

(79)

Here we introduce the following notation

$$\begin{aligned} \begin{aligned} \alpha =\left[ c\left( 1-\bar{u}^2\right) -\frac{b}{c}-4k\right] \quad \mathrm{and }\quad \gamma =1-\left( 1-\bar{u}^2\right) (b+2kc)+\frac{2bk}{c}+4k^2, \end{aligned} \end{aligned}$$

we have that the eigenvalues are

$$\begin{aligned} \lambda _1, \lambda _2=\frac{\alpha \pm \sqrt{\alpha ^2-4\gamma }}{2}. \end{aligned}$$

(80)

Thus, in order $\lambda _1$ and $\lambda _2$ being of the form $\lambda _1, \lambda _2=\frac{\alpha }{2}\pm i\frac{\beta }{2},$ we need

$$\begin{aligned} \alpha ^2-4\gamma {=}\left[ c\left( 1{-}\bar{u}^2\right) {-}\frac{b}{c}-4k\right] ^2-4\left[ 1{-}\left( 1{-}\bar{u}^2\right) (b+2kc)+\frac{2bk}{c}+4k^2\right] <0, \end{aligned}$$

which determines the shaded region in blue in Fig. 15.

Fig. 15

Region where are fulfilled all conditions to obtain a Hopf bifurcation

Additionally, we require the existence of a critical value $c^*,$ de c, where $\alpha (c^*)=0$ and $\beta (c^*)\ne 0.$ Such critical value is given by the expression

$$\begin{aligned} c^*=\frac{2k+\sqrt{4k^2+b\left( 1-\bar{u}^2\right) }}{1-\bar{u}^2}. \end{aligned}$$

(81)

If $c^*\in \mathbb {R},$ we require that the following conditions must be satisfied

$$\begin{aligned} \bar{u}^2\ne 1\quad \text {and}\quad 4k^2+b\left( 1-\bar{u}^2\right) \ge 0. \end{aligned}$$

Notice that the last one is automatically fulfilled provided $\bar{u}^2<1.$ We fix c, say $c=0.68.$ Then we have the constraints (32). In Fig. 16, from condition $0<-0.6+\frac{a}{b+2kc}$ it follows that $b<-\frac{a}{-0.6}-2kc$ corresponding to the region to the right of the curve $l_1.$

Condition $\frac{2k}{c}-1+\frac{1}{b+2kc}>0,$ that is, $b<\frac{c}{c-2k}-2kc,$ determines the region between the a axis and below the line $l_2.$

From definition $\bar{u},$ (33) and in order that $0<\bar{u}<1$ we obtain the region to the left of $l_3.$

Therefore, the region where all conditions are met is the intersection of these three areas and is shown in blue.

Fig. 16

Region where are fulfilled all conditions: $0<\bar{u}<1$ and the conditions (32)

1.3 For the Turing–Hopf Bifurcation

In order to determine the region where both bifurcations coexist, we consider the intersection of the shaded regions in blue in Figs. 14 and 15. This produces Fig. 14, representing the region where there is a Turing–Hopf bifurcation.

Appendix 3: Numerical Solution for the Discretized Laplace–Beltrami

The title of this appendix tells us its actual contents. Note this was used for deriving the CFL criterion for the numerical convergence.

For this, firstly we observe that the forward Euler method gives us the following approximation

$$\begin{aligned} \begin{aligned} \frac{\partial u}{\partial t} \approx \frac{u(n, m, j+1)-u(n, m, j)}{\Delta t}. \end{aligned} \end{aligned}$$

We recall that the discretization of the two-dimensional Laplacian operator via central finite differences is given by

$$\begin{aligned} \begin{aligned} \Delta u(n, m, j+1)\approx&\frac{u(n+1, m, j)+u(n-1, m, j)-2u(n, m, j)}{(\Delta \zeta )^2}\\&+\frac{u(n, m+1, j)+u(n, m-1, j)-2u(n, m , j)}{(\Delta \eta )^2}. \end{aligned} \end{aligned}$$

Thus, by using the previous approximations the first two terms of the first partial differential equation in (22)

$$\begin{aligned} \begin{aligned} u_t&=\frac{1}{h_1h_2}\left[ \left( \frac{h_2}{h_1}u_\zeta \right) _\zeta + \left( \frac{h_1}{h_2}u_\eta \right) _\eta \right] \\&=\frac{1}{h_1h_2}\left[ \frac{h_2}{h_1}u_{\zeta \zeta }+\left( \frac{h_2}{h_1}\right) _\zeta u_\zeta + \frac{h_1}{h_2}u_{\eta \eta }+\left( \frac{h_1}{h_2}\right) _\eta u_\eta \right] \end{aligned} \end{aligned}$$

(82)

can be discretized as follows

$$\begin{aligned} \begin{aligned} u&(n,m,j+1)=u(n,m,j)\\&+\frac{\Delta t}{h_1h_2}\left[ \left( \frac{h_2}{h_1}\right) \frac{u(n+1,m,j)+u(n-1,m,j)-2u(n,m,j)}{(\Delta \zeta )^2}\right. \\&+\left( \frac{h_1}{h_2}\right) \frac{u(n,m+1,j)+u(n,m-1,j)-2u(n,m,j)}{(\Delta \eta )^2}\\&+\left( \frac{h_2}{h_1}\right) _\zeta \frac{u(n+1,m,j)-u(n-1,m,j)}{2\Delta \zeta } \\&\left. +\left( \frac{h_1}{h_2}\right) _\eta \frac{u(n,m+1,j)-u(n,m-1,j)}{2\Delta \eta }\right] , \end{aligned} \end{aligned}$$

(83)

where $\Delta t, \Delta \zeta$ and $\Delta \eta$ are the time step, the $\zeta$ step and the $\eta$ step, respectively. We must observe that by (21), $h_1 = h_1(\zeta , \eta , j\Delta t)$ and $h_2 = h_2(\zeta , \eta , j\Delta t).$

For this discrete version, we propose solution of the form

$$\begin{aligned} u_{n,m,j}=\xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + m\Delta \eta )}, \end{aligned}$$

(84)

where $i=\sqrt{-1}$ and l is the wave number; meanwhile, $\xi ^{\Delta t}=\xi (l, j\Delta t),$ is the time dependence, this is a complex number that depends on l. The key fact is that the time dependence of a single eigenmode is nothing more than successive integer powers of the complex number $\xi ^{\Delta t}$; substituting in (83) and grouping appropriately, we obtain

$$\begin{aligned} \begin{aligned}&\xi ^{(j+1)\Delta t} \hbox {e}^{il(n\Delta \zeta + m\Delta \eta )}=\xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + m\Delta \eta )}+\frac{\Delta t}{h_1h_2}\bigg \{\\&\left( \frac{h_2}{h_1}\right) \frac{1}{(\Delta \zeta )^2}\left[ \xi ^{j\Delta t} \hbox {e}^{il((n+1)\Delta \zeta + m\Delta \eta )} +\xi ^{j\Delta t} \hbox {e}^{il((n-1)\Delta \zeta + m\Delta \eta )}-2\xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + m\Delta \eta )}\right] \\&+ \left( \frac{h_1}{h_2}\right) \frac{1}{(\Delta \eta )^2}\left[ \xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + (m+1)\Delta \eta )} +\xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + (m-1)\Delta \eta )}-2\xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + m\Delta \eta )}\right] \\&+\left( \frac{h_2}{h_1}\right) _\zeta \frac{1}{2\Delta \zeta } \left[ \xi ^{j\Delta t} \hbox {e}^{il((n+1)\Delta \zeta + m\Delta \eta )}-\xi ^{j\Delta t} \hbox {e}^{il((n-1)\Delta \zeta + m\Delta \eta )}\right] \\&\left. +\left( \frac{h_1}{h_2}\right) _\eta \frac{1}{2\Delta \eta } \left[ \xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + (m+1)\Delta \eta )}-\xi ^{j\Delta t} \hbox {e}^{il(n\Delta \zeta + (m-1)\Delta \eta )}\right] \right\} \end{aligned} \end{aligned}$$

after multiplying both sides of the equality for the inverse of (84) we obtain

$$\begin{aligned} \begin{aligned} \xi ^{\Delta t}&=1\\&\quad +\frac{\Delta t}{h_1h_2}\left\{ \left( \frac{h_2}{h_1}\right) \frac{1}{(\Delta \zeta )^2}\left[ \hbox {e}^{il\Delta \zeta } + \hbox {e}^{-il\Delta \zeta }-2\right] \right. \\&\quad + \left( \frac{h_1}{h_2}\right) \frac{1}{(\Delta \eta )^2}\left[ \hbox {e}^{il\Delta \eta } +\hbox {e}^{-il\Delta \eta }-2\right] \\&\quad \left. +\left( \frac{h_2}{h_1}\right) _\zeta \frac{1}{2\Delta \zeta } \left[ \hbox {e}^{il\Delta \zeta }- \hbox {e}^{-il\Delta \zeta }\right] +\left( \frac{h_1}{h_2}\right) _\eta \frac{1}{2\Delta \eta } \left[ \hbox {e}^{il\Delta \eta }- \hbox {e}^{-il\Delta \eta }\right] \right\} \\ \end{aligned} \end{aligned}$$

which, by using the equalities

$$\begin{aligned} \cos (\cdot )=\frac{\hbox {e}^{i(\cdot )}+\hbox {e}^{-i(\cdot )}}{2}\quad \text{ and }\quad \sin (\cdot )=\frac{\hbox {e}^{i(\cdot )}-\hbox {e}^{-i(\cdot )}}{2i}, \end{aligned}$$

can be written as follows

$$\begin{aligned} \begin{aligned} \xi ^{\Delta t}&=1+\frac{\Delta t}{h_1h_2}\left\{ \left( \frac{h_2}{h_1}\right) \frac{1}{(\Delta \zeta )^2}\left[ 2\cos (l\Delta \zeta )-2\right] \right. + \left( \frac{h_1}{h_2}\right) \frac{1}{(\Delta \eta )^2}\left[ 2\cos (l\Delta \eta )-2\right] \\&\left. \quad +\left( \frac{h_2}{h_1}\right) _\zeta \frac{1}{2\Delta \zeta } \left[ 2 i \sin (l\Delta \zeta )\right] +\left( \frac{h_1}{h_2}\right) _\eta \frac{1}{2\Delta \eta } \left[ 2 i \sin (l\Delta \eta )\right] \right\} . \end{aligned} \end{aligned}$$

Here, we use the trigonometric identity $\cos (s)-1=-2\sin ^2(s/2)$ to rewrite the previous equality as

$$\begin{aligned} \begin{aligned} \xi ^{\Delta t}&=1+\frac{\Delta t}{h_1h_2}\left\{ \left( \frac{h_2}{h_1}\right) \frac{1}{(\Delta \zeta )^2}\left[ -4\sin ^2\left( \frac{l\Delta \zeta }{2}\right) \right] \right. + \left( \frac{h_1}{h_2}\right) \frac{1}{(\Delta \eta )^2}\left[ -4\sin ^2\left( \frac{l\Delta \eta }{2}\right) \right] \\&\left. +\left( \frac{h_2}{h_1}\right) _\zeta \frac{1}{2\Delta \zeta } \left[ 2 i \sin (l\Delta \zeta )\right] +\left( \frac{h_1}{h_2}\right) _\eta \frac{1}{2\Delta \eta } \left[ 2 i \sin (l\Delta \eta )\right] \right\} . \end{aligned} \end{aligned}$$