Analytical Solution and Quasi-Periodic Behavior of a Charged Dilaton Black Hole

Abstract

With the vast breakthrough brought by the Event Horizon Telescope, the theoretical analysis of various black holes has become more critical than ever. In this paper, the second-order asymptotic analytical solution of the charged dilaton black hole flow in the spinodal region is constructed from the perspective of dynamics by using the two-timing scale method. Through a numerical comparison with the original charged dilaton black hole system, it is found that the constructed analytical solution is highly consistent with the numerical solution. In addition, several quasi-periodic motions of the charged dilaton black hole flow are numerically obtained under different groups of irrational frequency ratios, and the phase portraits of the black hole flow with sufficiently small thermal parameter perturbation display good stability. Finally, the final evolution state of black hole flow over time is studied according to the obtained analytical solution. The results show that the smaller the integral constant of the system, the greater the periodicity of the black hole flow.

1. Introduction

The black hole is a spacetime region where the gravity is so strong that any particle (including photons) close to it can be dragged to its center. The earliest paper can be traced back to the pioneering work of Oppenheimer and Snyder [1] in 1939. Visual observation of a real black hole is highly challenging, because it does not reflect light and is too far away from us. Therefore, plenty of researchers have investigated invisible companion stars, namely black holes, based on the visible companion stars of binary systems [2,3,4]. Nowadays, the research on black holes has involved gravitational waves, wormholes, modified gravitational theory, quantum regime, and several other related fields [5,6,7,8,9,10,11,12,13,14].

Black holes can emit thermal radiation. Researchers found a significant analogy between the mathematical form of their physical laws and the laws of thermodynamics as early as the 1970s [15,16], and then the thermodynamic properties of black holes received a lot of attention [17,18,19,20,21,22]. The similarity between their thermodynamic phase structure and van der Waals fluid system was further proven [23,24]. Moreover, there are a great number of studies on the phase transition of black holes [25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]. The critical behaviors of different black holes were discussed by depicting the P-V diagrams [25,26,27,28,29,30,31,32,33]. From these diagrams, one can identify the occurrence of the phase transition and the spinodal region, where the small-black-hole (SBH) and large-black-hole (LBH) coexist. Furthermore, the related homoclinic orbits in the extended phase space were also analyzed. Zhao et al. [34] mainly studied the phase transition process and demonstrated the SBH-LBH-phase coexistence curves’ boundary with different parameters in the charged topological dilaton AdS black hole. For the black holes in a higher dimension, such as Gauss-Bonnet-Born-Infeld AdS black holes, some interpretations on their critical behavior can be found in Refs. [35,44]. Dozens of researchers have studied the related critical behavior for the black hole in the extended phase space, where there are prosperous phase structures [11,27,35,43]. Furthermore, from the perspective of phase transition, some properties of the microstructure of the charged AdS black hole can be found in [46].

In order to identify the chaos motions of the Kerr-AdS black hole, Born-Infeld-AdS black hole, charged AdS and dilaton black hole, and charged Gauss-Bonnet AdS black hole, Chen et al. used the Melnikov method to study the temporal and spatial chaos of these black holes [47,48,49,50,51]. They found a critical value ${\delta }_c$ of the perturbation amplitude, which depends on other parameters in the black hole system; the temporal chaos exists in the case $\delta >{\delta }_c$, while the spatial chaos exists whatever the value of $\delta$ is. For the Schwarzschild black hole under the minimal length effects, the same method was also applied to investigate its chaotic behavior [52]. Moreover, there are several references on the chaos motion of the particles around the black holes, including the analyses of the dynamic behavior of the particles, Poincaré surface of section, and innermost stable circular orbits [53,54,55].

Inspired by the similarity of thermal phase structure and van der Waals fluid system and the literature of different black hole solutions, this paper will mainly use the two-timing scale method (see [56] for more details) and analysis techniques to study the dynamic behavior of a charged dilaton black hole and construct the corresponding analytical solutions from the perspective of dynamics. The two-timing scale method is one of the most effective methods in the quantitative analysis of nonlinear dynamics. It can describe periodic motion and disclose the attenuated vibration of the dissipative system. From the references mentioned earlier, there exist non-chaotic regions under certain conditions. Therefore, considering the temporal thermal perturbation, the quasi-periodic behavior of the charged dilaton black hole flow in the spinodal region will also be discussed and analyzed in the following sections. The layout of this paper is as follows. In Section 2, the equation model of the charged dilaton black hole will be introduced in the extended phase space. In Section 3, the two-timing scale method will be used to solve the dynamic equation of the black hole flow, and the numerical comparison will be carried out in Section 4, where there is a quasi-periodic motion. Accordingly, several kinds of quasi-periodic motion will be shown in Section 5, and the paper ends with the discussion and conclusion.

2. Thermodynamics and Dynamics in the Extended Phase Space

2.1. Equation of State

This section reviews the thermodynamic definition of a charged dilaton black hole solution in the extended phase space. The action of Einstein-Maxwell-dilation gravity in a four-dimensional spacetime is given by [25,37,48,57,58]

$$\begin{array}{c}\hfill I=\frac{1}{16\pi }\int d^{4}x\sqrt{-g}\left[R-2{\left(\nabla \varphi \right)}^2-2\Lambda e^{2\alpha \varphi }-\frac{2{\alpha }^2}{b^2\left({\alpha }^2-1\right)}{e}^{2\varphi /\alpha }-e^{-2\alpha \varphi }{F}_{\mu \nu }{F}^{\mu \nu }\right],\end{array}$$

(1)

where R is the Ricci scalar, $\varphi$ is the scalar field of dilaton, $\Lambda$ is the cosmological constant, $F_{\mu \nu }$ is the electromagnetic tensor related to vector potential $A_{\nu }$. The coupling parameter between the Maxwell and dilaton fields is defined by $\alpha$, and b is an arbitrary positive constant.

Following the action (1), the metric element describing the spherical symmetric black hole solution can be obtained, i.e., [48,57]

$$\begin{array}{c}\hfill d{s}^2=-f\left(r\right)d{t}^2+\frac{1}{f\left(r\right)}d{r}^2+r^{2}R{\left(r\right)}^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2),\end{array}$$

(2)

with

$$\begin{array}{cc}\hfill f\left(r\right)=& -\frac{{\alpha }^2+1}{{\alpha }^2-1}{\left(\frac{b}{r}\right)}^{-2\gamma }-\frac{m}{r^{1-2\gamma }}-\frac{3{\left({\alpha }^2+1\right)}^2{r}^2}{l^2\left({\alpha }^2-3\right)}{\left(\frac{b}{r}\right)}^{2\gamma }\hfill \\ \hfill & +\frac{q^2\left({\alpha }^2+1\right)}{r^2}{\left(\frac{b}{r}\right)}^{-2\gamma },\hfill \\ \hfill \varphi \left(r\right)=& \frac{\alpha }{{\alpha }^2+1}ln\left(\frac{b}{r}\right),R\left(r\right)={\left(\frac{b}{r}\right)}^{\gamma },\hfill \end{array}$$

where q is the charge of the black hole, m is related to the ADM mass M of (2) via $m=2({\alpha }^2+1)b^{-2\gamma }M$, and $\gamma ={\alpha }^2/({\alpha }^2+1)$. Note that when $\alpha =0$, the metric (2) simplifies to the R-N AdS black hole.

In the extended thermodynamic phase space, the thermodynamic pressure P and the volume V, which is the conjugate quantity of P, take the form

$$\begin{array}{cc}\hfill P& =-\frac{(3+{\alpha }^2)b^{2\gamma }\Lambda }{8\pi (3-{\alpha }^2)r_{+}^{2\gamma }},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill V& =\frac{4\pi (1+{\alpha }^2)b^{2\gamma }}{3+{\alpha }^2}{r}_{+}^{\frac{3+{\alpha }^2}{1+{\alpha }^2}},\hfill \end{array}$$

(4)

where $r_{+}$ represents the largest root of $f\left(r\right)=0$, which is the event horizon radius of the black hole. Concerning the first law of thermodynamics, P and V can be written as

$$\begin{array}{c}\hfill P=-\frac{\Lambda }{8\pi },V=\frac{4\pi r_{+}^3}{3}.\end{array}$$

(5)

Identifying the specific volume v with the event horizon radius of the charged dilaton black hole with

$$\begin{array}{c}\hfill v=\frac{2(1+{\alpha }^2)(3-{\alpha }^2)}{(3+{\alpha }^2)}{r}_{+},\end{array}$$

(6)

then the equation of state $P(v,T)$ becomes

$$\begin{array}{cc}\hfill P\left(v,T\right)=& \frac{T}{v}+\frac{b^{-2\gamma }}{2^{2\gamma }\pi v^{2(1-\gamma )}{({\alpha }^2+1)}^{2(\gamma -2)}}\hfill \\ \hfill & \times \left[\frac{{\left(3-{\alpha }^2\right)}^{1-2\gamma }}{2\left({\alpha }^2-1\right){\left({\alpha }^2+1\right)}^2{\left(3+{\alpha }^2\right)}^{1-2\gamma }}+\frac{2q^2{\left(3+{\alpha }^2\right)}^{2\gamma -3}}{v^2{\left(3-{\alpha }^2\right)}^{2\gamma -3}}\right].\hfill \end{array}$$

(7)

It describes the relationship between the three thermodynamic parameters (namely pressure P, specific volume v and Hawking temperature T) when the matter is in thermodynamic equilibrium. Note that there is a critical temperature

$$T_{crit}=\left({\alpha }^2+1\right){\left(3+{\alpha }^2\right)}^{\frac{2\gamma -3}{2}}/\left[\pi b^{2\gamma }{q}^{1-2\gamma }(1-{\alpha }^2){({\alpha }^2+2)}^{\frac{1-2\gamma }{2}}\right],$$

where the second-order phase transition happens (see Refs. [25,48] for more details). For $T_0<T_{crit}$, $P\left(v,T\right)$ have the characteristics:

(i)

$P_v\left(v,T\right)<0$ when $v\in (0,v_{\alpha })\cup (v_{\beta },\infty )$;

(ii)

$P_v\left(v,T\right)>0$ when $v\in (v_{\alpha },v_{\beta })$;

(iii)

$P_v\left(v_{\alpha },T_0\right)=P_v\left(v_{\beta },T_0\right)=0$.

2.2. Equation of Dynamics

Combined with the equation of state of the charged dilaton black hole, the thermodynamic equation of the black hole flow is considered in the spinodal region, which is affected by the temporal-periodic perturbation. Thus, the absolute temperature T with a weak time-periodic fluctuation is written as

$$\begin{array}{c}\hfill T=T_0+ϵ\delta cos\left(\omega t\right)cosM,\end{array}$$

(8)

where $ϵ\left(0<ϵ\ll 1\right)$ is a thermal perturbation parameter, $\delta$ is the perturbation amplitude relative to the small viscosity, $\omega$ is frequency of the absolute temperature, and $T_0$ value will be smaller than $T_{crit}$ value.

According to [48,49,50], there is a charged dilaton black hole flow, which is thermoelastic, slightly viscous, and isotropic, passes along the Eulerian coordinate x-axis in a tube with a fixed volume and unit cross-section. Moreover, the position x of the black hole is a function of the time t and the mass M of the column of fluid of unit cross-section between a fluid particle and the reference fluid particle. Thus, $v(M,t)=\partial x(M,t)/\partial M$ and $u(M,t)=\partial x(M,t)/\partial t$ are the specific volume and the velocity, respectively. Then, x satisfies the following equation

$$\begin{array}{c}\hfill \frac{{\partial }^{2}x}{\partial t^2}=-\frac{\partial P\left(v,T\right)}{\partial M}+ϵ{\mu }_{0}s\frac{{\partial }^{2}u}{\partial M^2}-A{s}^2\frac{{\partial }^{3}v}{\partial M^3},\end{array}$$

(9)

where the expression of $P\left(v,T\right)$ is shown as (10), ${\mu }_0$ is a positive parameter related to the viscosity, s is a positive parameter, and A is a positive constant (see Ref. [48] for more details).

$$\begin{array}{cc}\hfill P(v,T)=& P\left(v_0,T_0\right)+P_v\left(v_0,T_0\right)\left(v-v_0\right)+P_T\left(v_0,T_0\right)\left(T-T_0\right)\hfill \\ \hfill & +P_{vT}\left(v_0,T_0\right)\left(v-v_0\right)\left(T-T_0\right)+\frac{P_{vvv}\left(v_0,T_0\right)}{3!}{\left(v-v_0\right)}^3\hfill \\ \hfill & +\frac{P_{vvT}\left(v_0,T_0\right)}{2}{\left(v-v_0\right)}^2\left(T-T_0\right)+\cdots ,\hfill \end{array}$$

(10)

with

$$\begin{array}{cc}\hfill P_T\left(v_0,T_0\right)=& \frac{1}{v_0},P_{vT}\left(v_0,T_0\right)=-\frac{1}{v_0^2},P_{vvT}\left(v_0,T_0\right)=\frac{2}{v_0^3},\hfill \\ \hfill P_v\left(v_0,T_0\right)=& -\frac{T_0}{v_0^2}+\frac{{\left[\left(3-{\alpha }^2\right)\left({\alpha }^2+1\right)\right]}^{\frac{1-{\alpha }^2}{{\alpha }^2+1}}}{\pi \left(1-{\alpha }^2\right)4^{\frac{{\alpha }^2}{{\alpha }^2+1}}{b}^{\frac{2{\alpha }^2}{{\alpha }^2+1}}{\left({\alpha }^2+3\right)}^{\frac{{\alpha }^2+3}{{\alpha }^2+1}}{v}_0^{\frac{3{\alpha }^2+5}{{\alpha }^2+1}}}\hfill \\ \hfill & \times \left[4\left({\alpha }^4+{\alpha }^2-2\right){\left({\alpha }^4-2{\alpha }^2-3\right)}^2{q}^2+{\left({\alpha }^2+3\right)}^2{v}_0^2\right],\hfill \\ \hfill P_{vvv}\left(v_0,T_0\right)=& -\frac{6T_0}{v_0^4}+\frac{2{\left(3-{\alpha }^2\right)}^{\frac{1-{\alpha }^2}{1+{\alpha }^2}}\left({\alpha }^2+2\right)}{\pi \left(1-{\alpha }^2\right)4^{\frac{{\alpha }^2}{{\alpha }^2+1}}{b}^{\frac{2{\alpha }^2}{{\alpha }^2+1}}{\left({\alpha }^2+1\right)}^{\frac{3{\alpha }^2+1}{{\alpha }^2+1}}{\left({\alpha }^2+3\right)}^{\frac{{\alpha }^2+3}{{\alpha }^2+1}}{v}_0^{\frac{5{\alpha }^2+7}{{\alpha }^2+1}}}\hfill \\ \hfill & \times \left[{\left({\alpha }^2+3\right)}^3{v}_0^2+4\left({\alpha }^2-1\right)\left(2{\alpha }^2+3\right)\left(3{\alpha }^2+5\right){\left({\alpha }^4-2{\alpha }^2-3\right)}^2{q}^2\right].\hfill \end{array}$$

where $v_0$ is the inflection point in the spinodal region, which satisfies ${\partial }^{2}P/{\partial }^{2}v=0$ at $v_0$.

Expand the functions $v\left(M,t\right)$ and $u\left(M,t\right)$ in Fourier series with respect to $M\in \left[0,2\pi \right]$ near $v_0$ as

$$\begin{array}{cc}\hfill v\left(M,t\right)& =v_0+x_1\left(t\right)cosM+x_2\left(t\right)cos2M+x_3\left(t\right)cos3M+\dots ,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill u\left(x,t\right)& =u_1\left(t\right)sinM+u_2\left(t\right)sin2M+u_3\left(t\right)sin3M+\dots ,\hfill \end{array}$$

(12)

where $x_i\left(t\right)$ and $u_i\left(t\right)$ are regarded as the position and velocity of the hydrodynamical modes, respectively (see [49,50,59,60] for details). Consider the first two modes, i.e., $x_n\left(t\right)=u_n\left(t\right)=0$ when $n\ge 3$, the Equation (9) become (omit $(v_0,T_0)$)

$$\begin{array}{cc}\hfill {\dot{x}}_1=& u_1,\hfill \\ \hfill {\dot{x}}_2=& u_2,\hfill \\ \hfill {\dot{u}}_1=& \left(P_v-A{s}^2\right)x_1+ϵ\left(P_T+\frac{P_{vT}}{2}{x}_2+\frac{3P_{vvT}}{8}{x}_1^2+\frac{P_{vvT}}{4}{x}_2^2\right)\delta cos\omega t\hfill \\ \hfill & +\frac{P_{vvv}}{8}\left(x_1^3+2x_1{x}_2^2\right)-ϵ{\mu }_{0}s{u}_1,\hfill \\ \hfill {\dot{u}}_2=& \left(P_v-4A{s}^2\right)x_2+ϵ\left(\frac{P_{vT}}{2}{x}_1+\frac{P_{vvT}}{2}{x}_1{x}_2\right)\delta cos\omega t\hfill \\ \hfill & +\frac{P_{vvv}}{8}\left(x_2^3+2x_1^2{x}_2\right)-4ϵ{\mu }_{0}s{u}_2.\hfill \end{array}$$

(13)

Now we transform the dynamical equations of the charged dilaton black hole (13) into the form of second-order differential equations

$$\begin{array}{cc}\hfill {\ddot{x}}_1+ϵ{\mu }_{0}s{\dot{x}}_1& +\left(A{s}^2-P_v\right)x_1-\frac{P_{vvv}}{8}{x}_1^3-\frac{P_{vvv}}{4}{x}_1{x}_2^2\hfill \\ \hfill & =ϵ\left(P_T+\frac{P_{vT}}{2}{x}_2+\frac{3P_{vvT}}{8}{x}_1^2+\frac{P_{vvT}}{4}{x}_2^2\right)\delta cos\omega t,\hfill \\ \hfill {\ddot{x}}_2+4ϵ{\mu }_{0}s{\dot{x}}_2& +\left(4A{s}^2-P_v\right)x_2-\frac{P_{vvv}}{8}{x}_2^3-\frac{P_{vvv}}{4}{x}_1^2{x}_2\hfill \\ \hfill & =ϵ\left(P_{vT}{x}_1+\frac{P_{vvT}}{2}{x}_1{x}_2\right)\delta cos\omega t.\hfill \end{array}$$

(14)

3. Two-Timing Scale Method Solution

In this section, the asymptotic analytical solutions of the charged dilaton black hole are studied by using the two-timing scale method [56]. Suppose that the approximate solutions of the system (14) take the form

$$\begin{array}{cc}\hfill x_1\left(t,ϵ\right)& =ϵx_{11}\left(T_1,T_2\right)+{ϵ}^2{x}_{12}\left(T_1,T_2\right),\hfill \\ \hfill x_2\left(t,ϵ\right)& =ϵx_{21}\left(T_1,T_2\right)+{ϵ}^2{x}_{22}\left(T_1,T_2\right),\hfill \end{array}$$

(15)

where the two-timing scales are defined as $T_1=t$ and $T_2=ϵt$, respectively.

Denote $\begin{array}{cc}\hfill {\dot{x}}_{ij}& =\frac{\partial x_{ij}}{\partial T_1}\frac{\partial T_1}{\partial t}+\frac{\partial x_{ij}}{\partial T_2}\frac{\partial T_2}{\partial t}\hfill \\ \hfill & =\frac{\partial x_{ij}}{\partial T_1}+ϵ\frac{\partial x_{ij}}{\partial T_2}\hfill \end{array}$

$$\begin{array}{cc}\hfill & \triangleq {\partial }_{T_1}{x}_{ij}+ϵ{\partial }_{T_2}{x}_{ij},\hfill \\ \hfill {\ddot{x}}_{ij}& =\frac{\partial }{\partial t}\left(\frac{\partial x_{ij}}{\partial T_1}+ϵ\frac{\partial x_{ij}}{\partial T_2}\right)\hfill \\ \hfill & =\frac{{\partial }^2{x}_{ij}}{\partial T_1^2}\frac{\partial T_1}{\partial t}+\frac{{\partial }^2{x}_{ij}}{\partial T_1{T}_2}\frac{\partial T_2}{\partial t}+ϵ\frac{{\partial }^2{x}_{ij}}{\partial T_2{T}_1}\frac{\partial T_1}{\partial t}+ϵ\frac{{\partial }^2{x}_{ij}}{\partial T_2^2}\frac{\partial T_2}{\partial t}\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & \triangleq {\partial }_{T_1{T}_1}{x}_{ij}+2ϵ{\partial }_{T_1{T}_2}{x}_{ij}+{ϵ}^2{\partial }_{T_2{T}_2}{x}_{ij},\hfill \end{array}$$

(17)

where $i,j=1,2$. Then, substituting Equations (15)–(17) into Equation (14) and comparing the coefficients of powers of $ϵ$, one obtains

$$\begin{array}{cc}\hfill {ϵ}^1:& {\partial }_{T_1{T}_1}{x}_{11}+\left(A{s}^2-P_v\right)x_{11}-P_T\delta cos\omega T_1=0,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\partial }_{T_1{T}_1}{x}_{21}+\left(4A{s}^2-P_v\right)x_{21}=0,\hfill \\ \hfill {ϵ}^2:& {\partial }_{T_1{T}_1}{x}_{12}+\left(A{s}^2-P_v\right)x_{12}+2{\partial }_{T_1{T}_2}{x}_{11}+{\mu }_{0}s{\partial }_{T_1}{x}_{11}-\frac{P_{vT}}{2}{x}_{21}\delta cos\omega T_1=0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & {\partial }_{T_1{T}_1}{x}_{22}+\left(4A{s}^2-P_v\right)x_{22}+2{\partial }_{T_1{T}_2}{x}_{21}+4{\mu }_{0}s{\partial }_{T_1}{x}_{21}-P_{vT}{x}_{11}\delta cos\omega T_1=0,\hfill \\ \hfill {ϵ}^3:& {\partial }_{T_1{T}_1}{x}_{13}+\left(A{s}^2-P_v\right)x_{13}+{\partial }_{T_2{T}_2}{x}_{11}+2{\partial }_{T_1{T}_2}{x}_{12}+{\mu }_{0}s\left({\partial }_{T_2}{x}_{11}+{\partial }_{T_1}{x}_{12}\right)\hfill \\ \hfill & -\left(\frac{P_{vT}}{2}{x}_{22}+\frac{3P_{vvT}}{8}{x}_{11}^2+\frac{P_{vvT}}{4}{x}_{21}^2\right)\delta cos\omega T_1-\frac{P_{vvv}}{8}{x}_{11}^3-\frac{P_{vvv}}{4}{x}_{11}{x}_{21}^2=0,\hfill \\ \hfill & {\partial }_{T_1{T}_1}{x}_{23}+\left(4A{s}^2-P_v\right)x_{23}+{\partial }_{T_2{T}_2}{x}_{21}+2{\partial }_{T_1{T}_2}{x}_{22}+4{\mu }_{0}s\left({\partial }_{T_2}{x}_{21}+{\partial }_{T_1}{x}_{22}\right)\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & -\left(P_{vT}{x}_{12}+\frac{P_{vvT}}{2}{x}_{11}{x}_{21}\right)\delta cos\omega T_1-\frac{P_{vvv}}{8}{x}_{21}^3-\frac{P_{vvv}}{4}{x}_{11}^2{x}_{21}=0.\hfill \end{array}$$

(20)

Note that the works of temporal and spatial chaos of system (14) had been done in Ref. [59] when $\frac{P_v}{4A}<s^2<\frac{P_v}{A}$. We now consider the case $s^2\ge \frac{P_v}{A}$, i.e., $A{s}^2-P_v\ge 0$, and take the case of non-resonance into consideration, i.e., ${\omega }^2\ne {\lambda }_i^2\left(i=1,2\right)$ and $\mid {\lambda }_1\pm {\lambda }_2\mid \ne \omega$.

Let ${\lambda }_1=\sqrt{A{s}^2-P_v}$, ${\lambda }_2=\sqrt{4A{s}^2-P_v}$. Then, Equation (18) have the general solutions

$$\begin{array}{cc}\hfill x_{11}\left(T_1,T_2\right)=& G_1\left(T_2\right)e^{i{\lambda }_1{T}_1}+\overline{G_1\left(T_2\right)}{e}^{-i{\lambda }_1{T}_1}-\frac{P_T\delta }{2\left({\omega }^2-{\lambda }_1^2\right)}\left(e^{i\omega T_1}+e^{-i\omega T_1}\right),\hfill \\ \hfill x_{21}\left(T_1,T_2\right)=& G_2\left(T_2\right)e^{i{\lambda }_2{T}_1}+\overline{G_2\left(T_2\right)}{e}^{-i{\lambda }_2{T}_1},\hfill \end{array}$$

(21)

where $G_i\left(i=1,2\right)$ is a function of the time-scale $T_2$ and ${\overline{G}}_i$ denotes the conjugate quantity of $G_i$. Substituting Equation (21) into Equation (19), yields

$$\begin{array}{cc}\hfill {\partial }_{T_1{T}_1}{x}_{12}+{\lambda }_1^2{x}_{12}=& -2i{\lambda }_1{e}^{i{\lambda }_1{T}_1}{\partial }_{T_2}{G}_1+2i{\lambda }_1{e}^{-i{\lambda }_1{T}_1}{\partial }_{T_2}{\overline{G}}_1-{\mu }_{0}s\left[i{\lambda }_1{G}_1{e}^{i{\lambda }_1{T}_1}\right.\hfill \\ \hfill & \left.-\frac{P_T\delta i\omega }{2\left({\omega }^2-{\lambda }_1^2\right)}{e}^{i\omega T_1}-i{\lambda }_1{\overline{G}}_1{e}^{-i{\lambda }_1{T}_1}-\frac{P_T\delta i\omega }{2\left({\omega }^2-{\lambda }_1^2\right)}{e}^{-i\omega T_1}\right]\hfill \\ \hfill & +\frac{P_{vT}\delta }{4}\left[G_2{e}^{i\left({\lambda }_2+\omega \right)T_1}+G_2{e}^{i\left({\lambda }_2-\omega \right)T_1}+{\overline{G}}_2{e}^{i\left(\omega -{\lambda }_2\right)T_1}\right.\hfill \\ \hfill & \left.+{\overline{G}}_2{e}^{-i\left({\lambda }_2+\omega \right)T_1}\right],\hfill \\ \hfill {\partial }_{T_1{T}_1}{x}_{22}+{\lambda }_2^2{x}_{22}=& -2i{\lambda }_2{e}^{i{\lambda }_2{T}_1}{\partial }_{T_2}{G}_2+2i{\lambda }_2{e}^{-i{\lambda }_2{T}_1}{\partial }_{T_2}{\overline{G}}_2-4{\mu }_{0}s\left(i{\lambda }_2{e}^{i{\lambda }_2{T}_1}{G}_2\right.\hfill \\ \hfill & \left.-i{\lambda }_2{e}^{-i{\lambda }_2{T}_1}{\overline{G}}_2\right)+\frac{P_{vT}\delta }{2}\left[G_1{e}^{i\left({\lambda }_1+\omega \right)T_1}+G_1{e}^{i\left({\lambda }_1-\omega \right)T_1}\right.\hfill \\ \hfill & -\frac{P_T\delta }{2\left({\omega }^2-{\lambda }_1^2\right)}{e}^{2i\omega T_1}-\frac{P_T\delta }{2\left({\omega }^2-{\lambda }_1^2\right)}+{\overline{G}}_1{e}^{i\left(-{\lambda }_1+\omega \right)T_1}\hfill \\ \hfill & \left.+{\overline{G}}_1{e}^{-i\left({\lambda }_1+\omega \right)T_1}-\frac{P_T\delta }{2\left({\omega }^2-{\lambda }_1^2\right)}-\frac{P_T\delta }{2\left({\omega }^2-{\lambda }_1^2\right)}{e}^{-2i\omega T_1}\right].\hfill \end{array}$$

(22)

To eliminate the secular terms of the above equations, let

$$\begin{array}{c}\hfill -2i{\lambda }_1{\partial }_{T_2}{G}_1-{\mu }_{0}si{\lambda }_1{G}_1=0,\\ \hfill -2i{\lambda }_2{\partial }_{T_2}{G}_2-4{\mu }_{0}si{\lambda }_2{G}_2=0.\end{array}$$

Then the coefficients $G_1$ and $G_2$ are

$$\begin{array}{cc}\hfill G_1\left(T_2\right)& =c_1{e}^{-\frac{{\mu }_{0}s}{2}{T}_2},\hfill \\ \hfill G_2\left(T_2\right)& =c_2{e}^{-2{\mu }_{0}s{T}_2},\hfill \end{array}$$

where $c_1$ and $c_2$ are integration constants. Thus, the special solutions of Equation (18) become

$$\begin{array}{cc}\hfill x_{11}\left(T_1,T_2\right)& =2c_1{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}cos{\lambda }_1{T}_1-\frac{P_T\delta }{{\omega }^2-{\lambda }_1^2}cos\omega T_1,\hfill \\ \hfill x_{22}\left(T_1,T_2\right)& =2c_2{e}^{-2{\mu }_{0}s{T}_2}cos{\lambda }_2{T}_1.\hfill \end{array}$$

(23)

Therefore, Equation (22) are reduced to

$$\begin{array}{cc}\hfill {\partial }_{T_1{T}_1}{x}_{12}+{\lambda }_1^2{x}_{12}=& \frac{{\mu }_{0}s{P}_T\delta i\omega }{2\left({\omega }^2-{\lambda }_1^2\right)}\left(e^{i\omega T_1}-e^{-i\omega T_1}\right)+\frac{c_2{P}_{vT}\delta e^{-2{\mu }_{0}s{T}_2}}{4}\times \hfill \\ \hfill & \left[e^{i\left({\lambda }_2+\omega \right)T_1}+e^{i\left({\lambda }_2-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_2\right)T_1}+e^{-i\left({\lambda }_2+\omega \right)T_1}\right],\hfill \\ \hfill {\partial }_{T_1{T}_1}{x}_{22}+{\lambda }_2^2{x}_{22}=& \frac{c_1{P}_{vT}\delta e^{\frac{-{\mu }_{0}s}{2}{T}_2}}{2}\left[e^{i\left({\lambda }_1+\omega \right)T_1}+e^{i\left({\lambda }_1-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_1\right)T_1}\right.\hfill \\ \hfill & \left.+e^{-i\left({\lambda }_1+\omega \right)T_1}\right]-\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right)}\left(e^{2i\omega T_1}+e^{-2i\omega T_1}+2\right).\hfill \end{array}$$

(24)

Similarly, suppose that the solutions of Equation (24) are

$$\begin{array}{cc}\hfill x_{12}\left(T_1,T_2\right)=& K_1\left(T_2\right)e^{i{\lambda }_1{T}_1}+\overline{K_1\left(T_2\right)}{e}^{-i{\lambda }_1{T}_1}-\frac{{\mu }_{0}s{P}_T\delta i\omega }{2{\left({\lambda }_1^2-{\omega }^2\right)}^2}\left(e^{i\omega T_1}-e^{-i\omega T_1}\right)\hfill \\ \hfill & +\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{4\left[{\lambda }_1^2-{\left({\lambda }_2+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_2+\omega \right)T_1}+e^{-i\left({\lambda }_2+\omega \right)T_1}\right]\hfill \\ \hfill & +\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{4\left[{\lambda }_1^2-{\left({\lambda }_2-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_2-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_2\right)T_1}\right],\hfill \\ \hfill x_{22}\left(T_1,T_2\right)=& K_2\left(T_2\right)e^{i{\lambda }_2{T}_1}+\overline{K_2\left(T_2\right)}{e}^{-i{\lambda }_2{T}_1}+\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1+\omega \right)T_1}\right.\hfill \\ \hfill & \left.+e^{-i\left({\lambda }_1+\omega \right)T_1}\right]+\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_1\right)T_1}\right]\hfill \\ \hfill & -\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right)\left({\lambda }_2^2-4{\omega }^2\right)}\left(e^{2i\omega T_1}+e^{-2i\omega T_1}\right)-\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right){\lambda }_2^2}.\hfill \end{array}$$

(25)

Substituting Equations (23) and (25) into Equation (20), it follows

$$\begin{array}{cc}\hfill & {\partial }_{T_1{T}_1}{x}_{13}+{\lambda }_1^2{x}_{13}=-\frac{c_1{\mu }_0^2{s}^2{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4}\left(e^{i{\lambda }_1{T}_1}+e^{-i{\lambda }_1{T}_1}\right)-2i{\lambda }_1{\partial }_{T_2}{K}_1{e}^{i{\lambda }_1{T}_1}\hfill \\ \hfill & +2i{\lambda }_1{\partial }_{T_2}{\overline{K}}_1{e}^{-i\lambda T_1}+\frac{c_2{P}_T\delta {\mu }_{0}si\left({\lambda }_2+\omega \right)e^{-2{\mu }_{0}s{T}_2}}{{\lambda }_1^2-{\left({\lambda }_2+\omega \right)}^2}\left[e^{i\left({\lambda }_2+\omega \right)T_1}-e^{-i\left({\lambda }_2+\omega \right)T_1}\right]\hfill \\ \hfill & +\frac{c_2{P}_T\delta {\mu }_{0}si\left({\lambda }_2-\omega \right)e^{-2{\mu }_{0}s{T}_2}}{{\lambda }_1^2-{\left({\lambda }_2-\omega \right)}^2}\left[e^{i\left({\lambda }_2-\omega \right)T_1}-e^{i\left(\omega -{\lambda }_2\right)T_1}\right]-{\mu }_{0}s\left\{i{\lambda }_1{K}_1{e}^{i{\lambda }_1{T}_1}\right.\hfill \\ \hfill & -i{\lambda }_1{\overline{K}}_1{e}^{-i{\lambda }_1{T}_1}-\frac{c_1{\mu }_{0}s{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}}{2}\left(e^{i{\lambda }_1{T}_1}+e^{-i{\lambda }_1{T}_1}\right)+\frac{{\mu }_{0}s{P}_T\delta {\omega }^2}{2{\left({\lambda }_1^2-{\omega }^2\right)}^2}\times \hfill \\ \hfill & \left(e^{i\omega T_1}+e^{-i\omega T_1}\right)+\frac{c_2{P}_T\delta i\left({\lambda }_2+\omega \right)e^{-2{\mu }_{0}s{T}_2}}{4\left[{\lambda }_1^2-{\left({\lambda }_2+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_2+\omega \right)T_1}-e^{-i\left({\lambda }_2+\omega \right)T_1}\right]\hfill \\ \hfill & \left.+\frac{c_2{P}_T\delta i\left({\lambda }_2-\omega \right)e^{-2{\mu }_{0}s{T}_2}}{4\left[{\lambda }_1^2-{\left({\lambda }_2-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_2-\omega \right)T_1}-e^{i\left(\omega -{\lambda }_2\right)T_1}\right]\right\}\hfill \\ \hfill & +\frac{P_{vT}\delta }{4}\left(e^{i\omega T_1}+e^{-i\omega T_1}\right)\left\{K_2{e}^{i{\lambda }_2{T}_1}+{\overline{K}}_2{e}^{-i{\lambda }_2{T}_1}-\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right){\lambda }_2^2}\right.\hfill \\ \hfill & -\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right)\left({\lambda }_2^2-{\omega }^2\right)}\left(e^{2i\omega T_1}+e^{-2i\omega T_1}\right)\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_1\right)T_1}\right]\hfill \\ \hfill & \left.+\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1+\omega \right)T_1}+e^{-i\left({\lambda }_1+\omega \right)T_1}\right]\right\}\hfill \\ \hfill & +\frac{3P_{vvT}\delta }{16}\left(e^{i\omega T_1}+e^{-i\omega T_1}\right)\left\{c_1^2{e}^{-{\mu }_{0}s{T}_2}\left(e^{2i{\lambda }_1{T}_1}+e^{-2i{\lambda }_1{T}_1}+2\right)\right.\hfill \\ \hfill & -\frac{c_1{P}_T\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{{\omega }^2-{\lambda }_1^2}\left[e^{i\left({\lambda }_1+\omega \right)T_1}+e^{i\left({\lambda }_1-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_1\right)T_1}+e^{-i\left({\lambda }_1+\omega \right)T_1}\right]+\hfill \\ \hfill & \left.\frac{P_T^2{\delta }^2}{4{\left({\omega }^2-{\lambda }_1^2\right)}^2}\left(e^{2i\omega T_1}+e^{-2i\omega T_1}+2\right)\right\}+\frac{c_2^2{P}_{vvT}\delta e^{-4{\mu }_{0}s{T}_2}}{8}\left(e^{i\omega T_1}+e^{-i\omega T_1}\right)\times \hfill \\ \hfill & \left(e^{2i{\lambda }_2{T}_1}+e^{-2i{\lambda }_2{T}_1}+2\right)+\frac{P_{vvv}}{8}{x}_{11}^3-\frac{P_{vvv}}{4}{x}_{11}{x}_{21}^2,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\partial }_{T_1{T}_1}{x}_{23}+{\lambda }_2^2{x}_{23}=-4c_2{\mu }_0^2{s}^2{e}^{-2{\mu }_{0}s{T}_2}\left(e^{i{\lambda }_2{T}_1}+e^{-i{\lambda }_2{T}_1}\right)-2i{\lambda }_2{\partial }_{T_2}{K}_2{e}^{i{\lambda }_2{T}_1}\hfill \\ \hfill & +2i{\lambda }_2{\partial }_{T_2}{\overline{K}}_2{e}^{-i{\lambda }_2{T}_1}+\frac{c_1{P}_{vT}\delta {\mu }_{0}si\left({\lambda }_1+\omega \right)e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1+\omega \right)T_1}-e^{-i\left({\lambda }_1+\omega \right)T_1}\right]\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta {\mu }_{0}si\left({\lambda }_1-\omega \right)e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1-\omega \right)T_1}-e^{i\left(\omega -{\lambda }_1\right)T_1}\right]\hfill \\ \hfill & -4{\mu }_{0}s\left\{i{\lambda }_2{K}_2{e}^{i{\lambda }_2{T}_1}-i{\lambda }_2{\overline{K}}_2{e}^{-i{\lambda }_2{T}_1}-2c_2{\mu }_{0}s{e}^{-2{\mu }_{0}s{T}_2}\left(e^{i{\lambda }_2{T}_1}+e^{-i{\lambda }_2{T}_1}\right)\right.\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta i\left({\lambda }_1+\omega \right)e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1+\omega \right)T_1}-e^{-i\left({\lambda }_1+\omega \right)T_1}\right]\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta i\left({\lambda }_1-\omega \right)e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{4\left[{\lambda }_2^2-{\left({\lambda }_1-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_1-\omega \right)T_1}-e^{-i\left(\omega -{\lambda }_1\right)T_1}\right]\hfill \\ \hfill & \left.-\frac{P_{vT}{P}_T{\delta }^{2}i\omega }{2\left({\omega }^2-{\lambda }_1^2\right)\left({\lambda }_2^2-4{\omega }^2\right)}\left(e^{2i\omega T_1}-e^{-2i\omega T_1}\right)\right\}+\frac{P_{vT}\delta }{2}\left(e^{i\omega T_1}+e^{-i\omega T_1}\right)\times \hfill \\ \hfill & \left\{K_1{e}^{i{\lambda }_1{T}_1}+{\overline{K}}_1{e}^{-i{\lambda }_1{T}_1}+\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{4\left[{\lambda }_1^2-{\left({\lambda }_2+\omega \right)}^2\right]}\left[e^{i\left({\lambda }_2+\omega \right)T_1}+e^{-i\left({\lambda }_2+\omega \right)T_1}\right]\right.\hfill \\ \hfill & \left.+\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{4\left[{\lambda }_1^2-{\left({\lambda }_2-\omega \right)}^2\right]}\left[e^{i\left({\lambda }_2-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_2\right)T_1}\right]-\frac{{\mu }_{0}s{P}_T\delta i\omega }{2{\left({\lambda }_1^2-{\omega }^2\right)}^2}\left(e^{i\omega T_1}-e^{-i\omega T_1}\right)\right\}\hfill \\ \hfill & +\frac{c_2{P}_{vvT}\delta e^{-2{\mu }_{0}s{T}_2}}{4}\left(e^{i\left({\lambda }_2+\omega \right)T_1}+e^{i\left({\lambda }_2-\omega \right)T_1}+e^{i\left(\omega -{\lambda }_2\right)T_1}+e^{-i\left({\lambda }_2+\omega \right)T_1}\right)\times \hfill \\ \hfill & \left[c_1{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}\left(e^{i{\lambda }_1{T}_1}+e^{-i{\lambda }_1{T}_1}\right)-\frac{P_v\delta }{2\left({\omega }^2-{\lambda }_1^2\right)}\left(e^{i\omega T_1}+e^{-i\omega T_1}\right)\right]\hfill \\ \hfill & +\frac{c_2^3{P}_{vvv}{e}^{-6{\mu }_{0}s{T}_2}}{8}\left(e^{3i{\lambda }_2{T}_1}+3e^{i{\lambda }_2{T}_1}+3e^{-i{\lambda }_2{T}_1}+e^{-3i{\lambda }_2{T}_1}\right)+\frac{P_{vvv}}{4}{x}_{11}^2{x}_{21},\hfill \end{array}$$

(26)

where the complex expressions of $x_{11}^3$, $x_{11}{x}_{21}^2$ and $x_{11}^2{x}_{21}$ are shown in Appendix A.

To eliminate the secular terms of the Equation (26), we take

$$\begin{array}{cc}\hfill K_1\left(T_2\right)& =c_3{e}^{-\frac{{\mu }_{0}s}{2}{T}_2},\hfill \\ \hfill K_2\left(T_2\right)& =c_4{e}^{-2{\mu }_{0}s{T}_2},\hfill \end{array}$$

where $c_3$ and $c_4$ are integration constants. Thus, it leads to the special solutions of Equation (19) as

$$\begin{array}{cc}\hfill x_{12}\left(T_1,T_2\right)=& 2c_3{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}cos{\lambda }_1{T}_1+\frac{{\mu }_{0}s{P}_T\delta \omega }{{\left({\lambda }_1^2-{\omega }^2\right)}^2}sin\omega T_1\hfill \\ \hfill & +\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{2\left[{\lambda }_1^2-{\left({\lambda }_2+\omega \right)}^2\right]}cos\left({\lambda }_2+\omega \right)T_1\hfill \\ \hfill & +\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{2\left[{\lambda }_1^2-{\left({\lambda }_2-\omega \right)}^2\right]}cos\left({\lambda }_2-\omega \right)T_1,\hfill \\ \hfill x_{22}\left(T_1,T_2\right)=& 2c_4{e}^{-2{\mu }_{0}s{T}_2}cos{\lambda }_2{T}_1-\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right){\lambda }_2^2}\hfill \\ \hfill & -\frac{P_{vT}{P}_T{\delta }^2}{2\left({\omega }^2-{\lambda }_1^2\right)\left({\lambda }_2^2-4{\omega }^2\right)}cos2\omega T_1\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{2\left[{\lambda }_2^2-{\left({\lambda }_1+\omega \right)}^2\right]}cos\left({\lambda }_1+\omega \right)T_1\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{2\left[{\lambda }_2^2-{\left({\lambda }_1-\omega \right)}^2\right]}cos\left({\lambda }_1-\omega \right)T_1.\hfill \end{array}$$

(27)

Submitting the Equations (23) and (27) into the Equation (15), the solutions of the original system (14) can be obtained as follows

$$\begin{array}{cc}\hfill x_1=& ϵ\left(2c_1{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}cos{\lambda }_1{T}_1-\frac{P_T\delta }{{\omega }^2-{\lambda }_1^2}cos\omega T_1\right)+{ϵ}^2\left\{2c_3{e}^{-\frac{{\mu }_{0}s}{2}{T}_2}cos{\lambda }_1{T}_1\right.\hfill \\ \hfill & +\frac{{\mu }_{0}s{P}_T\delta \omega }{{\left({\lambda }_1^2-{\omega }^2\right)}^2}sin\omega T_1+\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{2[{\lambda }_1^2-{\left({\lambda }_2+\omega \right)}^2]}cos\left({\lambda }_2+\omega \right)T_1\hfill \\ \hfill & \left.+\frac{c_2{P}_T\delta e^{-2{\mu }_{0}s{T}_2}}{2[{\lambda }_1^2-{\left({\lambda }_2-\omega \right)}^2]}cos\left({\lambda }_2-\omega \right)T_1\right\},\hfill \\ \hfill x_2=& ϵ\left(2c_2{e}^{-2{\mu }_{0}s{T}_2}cos{\lambda }_2{T}_1\right)+{ϵ}^2\left\{2c_4{e}^{-2{\mu }_{0}s{T}_2}cos{\lambda }_2{T}_1-\frac{P_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right){\lambda }_2^2}\right.\hfill \\ \hfill & +\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{2\left[{\lambda }_2^2-{\left({\lambda }_1+\omega \right)}^2\right]}cos\left({\lambda }_1+\omega \right)T_1-\frac{P_{vT}{P}_T{\delta }^2}{2\left({\omega }^2-{\lambda }_1^2\right)\left({\lambda }_2^2-4{\omega }^2\right)}cos2\omega T_1\hfill \\ \hfill & \left.+\frac{c_1{P}_{vT}\delta e^{-\frac{{\mu }_{0}s}{2}{T}_2}}{2\left[{\lambda }_2^2-{\left({\lambda }_1-\omega \right)}^2\right]}cos\left({\lambda }_1-\omega \right)T_1\right\}.\hfill \end{array}$$

(28)

4. Numerical Comparison

In order to verify the validity of the calculation results, a numerical comparison is carried out in this section. Consider the case $T_0=0.0315$ and $ϵ=0.001$, and consequently $v_0=3.652$. The initial value of the charged dilaton black hole system is selected as $\left[x_1\left(0\right),u_1\left(0\right),x_2\left(0\right),u_2\left(0\right)\right]=\left[0.02345,0,0.02021,0\right]$, and take the aforementioned integration constants $c_1=c_2=0.1$ and $c_3=c_4=10000$. As shown in Figure 1, the solution (28) obtained by the two-timing scale method is almost the same as the solution of the original system (14) (so-called “exact solution”) at each time. Meantime, these two solutions display the same trend as time evolves. In summary, these reveal that the two-timing scale method effectively describes this complex nonlinear charged dilaton black hole system, and the obtained asymptotic solutions are resultful with some parameter values.

Moreover, at the same values of the above parameter, the phase portrait of the asymptotic solution is plotted as shown in Figure 2. Combining Figure 1 with Figure 2, it can be found that there exists a quasi-periodic motion, which is worthy of further study in the next section.

In order to explore the final state of the black hole flow, the following equations are considered by neglecting the non-periodic terms in the Equation (28), which are denoted by the “asymptotic periodic solutions”

$$\begin{array}{cc}\hfill x_1=& -\frac{ϵP_T\delta }{{\omega }^2-{\lambda }_1^2}cos\omega T_1+\frac{{ϵ}^2{\mu }_{0}s{P}_T\delta \omega }{{\left({\lambda }_1^2-{\omega }^2\right)}^2}sin\omega T_1,\hfill \\ \hfill x_2=& -\frac{{ϵ}^2{P}_{vT}{P}_T{\delta }^2}{2\left({\omega }^2-{\lambda }_1^2\right)\left({\lambda }_2^2-4{\omega }^2\right)}cos2\omega T_1-\frac{{ϵ}^2{P}_{vT}{P}_T{\delta }^2}{4\left({\omega }^2-{\lambda }_1^2\right){\lambda }_2^2}.\hfill \end{array}$$

(29)

Take $c_1=c_2=c_3=c_4={10}^{-6}$, a comparison between the asymptotic solution and asymptotic periodic solution is shown in Figure 3. It is clear that the non-periodic terms in Equation (28) have barely any effect on the obtained solutions when the order of magnitude of integration constants is small.

5. Quasi-Periodic Behavior

Quasi-periodicity is a new type of long-term behavior, which is different from fixed point, homoclinic orbit, heteroclinic orbit, and periodic orbit. In deep space exploration, the Lissajous orbit and quasi-halo orbit near the Lagrangian points that have practical applications are also quasi-periodic. Mathematically, the frequencies in different directions are incommensurable, which implies that when the value of ${\lambda }_1/{\lambda }_2$ in Section 3 is an irrational number, the corresponding trajectory is said to be quasi-periodic. Then, as time evolves, the trajectory will never close into itself, which means that time-domain solutions will never be repeated. This is because any closed trajectory is bound to rotate an integer circle about ${\lambda }_1$ and ${\lambda }_2$, the frequency ratio must be a rational number. On the contrary, each trajectory will be dense as time flows.

Based on the results in Section 4, it is worthy of studying the qualitative behavior of the “exact solution’’ of the charged dilaton black hole system (14). The quasi-periodic behavior characterized by the black hole flow is further studied in this section with different values of the parameters (see

Table 1. Parameter values in the charged dilaton black hole flow.

Parameters	q	$\mathit{\alpha }$	b	A	s
Group (a)	1	0.01	0.019836	0.2	0.04
Group (b)	1	0.01	1	0.3703	0.04
Group (c)	1	0.01	1	0.4884	0.04
Group (d)	1.0319	0.01	1	0.2	0.04
Group (e)	1	0.3	1	0.2	0.8
Group (f)	1	1.732	10,000	0.2	0.04

) and initial values of the black hole system. Note that the denominator of $P_v$ contains the irrational number $\pi$; then, it is obvious that there are constants $T_0,v_0,q,\alpha$ and b to make $P_v$ an irrational number. Therefore,

$$\frac{{\lambda }_1}{{\lambda }_2}=\frac{\sqrt{A{s}^2-P_v}}{\sqrt{4A{s}^2-P_v}}=\sqrt{1-\frac{3}{4-\frac{P_v}{A{s}^2}}}$$

is an irrational number for any rational constant $A{s}^2(>P_v)$. Then, several quasi-periodic motions are found and shown in Figure 4 and Figure 5. For example, take $T_0=0.0315$, $v_0=3.652$, $ϵ=0.001$, ${\mu }_0=0.1$, $\omega =0.01$, $\delta =0.004$, $q=1$, $\alpha =0.01$, $b=0.019836$, $A=0.2$ and $s=0.04$, and the initial value of the system as $\left[x_1\left(0\right),u_1\left(0\right),x_2\left(0\right),u_2\left(0\right)\right]=\left[-0.03226,0,0.01,0\right]$. Then, the phase portrait shows a “reticular shape” in Figure 4a, which characterizes the quasi-periodic motion of the black hole flow. In addition, seven more quasi-periodic motions are shown below, which are similar to “V-shape”, “8-shape” and “pillow-shape” and so on.

Considering that the value of the thermal perturbation parameter $ϵ$ is 0.000001, the related phase portraits of the system corresponding to the aforementioned Group (b) and Group (c) are shown in Figure 5. The phase trajectories of the system tend to be stable as time goes on. Moreover, the system’s motion tends to be periodic, and the vibration amplitude tends to be constant simultaneously.

6. Discussion and Conclusions

In this paper, the qualitative and quantitative analysis of charged dilation black holes is investigated. The two-timing scale method is applied to analyze the black hole flow equation and construct the second-order asymptotic analytical solution. According to these solutions, a numerical comparison with different system parameters values is carried out. By doing so, it can be found that these asymptotic solutions effectively describe the dynamical behavior of the black hole flow for a long time. At the same time, the relevant phase portraits are drawn to show quasi-periodic motion.

According to the obtained quasi-periodic motion, the quasi-periodic behavior of the black hole flow in the spinodal region is further studied when the frequency of ${\lambda }_1/{\lambda }_2$ is an irrational number. Several quasi-periodic motions with different parameter values are found. It is worth mentioning that when the thermal parameter perturbation takes a small value, the phase portraits of the black hole system exhibit better stability than the large value, and the vibration amplitude tends to be constant simultaneously.

Furthermore, to understand the final evolution state of the thermal dynamics of the black hole flow, the periodic solution is found according to the constructed analytical solution. The numerical comparison of these two solutions shows that they agree well for the small integral constant. In other words, the smaller the integral constant, the greater the extent of the black hole flow’s motion tends to be periodic.