Application of Symmetric Explicit Symplectic Integrators in Non-Rotating Konoplya and Zhidenko Black Hole Spacetime

Abstract

In this study, we construct symmetric explicit symplectic schemes for the non-rotating Konoplya and Zhidenko black hole spacetime that effectively maintain the stability of energy errors and solve the tangent vectors from the equations of motion and the variational equations of the system. The fast Lyapunov indicators and Poincaré section are calculated to verify the effectiveness of the smaller alignment index. Meanwhile, different algorithms are used to separately calculate the equations of motion and variation equations, resulting in correspondingly smaller alignment indexes. The numerical results indicate that the smaller alignment index obtained by using a global symplectic algorithm is the fastest method for distinguishing between regular and chaotic cases. The smaller alignment index is used to study the effects of parameters on the dynamic transition from order to chaos. If initial conditions and other parameters are appropriately chosen, we observe that an increase in energy E or the deformation parameter $\eta$ can easily lead to chaos. Similarly, chaos easily occurs when the angular momentum L is small enough or the magnetic parameter Q stays within a suitable range. By varying the initial conditions of the particles, a distribution plot of the smaller alignment in the X–Z plane of the black hole is obtained. It is found that the particle orbits exhibit a remarkably rich structure. Researching the motion of charged particles around a black hole contributes to our understanding of the mechanisms behind black hole accretion and provides valuable insights into the initial formation process of an accretion disk.

1. Introduction

Symplectic integrators [1,2,3,4,5] are geometric integrators that ensure the long-term stability of energy errors. By constructing a symmetric symplectic algorithm, truncation errors of odd orders can be effectively eliminated. Symplectic algorithms can be categorized into explicit symplectic algorithms and implicit symplectic algorithms. Explicit symplectic integrators have been widely used in black hole spacetime. The constructions of explicit symplectic integrators in Schwarzschild black holes and Reissner–Nordström black holes are very simple because their Hamiltonians can be easily split into four and five independently integrable parts, respectively [6,7]. Then, Wang et al. [8] applied this method to a Hamiltonian for the description of charged particles moving around a Reissner–Nordström–(anti)–de Sitter black hole with an external magnetic field and successfully constructed second- and fourth-order explicit symplectic integrators by splitting that Hamiltonian into six analytically solvable pieces. They discovered that these integrators exhibit exceptional long-term behavior in preserving the boundedness of Hamiltonian errors, irrespective of whether the orbits are ordered or chaotic, as long as appropriate step sizes are employed. For Kerr spacetime, in order to utilize an explicit symplectic integrator, a time transformation function needs to be applied to the Hamiltonian of Kerr geometry [9]. On this basis, Huang et al. [10] successfully constructed time-transformed explicit symplectic schemes in a magnetized deformed Schwarzschild black hole spacetime, and these schemes had good performance in stabilizing energy errors without secular drift. Recently, Zhang et al. [11] successfully constructed a time-transformed explicit symplectic integrator for a Hamiltonian system that describes the movement of charged particles around a non-Schwarzschild black hole that is surrounded by an external magnetic field.

As a subject of continuous exploration [12,13,14], the phenomenon of chaos is gradually becoming known. Until now, although there has been a clear understanding of the chaos characteristics of Newtonian systems, the chaos characteristics of relativistic gravity systems are quite vague. For example, a double fixed-center problem is integrable in Newtonian mechanics, but it has strong chaotic behavior in the relativistic framework [15,16,17,18]. Therefore, the study of chaotic dynamics has been a hot topic in recent years [19,20,21,22,23,24,25,26,27,28]. In the study conducted by Gonçalo A. S. Dias and José P. S. Lemos [29], they successfully constructed thin-shell electrically charged wormholes in d-dimensional general relativity with a cosmological constant. The energy conditions of the static wormhole solutions were analyzed by using the Darmois–Israel formalism, and a linearized stability analysis was carried out. According to the study of A. M. Al Zahrani et al. [30], they discovered that the trajectories of charged particles moving around a weakly magnetized static nonrotating black hole have three different types of asymptotic behavior, and the escape of a particle to infinity exhibits features similar to those of a diffusion process. In the research of Cao et al. [31], it was found that the presence of a fourth constant of motion that is similar to the Carter constant leads to regular behavior of charged particles in moving around a Kerr–Newman black hole that is surrounded by cloud strings, quintessence, and an electromagnetic field. Stuchlík, Z. et al. [32] presented a comprehensive review on the impact of cosmic repulsion and external magnetic fields on the accretion disks surrounding rotating black holes, as well as the jets associated with these rotating configurations. Their study revealed that in ionized Keplerian disks, an external magnetic field combined with the chaotic scattering process crucially influences the fate of the disks, enabling protons and ions with energy reaching $E={10}^{22}$ eV to exist simultaneously in the vicinity of rotating supermassive black holes with $M\sim {10}^{10}{M}_{\odot }$ while immersed in a sufficiently strong magnetic field with $B\sim {10}^{4}G$.

As the study of chaos progresses, numerous methods have been developed to identify chaos. The most typical ways are the Poincaré section method, the Lyapunov index [33], the local Lyapunov index and its spectral distribution [34], the fast Lyapunove index [35,36], the smaller alignment index [37], and the generalized alignment index [38]. Each method has its own unique strengths and weaknesses. For the Poincaré section, its advantage is that it directly reflects the dynamic properties of the system, while its limitation is that it is only suited for studying conservative systems with two degrees of freedom. However, the Lyapunov index measures the intensity of chaos by studying how the orbital deviation vectors change over time, and it can reflect the average separation ratio of two adjacent orbitals over time. This method is applicable to systems of any dimension, but it often requires a long calculation time. On the other hand, the fast Lyapunov index, as the name suggests, can identify chaos faster than the traditional Lyapunov index can. The smaller alignment index, which can accurately identify the possible alignment of two multidimensional vectors, is a fast and effective method for detecting chaos. The generalized alignment index describes the linear independence of deviation vectors, and it is a natural extension of the smaller alignment index for more than two deviation vectors. However, its calculation is considerably complex.

In this study, we construct an efficient explicit symplectic algorithm in a non-rotating Konoplya and Zhidenko black hole and employ it to calculate some chaos indicators. We found the most effective approach and applied it to examine the dynamics of charged particles moving around the black hole. The rest of this study is structured as follows. In Section 2, we construct an explicit symplectic scheme ($S4$) of the Hamiltonian in the non-rotating Konoplya and Zhidenko black hole spacetime and an explicit scheme of the Hamilton variational equations. In Section 3, the effectiveness of the SALI is verified, and the influences of different algorithms on the effectiveness of the SALI are investigated. In Section 4, we examine the influence of the parameter $\eta$. Section 5 is devoted to conclusions and discussions.

2. Constructions of Explicit Symplectic Schemes

Due to the highly nonlinear nature of the Hamiltonian function in relativity, it is difficult to directly employ efficient energy-preserving and explicit symplectic algorithms. In this article, the Hamiltonian function is split into multiple sub-Hamiltonians, each of which either has an analytical solution or can be treated with an explicit symplectic algorithm due to the separability of momentum and coordinates. A second-order explicit symplectic algorithm, $S2$, is constructed for the total Hamiltonian, which can further lead to a fourth-order explicit symplectic algorithm. This algorithm preserves both energy and the symplectic structure while improving computational efficiency. Although $RK8$ does not preserve energy and the symplectic structure and can accumulate computational errors during its long-term evolution, it offers high precision over short time intervals. Therefore, $RK8$ is commonly used as a universal accuracy benchmark for constructing geometric integration methods within a certain time range. In this article, $RK8$ is also adopted as the standard of accuracy for the algorithm.

2.1. The Konoplya and Zhidenko Black Hole

The metric of the rotating Konoplya and Zhidenko black hole is written as [39]

$$\begin{array}{cc}\hfill d{s}^2=-\left(1-\frac{2M{r}^2+\eta }{r\Sigma }\right)d{t}^2+& \frac{\Sigma }{\Delta }d{r}^2+\Sigma d{\theta }^2+si{n}^2\theta \left(r^2+a^2+\frac{(2M{r}^2+\eta )a^{2}si{n}^2\theta }{r\Sigma }\right)d{\varphi }^2\hfill \\ \hfill -\frac{2\left(2M{r}^2+\eta \right)asi{n}^2\theta }{r\Sigma }dtd\varphi .\end{array}$$

(1)

Note that $\Sigma =r^2+a^{2}co{s}^2\theta$, $\Delta =a^2+r^2-2Mr-\frac{\eta }{r}$. Here, the speed of light c and the constant of gravity G are geometric units: $c=G=1$. The parameter $\eta$ is used to measure deviations from the Kerr metric, while a represents the rotation parameter. Additionally, M denotes the mass of the black hole.

Roman Konoplya and Alexander Zhidenko obtained this metric by introducing a static deformation, $M\to M+\frac{\eta }{2r^2}$, into a Kerr spacetime. This deformation modifies the relationship between the black hole’s mass and the position of the event horizon, but preserves the asymptotic properties of the Kerr spacetime. This non-negligible deformation of the Kerr spacetime can result in similar frequencies of black hole ringing in the gravitational-wave signals detected by LIGO and VIRGO [40]. This means that at the existing level of experimental precision, there remains some possibility for alternative theories of gravity.

We let the mass M satisfy $M=1$; thus, dimensionless operations of $t\to tM$, $r\to rM$, and $\tau \to \tau M$ are performed. When $\eta =0$, this metric becomes the Kerr metric, and if we set the rotation parameter a to zero, the non-rotating form can be obtained, which gives:

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

(2)

where

$$f\left(r\right)=1-\frac{2}{r}-\frac{\eta }{r^3}.$$

Consider the non-rotating form surrounded by an external uniform magnetic field; the dimensionless four-vector potential is [41]

$$A_{\mu }=\left(0,0,0,\frac{1}{2}B{r}^{2}si{n}^2\theta \right),$$

(3)

where B is the strength of the magnetic field. A test particle moving around a black hole with charge q is described by the following Hamiltonian:

$$H=\frac{1}{2}{g}^{\mu \nu }\left(p_{\mu }-q{A}_{\mu }\right)\left(p_{\nu }-q{A}_{\nu }\right).$$

(4)

Here, $p_{\mu }$ is the generalized momentum, which is defined as

$$p_{\mu }=g_{\mu \nu }{\dot{x}}^{\nu }+q{A}_{\mu }.$$

(5)

In this equation, ${\dot{x}}^{\nu }$ is the four-velocity, and it is the derivative of the coordinate $x^{\nu }$ with respect to the proper time $\tau$. From the Hamiltonian equations ${\dot{p}}_{\mu }=-\partial H/\partial {\dot{x}}^{\mu }$, we can deduce that ${\dot{p}}_t={\dot{p}}_{\varphi }=0$. Consequently, we have two constant momentum components:

$$p_t=g_{tt}\dot{t}=-f\left(r\right)\dot{t}=-E,$$

(6)

$$p_{\varphi }=g_{\varphi \varphi }\dot{\varphi }+q{A}_{\varphi }=r^{2}si{n}^2\theta \dot{\varphi }+\frac{1}{2}qB{r}^{2}si{n}^2\theta =L.$$

(7)

In this way, the Hamiltonian can be rewritten as

$$H=\frac{1}{2}\left\{-\frac{E^2}{f\left(r\right)}+f\left(r\right)p_r^2+\frac{p_{\theta }^2}{r^2}+\frac{1}{r^{2}si{n}^2\theta }{\left[L-\frac{Q}{2}{r}^{2}si{n}^2\theta \right]}^2\right\},$$

(8)

where $Q=qB$, and, due to the resting mass of the charged particle in the time-like spacetime, the Hamiltonian is a conserved quantity, which is often given by

$$H=-\frac{1}{2}.$$

(9)

Here, the dimensionless operations for the related qualities are $L\to mML$, $B\to B/M$, $q\to mq$, $E\to mE$, $p_r\to m{p}_r$, $p_{\theta }\to mM{p}_{\theta }$, and $H\to m^{2}H$, where m stands for the mass of the particle.

2.2. Explicit Symplectic Methods for Hamiltonian Equations

Following the methods introduced by Wang et al. [6,7,8], the Hamiltonian (8) can be split into five parts:

$$H=H_1+H_2+H_3+H_4+H_5,$$

(10)

where these sub-Hamiltonians are

$$H_1=-\frac{E^2}{2f\left(r\right)}+\frac{1}{2r^{2}si{n}^2\theta }{\left(L-\frac{Q}{2}{r}^{2}si{n}^2\theta \right)}^2,$$

(11)

$$H_2=\frac{p_r^2}{2},$$

(12)

$$H_3=-\frac{p_r^2}{r},$$

(13)

$$H_4=-\frac{\eta p_r^2}{2r^3},$$

(14)

$$H_5=\frac{p_{\theta }^2}{2r^2}.$$

(15)

The equation of motion for the sub-Hamiltonian $H_1$ is $dr/d\tau =d\theta /d\tau =0$, and

$$\begin{array}{c}\hfill \frac{d{p}_r}{d\tau }=-\frac{\partial H_1}{\partial r}=p_r^{*}(r,\theta ),\\ \hfill \frac{d{p}_{\theta }}{d\tau }=-\frac{\partial H_1}{\partial \theta }=p_{\theta }^{*}(r,\theta ).\end{array}$$

(16)

Then, we can obtain the time-dependent analytical solutions of $H_1$:

$$\begin{array}{c}\hfill {\psi }_{\tau }^{H_1}:p_r\left(\tau \right)=p_{r_0}+\tau p_r^{*}(r,\theta ),\\ \hfill p_{\theta }\left(\tau \right)=p_{{\theta }_0}+\tau p_{\theta }^{*}(r,\theta ).\end{array}$$

(17)

The other sub-Hamiltonians also have the following equations of motion:

$$\begin{array}{ccc}\hfill H_2:\frac{\mathrm{d}r}{\mathrm{d}\tau }& =& p_r,\frac{\mathrm{d}{p}_r}{\mathrm{d}\tau }=0,\hfill \\ \hfill H_3:\frac{\mathrm{d}r}{\mathrm{d}\tau }& =& -\frac{p_r^2}{r^2},\frac{\mathrm{d}{p}_r}{\mathrm{d}\tau }=-\frac{2p_r}{r},\hfill \\ \hfill H_4:\frac{\mathrm{d}r}{\mathrm{d}\tau }& =& -\frac{\eta p_r}{r^3},\frac{\mathrm{d}{p}_r}{\mathrm{d}\tau }=-\frac{3\eta p_r^2}{2r^4},\hfill \\ \hfill H_5:\frac{\mathrm{d}\theta }{\mathrm{d}\tau }& =& \frac{2p_{\theta }}{r^2},\frac{\mathrm{d}{p}_r}{\mathrm{d}\tau }=\frac{2p_{\theta }^2}{r^3},\frac{\mathrm{d}r}{\mathrm{d}\tau }=\frac{\mathrm{d}{p}_{\theta }}{\mathrm{d}\tau }=0.\hfill \end{array}$$

(18)

Their analytical solutions are as follows:

$$\begin{array}{ccc}\hfill {\psi }_{\tau }^{H_2}:r\left(\tau \right)& =& r_0+\tau p_{r0},\hfill \\ \hfill {\psi }_{\tau }^{H_3}:r\left(\tau \right)& =& {[(r_0^2-3\tau p_{r0})/r_0]}^{1/3},\hfill \\ \hfill p_r\left(\tau \right)& =& p_{r0}{[(r_0^2-3\tau p_{r0})/r_0^2]}^{1/3},\hfill \\ \hfill {\psi }_{\tau }^{H_4}:r\left(\tau \right)& =& {(-5\tau \eta p_{r0}{r}_0^{-3/2}/2+r_0^{5/2})}^{2/5},\hfill \\ \hfill p_r\left(\tau \right)& =& p_{r0}{(-5\tau \eta p_{r0}^{8/3}/2r_0^4+p_{r0}^{5/3})}^{3/5},\hfill \\ \hfill {\psi }_{\tau }^{H_5}:\theta \left(\tau \right)& =& \tau p_{\theta 0}/r_0^2+{\theta }_0,\hfill \\ \hfill p_r\left(\tau \right)& =& \tau p_{\theta 0}^2/r_0^3+p_{r0}.\hfill \end{array}$$

(19)

All of these solutions are expressed as explicit functions of proper time $\tau$. Thus, we can construct a second-order explicit symplectic integrator, which is

$$S2\left(\tau \right)={\psi }_{\tau /2}^{H_5}\circ {\psi }_{\tau /2}^{H_4}\circ {\psi }_{\tau /2}^{H_3}\circ {\psi }_{\tau /2}^{H_2}\circ {\psi }_{\tau /2}^{H_1}\circ {\psi }_{\tau /2}^{H_2}\circ {\psi }_{\tau /2}^{H_3}\circ {\psi }_{\tau /2}^{H_4}\circ {\psi }_{\tau /2}^{H_5}.$$

(20)

Then, using the composition scheme of Yoshida (1990) [42], the fourth-order symplectic integrator can be written as

$$S4\left(\tau \right)=S2\left(\gamma \tau \right)\circ S2\left(\delta \tau \right)\circ S2\left(\gamma \tau \right),$$

(21)

where $\gamma =1/(2-\sqrt[3]{2})$ and $\delta =1-2\gamma$.

2.3. Symplectic Schemes of Hamiltonian Variational Equations

By using the method of Huang et al. [10], we can obtain the variational Hamiltonian of Equation (8):

$$\begin{array}{cc}\hfill \delta H=\delta H(\delta \overrightarrow{q},\delta \overrightarrow{p})=\frac{1}{2}{H}_{p_r{p}_r}\delta p_r^2+\frac{1}{2}{H}_{p_{\theta }{p}_{\theta }}\delta p_{\theta }^2+\frac{1}{2}{H}_{rr}\delta r^2& +\frac{1}{2}{H}_{\theta \theta }\delta {\theta }^2\hfill \\ \hfill +H_{r{p}_r}\delta r\delta p_r+H_{r{p}_{\theta }}\delta r\delta p_{\theta }+H_{r\theta }\delta r\delta \theta \end{array}$$

(22)

where $\delta H_{mn}={\partial }^{2}H/\partial m\partial n$. The remaining terms are zero, so they do not appear in the equation. This variational Hamiltonian can be split into two parts:

$$\begin{array}{ccc}\hfill \delta H_1& =& \frac{1}{2}{H}_{p_r{p}_r}\delta p_r^2+\frac{1}{2}{H}_{p_{\theta }{p}_{\theta }}\delta p_{\theta }^2+H_{r{p}_{\theta }}\delta r\delta p_{\theta }\hfill \\ \hfill & +& \frac{1}{2}{H}_{rr}\delta r^2+\frac{1}{2}{H}_{\theta \theta }\delta {\theta }^2+H_{r\theta }\delta r\delta \theta ,\hfill \\ \hfill \delta H_2& =& H_{r{p}_r}\delta r\delta p_r.\hfill \end{array}$$

(23)

By using the canonical equations $\delta \dot{\overrightarrow{q}}=\partial \delta H/\partial \delta \overrightarrow{p}$ and $\delta \dot{\overrightarrow{p}}=-\partial \delta H/\partial \delta \overrightarrow{q}$, we can derive the equations of motion for $\delta H_1$ and $\delta H_2$:

$$\begin{array}{ccc}\hfill \delta H_1:\delta \dot{r}& =& H_{p_r{p}_r}\delta p_r,\hfill \\ \hfill \delta \dot{\theta }& =& H_{p_{\theta }{p}_{\theta }}\delta p_{\theta }+H_{r{p}_{\theta }}\delta r,\hfill \\ \hfill \delta {\dot{p}}_r& =& -(H_{r{p}_{\theta }}\delta p_{\theta }+H_{rr}\delta r+H_{r\theta }\delta r),\hfill \\ \hfill \delta {\dot{p}}_{\theta }& =& -(H_{\theta \theta }\delta \theta +H_{r\theta }\delta r),\hfill \\ \hfill \delta H_2:\delta {\dot{p}}_r& =& -H_{rr}\delta p_r,\hfill \\ \hfill \delta \dot{r}& =& H_{r{p}_r}\delta r.\hfill \end{array}$$

(24)

The analytical solutions to the equations of motion (24) are

$$\begin{array}{ccc}\hfill {\psi }_{\tau }^{\delta H_1}:\delta r\left(\tau \right)& =& \delta r_0+\tau H_{p_r{p}_r}\delta p_{r0},\hfill \\ \hfill \delta \theta \left(\tau \right)& =& \delta {\theta }_0+\tau (H_{p_{\theta }{p}_{\theta }}\delta p_{\theta 0}+H_{r{p}_{\theta }}\delta r),\hfill \\ \hfill \delta p_r\left(\tau \right)& =& \delta p_{r0}-\tau (H_{r{p}_{\theta }}\delta p_{\theta 0}+H_{rr}\delta r+H_{r\theta }\delta r),\hfill \\ \hfill \delta p_{\theta }\left(\tau \right)& =& \delta p_{\theta 0}-\tau (H_{r\theta }\delta r+H_{\theta \theta }\delta \theta );\hfill \\ \hfill {\psi }_{\tau }^{\delta H_2}:\delta p_r\left(\tau \right)& =& e^{-\tau H_{r{p}_r}}\delta p_r,\hfill \\ \hfill \delta r\left(\tau \right)& =& e^{\tau H_{r{p}_r}}\delta r,\hfill \end{array}$$

(25)

and the second-order explicit symplectic scheme $S_2^{★}$ and a fourth-order explicit symplectic scheme $S_4^{★}$ can be written as

$$\begin{array}{ccc}\hfill S_2^{★}\left(\tau \right)& =& {\psi }_{\tau /2}^{\delta H_1}\circ {\psi }_{\tau }^{\delta H_2}\circ {\psi }_{\tau /2}^{\delta H_1},\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill S_4^{★}\left(\tau \right)& =& S_2^{★}\left(\gamma \tau \right)\circ S_2^{★}\left(\delta \tau \right)\circ S_2^{★}\left(\gamma \tau \right).\hfill \end{array}$$

(27)

2.4. The Carter Constant

For uncharged particles, the Hamiltonian is

$$H_0=\frac{1}{2}\left[-\frac{E^2}{f\left(r\right)}+f\left(r\right)p_r^2+\frac{p_{\theta }^2}{r^2}+\frac{L^2}{r^{2}si{n}^2\theta }\right].$$

(28)

By combining this with Equation (9), we obtain the following equation:

$$r^2-r^2\frac{E^2}{f\left(r\right)}+r^{2}f\left(r\right)p_r^2+p_{\theta }^2=\frac{L^2}{si{n}^2\theta }=\kappa ,$$

(29)

where $\kappa$ is the Carter constant. The existence of the Carter constant and the conservation of angular momentum and energy imply that the motion of uncharged particles can only follow orderly trajectories.

3. Numerical Evaluations

We consider the parameters $E=0.9915$, $L=4.6$, Q = 8.9 $\times \mathrm{ }{10}^{-4}$, $\eta =8$, and the proper time step $\tau =1$. The initial conditions are $r=11$, $\theta =\frac{\pi }{2}$, and $p_r=0$. The initial value of $p_{\theta }$ (positive) is determined by Equation (9). As shown in Figure 1, we obtain the Hamiltonian energy errors of different numerical algorithms. We can see that the Hamiltonian error of $S2$ is between ${10}^{-5}$ and ${10}^{-6}$, while that of $S4$ is between ${10}^{-8}$ and ${10}^{-9}$. Furthermore, these errors are bounded and stable. In contrast, the error of $RK4$ increases with the calculation time. This is a natural result because $RK4$ is not a geometric integrator, which means that it does not preserve the symplectic structure of Hamiltonian systems.

Then, we utilize some methods to determine the orbital dynamics of the charged test particles moving around the black hole. They are the Poincaré section, FLI, and SALI.

The definition of the FLI is expressed as [43]

$$\mathrm{FLI}\left(\tau \right)=lo{g}_{10}\frac{\left|\right|\mathit{\nu }\left(\tau \right)\left|\right|}{\left|\right|\mathit{\nu }\left(0\right)\left|\right|},$$

(30)

where $\left|\right|\mathit{\nu }\left(\tau \right)\left|\right|$ and $\left|\right|\mathit{\nu }\left(0\right)\left|\right|$ represent the Euclidean norms of the deviation vector $\mathit{\nu }$ at times $\tau >0$ and $\tau =0$, respectively. If the orbit is chaotic, the FLI will grow exponentially. Conversely, if it is regular, the FLI will exhibit linear growth.

The SALI is given by [37]

$$\mathrm{SALI}\left(\tau \right)=min\left\{\left|\right|{\widehat{\mathit{\nu }}}_1\left(\tau \right)\left|\right|-\left|\right|{\widehat{\mathit{\nu }}}_2\left(\tau \right)\left|\right|\right\},$$

(31)

where $\left|\right|{\widehat{\mathit{\nu }}}_i\left(\tau \right)\left|\right|=\frac{{\mathit{\nu }}_i\left(\tau \right)}{\left|\right|{\mathit{\nu }}_i\left|\right|}$ and $i=1,2$, are unit vectors. When the orbit is chaotic, the two unit vectors tend to align with each other, leading to SALI $\to 0$. However, for a regular orbit, the values of the SALI will exhibit bounded fluctuations around a positive number.

Figure 2a illustrates the time evolution of the FLIs for four different orbits, while Figure 2b depicts the corresponding Poincaré sections. We can see in Figure 2a that when $\eta =9.9$, the FLI increases exponentially over time, indicating that the corresponding orbit exhibits strong chaos; for the weaker chaotic orbit $\eta =7.7$, the FLI grows more slowly. For ordered orbits $\eta =8.8$ and $\eta =6$, the FLIs present linear growth. The Poincaré sections, which intuitively indicate whether an orbit is chaotic, are shown in Figure 2b. In the case of chaotic orbits, the points in the Poincaré sections are randomly distributed throughout the plane. In contrast, for ordered orbits, the Poincaré sections display one or more closed curves.

In Figure 3, we use different algorithms to obtain the SALIs of the system, whose initial conditions and parameters are the same as those in Figure 2. In Figure 3a, we adopt $S4$ to calculate the equations of motion and $RK8$ to calculate the variational equations ($S4$–$RK8$). The results indicate that the SALI of the strongly chaotic orbit with $\eta =9.9$ approaches zero, while the SALI of the weakly chaotic orbit with $\eta =7.7$ appears to be more stable than the former. For ordered orbits with $\eta =6$ and $\eta =8.8$, the SALIs fluctuate around two constants. A calculation time of $6\times {10}^5$ is needed here, and this is faster than the FLI method in Figure 2a.

In Figure 3b, we utilize the $RK8$–$RK8$ method to calculate the SALIs, and the results are nearly identical to those in Figure 3a. In Figure 3c,d, the SALIs are obtained by using global symplectic algorithms $S4$–$S_4^{★}$ and the $RK8$–three-particle method (due to the SALI requiring the computation of two deviation vectors and both deviation vectors needing triple integration by simultaneously using $RK8$, this is referred to as the three-particle method), respectively. For the global symplectic method, only a calculation time of $5\times {10}^4$ is needed to distinguish between chaos and order, making it the fastest approach. However, due to the rapid variation in the deviation vectors resulting from this calculation method, it becomes difficult to differentiate weakly chaotic orbits from strongly chaotic orbits. On the other hand, the SALIs obtained from the $RK8$–three-particle method lose their effectiveness.

Next, we are interested in examining how the parameters E, Q, L, and $\eta$ influence the orbital dynamics. Due to the computational advantage of SALIs in terms of the required calculation time, we employ the SALI and the Poincaré section to investigate the impacts of the four parameters on the dynamic transition from order to chaos, and the results are shown in Figure 4. In Figure 4(a1), it can be observed that if $E<0.9914$ or $0.992<E<0.9957$, the SALIs are greater than ${10}^{-3}$, which is the boundary between chaos and order in [10], indicating that the orbits are ordered. However, for $0.9914<E<0.992$ and $E>0.9958$, the corresponding orbits are chaotic. In Figure 4(a2), when $E=0.9927$ and $E=0.9954$, the points in the Poincaré sections form two rings, which are known as the Komogorov–Arnold–Moser (KAM) torus. This represents that these two orbits are orderly. When $E=0.9957$ and $E=0.9969$, the orbits appear to be clearly chaotic.

By changing the value of parameter $\eta$, we find that, in Figure 4(b1), the orbits are regular when $\eta$ $<7.7$ and $8.5<$ $\eta$ $<9.1$. The orbits transition into chaos when $7.7<$ $\eta$ $<8.5$. In addition, if $\eta$ exceeds 9.1, the SALI is smaller than ${10}^{-3}$, and as the value of $\eta$ increases, the SALI decreases, and the extent of chaos is strengthened. In Figure 4(b2), the motions of test particles are chaotic in $\eta =7.7$ and $\eta =9.9$ and regular in $\eta =6$ and $\eta =8.8$.

We now investigate whether the angular momentum of the test particles influences the dynamic transition from order to chaos. In Figure 4(c1), strong chaos is observed when $L<4.52$. When $L>4.52$, although there are some weakly chaotic orbits, most of the trajectories exhibit ordered behavior. In Figure 4(c2), the values of angular momentum corresponding to the four orbits are, respectively, $4.5$, $4.56$, $4.62$, and $4.92$. Notably, the orbit with $L=4.5$ is a chaotic orbit, while the orbit with $L=4.56$ is ordered. If we set $L=4.62$, the orbit seems to be a typical weak chaotic orbit, and for $L=4.92$, the points in the figure form a closed ring, which indicates an ordered orbit.

Finally, we explore the transition of orbits when increasing the value of the magnetic field parameter Q. In Figure 4(d1), it is observed that the orbits transition from ordered to chaotic when $Q>8\times {10}^{-4}$ and back to ordered when $Q>1.03\times {10}^{-3}$. The Poincaré section corresponding to $Q=6.8\times {10}^{-4}$ forms a closed torus, indicating a regular orbit. On the other hand, the Poincaré section of $Q=1.21\times {10}^{-4}$ becomes a complicated KAM torus, which is characterized by the presence of six small loops. These small loops belong to the same trajectory and form a chain of islands [13]. Such a torus is considered regular, but it can easily lead to the onset of resonance and chaos. The orbit corresponding to $Q=8.5\times {10}^{-4}$ is a weakly chaotic orbit, while the orbit for $Q=9.9$ is strongly chaotic. However, the strength of the chaos in the latter is weaker than that depicted in Figure 4(a2).

In order to conduct a more thorough investigation of the mixing of chaotic and regular motion, we focus on the exploration of chaotic particle dynamics in the X–Z plane. Figure 5 illustrates the chaotic nature of particle trajectories in certain X–Y planes. The step size for r is 0.05, while the step size for $\theta$ is $0.01$. Deeper shades of red are utilized to illustrate regions in which particles exhibit more prominent chaotic behavior, while lighter shades of blue are employed to indicate areas with less pronounced chaotic behavior. We can clearly observe that when the value of r falls within the range of approximately $5.8$ to $6.9$, the particles demonstrate a significant manifestation of chaotic behavior. However, when R exceeds $9.8$, the behavior of the particles becomes orderly and organized. It is noticeable that in certain areas, the chaotic behavior of particles sharing the same value of r changes as $\theta$ varies. The most prominent occurrence of this mixing of the chaotic and regular motion can be observed in the transitional region between chaos and order (roughly, when r falls within the range of 8.7 to 9.8).

4. The Effect of $\mathit{\eta }$

In this section, we examine the impact of $\eta$ on effective potentials and stable circular orbits and its influence on chaos dynamics through a comparison with the magnetized Schwarzschild black hole.

From Equations (8) and (9), we have

$$f^2\left(r\right)p_r^2+\frac{f\left(r\right)p_{\theta }^2}{r^2}=E^2-V^2,$$

(32)

where V represents the effective potential on the equatorial plane $\theta =\pi /2$, which characterizes the radial movement of a charged particle within an accretion disk encircling the black hole:

$$V^2=\frac{f\left(r\right)}{r^2}{\left[L-\frac{Q}{2}{r}^2\right]}^2+f\left(r\right).$$

(33)

The relationship between the effective potential and $\eta$ can be observed in Figure 6, which shows that as $\eta$ increases, the effective potential decreases—see, in particular, the black line with $\eta =0$. For the circular motion of the particles, the following conditions must be satisfied:

$$\begin{array}{ccc}\hfill \dot{r}& =& 0,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill \mathrm{d}V/\mathrm{d}r& =& 0.\hfill \end{array}$$

(35)

The presence of local extrema in the effective potential indicates the existence of circular orbits. If the second derivative of the effective potential is ${\mathrm{d}}^{2}V/\mathrm{d}{r}^2\ge 0$, the potential has local minimal values. These values correspond to the existence of stable circular orbits (SCOs). Specifically, when ${\mathrm{d}}^{2}V/\mathrm{d}{r}^2=0$, such a stable circular orbit is referred to as the innermost stable circular orbit (ISCO). The radii of some SCOs and ISCOs in Figure 6 are listed in

Table 1. Radii (r) of the stable circular orbits and innermost stable circular orbits.

η	0	2	4	6	8
SCO	17.62227	17.38088	17.12888	16.86473	16.58649
ISCO	5.99938	6.87223	7.53582	8.08965	8.57331

. To enhance the visual representation, we have drawn a rough sketch of the SCOs and ISCOs in Figure 7 for three different values of $\eta$: 0, 4, and 8. By referring to Table 1 and Figure 7, it is evident that as the parameter $\eta$ increases, the radius of the ISCO increases, while the radius of the SCO decreases.

By setting the value of $\eta$ to zero and maintaining the same initial conditions as those depicted in Figure 4, we obtain the correlations between the SALIs and the energy E, charge Q, and angular momentum L in a magnetized Schwarzschild black hole, and these are displayed in Figure 8. Comparing Figure 8a with Figure 4(a1), we can observe that the relationship between E and the SALI in Figure 8a is generally similar to that in (a1) in Figure 4. By comparing Figure 8b,c with (c1) and (d1) in Figure 4, respectively, we can see that when $\eta$ is set to zero, there is no chaotic behavior observed when the angular momentum L and charge Q are changed.

5. Conclusions and Discussion

In summary, compared with the traditional $RK4$ method, the fourth-order explicit symplectic scheme is a superior method in non-rotating Konoplya and Zhidenko black hole spacetime. It ensures that the energy errors remain below ${10}^{-8}$ and do not exhibit growth over time. The numerical results demonstrate that the SALI calculated with the global symplectic algorithm is the fastest method for distinguishing chaotic orbits. However, it is challenging to distinguish weak chaos when using this method. On the other hand, the $RK8$–$RK8$ and $S4$–$RK8$ methods are capable of distinguishing weak chaos, but a longer calculation time is required.

The SALI is employed to study how the parameters affect the dynamic transition from order to chaos. The overall results are dynamic properties of all orbital changes as these parameters change. Specifically, when $\eta$ and the energy E go beyond a certain value, the orbits tend to remain chaotic. Conversely, when the angular momentum L exceeds a certain threshold, while there may still be some chaotic orbits, the majority of the orbits are regular. If we set the magnetic field parameter Q within a suitable range, a chaotic orbit can be easily found. However, if Q is below or exceeds a certain value, the orbits tend to be mainly regular.

By altering the initial conditions of the particles, a distribution plot illustrating the smaller Lyapunov exponent on the X–Z plane of the black hole was generated. It was observed that the particle orbits manifest an exceptionally diverse and intricate structure, and when the orbital radius of the particles r falls within the range of $8.7$ to $9.8$, we can easily find the mixing of chaotic and regular motion.

The influence of the newly introduced parameter $\eta$ can cause a contraction in the SCO and an expansion in the ISCO. Furthermore, it can lead to chaotic behavior of the particles, which is not present in Schwarzschild black holes.

Researching the motion of charged particles around a black hole contributes to understanding the mechanisms of black hole accretion and provides insights into the initial formation process of accretion disks. Building upon this foundation, future considerations may involve studying the dynamic behavior of photons around a black hole, which can offer perspectives for investigating black hole shadows.