1 Introduction

The standard general relativistic black hole solutions, such as the Schwarzschild spacetime, Reissner–Nordström (RN) spacetime, Kerr spacetime and Kerr–Newman spacetime, are highly nonlinear. However, they are integrable. This integrability is due to the existence of four constants of motion in these spacetimes. In this case, the motion of photons or test particles around these black holes is regular and nonchaotic. The regular motion of particles around black holes was treated in the standard textbooks [1,2,3]. There are many integrable curved spacetimes in modified or alternative gravity theories [4,5,6,7,8,9].

When these black holes are surrounded by extra sources, the motions of photons, neutral test particles or charged test particles in the close vicinity of black hole horizons may be nonintegrable. A magnetic field, as an important extra source existing around a black hole, has been supported by observational evidence [10]. It may arise from the dynamo mechanism in collisionless plasma of an accretion disk around the central black hole [3]. However, a strong magnetic field around a supermassive black hole in the center of the Galaxy is not relevant to an accretion disc [11]. Such a magnetic field surrounding the black hole is an asymptotically uniform large-scale electromagnetic field with a complicated structure in the vicinity of a magnetar at large distance from the black hole.

If a magnetic field in the vicinity of a black hole has a small intensity satisfying the condition \(B\ll 10^{19}M_{\odot }/M\) Gauss, where \(M_{\odot }\) and M are the masses of the Sun and black hole, it has a negligible effect on the motion of neutral test particles. In other words, the gravitational background is not influenced by the magnetic field, and the metric tensor of the black hole spacetime has no modification. In spite of this, the magnetic field strongly influences the motion of charged test particles if the ratio of the particle charge to the particle mass is quite large. Even the dynamics of charged test particles is nonintegrable and probably chaotic. The motion of charged test particles around the RN black hole combined with an external asymptotically uniform magnetic field can be chaotic [12]. There have been a large variety of papers on the chaotic charged particle dynamics in combined black hole gravitational and external magnetic fields [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]. The authors of [16] showed that the chaotic charged particle dynamics close to the black hole horizon plays an important role in energy interchange between the translational and oscillatory modes. It is not necessarily correct that magnetic fields in the vicinity of black holes always lead to the nonintegrability of charged particle motion around the black holes. The dynamics of charged particles near the Kerr-Newman black hole immersed in an external magnetic field is integrable due to the existence of the Carter constant as a fourth constant of motion [31]. The regularity of the motion of charged particles in the Kerr–Newman–de Sitter dyon spacetimes was treated by Stuchlík [32]. When a magnetic charge of the Kerr–Newman black hole is included, the geodesic motion of charged test particles is still integrable [33]. The integrability is also suitable for charged particle motion around the Kerr–Newman black hole with quintessence, cloud of strings and external electromagnetic field [34].

When the strength of external magnetic field reaches the upper limit of magnetic field \(B= 10^{19}M_{\odot }/M\) Gauss, the magnetic field can significantly modify the black hole spacetime structure [16, 35, 36]. In this case, the metric tensor of the black hole spacetime needs an appropriate modification. Such a magnetized black hole spacetime is likely nonintegrable although the original nonmagnetized black hole spacetime is integrable. In Melvin’s magnetic universes, the Schwarzschild black hole, RN black hole, and Kerr–Newman black hole, which were derived from the coupled Einstein–Maxwell field equations by Ernst [37], are not integrable. This nonintegrability is due to the black hole immersed into the strong external magnetic field modifying the spacetime structure and causing the absence of the fourth motion constant. The result was shown by finding the chaoticity of neutral test particles in the magnetized Schwarzschild–Melvin spacetime [38, 39]. The chaotic motion of photons can give self-similar fractal structures to the shadows of Schwarzschild–Melvin and Kerr–Melvin black holes [40, 41]. For the motion of charged test particles, not only the external fields appear in the metrics, but also external electromagnetic fields are included in the Hamiltonian systems associated with these magnetized black hole metrics. Thus, the external magnetic fields have typical effects on the motion of neutral and charged test particles. Other properties of the RN-Melvin black hole solutions were investigated in [42, 43]. Note that the external magnetic field in [12] is not included in the RN black hole spacetime, but is added to the Hamiltonian system describing the motion of charged test particles around the RN black hole. Recently, the authors of [44] showed that the radii of the innermost stable circular orbits for neutral and charged test particles around the magnetized electric RN black hole could be strongly influenced by the combined effect of black hole electric charge and magnetic field. More recently, the tachyonic instability of RN-Melvin black holes in Einstein–Maxwell-scalar theory was considered in Ref. [45].

The self-force of the motion in combined gravitational and magnetic fields plays an important role in causing transitions from regular to chaotic motion (see e.g. [29, 30]). Besides the magnetic fields, other extra sources may make crucial contributions to the occurrence of chaos. Several works [46,47,48] have shown that the quadrupolar deformations of black hole masses are responsible for the existence of chaotic dynamics of test particles in rotating black hole solutions of Manko et al. [49]. The axially symmetric deformation described by the mass density parameter is necessary for the existence of chaotic dynamics in the Zipoy–Voorhees metric [50]. Spin effects of test particles in black hole spacetime backgrounds can induce chaos [51]. The general relativistic Poynting–Robertson effect shows a chaotic behavior of the geodesic motion of test particles orbiting around the Kerr black hole [52, 53].

The detection of the chaotical behavior needs very accurate long-time determination of the trajectories. The motions of test particles in many curved spacetimes can be described by Hamiltonian systems. The most appropriate methods for solving the Hamiltonian systems in the case of long-term integrations are symplectic schemes (see, e.g., [54,55,56,57]). Such integrators preserve the symplectic structure of Hamiltonian dynamics. Although they are unlike the energy-preserving integrators [58,59,60,61,62] that can exactly preserve energy for Hamiltonian systems, they have no secular drifts in errors of first integrals of the Hamiltonian systems. Because most of the Hamiltonians for curved spacetimes are nonseparable to the position and momentum variables or cannot be split into two explicitly integrable parts, explicit symplectic integrations had been seldom applicable for these nonseparable Hamiltonian problems. Of course, implicit symplectic integration schemes [14, 48] or explicit and implicit combined symplectic methods [62,63,64,65] are always available, and hence are computationally expensive. Recently, explicit symplectic integrators [12, 21, 22, 25, 26] have been made for nonseparable Hamiltonian problems of geodesics in some curved spacetimes, such as the Schwarzschild black hole spacetime. Their constructions are based on the Hamiltonians split into more parts, where the flow of each part can be integrated and represented in terms of explicit functions of time. For some other curved spacetimes (e.g., the Kerr black hole spacetime), their Hamiltonians have no such splits. However, these Hamiltonians have via appropriate time transformations and thus amenable for explicit symplectic integrations [23, 24, 27, 28, 66, 67].

In this paper we use an explicit symplectic integrator to investigate the regular and chaotic dynamics of neutral and charged test particles around the magnetized electric RN black holes [42]. We first briefly introduce the magnetized electric RN black hole metric. A Hamiltonian system for the motion of neutral and charged test particles around the black holes is presented. Next, we demonstrate how several explicit symplectic methods are constructed for this Hamiltonian system. We then evaluate the numerical performance of the proposed explicit symplectic methods and find the method satisfying a requirement for good long term behaviour. Finally, we study the motions of neutral and charged test particles. We particularly focus on the impact of varying the black hole electric charge and the particle electric charge on a dynamical transition from order to chaos. The effect of varying the other parameters on the dynamical transition is also considered. Explanations are given to the effects of the parameters on the dynamics.

2 Magnetized electric RN black hole

At first, a magnetized electric RN black hole metric is introduced. Then, the motion of a charged test particle around the RN black hole surrounded by an asymptotically uniform magnetic field is described in terms of a Hamiltonian formulation.

2.1 Black hole metric

In the Boyer–Lindquist dimensionless coordinates \(x^{\alpha }=(t, r, \theta , \phi )\), a magnetically charged RN black hole metric is written in Refs. [37, 42] as

$$\begin{aligned} \begin{aligned} ds^2&= g_{\alpha \beta }dx^\alpha dx^\beta \\&= g_{tt}dt^2+2g_{t\phi } dt d\phi +g_{rr}dr^2 \\&+g_{\theta \theta }d\theta ^2 +g_{\phi \phi }d\phi ^2; \\ g_{tt}&= -fF+\frac{\omega ^2}{F}r^2\sin ^2\theta , \quad g_{t\phi } = -\frac{\omega }{F}r^2\sin ^2\theta , \\ g_{rr}&= \frac{F}{f}, \quad g_{\theta \theta } = Fr^2, \quad g_{\phi \phi } = \frac{1}{F}r^2\sin ^2\theta . \end{aligned} \end{aligned}$$
(1)

Here, F, f and \(\omega \) are functions of r and \(\theta \) as follows:

$$\begin{aligned} F= & {} 1+\frac{1}{2}B^2(r^2\sin ^2\theta +3Q^2\cos ^2\theta ) \nonumber \\{} & {} +\frac{1}{16}B^4(r^2\sin ^2\theta +Q^2\cos ^2\theta )^2, \end{aligned}$$
(2)
$$\begin{aligned} f= & {} 1-\frac{2M}{r}+\frac{Q^2}{r^2}, \end{aligned}$$
(3)
$$\begin{aligned} \omega= & {} -\frac{2QB}{r}+\frac{1}{2}QB^3r(1+f\cos ^2\theta ). \end{aligned}$$
(4)

The speed of light c and the gravitational constant G take one geometric unit, \(c=G=1\). M is the mass of the black hole and Q denotes an electric charge of the black hole. B stands for the strength of an asymptotically uniform magnetic field in the black hole vicinity. The magnetic field is included in the spacetime geometry near the black hole because it is strong enough to distort the spacetime geometry. It gives gravitational effects rather than the Lorentz force contributions to neutral test particles. Of course, it must drastically affect the motion of neutral particles. In this sense, the RN spacetime geometry is magnetized.

If \(B=0\), the metric (1) corresponds to the RN black hole. If \(B\ne 0\) and \(Q=0\), the metric (1) is the Schwarzschild–Melvin magnetic universe [37]. When the electric charge of the black hole is nonzero, the metric (1) is not asymptotic to the static Melvin metric [42]. Although the gravitational effect of the magnetic field does not alter the even horizons of the RN black hole, it causes the spacetime (1) not to be asymptotically flat.

The presence of the term \(dtd\phi \) in the metric (1) is not due to the black hole rotating, but arises from the global SU(2, 1) symmetry group [42]. When a Kaluza–Klein reduction of the four-dimensional Einstein–Maxwell action is performed and the vector fields are dualized to scalars in three dimensions, the specific SU(2, 1) transformation can generate the magnetized solutions from nonmagnetized ones. If \(B=0\) or \(Q=0\), then \(\omega =0\) and the term \(dtd\phi \) is absent. It is clear that the term \(dtd\phi \) directly comes from the contribution of the magnetic field B and the electric charge Q. In fact, the black hole is not rotating at all.

If the test particle is charged, it suffers from the Coulomb force and Lorentz force given by an external electromagnetic field. The electromagnetic field is described by a four-vector potential with two non-zero covariant components [42]

$$\begin{aligned} A_\phi= & {} \frac{2}{B}-\frac{1}{F}\left[ \frac{2}{B} +\frac{B}{2}(r^{2}\sin ^2\theta +3Q^2\cos ^2\theta )\right] , \end{aligned}$$
(5)
$$\begin{aligned} A_t= & {} -\frac{Q}{r}+\frac{3}{4}QB^2r(1+f\cos ^2\theta )-\omega A_\phi . \end{aligned}$$
(6)

Besides the Coulomb force and Lorentz force, the gravity forces from the magnetized RN black hole are given to the charged particle.

2.2 Hamiltonian system

Now, let us consider the particle with charge q and mass m moving near the RN black hole surrounded by the external magnetic field B. The motion of charged particle can be described by a Hamiltonian system

$$\begin{aligned} H= & {} \frac{1}{2m}g^{\alpha \beta }(p_\alpha -qA_\alpha )(p_\beta -qA_\beta ) \nonumber \\= & {} \frac{f}{2mF}p_r^2+\frac{1}{2mFr^2}p_\theta ^2+H_1, \end{aligned}$$
(7)

where \(H_1\) is a function of r and \(\theta \):

$$\begin{aligned} H_1= & {} \frac{g^{tt}}{2m}(p_t -qA_t)^2+\frac{g^{\phi \phi }}{2m}(p_\phi -qA_\phi )^2 \nonumber \\{} & {} + \frac{g^{t\phi }}{m}(p_t -qA_t)(p_\phi -qA_\phi ). \end{aligned}$$
(8)

The non-zero contravariant components of the metric (1) are written as

$$\begin{aligned} g^{tt}= & {} \frac{g_{\phi \phi }}{g_{tt}g_{\phi \phi }-g^2_{t\phi }}=-\frac{1}{fF}, \\ g^{\phi \phi }= & {} \frac{g_{tt}}{g_{tt}g_{\phi \phi }-g^2_{t\phi }}=\frac{fF^2-\omega ^2r^2\sin ^2\theta }{fFr^2\sin ^2\theta }, \\ g^{t\phi }= & {} -\frac{g_{t\phi }}{g_{tt}g_{\phi \phi }-g^2_{t\phi }}=-\frac{\omega }{fF}. \end{aligned}$$

The external magnetic field in Eqs. (1) and (7) affects not only the spacetime geometry but also the motion of charged particles. The external magnetic field of [12] does not change the spacetime geometries, but has a nonnegligible effect on the motion of charged test particles in the gravitational backgrounds.

Because the Hamiltonian (7) does not explicitly depend on the coordinate time t, the momentum \(p_t\) is a motion constant related to the particle energy E with \(E=-p_t\). This Hamiltonian does not explicitly contain \(\phi \) or is axially symmetric, therefore, the momentum \(p_\phi \) corresponds to the particle conserved angular momentum \(L=p_\phi \). The two constants satisfy the relations

$$\begin{aligned} \dot{t}= & {} -\frac{g^{tt}}{m}(E +qA_t)+ \frac{g^{t\phi }}{m}(L -qA_\phi ), \end{aligned}$$
(9)
$$\begin{aligned} \dot{\phi }= & {} \frac{g^{\phi \phi }}{m}(L -qA_\phi )- \frac{g^{t\phi }}{m}(E+qA_t). \end{aligned}$$
(10)

Here, \(\dot{t}\) and \(\dot{\phi }\) as two components of the 4-velocity are derivatives of t and \(\phi \) with respect to the proper time \(\tau \). A third constant of motion is the conserved Hamiltonian. For time-like geodesic orbit, this constant is

$$\begin{aligned} H=-\frac{m}{2}. \end{aligned}$$
(11)

For simplicity, dimensionless operations are used via scale transformations to the related variables and parameters: \(r\rightarrow rM\), \(t\rightarrow tM\), \(\tau \rightarrow \tau M\), \(Q\rightarrow QM\), \(H \rightarrow mH\), \(E \rightarrow mE\), \(p_r \rightarrow mp_r\), \(p_\theta \rightarrow mMp_\theta \), \(L \rightarrow mML\), \(q \rightarrow mq\), and \(B \rightarrow B/M\). In this way, the two mass factors M and m in Eqs. (1)–(11) are eliminated. In this case, Eq. (8) is rewritten as

$$\begin{aligned} H_1= & {} \frac{g^{tt}}{2}(E+qA_t)^2+\frac{g^{\phi \phi }}{2}(L -qA_\phi )^2 \nonumber \\{} & {} - g^{t\phi }(E+qA_t)(L-qA_\phi ). \end{aligned}$$
(12)

Besides the three constants given in Eqs. (9)–(11), no fourth motion constant exists in the Hamiltonian system (7). The system is nonintegrable even if the particle has no charge, namely, the four-vector potential terms \(A_t\) and \(A_{\phi }\) are removed. The nonintegrability of the motion of neutral particle is due to the gravitational effect of the external magnetic field B in the gravitational background. Numerical techniques are convenient to solve such a nonintegrable system.

3 Construction of explicit symplectic integration algorithms

A symplectic integrator that preserves the symplectic structure of Hamiltonian dynamics is naturally a prior choice of an integrator for solving the Hamiltonian problem (7). The Hamiltonian has no separation of variables. It cannot be split into two parts with analytical solutions as explicit functions of proper time, either. In these cases, the construction of explicit symplectic integrators seems to be impossible. Recently, the authors of [12, 21, 22, 25, 26] showed that the Hamiltonians of some spacetimes like the Schwarzschild black hole have more than two explicitly interable splitting terms. Such splitting methods allow for the construction of explicit symplectic integrators based on splitting and composing. However, the Hamiltonian (7) has no such a direct multi-part splitting. Introducing appropriate time transformations to Hamiltonians of some other spacetimes such as the Kerr black hole, the authors of [23, 24, 27, 28, 66, 67] found that the obtained time-transformed Hamiltonians are suitable for the application of explicit symplectic methods. The Hamiltonian (7) belongs to Type 2 of the indirect splitting spacetimes in the latest paper [67]. Following this idea, we implement such algorithms for the Hamiltonian (7).

3.1 Algorithmic construction

Setting the proper time \(\tau \) as a new coordinate \(q_0=\tau \) and its corresponding momentum as \(p_0=-H =1/2\), we extend the phase space of the Hamiltonian (7) in the form

$$\begin{aligned} \mathcal {H} =H +p_0. \end{aligned}$$
(13)

The extended phase space Hamiltonian is always identical to zero for any proper time \(\tau \), i.e., \(\mathcal {H} =0\).

Following the time transformation method of Mikkola [68], we take a time transformation

$$\begin{aligned} d\tau =g(r,\theta )d w, \end{aligned}$$
(14)

where g is a time transformation function

$$\begin{aligned} g(r,\theta )=F. \end{aligned}$$
(15)

We then have a time transformation Hamiltonian

$$\begin{aligned} K=g \mathcal {H}. \end{aligned}$$
(16)

The time-transformed Hamiltonian is still equal to zero (\(K=0\)) for any new time w. It has five splitting pieces

$$\begin{aligned} K =K_1+K_2+K_3+K_4+K_5, \end{aligned}$$
(17)

where the five sub-Hamiltonians are expressed as

$$\begin{aligned} K_1= & {} g(H_1+p_0), \end{aligned}$$
(18)
$$\begin{aligned} K_2= & {} \frac{1}{2}p_r^2, \end{aligned}$$
(19)
$$\begin{aligned} K_3= & {} -\frac{1}{r}p_r^2, \end{aligned}$$
(20)
$$\begin{aligned} K_4= & {} \frac{p_\theta ^2}{2r^2}, \end{aligned}$$
(21)
$$\begin{aligned} K_5= & {} \frac{Q^2}{2r^2}p_r^2. \end{aligned}$$
(22)

The splitting of the time-transformed Hamiltonian K is similar to that of the Hamiltonian for the RN black hole with the external magnetic field in Ref. [12].

It is clear that the five sub-Hamiltonians are explicitly, analytically solvable. That is to say, their analytical solutions are explicit functions of the new time w. Suppose that \(\mathcal {K}_1\), \(\mathcal {K}_2\), \(\mathcal {K}_3\), \(\mathcal {K}_4\) and \(\mathcal {K}_5\) correspond to the analytical solvers of the five sub-Hamiltonians \(K_1\), \(K_2\), \(K_3\), \(K_4\) and \(K_5\), respectively. Let h be a time step of the new time w. Two first-order symplectic composing operators are defined as

$$\begin{aligned} \chi (h)= & {} \mathcal {K}_1(h)\times \mathcal {K}_2(h)\times \mathcal {K}_3(h)\nonumber \\{} & {} \times \mathcal {K}_4(h)\times \mathcal {K}_5(h), \end{aligned}$$
(23)
$$\begin{aligned} \chi ^{*} (h)= & {} \mathcal {K}_5(h)\times \mathcal {K}_4(h)\times \mathcal {K}_3(h)\nonumber \\{} & {} \times \mathcal {K}_2(h)\times \mathcal {K}_1(h). \end{aligned}$$
(24)

The two operators can symmetrically compose an explicit second-order symplectic algorithm

$$\begin{aligned} S_2(h)=\chi ^{*}\left( \frac{h}{2}\right) \times \chi \left( \frac{h}{2}\right) . \end{aligned}$$
(25)

A symmetric composition of three second-order methods easily raises a fourth-order method of Yoshida [69]

$$\begin{aligned} S_4(h)=S_2(\gamma h)\times S_2(\delta h)\times S_2(\gamma h), \end{aligned}$$
(26)

where \(\gamma =1/(1-\root 3 \of {2}) \) and \(\delta =1-2\gamma \). There is an optimized fourth-order partition Runge–Kutta (PRK) explicit symplectic integrator [70]:

$$\begin{aligned} PRK_64(h)= & {} \chi ^{*} (\alpha _{12} h)\times \chi (\alpha _{11} h)\times \cdots \nonumber \\{} & {} \times \chi ^{*} (\alpha _2 h) \times \chi (\alpha _1 h), \end{aligned}$$
(27)

The time coefficients of the algorithm are listed in Ref. [25] as follows:

$$\begin{aligned} \alpha _1= & {} \alpha _{12}= 0.079203696431196, \\ \alpha _2= & {} \alpha _{11}= 0.130311410182166, \\ \alpha _3= & {} \alpha _{10}= 0.222861495867608, \\ \alpha _4= & {} \alpha _9=-0.366713269047426, \\ \alpha _5= & {} \alpha _8= 0.324648188689706, \\ \alpha _6= & {} \alpha _7= 0.109688477876750. \end{aligned}$$

Equations (25)–(27) are the time-transformed explicit symplectic methods designed for the Hamiltonian (7). In these methods, fixed time steps are used for the new time w, while variant time steps may be considered for the proper time \(\tau \). The authors of [67] gave the time transformation function and Hamiltonian splitting form like those of Eqs. (15) and (17) to the Hamiltonian (13) with \(A_t=A_\phi =0\), but did not numerically test the established time-transformed explicit symplectic methods.

3.2 Evaluation of the algorithms

The parameters are the magnetic field strength \(B=6\times 10^{-4}\), black hole charge \(Q=0.3\), particle charge \(q=0.5\), particle angular momentum \(L=4.8\), and particle energy \(E=0.997\). The initial conditions are \(r=30\), \(\theta =\pi /2\) and \(p_r=0\). The initial value of \(p_{\theta }>0\) should satisfy Eq. (16).

Fig. 1
figure 1

Accuracies of the Hamiltonian K for the three explicit symplectic integrators \(S_2\), \(S_4\) and \(PRK_64\) with two step sizes \(h=1\) and \(h=3.5\). For the motion of a charged particle with \(q=0.5\), the other parameters are \(E=0.997\), \(L=4.8\), \(B=6\times 10^{-4}\) and \(Q=0.3\); the initial conditions are \(r=30\), \(p_r=0\), \(\theta = \pi /2\), and \(p_\theta >0\) given by Eq. (16). A secular drift occurs in the energy errors for \(PRK_64\) with \(h=1\), whereas does not occur for \(PRK_64\) with \(h=3.5\)

Full size image

Taking the time step \(h=1\), we plot Fig. 1, which shows accuracies of the Hamiltonian K yielded by the three methods \(S_2\), \(S_4\) and \(PRK_64\). The errors of K remain stable for \(S_2\) and \(S_4\), but have a secular drift for \(PRK_64\). The errors for \(S_4\) are about three orders of magnitude smaller than those for \(S_2\), but larger than those for \(PRK_64\). The secular error drift for \(PRK_64\) is due to the rapid growth of roundoff errors. It is absent when the time step increases to \(h=3.5\). The errors of \(PRK_64\) for the larger time step \(h=3.5\) are approximately the same as those of \(S_4\) for the smaller time step \(h=1\). The algorithm \(S_4\) with \(h=1\) has an advantage over \(PRK_64\) with \(h=3.5\) in computational efficiency, as is shown in Table 1.

Table 1 CPU times [units: minute (\(^\prime \)), second (\(^{\prime \prime }\))] for the three algorithms with two step sizes h. The initial separations of Orbit 1 and Orbit 2 are \(r=30\) and \(r=60\), respectively; the parameters and other initial conditions are the same as those of Fig. 1
Full size table

The relation between the new time w and the proper time \(\tau \) in Fig. 2 shows that the two times are almost the same. It means that the time step is fixed for the new time w, but the proper time steps appropriately remain invariant for the proper time \(\tau \).

Based on the accuracy and efficiency, the method \(S_4\) with \(h=1\) is used to survey the orbital dynamics in later discussions.

Fig. 2
figure 2

Relation between the proper time \(\tau \) and the new time w. The tested orbit is that of Fig. 1. The slope being 1 shows that the two times are almost the same

Full size image

4 Orbital dynamics

Using several chaotic indicators, we focus on the dynamics of neutral or charged particles moving around the magnetized electric RN black hole. The effects of varying one or two parameters on a transition from order to chaos are also considered.

4.1 Dynamics of neutral particles

For the motion of neutral particles with \(q=0\), the terms \(A_t\) and \(A_{\phi }\) are dropped in Eqs. (8)–(10). The parameters are taken as \(B=6\times 10^{-4}\), \(L=4.8\), \(Q=0.3\) and \(E=0.997\). The initial conditions are \(\theta = \pi /2\) and \(p_r = 0\). The initial radii have various choices, and the initial values \(p_\theta >0\) are calculated by Eq. (13). Figure 3a relates to the Poincaré map at the plane \(\theta = \pi /2\) with \(p_\theta >0\), which represents intersections \((r, p_r)\) of the particle trajectories with the surface of section. Orbit 1 with the initial radius \(r=30\) is chaotic because the intersections are randomly distributed points in an area. However, Orbit 2 with the initial radius \(r=60\) has intersection points forming a closed curve. As a result, the motion is regular.

In addition to the Poincaré section method, the largest Lyapunov exponent for measuring the average deviation between two adjacent orbits is often used to distinguish between ordered and chaotic motions. It is defined in [71] by

$$\begin{aligned} \lambda =\lim _{w \rightarrow \infty } \frac{1}{w} \ln \frac{d(w)}{d(0)}, \end{aligned}$$
(28)

where d(w) and d(0) are the distances between two adjacent orbits at the new time w and the starting time, respectively. When the integration time reaches \(w=1\times 10^8\) in Fig. 3b, the Lyapunov exponent \(\lambda \) of bounded orbit 1 tending to a stabilizing value \(10^{-3.761}\) indicates the chaoticity of Orbit 1. The Lyapunov exponent \(\lambda \) of bounded orbit 2 tending to zero indicates the regularity of Orbit 2.

The fast Lyapunov indicator (FLI) [72] is a faster and more sensitive tool to detect chaos from order than the largest Lyapunov exponent. It is defined in [73] by

$$\begin{aligned} FLI =\log _{10} \frac{d(w)}{d(0)}. \end{aligned}$$
(29)

Different growth rates of the deviation vectors are used to distinguish between the regular and chaotic two cases. The the exponential growth of the FLI of Orbit 1 with time shows that of chaotic orbit 1, whereas the algebraical growth of the FLI of Orbit 2 with time \(\log _{10} w\) describes the characteristic of regular orbit 2 in Fig. 3c.

Fig. 3
figure 3

a Poincaré sections at the plane \(\theta = \pi /2\) with \(p_\theta >0\) for the motions of neutral particles with \(q=0\). The other parameters are those of Fig. 1. Orbit 1 with the initial separation \(r=30\) is chaotic, but Orbit 2 with the initial separation \(r=60\) is regular. These results are supported by the largest Lyapunov exponents in panel (b) and fast Lyapunov indicators (FLIs) in panel (c)

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Fig. 4
figure 4

Dependence of FLIs on two parameters. Each of the FLI values is obtained after the integration time \(w=10^6\). The FLIs \(\le \) 5 show the regularity of bounded orbits, but the FLIs > 5 describe the chaoticity of bounded orbits. a The two parameters are the magnetic field B and the black hole charge Q, and the other parameters are \(q=0\), \(E=0.998\), and \(L=4.5\); the initial separation is \(r=30\). b The two parameters are the particle energy E and the black hole charge Q, and the other parameters are \(q=0\), \(L=4.7\), and \(B=5 \times 10^{-4}\); the initial separation is \(r=30\). c The two parameters are the particle angular momentum L and the black hole charge Q, and the other parameters are \(q=0\), \(E=0.998\) and \(B=4.5 \times 10^{-4}\); the initial separation is \(r=45\). The three panels clearly show that the degree of chaos increase with the increase of B and E or the decrease of L. However, a change of the black hole charge Q has no explicit effect on the dynamical transition from order to chaos

Full size image
Fig. 5
figure 5

Poincaré sections for the motions of neutral particles. The parameters are \(E=0.998\), \(L=4.5\) and \(Q=0.4\) in (a–c), but the magnetic fields are \(B=5\times 10^{-5}\) in (a), \(B=3.5\times 10^{-4}\) in (b), and \(B=6.5\times 10^{-4}\) in (c). (d-i) The parameters are \(E=0.998\) and \(L=4.5\). Given the magnetic field \(B=4\times 10^{-4}\), the black hole charges are \(Q=0.2\) in (d), \(Q=0.6\) in (e), and \(Q=0.8\) in (f). Unlike panels (d)–(f), panels (g)–(i) replace the magnetic field with \(B=5.5\times 10^{-5}\). The method of Poincaré sections shows the effect of one of the varying parameters B and Q on the dynamical transition, as the method of FLIs does in Fig. 4

Full size image
Fig. 6
figure 6

Figure 5 continued. The parameter values are \(B=5 \times 10^{-4} \), \(Q=0.4\) and \(L=4.7\), but the energy values are a \(E=0.996\), b \(E=0.9968\) and c \(E=0.998\). The parameter values are \(E=0.998 \), \(Q=0.4\), and \(B=4.5 \times 10^{-4}\), while the angular momentum values are d \(L=4.5\), e \(L=5.5\), and f \(L=6\). It is clear that the degree of chaos increases with the energy increasing, whereas decreases with the angular momentum increasing

Full size image
Fig. 7
figure 7

Same as Fig. 4, but the particle charge \(q=0\) is replaced with \(q=0.5\). a–c Correspond to Fig. 4a–c, respectively. Under some circumstances, chaos gets stronger as the magnetic field strength B and energy E increase, but weaker when the black hole charge Q or the particle angular momentum L increases

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Fig. 8
figure 8

Poincaré sections. The parameters are the particle energy \(E=0.998\) and the particle charge \(q=0.5\). The black hole charges are \(Q=0.2\) in panels (a), (d) and (g), \(Q=0.6\) in panels (b), (e) and (h), and \(Q=0.8\) in panels (c), (f) and (i). a–c The magnetic field strength is \(B=4\times 10^{-4}\), and the other parameters are those of Fig. 7a. d–f The energy is \(E=0.998\), and the other parameters are those of Fig. 7b. g–i The particle angular momentum is \(L=4.9\), and the other parameters are those of Fig. 7c. For the three cases, chaos becomes weaker as the black hole charge increases

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Fig. 9
figure 9

a The dependence of FLI on the particle positive charge q. The initial separation is \(r=40\), and the parameters are \(E=0.998\), \(L=5.7\), \(B=4\times 10^{-4}\), and \(Q=0.1\). b–d Poincaré sections. The particle charges are a \(q=0.15\), b \(q=0.5\) and c \(q=0.8\). With the increase of q, more orbits can be chaotic

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Fig. 10
figure 10

a FLIs for the particle negative charges q and the black hole positive charges Q. The other parameters and the initial separation are the same as those in Fig.  9. b–d Poincaré sections. The particle charge is \(q=-0.5\). The black hole positive charges are b \(Q=0.2\), c \(Q=0.5\), and d \(Q=0.8\). With the increase of Q, more chaotic orbits appear

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Fig. 11
figure 11

Poincaré sections. The black hole positive charge is \(Q=0.5\), and the particle negative charges are a \(q=-0.8\), b \(q=-0.5\) and c \(q=-0.15\). The other parameters are the same as those of Fig. 10. Chaos easily occurs with the magnitude of particle negative charge increasing

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The FLI is convenient to find chaos by scanning one or two parameter spaces. Taking the initial conditions \(\theta =\pi /2\), \(p_r=0\), \(r=30\) and the parameters \(E=0.998\), \(L=4.5\), we plot the dependence of FLIs on the parameters Q and B in Fig. 4a. When a pair of the values Q and B are given, each of the FLIs is obtained after the integration time \(w=1\times 10^6\). The FLIs not more than 5 correspond to ordered orbits, while those larger than 5 show chaotic orbits. The dynamical transition is sensitively dependent on varying the magnetic field strength B. As B increases, the degree of chaos is typically enhanced. However, the dynamical transition to chaos exhibits no sensitive dependence on a change of the black hole charge Q. This result is also shown in Fig. 4b, c. In addition, the occurrence of chaos is easier when the energy E increases or the angular momentum L decreases.

The Poincaré sections in Fig. 5 are used to check the results of Fig. 4. More orbits become chaotic and the strength of chaos increases when the magnetic field strength increases from \(B=5\times 10^{-5}\) in Fig. 5a to \(B=3.5\times 10^{-4}\) in Fig. 5b, and to \(B=6.5\times 10^{-4}\) in Fig. 5c. If a larger magnetic field \(B=4\times 10^{-4}\) is given, an increase of Q seems to have no typical effect on the dynamical transition from order to chaos in Fig. 5d–f. When a smaller magnetic field \(B=5.5\times 10^{-5}\) is fixed, the changes of Q in Fig. 5g–i seem to exert no explicit influences on the dynamics of neutral particles. The method of Poincaré sections in Fig. 6 is used to check the results of Fig. 4 regarding the influence of varying the particle energy E and angular momentum L on chaos. The increase of E in Fig. 6a–c leads to enhancing the extent of chaos, while that of L in Fig. 6a–c results in weakening the chaoticity of neutral particles. The results support those of Fig. 4.

In a word, the methods of FLIs and Poincaré sections give the consistent results regarding the effects of varying one or two parameters on the dynamical transition. The dependence of chaos on different parameters behaves well in the spirit of KAM (Kolmogorov–Arnold–Moser) theorem that is crucial for the transitions between the chaos and regularity. The regularity has to be related to the local minima of the effective potential of the motion in the equatorial plane \(\theta =\pi /2\). It is because the local minima correspond to the existence of stable circular orbits. The values in the vicinity of the local minima of the effective potential correspond to regular KAM tori. See Ref. [17] for more details on the minima of the effective potential. The astrophysical origin and role of these minima were discussed in [74]. As the parameters gradually increase, the KAM tori are twisted and some tori are destroyed. The absence of some tori brings the possibility for the occurrence of chaos. Of course, the fundamental reason for the chaoticity of neutral particles is that the magnetic field as gravitational effects causes the non-integrability of the spacetime geometry.

4.2 Dynamics of charged particles

For the motion of charged particles with \(q\ne 0\), the terms \(A_t\) and \(A_{\phi }\) are present in Eqs. (8)–(10). In this case, not only the Coulomb force and Lorentz force from the external electromagnetic field but also the gravitational forces from the magnetized black hole affect the motion of charged particles. Even the Lorentz force has an important contribution to the motion of charged particles around the black holes.

4.2.1 Particles with positive charges

For \(q=0.5\), the initial conditions and other parameters of Fig. 7a–c for using the FLIs to scan the two-dimensional parameters correspond to those of Fig. 4a–c, respectively. The degree of chaos is still strengthened when B and E increase or L decreases in Fig. 7a–c, as is in Fig. 4a–c.

The FLIs with respect to the two parameters in Fig. 7 show that the increase of Q suppresses the occurrence of chaos. The increase of Q weakening and suppressing the chaoticity of positive charged particles is also shown by the method of Poincaré sections in Fig. 8.

Unlike the increase of the black hole charge Q, the increase of the particle positive charge q can easily induce the occurrence of chaos. This result can be described clearly by the methods of FLIs and Poincaré sections in Fig. 9.

4.2.2 Particles with negative charges

Now, let us consider the motion of negative charged particles with \(q<0\).

Seen from the FLIs with respect to varying the black hole charge Q in Fig. 10a, chaos becomes stronger as Q increases for the particle having an appropriate negative charge. This result is also supported by the method of Poincaré sections in Fig. 10b–d, which take three different values of \(Q=\)0.2, 0.5 and 0.8, and give q the same value \(-0.5\).

When the black hole charge Q is given an appropriate value, the FLIs in Fig. 10a show that a larger absolute value of the particle negative charge q brings stronger chaos. For \(Q=0.5\), \(q=-0.15\), \(-0.5\), \(-0.8\) in Fig. 11 correspond to the degree of chaos from weak to strong, as shown through the method of Poincaré sections.

4.3 Theoretical analysis and explanations

The comparison among Figs. 4, 7 and  10a shows that the increase of the black hole charge Q exerts different influences on the dynamical transition from order to chaos for neutral, positive charged and negative charged particles. For neutral particles, a change of the black hole charge Q does not sensitively cause the dynamical transition from order to chaos under some circumstances. When the black hole charge Q increases, the degree of chaos is weakened for positive charged particles, whereas strengthened for negative charged particles. An increase of the magnitude of particle charge q can easily induce the occurrence of chaos regardless of whether the particle charges are positive or negative. As the magnetic field and the particle energy increase or the particle angular momentum decreases, chaos is always stronger for any one of the three types of test particles. In what follows, we analytically interpret the effects of varying one or two parameters on the dynamical transition from order to chaos.

For simplicity, the Hamiltonian K (17) for the description of the equatorial zero velocity motions of neutral or charged particles is considered, where \(\theta =\pi /2\) and \(p_r=p_\theta =0\). In this case, attractive forces balance repulsive forces. In addition, the dynamics of the Hamiltonian K is that of the sub-Hamiltonian \(K_1\) (18). Considering \(r\gg 2\), we expand \(K_1\) as follows:

$$\begin{aligned} K_1\approx & {} \frac{1}{2}\left( 1-E^2+B^2L^2-BqL\right) -\frac{E^2}{r}\left( 1+\frac{2}{r}\right) \nonumber \\{} & {} +\frac{1}{4}B^2r^2+\frac{1}{8}B^2q^2r^2 +\frac{2}{r}BQEL+\frac{1}{r}EQq \nonumber \\{} & {} +\frac{L^2}{2r^2}+\frac{1}{2r^2}E^2Q^2 +\frac{4}{r^2}BQEL +\frac{1}{2r^2}q^2Q^2 \nonumber \\{} & {} +\frac{2}{r^2}EqQ. \end{aligned}$$
(30)

The second term describes that the black hole gives an attractive force to a particle.

For the motion of a neutral particle with \(q=0\), the third term \(B^2r^2/4\) acts as an attractive force from the magnetic field. The terms with repulsive force contributions to the particle are the fifth term 2BQEL/r, seventh term \(L^2/(2r^2)\), eighth term \(E^2Q^2/(2r^2)\), and ninth term \(4BQEL/r^2\). Because \(E\sim 1\) (\(E<1\)), \(L>3\), \(0.1< Q\le 1\), and \(0\le B\ll 1\), the eighth term is more important than the fifth, ninth terms, but is denominated by the seventh term. An increase of the energy E or the magnetic field B leads to that of attractive forces, and therefore chaos is easily induced under some circumstances. As the angular momentum L increases, the repulsive force increases, and then the degree of chaos is weakened. With the black hole charge Q increasing, the eighth term exerts a small influence on the particle motion compared with the seventh term. This fact is why the dynamical transition of neutral particles from order to chaos does not sensitively depend on a change of the black hole charge.

When the particle positive charge q with \(q>0\) increases, the Lorentz force as an attractive force from the magnetic field of the fourth term \(B^2q^2r^2/8\) also increases. Therefore, chaos occurs easily. If Q increases for a given positive charge q, the Coulomb force as a repulsive force from the sixth term EQq/r increases, and is larger than the repulsive forces from the eighth term \(E^2Q^2/(2r^2)\), tenth term \(q^2Q^2/(2r^2)\) and eleventh term \(2EqQ/r^2\). As a result, chaos becomes weaker.

When the magnitude of particle negative charge q with \(q<0\) increases, the Coulomb forces as the attractive forces from the sixth, eleventh terms increase. Thus, chaos becomes stronger. Clearly, this result is also suitable for the increase of the black hole positive charge Q for a given negative charge q.

5 Conclusions

The Hamiltonian for describing the motion of neutral, or charged particles around the magnetized RN black holes cannot be split into several parts, which have analytical solutions as explicit functions of time. However, there are five explicitly integrable splitting pieces through an appropriate time transformation to the Hamiltonian. In this way, explicit symplectic integrators can be designed for the time-transformed Hamiltonian. These symplectic methods perform good numerical performance in long-term stabilized behavior of energy errors for suitable choices of new time steps. One of the integrators with the best performance is used to provide some insight into the dynamics of particles.

The dynamics of neutral particles around magnetized RN black holes is nonintegrable. This nonintegrability is because the external magnetic field reaches the upper limit of magnetic field modifying the spacetime structure, and acts as a gravitational effect destroying the integrability of the RN spacetime. It can be chaotic under some circumstances. With the magnetic field and the particle energy increasing or the particle angular momentum decreasing, chaos is easily induced. This result is also suitable for the motion of charged particles.

The effect of varying the black hole positive charge on the dynamical transition from order to chaos is dependent on the electric charges of test particles. A change of the black hole charge does not sensitively affect the dynamical transition of neutral particles. An increase of the black hole charge leads to weakening the chaoticity of positive charged particles, but to enhancing the chaoticity of negative charged particles. As the magnitude of particle charge increases, chaos always gets stronger regardless of whether the particle charges are positive or negative.