A new type of multi-branch periodic orbits in dyonic black holes

TL;DR

This paper uncovers a new class of multi-branch periodic orbits in dyonic black holes, revealing how non-monotonic metric functions lead to complex orbital structures and multiple bound orbit branches.

Contribution

It introduces the discovery of multi-branch periodic orbits linked to non-monotonic metric functions in dyonic black holes, a novel phenomenon not previously documented.

Findings

Non-monotonic metric functions enable multiple bound orbit branches.

Bound orbits with energy greater than unity are possible when potential barriers exceed one.

Outer orbit branches show increased eccentricity with higher energy or angular momentum.

Abstract

In this work, we investigate bound periodic orbits of timelike particles in the spacetime of dyonic black holes arising from quasi-topological electromagnetic theory. By varying the coupling parameter $\alpha_1$, the corresponding black hole solutions exhibit diverse horizon structures, including naked singularities and black holes with one to four horizons. We find that for sufficiently small $\alpha_1$, the metric function $f(r)$ becomes non-monotonic outside the event horizon in spacetimes with one or two horizons, while in all other cases, $f(r)$ remains strictly monotonic. In the non-monotonic regime, the radial effective potential develops a double-barrier structure, allowing the emergence of multiple marginally bound orbits and multiple branches of periodic orbits associated with the same rational number $l$. Although differing in radial structure, these orbit branches are topologically equivalent. Remarkably, when the outer potential barrier exceeds unity, bound orbits with energy $E>1$ become possible, in addition to the standard $E<1$ branches. When the peak reaches $E=1$, up to three distinct bound orbit branches may coexist. We also identify a novel eccentricity behavior, the innermost branch becomes increasingly circular with increasing energy or angular momentum, while outer branches exhibit greater eccentricity and a larger apastron-periastron separation. These features, absent in previous studies, are unique signatures of non-monotonic metric functions. In contrast, monotonic cases yield a single-well potential, a unique marginally bound orbit, and a single periodic orbit branch per $q$, consistent with earlier findings. Our results highlight the critical role of the metric function's shape in determining the orbital structure around dyonic black holes.

Tables

Table 1. Table 1: The particle energy E E corresponding to orbital trajectories with various ( z , w , v ) (z,w,v) configurations for L = L ISCO + ϵ ​ ( L MBO2 − L ISCO ) L=L_{\text{ISCO}}+\epsilon(L_{\text{MBO2}}-L_{\text{ISCO}}) with α 1 \alpha_{1} fixed at 1.025775. Each q q configuration is listed in the column headers, and the corresponding values of L L are given in the second column. A cross (✗) indicates that no valid orbit exists for the given parameters.

		$(1,2,0)$		$(1,3,0)$		$(2,2,1)$		$(2,3,1)$		$(3,2,2)$		$(4,2,3)$
$ϵ$	$L$	$E_1$	$E_2$	$E_1$	$E_2$	$E_1$	$E_2$	$E_1$	$E_2$	$E_1$	$E_2$	$E_1$	$E_2$
0.1	2.887166748	0.935814492	0.985785156	0.937080052	0.948715979	0.936829058	0.960301539	0.937143091	0.943093126	0.936953377	0.955441636	0.936997122	0.953435488
0.3	3.203663498	0.995674684	✗	0.996543502	✗	0.996404390	✗	✗	✗	0.996474484	✗	0.996500038	✗
0.5	3.520160249	1.062037703	✗	1.062811882	✗	1.062694729	✗	✗	✗	1.062754589	✗	1.062777002	✗
0.7	3.836656999	1.132310730	✗	1.133068444	✗	1.132955503	✗	✗	✗	1.133015013	✗	1.133034045	✗

Table 2. Table 2: The particle energy E E corresponding to orbital trajectories with various ( z , w , v ) (z,w,v) configurations for L = L ISCO + ϵ ​ ( L MBO3 − L ISCO ) L=L_{\text{ISCO}}+\epsilon(L_{\text{MBO3}}-L_{\text{ISCO}}) with α 1 \alpha_{1} fixed at 1.025775. Each q q configuration is listed in the column headers, and the corresponding values of L L are given in the second column.

		$(1,2,0)$			$(1,3,0)$			$(2,2,1)$			$(2,3,1)$			$(3,2,2)$			$(4,2,3)$
$ϵ$	$L$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
0.1	2.778089089	0.917706569	0.918695150	0.962500833	0.919426607	0.919522284	0.929805338	0.919025550	0.919323100	0.939783105	0.919554320	0.919586099	0.925081318	0.919212002	0.919415278	0.935558157	0.919282106	0.919449845	0.933832329
0.3	2.876430523	0.933956552	0.934749064	0.983481552	0.935253068	0.935301979	0.946811772	0.934992438	0.935191004	0.958256880	0.935318912	0.935332332	0.941259375	0.935120280	0.935244504	0.953453718	0.935163911	0.935264201	0.951472934
0.5	2.974771956	0.951473786	0.952160555	✗	0.952560004	0.952594514	0.964646195	0.952361091	0.952514085	0.977268010	0.952604995	✗	0.958557602	0.952460002	0.952554235	0.971977865	0.952494009	0.952568120	0.969792659
0.7	3.073113389	0.969959415	0.970582989	✗	0.970925007	0.970952015	0.983321465	0.970758972	0.970887129	0.996902526	✗	✗	0.976862452	0.970842031	0.970920062	0.991202146	0.970871005	0.970931232	0.988849501

Table 3. Table 3: The particle angular momentum L L corresponding to the orbital trajectories with various ( z , w , v ) (z,w,v) configurations for α 1 = 1.025755 \alpha_{1}=1.025755 . Each configuration q = q ​ ( z , w , v ) q=q(z,w,v) is listed in the column headers, and the corresponding energy values E E are shown in the first column.

	$(1,2,0)$		$(1,3,0)$		$(2,1,1)$		$(2,2,1)$		$(3,1,2)$		$(4,1,3)$
$E$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$
0.94	2.911065711	2.906791852	2.904131383	2.903860301	2.934562008	2.916424168	2.905491329	2.904436514	2.922730716	2.911626624	2.918685193	2.909963741
0.96	3.020700283	3.017204550	3.015210699	3.015045650	3.041666857	3.025306516	3.016195715	3.015440492	3.030890304	3.021177329	3.027285085	3.019777472
0.98	3.124822463	3.121750702	3.120096321	3.119973702	3.144268795	3.128983415	3.120894035	3.120279654	3.130806891	3.125238559	3.134147891	3.124002511
1	3.225074834	✗	3.220823184	✗	3.243570102	✗	3.221519980	✗	3.233851779	✗	3.230686335	✗
1.04	3.417512803	✗	3.413778979	✗	3.434947094	✗	3.414358748	✗	3.425683250	✗	3.422703440	✗
1.12	3.782175771	✗	3.778809934	✗	3.799028155	✗	3.779325975	✗	3.789956072	✗	3.787088046	✗

Table 4. Table 4: The particle energy E E corresponding to the orbital trajectories with various ( z , w , v ) (z,w,v) configurations. The angular momentum is defined by L = L ISCO + ϵ ​ ( L MBO2 − L ISCO ) L=L_{\text{ISCO}}+\epsilon(L_{\text{MBO2}}-L_{\text{ISCO}}) with α 1 \alpha_{1} =1.0494625. Each q q configuration is listed in the column headers, and the corresponding values of L L are given in the second column.

		$(1,2,0)$			$(1,3,0)$			$(2,2,1)$			$(2,3,1)$			$(3,2,2)$			$(4,2,3)$
$ϵ$	$L$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
0.1	3.056459520	0.961986182	0.962863298	✗	0.963161895	0.963199952	0.976458625	0.962960199	0.963141602	0.991538788	0.963204233	✗	0.969465858	0.963063815	0.963171552	0.985150758	0.963098545	0.963181595	0.982538037
0.3	3.659770325	1.084129625	✗	✗	1.084983352	✗	✗	1.084862758	✗	✗	✗	✗	✗	1.084928135	✗	✗	1.084948735	✗	✗
0.5	4.263081130	1.219086205	✗	✗	1.219921675	✗	✗	1.219810614	✗	✗	✗	✗	✗	1.219871637	✗	✗	1.219890395	✗	✗
0.7	4.866391935	1.360553203	✗	✗	1.361431569	✗	✗	1.361318179	✗	✗	✗	✗	✗	1.361380825	✗	✗	1.361400045	✗	✗

Table 5. Table 5: The particle energy E E corresponding to the orbital trajectories with various ( z , w , v ) (z,w,v) configurations for L = L ISCO + ϵ ​ ( L MBO3 − L ISCO ) L=L_{\text{ISCO}}+\epsilon(L_{\text{MBO3}}-L_{\text{ISCO}}) with α 1 \alpha_{1} =1.0494625. Each q q configuration is listed in the column headers, and the corresponding values of L L are given in the second column.

		$(1,2,0)$			$(1,3,0)$			$(2,2,1)$			$(2,3,1)$			$(3,2,2)$			$(4,2,3)$
$ϵ$	$L$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$	$E_1$	$E_2$	$E_3$
0.1	2.804294152	✗	0.920238738	0.970819502	0.920865882	0.920990212	0.932080884	0.920415550	0.920811935	0.943546088	0.921007423	✗	0.926839358	0.920627138	0.920894848	0.938640080	0.920705270	0.920925804	0.936654695
0.3	2.903274222	0.934895209	0.935979503	0.992153394	0.936403295	0.936469124	0.948903402	0.936108757	0.936371603	0.962001343	0.936477123	✗	0.942757533	0.936254558	0.936419528	0.956445532	0.936305446	0.936436444	0.954178025
0.5	3.002254292	0.952132783	✗	✗	0.953396152	0.953440678	0.966486906	0.953171454	0.953373042	0.980930592	0.953445795	✗	0.959740666	0.953285475	0.953407352	0.974814667	0.953324316	0.953419041	0.972313317
0.7	3.101234361	0.970315901	0.971155136	✗	0.971433341	0.971467522	0.984869328	0.971246555	0.971414858	✗	✗	✗	0.977701762	0.971343225	0.971442088	0.993833886	0.971375366	0.971451144	0.991136896

Table 6. Table 6: The particle angular momentum L L corresponding to orbital trajectories with various ( z , w , v ) (z,w,v) configurations for α 1 = 1.0494625 \alpha_{1}=1.0494625 . Each configuration q = q ​ ( z , w , v ) q=q(z,w,v) is listed in the column headers, and the corresponding energy E E is shown in the first column.

	$(1,2,0)$		$(1,3,0)$		$(2,1,1)$		$(2,2,1)$		$(3,1,2)$		$(4,1,3)$
$E$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$	$L_1$	$L_2$
0.94	2.933241600	2.927184798	2.924837584	2.924483753	2.962868216	2.936299105	2.926464096	2.925012509	2.947806223	2.931733665	2.942714947	2.930160432
0.96	3.045647950	3.040774987	3.039082022	3.038856906	3.072064747	3.048251620	3.040229385	3.039198367	3.058336392	3.044406231	3.053808659	3.043120311
0.98	3.152272147	3.148065496	3.146691474	3.146527455	3.176756286	3.154616778	3.147608345	3.146781257	3.163850512	3.151194916	3.159667943	3.150069695
1	3.254881142	✗	3.249884868	✗	3.278075887	✗	3.250668624	✗	3.265721158	✗	3.261765158	✗
1.04	3.451642293	✗	3.447323705	✗	3.473371643	✗	3.447961785	✗	3.461646717	✗	3.457953020	✗
1.12	3.824198510	✗	3.820306718	✗	3.844812199	✗	3.820833733	✗	3.833444448	✗	3.829965241	✗

Table 7. Table 7: The orbital energy E E and angular momentum L L for trajectories with different ( z , w , v ) (z,w,v) configurations. The first three rows show the values of E E for α 1 = 1.07317 \alpha_{1}=1.07317 , 1.08 1.08 , and 1.09859 1.09859 , respectively. The last three rows show the values of L L at E = 0.98 E=0.98 for the same α 1 \alpha_{1} . L α 1 = 1.07317 = 3.027183930 L_{\alpha_{1}=1.07317}=3.027183930 , L α 1 = 1.8 = 3.033956704 L_{\alpha_{1}=1.8}=3.033956704 and L α 1 = 1.09859 = 3.051567153 L_{\alpha_{1}=1.09859}=3.051567153 with ϵ = 0.5 \epsilon=0.5 .

$ϵ$	$E_{(1,1,0)}$	$E_{(1,2,0)}$	$E_{(1,3,0)}$	$E_{(2,1,1)}$	$E_{(2,2,1)}$	$E_{(3,1,2)}$	$E_{(3,2,2)}$	$E_{(4,0,3)}$	$E_{(4,1,3)}$
0.5	0.945878325	0.953847528	0.954177589	0.952542213	0.954119707	0.953216520	0.954149305	0.934681118	0.953440466
0.5	0.946215325	0.954054116	0.954373572	0.952779068	0.954317989	0.953438136	0.954346452	0.935131614	0.953657415
0.5	0.947065086	0.954582061	0.954875166	0.953378385	0.954824890	0.954003376	0.954850770	0.936265965	0.954210483
$E$	$L_{(1,1,0)}$	$L_{(1,2,0)}$	$L_{(1,3,0)}$	$L_{(2,1,1)}$	$L_{(2,2,1)}$	$L_{(3,1,2)}$	$L_{(3,2,2)}$	$L_{(4,0,3)}$	$L_{(4,1,3)}$
0.98	3.211221386	3.171898546	3.170540253	3.177923360	3.170755536	3.174749562	3.170642860	3.269318098	3.173709928
0.98	3.217062611	3.178348135	3.177050180	3.184270619	3.177253951	3.181149616	3.177147939	3.274821300	3.180129621
0.98	3.232663461	3.195143399	3.193950308	3.200784718	3.194138008	3.197783780	3.194038503	3.289379631	3.196824862

Table 8. Table 8: Orbital energy E E and specific angular momentum L L for trajectories with different ( z , w , v ) (z,w,v) configurations under a fixed parameter. The cases α 1 = 1.5 \alpha_{1}=1.5 and α 1 = 10 \alpha_{1}=10 , corresponding to four-horizon black holes, are listed in the first and second rows, respectively. L α 1 = 1.5 = 3.276343756 L_{\alpha_{1}=1.5}=3.276343756 and L α 1 = 10 = 3.555852800 L_{\alpha_{1}=10}=3.555852800 with ϵ = 0.5 \epsilon=0.5 .

$ϵ$	$E_{(1,1,0)}$	$E_{(1,2,0)}$	$E_{(1,3,0)}$	$E_{(2,1,1)}$	$E_{(2,2,1)}$	$E_{(3,1,2)}$	$E_{(3,2,2)}$	$E_{(4,0,3)}$	$E_{(4,1,3)}$
0.5	0.955410041	0.960297323	✗	0.959623897	0.960411281	0.959987512	0.960421785	0.947327593	0.960102089
0.5	0.962210601	0.965660402	✗	0.965228841	0.965725559	0.965466948	✗	0.955976069	0.965540382
$E$	$L_{(1,1,0)}$	$L_{(1,2,0)}$	$L_{(1,3,0)}$	$L_{(2,1,1)}$	$L_{(2,2,1)}$	$L_{(3,1,2)}$	$L_{(3,2,2)}$	$L_{(4,0,3)}$	$L_{(4,1,3)}$
0.98	3.435789732	3.408150797	3.407547608	3.411616491	3.407624314	3.409710362	3.415311555	3.483832955	3.409103862
0.98	3.693518265	3.670467517	✗	3.673110376	3.670113317	3.671590231	3.670087012	3.736994345	3.671159526

Equations

$$d{s}^2=-f(r)d{t}^2+\frac{1}{g(r)}d{r}^2+r^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right),$$

$$S=\frac{1}{16\pi }\int \sqrt{-g}{d}^{4}x\left[R-{\alpha }_1{F}^2-{\alpha }_2\left((F^2{)}^2-2F^{(4)}\right)\right],$$

$$f(r)=g(r)$$

$$\mathcal{L}=\frac{1}{2}{g}_{\mu \nu }\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }=-1,$$

$$p_{\mu }=\frac{\partial \mathcal{L}}{\partial x^{\mu }}.$$

$$p_t=f(r)\dot{t}=-E,p_{\varphi }=r^2\dot{\varphi }=L,$$

$$g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-1.$$

$$\dot{t}=\frac{E}{f(r)},\dot{\varphi }=\frac{L}{r^2},{\dot{r}}^2=E^2-V_{\mathrm{eff}}.$$

$$V_{\mathrm{eff}}=f(r)\left(1+\frac{L^2}{r^2}\right).$$

$$V_{\mathrm{eff}}=1,{\partial }_r{V}_{\mathrm{eff}}=0.$$

$$V_{\mathrm{eff}}=E^2,{\partial }_r{V}_{\mathrm{eff}}=0,{\partial }_{r,r}{V}_{\mathrm{eff}}=0.$$

$${\omega }_r=\frac{2\pi }{T_r},$$

$${\omega }_{\varphi }=\frac{1}{T_r}{\int }_0^{T_r}\frac{d\varphi }{dt}dt=\frac{\Delta {\varphi }_r}{T_r},$$

$$\frac{{\omega }_{\varphi }}{{\omega }_r}=\frac{\Delta {\varphi }_r}{2\pi }=1+q,$$

$$\Delta {\varphi }_r$$

$$q=w+\frac{v}{z},$$

$$1\le v\le z-1.$$

$$\Delta {\varphi }_r=2\pi (1+w+\frac{v}{z})=\frac{\Delta \varphi }{z},$$

$$L=L_{\mathrm{ISCO}}+ϵ(L_{\mathrm{MBO}}-L_{\mathrm{ISCO}}),$$

$$n=\frac{\Delta \varphi }{2\pi }=z(1+w+\frac{v}{z})=z(1+q).$$

Full text

Quasi-topological electromagnetic, Dyonic black, Periodic orbits

I Introduction

The dynamics of particle orbits around black holes not only reflect the geometric structure of the surrounding spacetime, but also give rise to a variety of observable gravitational phenomena, such as gravitational waves [1, 2] and black hole shadows [3, 4]. These observations provide vital tools for testing general relativity and probing alternative theories of gravity, especially in the strong-field regime. In both weak and strong gravity regions, understanding the fundamental nature of spacetime near compact objects critically depends on the study of a test particle’s geodesic motion.

In recent years, attention has been drawn to various novel orbital structures in non-Kerr backgrounds, such as rotating boson stars [5, 6, 7], wormholes [8], hairy black holes [9, 10], and higher-dimensional black holes [11, 12, 13]. These studies have revealed the existence of exotic features, including semi-orbits, pointypetal orbits, and static light points, which may leave detectable imprints on observational signatures. A growing body of literature [14, 15] highlights the potential of such structures to serve as novel probes of spacetime geometry.

Among these, periodic orbit trajectories of massive particles that return exactly to their initial positions after a finite time play a central role in gravitational dynamics [17]. These orbits are characterized by rational frequency ratios between radial and angular motion, enabling a unique labeling scheme via a rational number $q$. The periodic nature of such orbits not only facilitates analytical treatment, but also serves as a foundation for understanding general orbital behavior through small perturbations [17]. For instance, zoom-whirl phenomena observed near strong gravity regions can be effectively interpreted using periodic orbits, offering insights into the relativistic regime with relatively low computational cost.

Furthermore, periodic orbits are of critical importance in the context of extreme mass ratio inspirals (EMRIs), where they serve as transition states in the inspiral trajectory. In the adiabatic limit, they provide an efficient means to model EMRIs’ gravitational waveforms [16, 18]. Notably, Healy et al. [19] demonstrated that higher mass ratios prolong the premerger orbital phase, producing more frequent and sharper zoom-whirl features in the waveform. The characteristic peaks in these waveforms are directly associated with the rational parameter $q$, suggesting that algorithms optimized for burst searches may be employed to extract these signatures from data [19, 20]. As such, periodic orbit theory has found widespread application in a variety of backgrounds, including Schwarzschild [17, 21], Kerr [22, 23, 24, 25], charged black holes [26], black holes with dark matter halo [27], black bounce [28], Kehagias-Sfetsos black holes in deformed Hor̆ava-Lifshitz gravity [29], quantum black holes [30, 31, 32], and naked singularities [33]. Recent efforts continue to expand this framework to more exotic gravity theories [34, 35, 36, 38, 37], with a comprehensive summary provided in ref. [20].

A promising extension of this framework emerges in the context of quasi-topological electromagnetism, a modified theory that serves as a physically viable alternative to Maxwell electrodynamics, particularly in regimes inaccessible to the Earth laboratory tests [39]. In this setting, a novel class of black hole solutions known as dyonic black holes arises, characterized by both electric and magnetic charges and additional coupling parameters. These solutions exhibit rich causal and geometrical structures, making them a fertile ground for investigating new gravitational phenomena.

Intriguingly, dyonic black holes can support up to four event horizons and three photon spheres under certain parameter choices [39, 40], a marked departure from standard black hole spacetimes. The existence of multiple photon spheres significantly alters the optical appearance of the black hole, producing multiple bright rings in the shadow image [41] and enhancing total flux [42]. Moreover, static point orbits—locations where a massive particle remains at rest relative to a static observer at infinity can form in these spacetimes, resulting in dyson-like structures [44]. The corresponding Aschenbach effect has also been verified in dyonic black holes [45]. On one hand, topological arguments also demonstrate that stable and unstable static time-like circular orbits always appear in pairs [46]. In addition, geometrically thick equilibrium tori have been shown to exist in these backgrounds, with their structure strongly influenced by the interplay between electric and magnetic charges and the quasi-topological coupling parameter [47]. These systems may also exhibit violations of the universal chaos bound [48] and produce gravitational echoes without requiring artificial boundary conditions [49]. From a thermodynamic perspective, the presence of a triple point phase structure further underscores the rich microphysics of these black holes [50, 51].

These theoretical developments collectively suggest that dyonic black holes governed by quasi-topological electromagnetism provide a unique and compelling setting in which to study periodic orbits and their associated observational signatures. The interplay between novel spacetime structure, modified field equations, and complex orbital dynamics opens new possibilities for testing gravity in the strong-field regime.

Motivated by this, the present work aims to investigate the properties of periodic orbits of a massive particle in the background of dyonic black holes within quasi-topological electromagnetism. By analyzing the geometric structure and associated rational parameters of the periodic orbits, we seek to explore the potential observational imprints in gravitational waveforms and orbits of the particle. This study not only contributes to the theoretical understanding of modified gravity scenarios but also provides valuable templates for future observational efforts targeting EMRIs.

The paper is organized as follows. In Sec. II, we introduce the dyonic black hole solutions within quasi-topological electromagnetic theory and classify the black hole configurations according to their horizon structure. The geodesic equations and corresponding effective potentials for massive particles are also provided. Sec. III briefly reviews the concept and classification of periodic orbits. Sec. IV contains our main results, structured by the number of horizons, and explores the characteristics of periodic orbits in each case. Sec. V concludes with a summary and discussion of the implications of our results.

II Black Hole Solution and Geodesics

II.1 Black hole with quasi-topological electromagnetism

We begin with the general line element for a static, spherically symmetric black hole spacetime, given by

[TABLE]

where the metric functions $f(r)$ and $g(r)$ depend only on the radial coordinate $r$.

The spacetime considered in this work is the dyonic black hole solution proposed in Ref. [39], which arises within the framework of quasi-topological electromagnetism. The action governing an asymptotically flat solution is given by

[TABLE]

where $\alpha_{1}$ and $\alpha_{2}/M^{2}$ are non-negative dimensionless coupling parameters associated with the standard Maxwell term $F^{2}=F^{\mu\nu}F_{\mu\nu}$ and the quasi-topological correction, respectively. The higher-order term involves $F^{(2)}=F^{\mu}_{\;\;\;\nu}F^{\nu}_{\;\;\;\mu}=-F^{2}$ and $F^{(4)}=F^{\mu}_{\;\;\;\nu}F^{\nu}_{\;\;\;\rho}F^{\rho}_{\;\;\;\sigma}F^{\sigma}_{\;\;\;\mu}$. All quantities are expressed in geometric units, where $c=G=1$, making the action dimensionless. This theory introduces higher-order electromagnetic corrections to the standard Maxwell action. Under the assumption of global polarization, the quasi-topological term does not affect the Maxwell equations or the energy-momentum tensor, but it modifies the spacetime geometry nontrivially. In this background, the metric function is given by

[TABLE]

where ${}_{2}F_{1}$ denotes the hypergeometric function. The solution contains three integration constants: $M$, $q$, and $p$, corresponding to the black hole mass, electric charge, and magnetic charge, respectively. As discussed in ref. [39], the null, weak, and dominant energy conditions are satisfied when both $\alpha_{1}$ and $\alpha_{2}/M^{2}$ are positive, while the strong energy condition is violated. Note that when $\alpha_{1}=1$ and $\alpha_{2}/M^{2}=0$, the solution reduces to the Reissner–Nordström black hole.

Due to the presence of the hypergeometric term, the dyonic black hole exhibits a remarkably rich horizon structure. To explore this behavior, we fix the parameters as $M=1,~\alpha_{2}=2.76,~p=0.15,~q=1.05$ and vary the coupling constant $\alpha_{1}$ only. The representative behaviors of the metric function $f(r)$ under different values of $\alpha_{1}$ are shown in FIG. 1. The horizon structures can be categorized as follows:

• (1)

Naked singularity ($\alpha_{1}<1.025755$ or $\alpha_{1}>43.31557$): In these regimes, $f(r)>0$ for all $r>0$, indicating the absence of an event horizon. These cases are excluded from our further consideration and are shaded gray in the figure.

• (2)

Single-horizon black holes: At the critical values $\alpha_{1}=1.025755$ and $\alpha_{1}=43.31557$, $f(r)$ has a single zero, corresponding to a black hole with one degenerate event horizon.

• (3)

Two-horizon black holes: Two distinct parameter ranges yield double-horizon configurations. The first is $1.025755<\alpha_{1}<1.07317$, marked in light green, where the function $f(r)$ is non-monotonic outside the outer horizon and exhibits two local minima. The second is $1.09859<\alpha_{1}<43.31447$ marked in light orange, where $f(r)$ is monotonic outside the outer horizon.

• (4)

Three-horizon black holes: At $\alpha_{1}=1.07317$ and $\alpha_{1}=1.09859$, $f(r)$ admits three real positive roots, corresponding to black holes with three horizons. These are indicated by the dark teal blue and cyan curves in the figure.

• (5)

Four-horizon black holes ($1.07317<\alpha_{1}<1.09859$): In this narrow window, $f(r)$ has three extrema intersecting the axis $f(r)=0$, yielding four horizons. This regime is shaded in lavender blue, with a representative curve shown in blue.

From FIG. 1, one observes that in the range $\alpha_{1}\leq 1.025755<1.07317$, the metric function $f(r)$ is non-monotonic outside the event horizon, with two local minima. In contrast, for $\alpha_{1}>1.07317$, $f(r)$ becomes monotonic outside the outermost horizon. These transitions reflect the rich geometrical and causal structures of the dyonic black hole, governed by the coupling parameter $\alpha_{1}$.

The above analysis highlights the diversity of horizon structures in the dyonic black hole arising from quasi-topological electromagnetism. The number and nature of horizons depend sensitively on the coupling parameter $\alpha_{1}$, providing a dynamic platform to study black hole thermodynamics, causal structure, and geodesic behavior. In the following sections, we will investigate the properties of periodic time-like geodesics in this background, with particular focus on their potential signatures in periodic orbits.

II.2 Geodesics and effective potential

To study the motion of a test particle in the dyonic black hole background, we begin with the Lagrangian describing a freely falling massive particle,

[TABLE]

where $\tau$ denotes the particle’s proper time. Due to the spherical symmetry of the spacetime, without loss of generality, we can confine the motion to the equatorial plane by setting $\theta=\pi/2$. Then the generalized momentum $p_{\mu}$ are defined by

[TABLE]

Given the static, spherically symmetric nature of the spacetime, the metric (1) admits two Killing vectors: $\xi^{\mu}=(\partial_{t})^{\mu}$ and $\psi^{\mu}=(\partial_{\phi})^{\mu}$. Consequently, there are two conserved quantities along geodesics: the energy $E$ and angular momentum $L$ per unit mass, given by

[TABLE]

where the dot denotes the derivative with respect to the proper time. For a massive particle, the normalization condition of the four-velocity leads to

[TABLE]

Combining these relations yields the first-order equations of motion in the $t$, $r$, and $\phi$ directions:

[TABLE]

Here the radial effective potential $V_{\mathrm{eff}}$ reads

[TABLE]

To understand the bound motion of the particles, one needs to analyze the structure of the effective potential. In particular, we focus on two critical types of circular orbits. (1) Marginally bound orbit (MBO): This orbit represents the unstable circular trajectory at the threshold between bound and unbound motion. It satisfies the following requirements

[TABLE]

(2) Innermost stable circular orbit (ISCO): This orbit marks the innermost radius at which stable circular motion is possible. The conditions for ISCO are

[TABLE]

These characteristic orbits provide a basis for understanding the dynamics of massive particles in strong gravitational fields and are particularly relevant in the analysis of periodic orbits in the dyonic black hole spacetime.

III Periodic orbits

Periodic orbits are a special class of bound geodesics in which a test particle returns exactly to its initial position after a finite proper time [17]. In the context of a static, spherically symmetric black hole, where the motion is confined to the equatorial plane, $\theta=\pi/2$ , the particle’s trajectory involves only the radial $r$ and azimuthal $\phi$ directions. Each non-circular (eccentric) equatorial orbit is characterized by two fundamental orbital frequencies, $\omega_{r}$ and $\omega_{\phi}$. The radial frequency $\omega_{r}$, which quantifies the periodic motion between the periastron and apastron, is given by

[TABLE]

where $T_{r}$ is the radial period—the time taken to travel from apastron to periastron and back. The azimuthal frequency $\omega_{\phi}$, defined as the time-averaged rate of angular advance in the $\phi$-direction over one radial period, is expressed as

[TABLE]

where $\Delta\phi_{r}$ is the accumulated azimuthal angle during a single radial cycle.

For generic orbits, the ratio $\omega_{\phi}/\omega_{r}$ is irrational. However, for periodic orbits, this frequency ratio must be rational, and can be written in the form [17]

[TABLE]

where $q$ is a positive rational number known as the rational number label of the orbit. The total azimuthal deviation during one radial cycle is $\Delta\phi_{r}$

[TABLE]

where $r_{a}$ and $r_{p}$ are the apastron and periastron radii, respectively.

On the other hand, the rational number $q$ can be uniquely expressed in the following canonical form:

[TABLE]

where $w\geq 0$ is the integer part and $v/z$ is the irreducible fractional part with coprime integers $v$ and $z$ satisfying

[TABLE]

Each periodic orbit is thus uniquely specified by a triplet $(z,~w,~v)$, or equivalently, by the rational number $q$, which captures the amount of azimuthal advance per radial cycle. This leads to the geometric relation [17]

[TABLE]

where $\Delta\phi$ is the total azimuthal angle accumulated over one complete orbit. The relation $\Delta\phi=z\Delta\phi_{r}$ implies that the particle completes exactly $z$ radial oscillations while executing an integral number of revolutions in the azimuthal direction.

Periodic orbits serve as fundamental building blocks in understanding the geodesic structure of strong gravitational fields. They not only capture the core dynamics of bound motion, but also provide key insights into the generation of gravitational radiation in EMRIs and the geometric fingerprints observable in black hole shadows and relativistic precession phenomena.

IV Result

In what follows, we investigate the periodic orbits of massive test particles in the dyonic black hole background by systematically varying the coupling parameter $\alpha_{1}$, which controls the number of horizons of the black hole. As shown in FIG. 1, different ranges of $\alpha_{1}$ correspond to black holes with one, two, three, or four horizons. These structural changes in the causal geometry have a profound impact on the geodesic dynamics.

For each horizon configuration, we analyze in detail the corresponding properties of bound geodesic orbits, including the positions of the horizons, the location and angular momentum of the MBO and the ISCO, and the parameter space $(E,L)$ that allows for bound orbits. We further identify the rational number $q$ associated with periodic orbits and illustrate representative trajectories.

This classification highlights how the evolution of horizon structure driven by variations in $\alpha_{1}$ affects the existence and characteristics of periodic orbits. These results deepen our understanding of relativistic motion in quasi-topological dyonic black hole spacetimes and provide potential theoretical support for identifying observable signatures in strong gravity regimes.

IV.1 One and Two-Horizon: Case A $1.025755\leq\alpha_{1}<1.07317$

We begin our analysis with the cases where the dyonic black hole possesses either a single horizon or two distinct horizons. As shown in FIG. 1, the black hole admits a single event horizon at $\alpha_{1}=1.025755$, while for $1.025755<\alpha_{1}<1.07317$, the spacetime transitions to a two-horizon configuration. In this regime, the metric function $f(r)$ becomes non-monotonic in the region outside the outer event horizon, giving rise to richer geodesic structures.

To characterize the horizon behavior, FIG. 2 illustrates the evolution of the outer event horizon radius $r_{h}$ as a function of $\alpha_{1}$, with other parameters fixed as $M=1,~\alpha_{2}=2.76,~p=0.15$, and $q=1.05$. The red dot on the left marks the one-horizon configuration at $\alpha_{1}=1.025755$, while the blue curve corresponds to two-horizon black holes for $1.025755<\alpha_{1}<1.07317$. The red open circle on the right marks the threshold $\alpha_{1}=1.07317$, beyond which a third horizon emerges, and the corresponding value of $r_{h}$ lies outside the plotted range.

With the horizon structure established, we proceed to examine the behavior of bound orbits in these cases. First, we consider the MBO, which are determined by Eq. (10). FIG. 3 shows the dependence of the radial position $r_{\mathrm{MBO}}$ and angular momentum $L_{\mathrm{MBO}}$ on the parameter $\alpha_{1}$ in the interval $1.025755\leq\alpha_{1}<1.07317$. The red point at the left end indicates the one-horizon case, while the solid curves represent the multi-branch structure of MBOs in the two-horizon regime. The red open circle denotes the endpoint at $\alpha_{1}=1.07317$, which is not included in the numerical integration.

Due to the non-monotonic nature of $f(r)$ outside the horizon, three distinct branches of marginally bound orbits emerge. The first branch (orange curves) corresponds to the smallest $r_{\mathrm{MBO}}$ and $L_{\mathrm{MBO}}$. Interestingly, the second branch (red curves) has a larger radius than the first but a higher angular momentum than the third branch (blue curves), which has the largest radius. As $\alpha_{1}$ increases, both $r_{\mathrm{MBO1}}$ and $r_{\mathrm{MBO3}}$ shift outward, while $r_{\mathrm{MBO2}}$ decreases. Nonetheless, all three corresponding angular momenta increase monotonically with $\alpha_{1}$.

Next, we analyze the ISCO, which are determined by Eq. (11). FIG. 4 shows the behavior of the ISCO radius $r_{\mathrm{ISCO}}$, angular momentum $L_{\mathrm{ISCO}}$, and energy $E_{\mathrm{ISCO}}$ as functions of $\alpha_{1}$. For the single-horizon case, we find $r_{\mathrm{ISCO}}=3.667453494$, $L_{\mathrm{ISCO}}=2.728918373$, and $E_{\mathrm{ISCO}}=0.912789070$. As $\alpha_{1}$ increases, leading to the appearance of a second horizon, all three quantities $r_{\mathrm{ISCO}}$, $L_{\mathrm{ISCO}}$, and $E_{\mathrm{ISCO}}$ increase monotonically.

These numerical results confirm that the emergence of a second horizon and the associated non-monotonicity of the metric function significantly enrich the structure of bound orbits, leading to multi-branched solutions for marginally bound orbits and a smooth shift in ISCO properties. These features are expected to leave distinctive imprints on the dynamics of periodic orbits.

As evident from Eq. (9), the behavior of the effective potential $V_{\mathrm{eff}}$ is governed not only by the metric function $f(r)$, but also, crucially, by the orbital angular momentum $L$ of the test particle. Once the angular momenta corresponding to the MBO and the ISCO are known, the complete structure of the effective potential can be reconstructed.

In FIG. 5, we present the effective potentials $V_{\mathrm{eff}}$ for the dyonic black hole in the case where the metric function exhibits non-monotonic behavior outside the event horizon ($1.025755\leq\alpha_{1}<1.07317$). By substituting into Eq. (9), the potentials are computed using the three branches of angular momentum $L_{\mathrm{MBO}}$ obtained from FIG. 3(b), and the corresponding $L_{\mathrm{ISCO}}$ values from FIG. 4(b). The horizontal axis represents the normalized radial coordinate $r$, and the vertical axis denotes $V_{\mathrm{eff}}$.

The first and second rows of FIG. 5 correspond to $\alpha_{1}=1.025755$ (single-horizon) and $\alpha_{1}=1.0494625$ (two-horizon), respectively. Each column shows the effective potential associated with a different branch of $L_{\mathrm{MBO}}:~L_{\mathrm{MBO1}},~L_{\mathrm{MBO2}},~L_{\mathrm{MBO3}}$, from left to right. In each panel, the red curve denotes $L=L_{\mathrm{ISCO}}$, the blue curve corresponds to $L=L_{\mathrm{MBO}}$, and intermediate colored curves represent varying $L$ in the range $L_{\mathrm{ISCO}}<L<L_{\mathrm{MBO}}$. Insets zoom into the near-horizon and asymptotic regions, highlighting the fine structure of the potential landscape.

A notable contrast is observed in the first column, where the effective potential exhibits inverse behavior compared to the second and third columns: $V_{\mathrm{eff}}$ decreases with increasing $L$. This arises from the atypical ordering $L_{\mathrm{MBO1}}<L_{\mathrm{ISCO}}$, which does not occur in standard black holes with monotonic $f(r)$. In contrast, for $L_{\mathrm{MBO2}},~L_{\mathrm{MBO3}}>L_{\mathrm{ISCO}}$, the effective potential increases with $L$ as expected.

Due to this anomaly, our analysis in the following will focus only on the second and third branches, which exhibit more standard behavior. In these cases, the effective potentials show the emergence of multiple extrema: two local maxima and two local minima, corresponding to two unstable and two stable timelike circular orbits, respectively. This structure is consistent with the topological classification of circular orbits presented in Ref. [46]. It is important to emphasize, however, that the inner local minimum does not correspond to an ISCO, as it does not satisfy the condition $\partial_{r,r}V_{\mathrm{eff}}=0$. It is therefore classified as a marginally stable circular orbit [46].

Further insights can be gained by examining the energy landscape. In both the second and third columns, the effective potential forms double-well structures separated by twin barriers. The outer potential barrier is generally lower than the inner one. For the configuration constructed from $L_{\mathrm{ISCO}}$ to $L_{\mathrm{MBO2}}$, the outer barrier can exceed unity, allowing bound orbits to exist in both wells when $E<1$. However, when $E>1$ only the inner well can support bound motion, since the potential asymptotically approaches $V_{\mathrm{eff}}\to 1$ at spatial infinity.

For the configuration associated with $L_{\mathrm{MBO3}}$ and $L_{\mathrm{ISCO}}$, the outer barrier reaches $V_{\mathrm{eff}}=1$ only when $L=L_{\mathrm{MBO3}}$, suggesting a narrower energy range for trapping particles in the outer well. Nevertheless, for sufficiently small angular momentum $L$ , the effective potential can develop three distinct bound regions: two near the two wells, and an additional one in between, as detailed in FIG. 5(b). This rich structure implies that the system can support up to three coexisting bound orbits for certain values of energy and angular momentum.

In conclusion, the non-monotonic behavior of the metric function $f(r)$ outside the horizon leads to the formation of twin potential wells and barriers in the effective potential. These, in turn, give rise to multiple branches of bound orbits, including multiple photon rings [39, 40] and stable/unstable time-like circular orbits [44, 45, 46]. Such features are not present in spacetimes with monotonic metric functions and play a crucial role in determining the diversity of periodic bound orbits. It is therefore reasonable to expect the appearance of multiple families of periodic orbits in this regime, as will be demonstrated in subsequent sections.

With the effective potential analysis established, we now determine the allowed parameter space for bound orbits of massive test particles. By combining the results from FIGs. 3 and 4, we identify the regions in the $(E,L)$ plane that admit bound motion. Specifically, bound orbits occur only when the particle’s energy and angular momentum lie within a triangular region defined by the MBOs and ISCOs. Outside this region, no bound states are permitted.

In particular, for a fixed value of $L$, the allowed energy range for bound orbits is finite. Furthermore, this energy width increases monotonically with increasing $L$. Based on conditions $\dot{r}=0$ and $\partial_{r}\dot{r}=0$, we numerically extract the domains permissible $(E,L)$ for the two branches defined by $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$ and $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$, as illustrated in FIG. 6.

An especially intriguing feature arises in FIGs. 5(b) and 5(e), where the outer potential barrier is low enough to allow the formation of a potential well even when the particle’s energy is greater than 1. The range of allowed energies in FIG. 6(a) is well in agreement with the height of the outer potential barrier in these figures, which supports the physical interpretation. In addition, the angular momentum values corresponding to the bound states in FIGs. 3 and 4. For example, at $\alpha_{1}=1.025755$, the range of allowed angular momentum lies between $L_{\mathrm{ISCO}}=2.728918373$ and $L_{MB02}=4.311402125$, consistent across all relevant figures. On the other hand, for the branch $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$, as shown in FIG. 3, $L_{MBO3}=3.220625539$, in agreement with the bound region shown in FIG. 6(b).

In summary, the non-monotonic behavior of the metric function outside the horizon leads to a rich structure in the effective potential, enabling the existence of multiple bound orbit branches, each characterized by distinct energy and angular momentum intervals. Moreover, the possibility of bound orbits with energy exceeding unity further highlights the exotic nature of the dyonic black hole in this regime, setting it apart from more conventional black hole solutions.

We now turn to the behavior of periodic orbits in the dyonic black hole spacetime with one or two horizons. In particular, we focus on the dependence of the rational number $q$, which characterizes periodic orbits, on the particle’s energy $E$ and angular momentum $L$. According to Eq. (14), $q$ can be written as $q=\Delta\phi_{r}/2\pi-1$, where $\Delta\phi_{r}$ is given by Eq. (15) and depends explicitly on the conserved quantities of the particle and the background metric function $f(r)$. Since all other parameters are fixed, variations in $f(r)$ are controlled solely by the parameter $\alpha_{1}$.

To explore the structure of periodic orbits, we examine the behavior of $q$ as a function of the particle’s energy $E$ for different angular momenta. The angular momentum is parametrized using

[TABLE]

where the parameter $\epsilon\in(0,1)$ corresponds to bound orbits. The limits $\epsilon=0$ and $\epsilon=1$ correspond to $L=L_{\mathrm{ISCO}}$ and $L=L_{\mathrm{MBO}}$, respectively. For $\epsilon>1$ , bound orbits no longer exist.

In FIG. 7, we illustrate the behavior of $q$ as a function of energy $E$ for various values of $\epsilon$ and two representative cases of $\alpha_{1}$, corresponding to single- and double-horizon configurations. The left column shows the case $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$, while the right column represents $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$.

Several notable features emerge. First, for lower angular momentum (i.e., smaller $\epsilon$), multiple branches of $q$ appear for the same energy, especially when $E<1$. This is a direct manifestation of the non-monotonicity of the metric function $f(r)$ outside the event horizon, which induces a double-well structure in the effective potential. This behavior allows the existence of up to three distinct branches of $q$ for the same energy, indicating multiple bound periodic orbits—a feature absent in spacetimes with monotonic $f(r)$.

Secondly, in the case $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$, we observe that bound orbits with $E>1$ are also permitted, consistent with the allowed regions in the $(L,E)$ plane for the bound orbits shown earlier in FIG. 6(a). Furthermore, beyond the standard monotonic growth of $q$ with $E$, we also observe branches where $q$ decreases as energy increases. These inverted branches tend to converge to a particular value of energy as $q\to\infty$, suggesting clustering of periodic orbits in energy space.

On the other hand, the complementary view of $q$ as a function of angular momentum for fixed energy is provided in FIG. 8. For energies $E<1$, two distinct branches of $q(L)$ exist, while for $E\geq 1$, only one branch is observed. In all cases, $q$ decreases monotonically with increasing $L$, further reflecting the intricate structure of bound motion enabled by the non-monotonicity of $f(r)$.

These results demonstrate that the rational number $q$, which labels periodic orbits, encodes rich information about the underlying geometry and the allowed orbital configurations. The presence of multiple branches of $q$ underscores the existence of multiple classes of periodic orbits, consistent with the presence of multiple effective potential wells. This provides strong evidence that the non-monotonicity of the metric function plays a fundamental role in shaping the landscape of periodic motion in dyonic black holes.

With the numerical results of $E-q$ and $L-q$ in hand, we can now identify specific rational numbers $q$ by choosing the appropriate triplets $(z,w,v)$ according to Eq. (16). Each rational value of $q$ uniquely determines a class of periodic orbits, characterized by a zoom–whirl–vertex structure in the particle’s trajectory [17]. These structural features are intimately related to the number of zooms (radial oscillations), whirls (near-circular angular loops), and vertices (turning points in the orbital path) during one full orbit.

To clearly illustrate how these periodic trajectories behave in the case of non-monotonic metric functions with one or two horizons, we examine the trajectories in two separate subsections. These will correspond to configurations with angular momentum ranges $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$ and $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$, respectively. By selecting representative rational numbers $q=w+v/z$ for each scenario, we visualize the corresponding periodic orbits and analyze their dynamical properties in detail.

This subsection will present the explicit trajectory structures of bound timelike periodic orbits in dyonic black hole backgrounds with one and two horizons, focusing on cases where the metric function is non-monotonic outside the event horizon. The results reveal the rich phenomenology induced by the interplay between the black hole geometry and the orbital dynamics of the test particle.

IV.1.1 Trajectories of periodic orbits with single horizon

• •

Periodic orbits for $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$ in the single-horizon case

We begin by analyzing periodic trajectories of massive particles in the dyonic black hole spacetime with a single horizon, corresponding to the case $\alpha_{1}=1.025755$. Specifically, we examine the configuration where the particle’s angular momentum lies within the range $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$, as indicated in FIGs. 5(b) and 6(a). In this angular momentum range, bound orbits with energy $E>1$ are allowed.

Based on the $E-q$ diagram in FIG. 7(a), the values of energy corresponding to selected rational numbers $q$ labeled by $(z,w,v)$ for several values of the interpolation parameter $\epsilon\in(0,1)$, which parametrizes angular momentum via Eq. (19) are shown in Tab. 1. When $\epsilon$ is small (e.g. $\epsilon=0.1$), two distinct values of energy exist for each rational $q$, corresponding to two distinct branches of bound periodic orbits; these are denoted as $E_{1}$ and $E_{2}$ in the table. In contrast, at higher values of $\epsilon$, only a single branch persists, and the second energy solution ceases to exist, as marked by ✗. Furthermore, the data indicate that for a fix $q$, the energy required for a periodic orbit increases monotonically with increasing angular momentum.

Using the values of $L$ and $E$ listed in Tab. 1, the particle trajectories are computed by solving the equations of geodesic Eq. (8). The resulting orbits are displayed in FIG. 9, where the radial and angular positions are mapped onto the Cartesian coordinates $(rcos\phi,~rsin\phi)$. This figure illustrates how, even for energies $E>1$, the orbits remain closed and periodic. When $\epsilon=0.1$, two distinct orbits can be observed for the same rational $q$, while for larger angular momentum values, only one trajectory exists.

To further investigate the difference between the two branches of orbits at $\epsilon=0.1$, we decompose the trajectories into their radial and angular components. In FIG. 10, we plot the radial coordinate $r$ as a function of the accumulated azimuthal angle $\Delta\phi/2\pi$. The two panels correspond to the two branches: (a) for energy $E_{1}$ and (b) for energy $E_{2}$. Despite the distinct values of $E$ , the total accumulated angle $\Delta\phi$ emains identical, confirming that both orbits share the same rational number $q$, and hence the same topological structure.

This observation is consistent with the fundamental structure of periodic orbits: for each rational number $q=w+v/z$, the total angular advance over one full radial cycle is $\Delta\phi=z2\pi(1+q)$, which implies that the number of full revolutions around the black hole during one complete orbit is

[TABLE]

For example, $q=(2,2,1)$, yields $n=7$ turns, while $q=(4,2,3)$ gives $n=15$, consistent with the total azimuthal phase observed in FIG. 10.

This confirms that although two different energies can support orbits with the same rational number $q$, they correspond to trajectories of identical winding number and topological type. The presence of these multiple branches arises from the non-monotonic behavior of the metric function outside the horizon, which induces a multi-well structure in the effective potential and allows for multiple bound periodic solutions with the same orbital classification.

• •

Periodic orbits for $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$ in the single-horizon case

We now consider periodic orbits in the single-horizon dyonic black hole spacetime with angular momentum in the extended range $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$. As shown in FIG. 7(b), we list in Tab. 2 the energy $E_{1}$, $E_{2}$, and $E_{3}$ for several representative rational numbers $q=q(z,w,v)$ at varying angular momentum levels (parametrized by $\epsilon\in(0,1)$).

The table clearly shows that for fixed $q$, the energy increases with increasing angular momentum $L$, and that each $E_{i}$ represents one of the multiple branches of bound orbits allowed by the non-monotonic effective potential. This three-branch structure is a direct manifestation of the double-well (or double-barrier) nature of the potential in the non-monotonic geometry.

FIG. 11 illustrates the corresponding periodic orbits for the parameter sets in Tab. 2, focusing on $\epsilon=0.1$ and $\epsilon=0.3$. For clarity, orbits are plotted using different styles, and for $\epsilon=0.1$, black dotted segments are used to highlight one radial period (excluding single leaves) of motion. This plotting style enables direct visualization of the zoom, whirl, and vertex characteristics of the orbits, as encoded by the integers $(z,w,v)$. Our classification here follows the scheme proposed in Ref. [17].

(1) Zoom number $z$: The number of leaves (aphelia) in the orbit. For example, $z=1$ indicates a single-leaf orbit (e.g., $(1,~2,~,0)$, $(1,~3,~0)$, while $z=2,~3,~4$ correspond to orbits with 2, 3, or 4 aphelia, respectively. These are clearly visible in the $r-\phi$ plane. (2) Whirl number $w$: The number of rapid revolutions around the center between two aphelia. This can be seen from the spiral-like motion near perihelion. For example, the orbit labeled $(1,3,0)$ completes three full whirls within one radial period. The dotted-line segments in the figures help track these revolutions. (3) Vertex number $v$: Indicates the position of the next aphelion (vertex of a polygon) in the sequence, counting from the initial one labeled $v=0$. When $z\geq 2$, the orbit can be thought of as tracing a regular polygon with $z$ vertices. For example, in the $(3,2,2)$ orbit, the particle proceeds from the initial vertex to the second (out of three total), while in $(4,2,3)$, it reaches the last vertex in a four-vertex structure within one radial cycle. These definitions align with Eq. (17), which constrains $v\in[0,~z-1]$ (For the single leaf case, since $z=1$, then $v$ can only be 0).

The above classification ensures that all orbits with the same rational number $q$ — regardless of which energy branch $E_{i}$ they belong to — share the same topological structure. Although their geometrical shapes may appear different (e.g., more circular or more eccentric), the number of leaves, whirls, and polygonal vertices remain unchanged. This observation is further supported by FIG. 12 and Eq. (20), which represents the number of full revolutions completed by the particle during one complete orbit.

Furthermore, as evident from Tab. 2, for a given $q$, the sequence $E_{1}<E_{2}<E_{3}$ holds consistently, with increasing energy associated with larger radial excursion (i.e., greater eccentricity). Likewise, for each fixed branch $E_{i}$, increasing $\epsilon$ leads to higher energy and larger angular momentum, indicating that all branches shift consistently with the particle’s orbital parameters.

In summary, the presence of three distinct but topologically equivalent orbits for the same rational number $q$ is a direct consequence of the multi-well structure of the effective potential in dyonic black holes with non-monotonic geometries. The $(z,w,v)$ classification provides a powerful and precise language to describe and distinguish the zoom-whirl structures of these periodic trajectories.

• •

Periodic orbits at fixed energy $E$ in the single-horizon case

To further examine the rationality of the angular momentum range used in FIG. 8(a), we list in Tab. 3 the corresponding values of $L$ for each $q$ configuration at selected energies. When the particle energy satisfies $E<1$, two distinct values of angular momentum $L_{1}$ and $L_{2}$ are generally found for a given $q$, corresponding to the inner and outer branches of the effective potential well. For $E>1$ , only a single value of $L$ exists, as the effective potential supports only one turning point in this regime. From the data in Tab. 3, we observe that for a fixed rational number $q$, the required angular momentum increases monotonically with energy. This aligns with physical intuition: higher-energy particles require larger angular momentum to maintain periodic orbits with the same orbital structure.

FIG. 13 illustrates representative periodic trajectories for various $q$ at fixed energies, based on the tabulated values. For a given energy level, the two branches of orbits exhibit distinct spatial characteristics. The inner branch, associated with higher angular momentum $L_{1}$ , lies closer to the black hole and exhibits smaller radial excursions between perihelion and aphelion. As a result, these orbits appear more circular. In contrast, the outer branch, corresponding to lower angular momentum $L_{2}$, lies farther from the black hole and shows larger radial variation, manifesting as more eccentric trajectories.

To examine the angular and radial dynamics in greater detail, the orbital phase evolution for energies $E=0.94$,0.96, 0.98 is presented in FIG. 14. This consistency is in precise agreement with the theoretical prediction of Eq. (20), which determines the angular advancement over one radial cycle solely based on the triplet $(z,w,v)$.

In summary, at fixed energy $E<1$, each rational number $q$ admits two topologically equivalent but geometrically distinct periodic orbits due to the double-well nature of the effective potential. Their angular momentum values differ, leading to different radial extents, yet their angular motion and zoom-whirl structure remain invariant. This again highlights the robustness and completeness of the $(z,w,v)$ classification scheme for periodic motion in curved spacetimes.

IV.1.2 Trajectories of periodic orbits with two horizons

• •

Periodic orbits for $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$ in the two-horizon case

As shown in FIG. 1, the metric function $f(r)$ of the dyonic black hole is non-monotonic outside the event horizon not only in the single-horizon case at $\alpha_{1}=1.025755$, but also in the range $1.025755<\alpha_{1}<1.07317$, where the spacetime possesses two horizons. To examine the implications of this non-monotonic behavior in the two-horizon regime, we focus on a representative example with $\alpha_{1}=1.0494625$, and investigate the properties of periodic orbits for massive test particles.

Based on the allowed region in the $E-L$ parameter space shown in FIG. 7(c), we display in Tab. 4 the energy branches $E_{i}$ for different combinations of $(z,w,v)$, under the effective potential structure for angular momentum values $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$ (cf. FIG. 5(e)). Due to the richer potential structure in the two-horizon scenario, an additional periodic orbit solution arises in the low angular momentum regime ($\epsilon=0.1$), compared to the single-horizon case. However, as the angular momentum increases ($\epsilon=0.3,~0.5,~0.7$), the number of viable periodic orbits decreases, and eventually only a single branch remains. From the table, we observe a general trend: the energy corresponding to a given $q=q(z,w,v)$ increases with increasing angular momentum, consistent with the behavior of the effective potential.

Utilizing the data from Tab. 4, we numerically integrate the geodesic equations to construct the trajectories of periodic orbits for each $q$. The results are shown in FIG. 15. For lower angular momentum values $\epsilon=0.1$, we observe three distinct periodic orbits for each rational $q$, associated with energies $E_{1}<E_{2}<E_{3}$. As the energy increases, the radial excursion of the orbit becomes more pronounced, and the eccentricity grows correspondingly.

A particularly notable case arises for $(z,w,v)=(2,2,1)$ at $\epsilon=0.1$, where the third energy branch $E_{3}=0.991538788$ corresponds to an orbit with a remarkably large radial range, stretching from a periastron at $r_{p}=0.709311641M$ to an apastron at $r_{a}=113.319643083M$. This reflects the highly non-monotonic structure of the effective potential in this parameter range. Moreover, due to the depth and width of the potential well, we find that closed periodic orbits can still exist even for energies $E>1$, particularly at higher values of $\epsilon$, such as 0.3, 0.5, 0.7.

Although the appearance of multiple periodic solutions for the same $q$ only occurs in the low angular momentum regime ($\epsilon=0.1$), we analyze the structure of these orbits in more detail. In FIG. 16, we present the radial and angular evolution of orbits corresponding to the three energy branches $E_{i}$ for selected $q$. The change in the direction of $\phi$ consistent with the value of $n$ derived from Eq. (20), which depends solely on the integers $(z,w,v)$.

• •

Periodic Orbits for $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$ in the two-horizon case

We now turn to the case of $\alpha_{1}=1.0494625$, where the dyonic black hole possesses two horizons and exhibits non-monotonic behavior of the metric function $f(r)$ outside the outer horizon. In this range, we consider periodic orbits of massive particles with angular momentum constrained to the interval $L\in[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$. According to FIG. 6(b), the particle energy must satisfy $E<1$ for bound orbits to exist in this configuration.

Following the numerical results shown in FIG. 7(d), we extract the energy levels $E_{i}$ associated with $q$ for different values of the angular momentum parameter $\epsilon$. These values are listed in Tab. 5. As expected, for a fixed $q$, the energies $E_{i}$ increase monotonically with increasing angular momentum, consistent with the shape of the effective potential in this domain.

Using the data given in Tab. 5, we compute the periodic orbit trajectories for various $(z,w,v)$ configuration in the two-horizon background with $\alpha_{1}=1.0494625$, as shown in FIG. 17. From the top row of FIG. 17 with $\epsilon=0.1$, we observe that, when three periodic solutions exist for the same $q$, the solution closest to the black hole (i.e., with smallest radial extent) corresponds to the lowest energy and lowest eccentricity. As the particle energy increases, the radial extent of the orbit grows, and so does the orbital eccentricity.

To further analyze these orbits, we plot in FIG. 18 the radial and angular evolutions over a full cycle for all three branches at two representative angular momentum values $L=2.804294152$ and $L=2.903274222$. We confirms the topological equivalence of the motion despite their geometric differences by Eq. (20).

Combining FIGs. 17 and 18, we observe that increasing the particle energy at fixed angular momentum leads to a significant increase in the radial span of the orbit. For instance, at $\epsilon=0.1$, the radial excursion increases from approximately $r\in[0.6~M,~3~M]$ for $E_{1}$, to $r\in[3.2~M,~6~M]$ for $E_{2}$, and up to $[1~M,~30~M]$ for $E_{3}$.

Moreover, as the angular momentum increases (e.g., from $\epsilon=0.1$ to $\epsilon=0.3$), we observe a nontrivial shift in the radial behavior of the orbits. Specifically, the radial range for the lowest energy branch $E_{1}$ becomes narrower, whereas those for $E_{2}$ and $E_{3}$ expand significantly. Taking $E_{1}$ as an example, the radial range shifts from roughly $[0.7~M,~3~M]$ at $\epsilon=0.1$ to $[0.7~M,~2.8~M]$ at $\epsilon=0.3$. However, for $E_{2}$, the radial domain changes from approximately $[3.2~M,~6~M]$ at $\epsilon=0.1$ to $[3~M,~10~M]$ at $\epsilon=0.3$; similarly, for $E_{3}$ the radial range grows from $[0.7~M,~12~M]$ to nearly $[0.7~M,~120~M]$. This indicates a rich and nontrivial dependence of the orbital structure on the particle’s angular momentum, particularly in the multi-branch regime (refer to FIG. 12).

• •

Periodic orbits at fixed energy $E$ in the two-horizon case

In this final subsection, we investigate periodic orbits in the dyonic black hole background with two horizons by fixing the particle’s energy $E$. Based on the $L-q$ relation shown in FIG. 8(b), we extract the corresponding values of angular momentum $L$ for various rational numbers $q=q(z,w,v)$, as summarized in Tab. 6. It is evident that two distinct periodic orbits exist only when $E<1$; for $E>1$, only a single periodic orbit is supported for each $q$. Moreover, for a fixed $q$, the angular momentum increases monotonically with the particle’s energy.

Using the energy and angular momentum values listed in Tab. 6, we compute the periodic trajectories in the $r-\phi$ plane for various $q$ -configurations at $\alpha_{1}=1.0494625$. These results are presented in FIG. 19. The figure demonstrates the spatial structure of the orbits at fixed energies for different rational numbers $q$, covering both the multi-branch ($E<1$) and single-branch ($E>1$) regimes.

To further elucidate the orbital dynamics, we show in FIG. 20 the angular phase evolution of the trajectories from FIG. 19, again for different $(z,w,v)$ combinations. According to Eq. (20), the number of revolutions $n$ remains invariant for a given $q$, regardless of whether $E<1$ or $E>1$, confirming that the topology of the orbits in the $r-\phi$ plane is solely determined by $q$.

From FIGs. 19 and 20, several interesting properties emerge. For fixed particle energy $E<1$, two distinct periodic orbits exist for the same $q$. Among these, the orbit with smaller angular momentum $L_{2}$ corresponds to a broader radial extent and higher eccentricity, while the orbit with larger angular momentum $L_{1}$ has a narrower radial range and lower eccentricity. For example: at $E=0.94$, the orbit with $L_{1}$ spans roughly $r\in[0.7~M,~2.7~M]$, while $L_{2}$ gives $r\in[2.5~M,~10.5~M]$. At $E=0.96$, $L_{1}$ corresponds to $r\in[0.7~M,~2.6~M]$, while $L_{2}$ gives $r\in[3~M,~19~M]$. At $E=0.98$, $L_{1}$ yields $r\in[0.7~M,~2.5~M]$, and $L_{2}$ expands to $r\in[2~M,~45~M]$.

Additionally, for a fixed rational number $q$, we observe contrasting trends in the radial excursion depending on which orbit branch is followed. For the $L_{1}$ branch, as the particle energy increases, the radial motion range narrows, and the eccentricity decreases. In contrast, for the $L_{2}$ branch, higher particle energy leads to a broader radial motion range and an increase in eccentricity. Referring to FIGs. 14 and 19. This dichotomy mirrors the contrasting trends observed earlier when the angular momentum was held fixed: in the $L_{1}$ branch, increasing $E$ leads to the reduction of the radial range, while in the $L_{2}$ branch, increasing $E$ widens the radial domain. Together, these phenomena underscore the distinctive orbital structure arising from the non-monotonic nature of the metric function $f(r)$ outside the outer horizon in the two-horizon dyonic black hole. Although the precise physical mechanism behind this reversed trend in radial motion remains unclear, the existence of such behavior is a notable and intriguing feature that warrants further theoretical investigation.

We conclude Sec. IV.1 with a summary of the periodic orbits in the dyonic black hole background characterized by a non-monotonic metric function $f(r)$ outside the event horizon. Specifically, for the parameter range $1.025755\leq\alpha_{1}<1.07317$, the function $f(r)$ exhibits non-monotonic behavior beyond the outer horizon, leading to novel orbital phenomena not present in the monotonic case. We first determine the values of the event horizon $r_{h}$ as a function of $\alpha_{1}$ shown in FIG. 2. Due to the non-monotonicity of $f(r)$, the analysis within this parameter range reveals the existence of three distinct MBO solutions, as shown in FIG. 3. These solutions enable the construction of multi-well effective potentials, in contrast to the single-well structure found in standard black hole spacetimes. By combining the MBO structure with the behavior of the ISCO, presented in FIG. 4, we identified two physically distinct effective potential configurations that are relevant for periodic orbits: $L=[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$ and $L=[L_{\mathrm{ISCO}},L_{\mathrm{MBO3}}]$, both supporting double-well effective potentials (see FIG. 5). The effective potential configuration for $L=[L_{\mathrm{MBO1}},L_{\mathrm{ISCO}}]$, although mathematically interesting, exhibits potential shapes not associated with normal closed bound orbits and was therefore not discussed further in this work (FIGs. 5(a) and 5(d)). Our study focused on the dynamical consequences of the double-well potentials, particularly when the particle energy satisfies $E<1$. Under this condition, up to three distinct periodic orbits may exist for a given rational number $q=w+v/z$. This multi-valued structure is clearly reflected in the $E-q$ diagrams as well as the corresponding orbital trajectories (FIGs. 9, 11, 13, 15, 17, and 19). Such behaviors—namely, the coexistence of multiple periodic solutions for the same rational number $q$, and their distinct radial profiles under varying angular momentum—are exclusive to the $f(r)$ non-monotonic scenario. They highlight the profound influence of the black hole’s spacetime geometry on the dynamics of test particles, and underscore the importance of considering full multiparameter families of solutions when characterizing periodic motion in strong gravitational fields. To characterize the orbital structure, we introduced the winding number $n$, defined in Eq. (20), which counts the number of revolutions the particle completes in the azimuthal direction over one radial cycle. We verified through essential taxonomy reference [17] and $n-r$ $(n=\Delta\phi/2\pi)$ diagrams that all orbits corresponding to the same $q$ share identical topological structures in the $r-\phi$ plane, regardless of whether the energy is below or above unity. Partial orbit segments between successive apocenters were highlighted using dashed black lines in several trajectory figures to visualize individual radial cycles, following the classification scheme of Ref. [17]. The roles of the integers $z$, $w$, $v$ in characterizing the zoom, whirl, and vertex behaviors of the orbits were also clarified. An additional notable feature arises in the configuration $L=[L_{\mathrm{ISCO}},L_{\mathrm{MBO2}}]$, where periodic orbits with $E>1$ are also admitted. The parameter space for such high-energy periodic orbits is presented in FIG. 6(a). Despite the topological consistency among orbits with the same $q$, our analysis reveals a remarkable inversion in the radial extent trends among different branches of periodic solutions. Specifically, for the innermost branch (labeled $E_{1}$ or $L_{1}$), the radial range (from periastron to apastron) decreases as either the energy or angular momentum increases. In contrast, the outer branches (e.g., $E_{2}$, $L_{2}$ and $E_{3}$, $L_{3}$) exhibit the expected trend of increasing radial range with rising energy or angular momentum. This behavior is demonstrated in multiple FIGs (e.g., FIGs. 12 and 18) and was also observed under fixed angular momentum scenarios (e.g., FIGs. 14 and 19). This inverse radial trend—wherein the innermost orbit contracts as energy or angular momentum increases—is a distinctive consequence of the non-monotonic metric structure in the outer region of the two-horizon dyonic black hole. While the underlying mechanism driving this phenomenon remains unclear, it represents an intriguing deviation from classical expectations and merits further investigation in future work.

IV.2 Three and Four-Horizon Case: $1.07317\leq\alpha_{1}\leq 1.09859$

In this subsection, we investigate the properties of periodic orbits of massive particles around dyonic black holes with three or four horizons. According to FIG. 1, when the parameter $\alpha_{1}$ takes values 1.07317 and 1.09859, the black hole has three horizons; for $1.07317<\alpha_{1}\leq 1.09859$, it possesses four horizons. Notably, in the 3-/4- horizon regime, the metric function $f(r)$ is monotonic outside the event horizon.

First, the relationship between the event horizon radius and the parameter $\alpha_{1}$ in the range $1.07317\leq\alpha_{1}\leq 1.09859$ is depicted in FIG. 21. The hollow points on the right side of FIG. 2 corresponds to the left endpoint in FIG. 21, marked by red dots. We observe a jump in the horizon radius at $\alpha_{1}=1.07317$, where $r_{h}=0.868635078~M$, indicating a transition to the three-horizon black hole. The blue curve in FIG. 21 traces the horizon radius variation for four-horizon configurations within $1.07317<\alpha_{1}<1.09859$. The right endpoint marked by a red dot corresponds to $\alpha_{1}=1.09859$, with three horizons and event horizon radius $r_{h}=1.063747094~M$.

Next, we focus on MBOs, which are among the critical circular orbits with maximal energy. Using the definition in Eq. (10), FIG. 22 shows the radial positions and corresponding angular momenta of MBOs for $1.07317\leq\alpha_{1}<1.09859$. The red points on both sides represent MBO radii and angular momenta for the three-horizon black hole, while the interior points correspond to four-horizon configurations. Both $r_{\mathrm{MBO}}$ and $L_{\mathrm{MBO}}$ increase with $\alpha_{1}$.

Since $f(r)$ is monotonic outside the horizon and MBO corresponds to the maximal bound orbit energy $E=1$, bound orbits with $E<1$ reside outside the MBO radius, i.e., at $r>r_{\mathrm{MBO}}$. These orbits include stable and unstable bound orbits, distinguished by the sign of the second radial derivative of the effective potential, ${\partial_{r,r}V_{\mathrm{eff}}}$. Stability is ensured for ${\partial_{r,r}V_{\mathrm{eff}}}>0$, instability otherwise. The critical condition ${\partial_{r,r}V_{\mathrm{eff}}}=0$ defines the ISCO, whose radius $r_{\mathrm{ISCO}}$ satisfies Eq. (11). In FIG. 23, we plot $r_{\mathrm{ISCO}}$, $L_{\mathrm{ISCO}}$, and $E_{\mathrm{ISCO}}$ as functions of $\alpha_{1}$ for the 3-/4- horizon dyonic black hole, showing monotonic increase with $\alpha_{1}$.

Combining these results with the effective potential expression (9), FIGs. 24(a), 24(b), and 24(c) display radial effective potentials for various angular momenta, corresponding to three- and four-horizon cases. Unlike the more complex multi-well potentials in earlier sections, these effective potentials feature single-peaked structures. The maximum value equals one and corresponds to the MBO, consistent with typical behavior in single-photon-sphere black holes.

FIG. 25 illustrates the allowed regions in the $E-L$ plane for massive particle bound orbits near dyonic black holes with 3-/4- horizons. The red and magenta triangular areas denote parameter ranges for three-horizon black holes, while the remaining regions correspond to four-horizon cases. As $\alpha_{1}$ increases continuously from 1.07317 to 1.09859, the permitted angular momentum range broadens and the energy range narrows.

We then analyze periodic orbits characterized by the rational number $q$, describing the relation between particle energy $E$, angular momentum $L$, and orbit topology. FIG. 26 presents $q$ as a function of $E$ (top row) for angular momentum parameters $\epsilon=$0.1, 0.3, 0.5, 0.7, 0.9 and as a function of $L$ (bottom row) for fixed $E$, within the parameter space shown in FIG. 25. We observe that $q$ increases slowly with $E$ until near the maximum energy, where it sharply diverges; this maximum increases with $\epsilon$. In contrast, $q$ decreases monotonically with increasing $L$, diverging positively as $L$ approaches its minimum, which itself grows with $E$.

Specific periodic orbits corresponding to rational values $q=q(z,w,v)$ are extracted from FIG. 26 and summarized in Tab. 7. For fixed $\epsilon=0.5$, energies and angular momenta increase with $\alpha_{1}$ for the same $q$, reflecting the continuous deformation of black hole parameters.

FIG. 27 visualizes periodic orbits at $\alpha_{1}=1.08$, with $\epsilon=0.5$ and $E=0.98$, for various $(z,w,v)$ triplets. The number of leaves in the orbital pattern corresponds to $z$, increasing complexity with larger $z$. The black dashed segments highlight partial trajectories within one radial period, illustrating zoom, whirl, and vertex behaviors consistent with the classification of Ref. [17]. Orbits at fixed angular momentum have narrow radial excursions and small eccentricities; those at fixed energy display wider radial ranges and larger eccentricities. Similar trajectory structures arise for $\alpha_{1}=1.07317$ and $\alpha_{1}=1.09859$ but are omitted for brevity.

Finally, FIG. 28 shows radial variation versus azimuthal angle for periodic orbits under fixed $\epsilon$ or fixed $E$. The radial motion range depends sensitively on the choice of fixed parameter. Nonetheless, from Eq. (20) and these orbital phase plots, it is clear that even in the four-horizon dyonic black hole background, periodic orbit trajectories are fully and uniquely characterized by the rational number $q=q(z,w,v)$.

We conclude this subsection with the following summary. Within the parameter range $1.07317\leq\alpha_{1}\leq 1.09859$, although the dyonic black hole may exhibit configurations with either three or four horizons, the analysis is significantly simplified by the fact that the metric function $f(r)$ remains monotonic outside the event horizon. In this section, we first determined the location of the event horizon as a function of $\alpha_{1}$, followed by computations of the MBO radius $r_{\mathrm{MBO}}$ and the ISCO radius $r_{\mathrm{ISCO}}$. We then constructed the radial effective potential $V_{\mathrm{eff}}(r)$ within the range $L\in[L_{\mathrm{ISCO}},~L_{\mathrm{MBO}}]$, and identified the corresponding allowed region in the $E-L$ parameter space for bound orbits. Subsequently, we analyzed the dependence of the rational number $q$, characterizing periodic orbits, on the particle’s energy $E$ and angular momentum $L$. For selected sets of integers $(z,w,v)$, we extracted the corresponding values of $E$ and $L$, and plotted the associated orbital trajectories. Finally, by examining the radial–angular phase profiles in $r-\Delta\phi/2\pi$ diagrams, we verified that, under both fixed-energy and fixed-angular-momentum conditions, periodic orbits associated with the same rational number $q$ are topologically equivalent in the $r-\phi$ plane.

IV.3 One and Two-Horizon: Case B $1.09859<\alpha_{1}\leq 43.31447$

In this subsection, we investigate the properties of periodic orbits for dyonic black holes in the orange-shaded parameter region of FIG. 1. Our focus here is restricted to the range $1.09859<\alpha_{1}\leq 43.31447$. We begin by examining the behavior of the event horizon radius $r_{h}$ as a function of $\alpha_{1}$, as shown in FIG. 29. The left red endpoint corresponds to $\alpha_{1}=1.09859$, where the black hole possesses three horizons, while the right red endpoint marks $\alpha_{1}=43.31447$, corresponding to a single-horizon black hole with $r_{h}=1.000318866~M$. The blue curve in between represents black holes with two horizons. Within this range, the horizon radius $r_{h}$ exhibits non-monotonic dependence on $\alpha_{1}$: it increases with $\alpha_{1}$ up to a maximum $r_{h}=1.827707121~M$ at $\alpha_{1}=6.995555544$, and subsequently decreases as $\alpha_{1}$ increases. At $\alpha_{1}=43.31447$, the black hole transitions from two horizons to one, and the metric function $f(r)$ becomes monotonic outside the event horizon, in contrast to the non-monotonic behavior observed near $\alpha_{1}=1.025755$.

Using the definition of the MBOs in Eq. (10), we computed the corresponding orbital radius $r_{\mathrm{MBO}}$ and angular momentum $L_{\mathrm{MBO}}$ across the full parameter range. Their dependence on $\alpha_{1}$ is plotted in FIG. 30, where both quantities also display non-monotonic behavior.

Similarly, following Eq. (11), we obtained the ISCO radius $r_{\mathrm{ISCO}}$, angular momentum $L_{\mathrm{ISCO}}$, and energy $E_{\mathrm{ISCO}}$, all as functions of $\alpha_{1}$. As shown in FIG. 31, these quantities also vary non-monotonically with $\alpha_{1}$.

To further illustrate the effective potential landscape, we selected three representative values: $\alpha_{1}=1.5$, $10$, and $43.31447$ (the single-horizon limit). The radial effective potential $V_{\mathrm{eff}}(r)$ for angular momenta $L\in[L_{\mathrm{ISCO}},~L_{\mathrm{MBO}}]$ is plotted in FIG. 32. All three cases exhibit qualitatively similar single-peak profiles, with a single extremum at $L=L_{\mathrm{ISCO}}$ corresponding to the ISCO, and a local maximum reaching $V_{\mathrm{eff}}(r_{\mathrm{MBO}})=1$ at $L=L_{\mathrm{MBO}}$, indicative of the MBO.

The allowed regions in the $(L,E)$ plane for bound orbits are displayed in FIG. 33. In FIG. 33(a), we present the allowed regions for $\alpha_{1}$ ranging from 1.1 to 6.99556. As $\alpha_{1}$ increases, the corresponding allowed parameter space contracts monotonically. In contrast, FIG. 33(b) shows the case for $\alpha_{1}$ values from 10 to 43.31447, where the allowed region expands with increasing $\alpha_{1}$.

Next, we turn to study the behavior of the rational number $q$, which characterizes periodic orbits, as a function of particle energy $E$ and angular momentum $L$. Results for $\alpha_{1}=1.5$ and $\alpha_{1}=10$ are shown in FIG. 34. In both cases, the trends of $E-q$ and $L-q$ curves closely resemble those observed in the 3-, 4-horizon configurations discussed in the previous section.

Tab. 8 presents numerical values of energy $E$ and angular momentum $L$ for several periodic orbits labeled by different $(z,w,v)$ combinations. These values are computed for fixed angular momentum parameter $\epsilon=0.5$ and energy $E=0.98$, under two parameter choices: $\alpha_{1}=1.5$ and $\alpha_{1}=10$. As the orbital structures at $\alpha_{1}=1.5$ and $\alpha_{1}=10$ resemble those already depicted in FIG. 27, we do not present the trajectories again. Likewise, for $\alpha_{1}=43.31447$, the monotonicity of $f(r)$ ensures that the behaviors of $E-q$, $L-q$, and corresponding trajectories remain consistent with the generic single-horizon case, and are therefore omitted.

In this section, we analyzed periodic orbits in dyonic black hole spacetimes within the parameter range $1.09859<\alpha_{1}\leq 43.31447$, which includes both two-horizon and single-horizon configurations. As $\alpha_{1}$ increases, the horizon radius $r_{h}$ exhibits a non-monotonic dependence on $\alpha_{1}$, leading to similar non-monotonic trends in the associated orbital quantities such as $r_{\mathrm{MBO}}$, $L_{\mathrm{MBO}}$, $r_{\mathrm{ISCO}}$, $L_{\mathrm{ISCO}}$, and $E_{\mathrm{ISCO}}$. The energy–angular momentum parameter space for bound orbits also varies non-monotonically with $\alpha_{1}$. We further examined periodic orbits for $\alpha_{1}=1.5$ and $\alpha_{1}=10$, and found that the behavior of these orbits is qualitatively similar to that observed in the 3-/4- horizon case.

V Conclusions and discussions

In this work, we have investigated the properties of periodic orbits for massive particles in the background of dyonic black holes arising from quasi-topological electrodynamics. Focusing on a family of black hole solutions parametrized by $\alpha_{1}$, we varied this parameter continuously from $\alpha_{1}=1.025755$ to $\alpha_{1}=43.31447$, thereby spanning configurations with 1-/2-, 3-/4-, and again 1-/2- horizon structures (as illustrated in FIG. 1). This range gives rise to a rich set of dyonic black hole geometries, offering fertile ground for exploring periodic orbital motion.

We began by analyzing the geodesic motion of massive particles confined to the equatorial plane. The radial equation of motion was recast in terms of an effective potential $V_{\mathrm{eff}}(r)$, which can be used to define the MBOs and ISCOs. By constraining the angular momentum $L$ of particles within the range $L\in[L_{\mathrm{ISCO}},~L_{\mathrm{MBO}}]$, we ensured that the motion of the test massive particle was bounded between the potential wells, making periodic orbits possible. These orbits were then studied in the framework developed by Levin et al. [17], using rational numbers $q=q(z,w,v)=w+v/z$ that relate the radial and azimuthal oscillation frequencies through three integers $(z,w,v)$. The corresponding geodesic equations governing such periodic orbits were also established.

We then examined the behavior of periodic orbits across the different horizon structures identified by varying $\alpha_{1}$:

1. For $1.025755\leq\alpha_{1}<1.07317$ (1-/2- horizon black holes): The metric function $f(r)$ exhibits non-monotonicity outside the event horizon, resulting in a double-barrier effective potential structure. In this regime, three MBO solutions emerge, though we only focused on the two outer ones, $L_{\mathrm{MBO2}}$ and $L_{\mathrm{MBO3}}$, which define the dual potential wells. The inner potential barrier can exceed the critical value $V_{\mathrm{eff}}=1$, while the outer barrier reaches exactly $V_{\mathrm{eff}}=1$ at $L=L_{\mathrm{MBO3}}$. These double-well structures support multiple bound trajectories for $E<1$, and remarkably, also allow bound orbits for $E>1$, which is uncommon in single-well potentials. This multi-valued behavior is reflected in the non-monotonic $E-q$ and $L-q$ curves (see FIGs. 7 and 8), leading to multiple periodic orbits associated with the same rational number $q$. We presented explicit trajectories corresponding to both the multi-branch $q$ values and the $E>1$ single-branch cases. Moreover, using the triplet $(z,w,v)$, we defined a new winding number $n$ (20) characterizing the number of azimuthal loops per radial cycle, and plotted the radial turning behavior via $r-n$ diagrams. These results demonstrate that all branches corresponding to the same rational number $q$ are topologically equivalent, whether $L$ or $E$ is held fixed. Interestingly, the radial oscillation width behaves oppositely in the inner and outer periodic orbits as $E$ or $L$ varies. These features—multiple periodic orbits for the same $q$, and the existence of $E>1$ bound orbits—are uniquely tied to the non-monotonicity of $f(r)$, much like the emergence of multiple photon spheres or classes of time-like circular orbits in similar double-barrier systems [40, 46].

2. For $1.07317\leq\alpha_{1}\leq 1.09859$ (3-/4- horizon black holes): The metric function becomes monotonic outside the event horizon. The effective potential is of the standard single-well type, supporting only one MBO solution. Consequently, only a single branch of periodic orbits is permitted, all with energy $E<1$. These periodic orbits exhibit regular behavior, with $E-q$ and $L-q$ curves being monotonic.

3. For $1.09859<\alpha_{1}\leq 43.31447$ (1-/2- horizon black holes): Although the black hole possesses only one or two horizons, the function $f(r)$ remains monotonic outside the horizon. However, the radius of the event horizon itself varies non-monotonically with $\alpha_{1}$, which in turn leads to non-monotonic variations in the MBO radius $r_{\mathrm{MBO}}$, ISCO radius $r_{\mathrm{ISCO}}$, and the associated $E-L$ parameter regions. Still, due to the monotonicity of $f(r)$, only a single MBO solution exists, and the effective potential remains single-peaked. Periodic orbits are only supported for $L\in[L_{\mathrm{ISCO}},~L_{\mathrm{MBO}}]$ and $E<1$, resulting again in a unique periodic orbit per rational $q$.

In summary, our study reveals that the structure of periodic orbits in dyonic black hole spacetimes is highly sensitive to the horizon structure and the behavior of the metric function $f(r)$. In particular, non-monotonicity of $f(r)$ outside the horizon—possible in multi-horizon configurations—gives rise to novel phenomena such as multi-branch periodic orbits and bound orbits with energy exceeding unity. These results offer new insights into particle dynamics and potential observables in modified gravity theories involving nontrivial black hole geometries.

Acknowledgements.

This work was supported by the National Natural Science Foundation of China (Grants No. 12475055, and No. 12247101), the Basic Research Foundation of Central Universities (Grant No. lzujbky-2024-jdzx06), the Natural Science Foundation of Gansu Province (Grant No. 22JR5RA389), and the ‘111 Center’ under Grant No. B20063. Yu-Peng Zhang was supported by the “Talent Scientific Fund of Lanzhou University”. Tao Zhu is supported by the National Natural Science Foundation of China under Grants No. 12275238, the Zhejiang Provincial Natural Science Foundation of China under Grants No. LR21A050001 and No. LY20A050002, the National Key Research and Development Program of China under Grant No. 2020YFC2201503, and the Fundamental Research Funds for the Provincial Universities of Zhejiang in China under Grant No. RF-A2019015.