The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits

Niraj Pathak; V. O. Thomas; Elbaz I. Abouelmagd

doi:10.3934/dcdss.2019057

Article Contents

2019, Volume 12, Issue 4&5: 849-875. Doi: 10.3934/dcdss.2019057 open access

This issue Previous Article A mathematical analysis for the forecast research on tourism carrying capacity to promote the effective and sustainable development of tourism Next Article An independent set degree condition for fractional critical deleted graphs

The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits

1.
Department of Mathematics, Dharmsinh Desai University, Nadiad, Gujarat 3870001, India
2.
Department of Mathematics, The Maharaja Sayajirao University of Baroda, Vadodara, 390002 Gujarat, India
3.
Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, King Abdulaziz University, Jeddah, Saudi Arabia
4.
Celestial Mechanics Unit, Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan 11421, Cairo, Egypt

* Corresponding author: Elbaz I. Abouelmagd
* Corresponding author: Elbaz I. Abouelmagd

Received: June 2017

Revised: November 2017

Published: March 2019

Abstract / Introduction Full Text(HTML) Figure(25) / Table(12) Related Papers Cited by

Abstract

In the framework of the perturbed photo-gravitational restricted three-body problem, the first order exterior resonant orbits and the first, third and fifth order interior resonant periodic orbits are analyzed. The location, eccentricity and period of the first order exterior and interior resonant orbits are investigated in the unperturbed and perturbed cases for a specified value of Jacobi constant C.

It is observed that as the number of loops increases successively from one loop to five loops, the period of infinitesimal body increases in such a way that the successive difference of periods is either 6 or 7 units. It is further observed that for the exterior resonance, as the number of loops increases, the location of the periodic orbit moves towards the Sun whereas for the interior resonance as the number of loops increases, location of the periodic orbit moves away from the Sun. Thereby we demonstrate that the location of resonant orbits of the given order moves away from the Sun when perturbation is included.

The evolution of interior first order resonant orbit with three loops is studied for different values of Jacobi constant C. It is observed that when the value of C increases, the size of the loop decreases and degenerates finally into a circle, the eccentricity of periodic orbit decreases and location of the periodic orbit moves towards the second primary body.

Keywords:

Photogravitational restricted three-body problem,
periodic orbits,
interior resonance,
exterior resonance,
oblateness,
Poincaré surface of section.

Mathematics Subject Classification: 37N05, 70F07, 70F15.

Citation:

FullText(HTML)

Figure 1. Exterior first order resonant single loop orbit and PSS for $C = 2.93$ in the Sun - Earth system

Download: Full-size image PowerPoint slide

Figure 2. Exterior first order resonant two loops orbit and PSS for $C = 2.93$ in the Sun - Earth system

Download: Full-size image PowerPoint slide

Figure 3. Interior first order resonant two loops orbit and PSS for $C = 2.93$ in the Sun - Earth system

Download: Full-size image PowerPoint slide

Figure 4. Variation in three loops orbit due to interior first order resonant when $q = 0.9845$ and ${A}_{2} = 0.0001$ in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 5. Variation in PSS of three loops orbit due to interior first order resonant when $q = 0.9845$, $A_2 = 0.0001$ in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 6. Variation in location of the periodic orbit of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case ($q = 1$, ${A}_{2} = 0$) for the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 7. Variation in eccentricity of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case ($q = 1$, ${A}_{2} = 0$) cases in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 8. Variation in location and eccentricity of first order interior three loops orbit when $q = 0.9845$ and ${A}_{2} = 0.0001$ in Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 9. Variation in location of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case($q = 1$, ${A}_{2} = 0$) in the Sun-Mars system

Download: Full-size image PowerPoint slide

Figure 10. Variation in eccentricity of the first order interior and exterior resonant periodic orbit for $C$ = 2.93 in perturbed case ($q = 0.9845$, ${A}_{2} = 0.0001$) and ideal case ($q = 1$, ${A}_{2} = 0$) in the Sun-Mars system

Download: Full-size image PowerPoint slide

Figure 11. Variation in location and eccentricity of interior first order three loops orbit for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Mars system

Download: Full-size image PowerPoint slide

Figure 12. Variation in interior third order resonant seven loops orbit for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun - Earth system

Download: Full-size image PowerPoint slide

Figure 13. PSS of interior third order resonant seven loops orbit of family Ⅰ for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun - Earth system

Download: Full-size image PowerPoint slide

Figure 14. Family Ⅰ interior third order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 15. PSS of interior third order resonant seven loops orbits from family Ⅱ for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 16. Family Ⅱ interior third order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 17. PSS of interior fifth order resonant eleven loops orbits from Family Ⅱ for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 18. Family Ⅰ interior fifth order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 19. Family Ⅱ interior fifth order resonant orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 in the Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 20. Variation in location of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 21. Variation in eccentricity of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 22. Variation in period of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Earth system

Download: Full-size image PowerPoint slide

Figure 23. Variation in location of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Mars system

Download: Full-size image PowerPoint slide

Figure 24. Variation in eccentricity of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Mars system

Download: Full-size image PowerPoint slide

Figure 25. Variation in period of the interior third and interior fifth order resonant periodic orbits for $q = 0.9845$, ${A}_{2} = 0.0001$ and $C$ = 2.93 Sun-Mars system

Download: Full-size image PowerPoint slide

Table 1. Analysis of exterior first order resonance in the perturbed Sun-Earth system

$FA$	$SR$	$OB$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	1	0	1	0.93904	1	1:2	0.40895	13	0.49936
			2	0.88740		2:3	0.32301	19	0.66633
			3	0.85623		3:4	0.29337	26	0.74943
			4	0.83547		4:5	0.28015	32	0.79977
			5	0.82075		5:6	0.27331	38	0.83313
Ⅱ	1	0.0001	1	0.93877	1	1:2	0.40904	13	0.49945
			2	0.88710		2:3	0.32319	19	0.66641
			3	0.85592		3:4	0.29358	26	0.74979
			4	0.83516		4:5	0.28039	32	0.79982
			5	0.82044		5:6	0.27355	38	0.83318
Ⅲ	0.9845	0	1	0.97895	1	1:2	0.36044	13	0.52805
			2	0.93800		2:3	0.26118	19	0.69904
			3	0.91210		3:4	0.22428	26	0.78432
			4	0.89403		4:5	0.20685	32	0.83562
			5	0.88075		5:6	0.19740	38	0.86990
Ⅳ	0.9845	0.0001	1	0.97870	1	1:2	0.36081	13	0.52779
			2	0.93764		2:3	0.26136	19	0.69919
			3	0.91172		3:4	0.22453	26	0.78443
			4	0.89363		4:5	0.20713	32	0.83574
			5	0.88035		5:6	0.19771	38	0.86999

| Show Table

DownLoad: CSV

Table 2. Analysis of exterior first order resonance in the perturbed Sun-Mars system

$FA$	$SR$	$OB$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	1	0	1	0.939000	1	1:2	0.40852	13	0.50015
			2	0.887370		2:3	0.32284	19	0.66692
			3	0.856190		3:4	0.29324	26	0.75031
			4	0.835433		4:5	0.28006	32	0.80033
			5	0.820715		5:6	0.27323	38	0.83368
Ⅱ	1	0.0001	1	0.93875	1	1:2	0.40866	13	0.50017
			2	0.88708		2:3	0.32302	19	0.66697
			3	0.85588		3:4	0.29346	26	0.75037
			4	0.83512		4:5	0.28029	32	0.80039
			5	0.82040		5:6	0.27347	38	0.83374
Ⅲ	0.9845	0	1	0.97891	1	1:2	0.35887	13	0.53027
			2	0.93795		2:3	0.26069	19	0.70019
			3	0.91204		3:4	0.22397	26	0.78521
			4	0.89397		4:5	0.20662	32	0.83643
			5	0.88069		5:6	0.19722	38	0.87067
Ⅳ	0.9845	0.0001	1	0.97861	1	1:2	0.35895	13	0.53041
			2	0.93761		2:3	0.26090	19	0.70018
			3	0.91167		3:4	0.22423	26	0.78529
			4	0.89358		4:5	0.20691	32	0.83652
			5	0.88029		5:6	0.19752	38	0.87076

| Show Table

DownLoad: CSV

Table 3. Analysis of interior first order resonance in the perturbed Sun-Earth system

$FA$	$SR$	$OB$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	1	0	2	0.29385	1	2:1	0.53353	07	2.00000
			3	0.47692		3:2	0.37506	13	1.50000
			4	0.55735		4:3	0.32483	19	1.33330
			5	0.60105		5:4	0.30258	26	1.24991
			6	0.62815		6:5	0.29074	32	1.19981
			7	0.64650		7:6	0.28366	38	1.16633
			8	0.66000		8:7	0.27897	44	1.14185
Ⅱ	1	0.0001	2	0.29375	1	2:1	0.53367	07	2.00015
			3	0.47675		3:2	0.37525	13	1.50009
			4	0.55713		4:3	0.32506	19	1.33339
			5	0.60080		5:4	0.30283	26	1.25001
			6	0.62788		6:5	0.29100	32	1.19992
			7	0.64627		7:6	0.28391	38	1.16635
			8	0.65980		8:7	0.27921	44	1.14180
Ⅲ	0.9845	0	2	0.31234	1	2:1	0.47832	07	2.15851
			3	0.50990		3:2	0.30776	13	1.58182
			4	0.59888		4:3	0.25104	19	1.39854
			5	0.64770		5:4	0.22523	26	1.30827
			6	0.67801		6:5	0.21135	32	1.25452
			7	0.69851		7:6	0.20301	38	1.21879
			8	0.71327		8:7	0.19758	44	1.19323
Ⅳ	0.9845	0.0001	2	0.31222	1	2:1	0.47848	07	2.15875
			3	0.50970		3:2	0.30799	13	1.58195
			4	0.59861		4:3	0.25133	19	1.39869
			5	0.64740		5:4	0.22554	26	1.30839
			6	0.67768		6:5	0.21168	32	1.25465
			7	0.69819		7:6	0.20333	38	1.21887
			8	0.71290		8:7	0.19793	44	1.19338

| Show Table

DownLoad: CSV

Table 4. Analysis of interior first order resonance in the perturbed Sun-Mars system

$FA$	$SR$	$OB$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	1	0	2	0.29386	1	2:1	0.53352	07	2.00090
			3	0.47693		3:2	0.37504	13	1.50067
			4	0.55734		4:3	0.32482	19	1.33393
			5	0.60102		5:4	0.30258	26	1.25055
			6	0.62808		6:5	0.29075	32	1.20005
			7	0.64637		7:6	0.28369	38	1.16792
			8	0.65954		8:7	0.27911	44	1.14323
Ⅱ	1	0.0001	2	0.293750	1	2:1	0.533670	07	2.00105
			3	0.476745		3:2	0.375250	13	1.50079
			4	0.557125		4:3	0.325050	19	1.33402
			5	0.600770		5:4	0.302830	26	1.25065
			6	0.627830		6:5	0.291011	32	1.20058
			7	0.646100		7:6	0.283950	38	1.16722
			8	0.659270		8:7	0.279370	44	1.14331
Ⅲ	0.9845	0	2	0.31235	1	2:1	0.47831	07	2.15947
			3	0.50991		3:2	0.30774	13	1.58251
			4	0.59887		4:3	0.25103	19	1.39922
			5	0.64768		5:4	0.22522	26	1.30892
			6	0.67797		6:5	0.21134	32	1.25519
			7	0.69843		7:6	0.20301	38	1.21952
			8	0.71309		8:7	0.19761	44	1.19412
Ⅳ	0.9845	0.0001	2	0.31223	1	2:1	0.478470	07	2.15970
			3	0.50971		3:2	0.307980	13	1.58266
			4	0.59860		4:3	0.251320	19	1.39936
			5	0.64738		5:4	0.225530	26	1.30905
			6	0.67764		6:5	0.211567	32	1.25532
			7	0.69809		7:6	0.203340	38	1.21963
			8	0.71273		8:7	0.197960	44	1.19426

| Show Table

DownLoad: CSV

Table 5. Variation in three loops orbit due to variation in $C$ for $q = 0.9845$ and ${A}_{2} = 0.0001$ in the Sun-Earth system

$JC$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
2.93	0.50970	1	3:2	0.30799	13	1.58195
2.95	0.53653			0.27320		1.57665
2.97	0.56750			0.23304		1.57112
2.99	0.60501			0.18439		1.56524
3.01	0.65610			0.11817		1.55825
3.02	0.69590			0.06656		1.55357
3.03	0.75300			0.01658		1.56821

| Show Table

DownLoad: CSV

Table 6. Variation in three loops orbit due to variation in $C$ for $q = 0.9845$ and ${A}_{2} = 0.0001$ in the Sun-Mars system

$JC$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
2.93	0.50971	1	3:2	0.30798	13	1.58266
2.97	0.56750			0.23303		1.57184
3.01	0.65608			0.11815		1.55901
3.02	0.69590			0.06651		1.55431
3.03	0.75200			0.01514		1.56908

| Show Table

DownLoad: CSV

Table 7. Variation in third order interior resonant seven loops orbit due to variation in $C$ in the Sun-Earth system

$JC$	$LO$	$NI$	$RS$	$EC$	$TP$	$RP$
2.93	0.39923	3	7:4	0.39522	26	1.86449
2.96	0.42824			0.35353		1.85476
2.98	0.44991			0.32238		1.84836

| Show Table

DownLoad: CSV

Table 8. Variation in third order interior resonant seven loops orbit due to variation in $C$ in the Sun-Mars system

$JC$	$LO$	$NI$	$RS$	$EC$	$TP$	$RP$
2.93	0.39923	3	7:4	0.39521	26	1.86534
2.96	0.42823			0.35353		1.85564
2.98	0.44991			0.32232		1.84921

| Show Table

DownLoad: CSV

Table 9. Third order interior resonance in the Sun-Mars system

$FA$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	7	0.39923	3	7:4	0.39522	26	1.86449
	8	0.46231		8:5	0.34307	32	1.69384
	10	0.54635		10:7	0.28318	44	1.50281
	11	0.57515		11:8	0.26511	51	1.44432
	13	0.61796		13:10	0.24063	63	1.36221
	14	0.63358		14:11	0.23244	70	1.33343
Ⅱ	7	0.56160	3	7:4	0.27346	32	1.47147
	9	0.62620		9:6	0.23625	44	1.34696
	11	0.66415		11:8	0.21766	57	1.27850
	13	0.68886		13:10	0.20702	70	1.23510

| Show Table

DownLoad: CSV

Table 10. Third order interior resonance in the Sun-Mars system

$FA$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	7	0.39923	3	7:4	0.39521	26	1.86534
	8	0.46235		8:5	0.34303	32	1.69452
	10	0.54630		10:7	0.28320	44	1.50361
	11	0.57521		11:8	0.26506	51	1.44487
	13	0.61783		13:10	0.24068	63	1.36310
	14	0.63380		14:11	0.23231	70	1.33366
Ⅱ	7	0.56158	3	7:4	0.27346	32	1.47220
	9	0.62616		9:6	0.23626	44	1.34767
	11	0.66413		11:8	0.21764	57	1.2794
	13	0.68878		13:10	0.20702	70	1.23584

| Show Table

DownLoad: CSV

Table 11. Fifth order interior resonance in the Sun-Earth system

$FA$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	11	0.36792	5	11:6	0.42357	38	1.96103
	12	0.41350		12:7	0.38286	44	1.82332
	13	0.45107		13:8	0.35190	51	1.72225
	14	0.48277		14:9	0.32751	57	1.64406
	16	0.53281		16:11	0.29211	70	1.53139
	17	0.55295		17:12	0.27893	76	1.48915
Ⅱ	15	0.58150	5	15:10	0.26122	70	1.43180
	17	0.61344		17:12	0.24307	82	1.37065
	19	0.63733		19:14	0.23053	95	1.32660
	21	0.65640		21:16	0.22124	107	1.29228
	23	0.67095		23:18	0.21460	120	1.26649

| Show Table

DownLoad: CSV

Table 12. Fifth order interior resonance in the Sun-Mars system

$FA$	$NL$	$LO$	$NI$	$RO$	$EC$	$TP$	$RP$
Ⅰ	11	0.36790	5	11:6	0.42359	38	1.96199
	12	0.41345		12:7	0.38289	44	1.82430
	13	0.45118		13:8	0.35180	51	1.72276
	14	0.48285		14:9	0.32744	57	1.64463
	16	0.53273		16:11	0.29216	70	1.53227
	17	0.55264		17:12	0.27912	76	1.49047
Ⅱ	15	0.58152	5	15:10	0.26127	70	1.43243
	17	0.61335		17:12	0.24310	82	1.37146
	19	0.63740		19:14	0.23048	95	1.32710
	21	0.65623		21:16	0.22130	107	1.29319
	23	0.67119		23:18	0.21447	120	1.26667

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

	E. I. Abouelmagd , L. G. Guirao Juan and A. Mostafa , Numerical integration of the restricted thee-body problem with Lie series, Astrophysics Space Science, 354 (2014) , 369-378.
	E. I. Abouelmagd and M. A. Sharaf , The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophys. Space Sci., 344 (2013) , 321-332.
	E. I. Abouelmagd , F. Alzahrani , J. L. G. Guiro and A. Hobiny , Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl. (JNSA), 9 (2016) , 1716-1727. doi: 10.22436/jnsa.009.04.27.
	E. I. Abouelmagd , M. S. Alhothuali , L. G. Guirao Juan and H. M. Malaikah , Periodic and secular solutions in the restricted three ody problem under the effect of zonal harmonic parameters, Applied Mathematics & Information Science, 9 (2015) , 1659-1669.
	E. I. Abouelmagd , J. L. G. Guirao , A. Hobiny and F. Alzahrani , Stability of equilibria points for a dumbbell satellite when the central body is oblate spheroid, Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 8 (2015) , 1047-1054. doi: 10.3934/dcdss.2015.8.1047.
	E. I. Abouelmagd and J. L. G. Guirao , On the perturbed restricted three-body problem, Applied Mathematics and Nonlinear Sciences, 1 (2016) , 123-144.
	E. I. Abouelmagd , J. L. G. Guirao and J. A. Vera , Dynamics of a dumbbell satellite under the zonal harmonic effect of an oblate body, Commun Nonlinear Sci Numer Simulat., 20 (2015) , 1057-1069. doi: 10.1016/j.cnsns.2014.06.033.
	E. I. Abouelmagd, D. Mortari and H. H. Selim, Analytical study of periodic solutions on perturbed equatorial two-body problem, International Journal of Bifurcation and Chaos, 25 (2015), 1540040, 14pp. doi: 10.1142/S0218127415400404.
	E. I. Abouelmagd , Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophys Space Sci., 342 (2012) , 45-53.
	E. I. Abouelmagd , Stability of the triangular points under combined effects of radiation and oblateness in the restricted three-body problem, Earth Moon Planets, 110 (2013) , 143-155.
	E. I. Abouelmagd , The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophys Space Sci., 346 (2013) , 51-69.
	E. I. Abouelmagd , M. S. Alhothuali , L. G. Guirao Juan and H. M. Malaikah , The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Advances in Space Research, 55 (2015) , 1660-1672.
	E. Balint , R. Renata , S. Zsolt and F. Emese , Stability of higher order resonances in the restricted-three body problem, Celest. Mech. Dyn. Astron., 113 (2012) , 95-112. doi: 10.1007/s10569-012-9420-4.
	F. Cachucho , P. M. Cincotta and S. Ferraz-Mello , Chirikov diffusion in the asteroidal three-body resonance (5, -2, -2), Celest. Mech. Dyn. Astron., 108 (2010) , 35-58. doi: 10.1007/s10569-010-9290-6.
	E. I. Chiang , J. Lovering , R. I. Millis , M. W. Buie , L. H. Wasserman and K. J. Meech , Resonant and secular families of the Kuiper belt, Earth Moon Planets, 92 (2003) , 49-62.
	C. Douskos, V. Kalantonis and P. Markellos, Effects of resonances on the stability of retrograde satellites, Astrophys. Space Sci., 310, 245-249.
	R. Dvorak , A. Bazs and L.-Y. Zhou , Where are the Uranus Trojans?, Celest. Mech. Dyn. Astron., 107 (2010) , 51-62. doi: 10.1007/s10569-010-9261-y.
	V. V. Emel'yanenko and E. L. Kiseleva , Resonant motion of trans-Neptunian objects in high-eccentricity orbits, Astron. Lett., 34 (2008) , 271-279.
	J. Gayon , E. Bois and H. Scholl , Dynamics of planets in retrograde mean motion resonance, Celest. Mech. Dyn. Astron., 103 (2009) , 267-279. doi: 10.1007/s10569-009-9191-8.
	J. D. Hadjidemetriou , D. Psychoyos and G. Voyatzis , The 1:1 resonance in extrasolar planetary systems, Celest. Mech. Dyn. Astron., 104 (2009) , 23-38. doi: 10.1007/s10569-009-9185-6.
	J. D. Hadjidemetriou and G. Voyatzis , On the dynamics of extrasolar planetary systems under dissipation: Migration of planets, Celest. Mech. Dyn. Astron., 107 (2010) , 3-19. doi: 10.1007/s10569-010-9260-z.
	M. J. Holman and N. W. Murray , Chaos in high-order mean motion resonances in the outer asteroid belt, Astron. J., 112 (1996) , 1278-1293.
	A. S. Libert and K. Tsiganis , Trapping in three-planet resonances during gas-driven migration, Celest. Mech. Dyn. Astron., 111 (2011) , 201-218.
	E. Kolmen , N. J. Kasdin and P. Gurfil , Quasi-periodic orbits of the restricted three body problem made easy, AIP Conference Proceedings, 886 (2007) , 68-77.
	V. V. Markellos , K. E. Papadakis and E. A. Perdios , Non-linear stability zones around triangular equilibria in the plane circular restricted three-body problem with oblateness, Astrophys Space Sci., 245 (1996) , 157-164.
	F. Migliorini , P. Michel , A. Morbidelli , D. Nesvorn and V. Zappal , Origin of multi kilometer Earth and Mars-crossing asteroids: A quantitative simulation, Science, 281 (1998) , 2022-2024.
	A. Morbidelli , V. Zappala , M. Moons , A. Cellino and R. Gonczi , Asteriod families close to mean motion resonances: Dynamical effects and physical implications, Icarus, 118 (1995) , 137-154.
	C. D. Murray and S. F. Dermot, Solar System Dynamics, Cambridge University Press, 1999.
	N. M. Pathak , R. K. Sharma and V. O. Thomas , Evolution of periodic orbits in the Sun- Saturn system, International Journal of Astronomy and Astrophysics, 6 (2016) , 175-197.
	N. M. Pathak and V. O. Thomas , Evolution of the f Family Orbits in the Photo-Gravitational Sun-Saturn System with Oblateness, International Journal of Astronomy and Astrophysics, 6 (2016) , 254-271.
	N. M. Pathak and V. O. Thomas , Analysis of effect of oblateness of smaller primary on the evolution of periodic orbits, International Journal of Astronomy and Astrophysics, 6 (2016) , 440-463.
	N. M. Pathak and V. O. Thomas , Analysis of effect of solar radiation pressure of bigger primary on the evolution of periodic orbits, International Journal of Astronomy and Astrophysics, 6 (2016) , 464-493.
	E. A. Perdios and V. S. Kalantonis , Self-resonant bifurcations of the Sitnikov family and the appearance of 3D isolas in the restricted three-body problem, Celest. Mech. Dyn. Astron., 113 (2012) , 377-386. doi: 10.1007/s10569-012-9424-0.
	H. Poincaré, Les Méthodes Nouvelles de la Méchanique, Celeste. Gauthier- Villas, Paris., 1987.
	N. Pushparaj and R. K. Sharma , Interior resonance periodic orbits in photogravitational restricted three-body problem, Advances in Astrophysics, 2 (2017) , 263-272.
	A. E. Roy and M. W. Ovenden , On the occurrence of commensurable mean motions in the solar system, Monthly Notices of the Royal Astronomical Society, 114 (1954) , 232-241.
	R. K. Sharma and P. V. Subbarao , A case of commensurability induced by oblateness, Celest. Mech., 18 (1978) , 185-194.
	R. K. Sharma , The linear stability of libration points of the photo gravitational restricted three body problem when the smaller primary is an oblate spheroid, Astrophysics and Space Science, 135 (1987) , 271-281.
	P. P. Stor , J. Kla cka and L. K. mar , Motion of dust in mean motion resonance with planets, Celest. Mech. Dyn. Astron., 103 (2009) , 343-364. doi: 10.1007/s10569-009-9202-9.
	E. W. Thommes , A safety net for fast migrators: Interactions between gap-opening and sub ap-opening bodies in a protoplanetary disk, Astrophys. J., 626 (2005) , 1033-1044.