# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2021031
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit

 1 Departamento de Ciências Exatas, Centro de Ciências Aplicadas e Educação, Universidade Federal da Paraíba, Rio Tinto, Brazil 2 Departamento de Matemática, Centro de Ciências Exatas e Tecnologia, Universidade Federal de Sergipe, São Cristovão, Brazil 3 Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Concepción, Chile

* Corresponding author: José Laudelino de Menezes Neto

Received  September 2021 Early access January 2022

We consider the planar double pendulum where its center of mass is attached in an elliptic orbit. We consider the case where the rods of the pendulum have variable length, varying according to the radius vector of the elliptic orbit. We make an Hamiltonian view of the problem, find four linearly stable equilibrium positions and construct the boundary curves of the stability/instability regions in the space of the parameters associated with the pendulum length and the eccentricity of the orbit.

Citation: José Laudelino de Menezes Neto, Gerson Cruz Araujo, Yocelyn Pérez Rothen, Claudio Vidal. Parametric stability of a double pendulum with variable length and with its center of mass in an elliptic orbit. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021031
##### References:
 [1] V. S. Aslanov, Orbital oscillations of an elastic vertically-tethered satellite, Mech. Solids, 46 (2011), 657-668.  doi: 10.3103/S0025654411050013. [2] L. Blitzer, Equilibrium and stability of a pendulum in an orbiting spaceship, Am. J. Phys., 47 (1979), 241-246.  doi: 10.1119/1.11561. [3] A. A. Burov, Oscillations of a vibrating dumbbell on an elliptic orbit, Dokl. Akad. Nauk, 437 (2011), 186-189. [4] A. A. Burov, A. D. Guerman and I. I. Kosenko, On plane oscillations of a pendulum with variable length suspended on the surface of a planet's satellite, Cosmic Research, 52 (2014), 289-294.  doi: 10.1134/S0010952514040029. [5] A. A. Burov and I. I. Kosenko, Planar vibrations of a solid with variable mass distribution in an elliptic orbit, Dokl. Phys., 56 (2011), 760-764. [6] A. A. Burov and I. I. Kosenko, Planar oscillations of a dumb-bell of a variable length in a central field of newtonian attraction, exact approach, Int. J. of Non-Linear Mech., 72 (2015), 1-5.  doi: 10.1016/j.ijnonlinmec.2015.01.011. [7] A. Burov, I. Kosenko and A. Guerman, Dynamics of a moon-anchored tether with variable lenght, Adv. Astronautical Sci., 142 (2012), 3495-3507. [8] G. Cruz Araujo and H. E. Cabral, Parametric stability in a $P +2$-body problem, J. Dyn. Diff. Equat., 30 (2018), 719-742.  doi: 10.1007/s10884-017-9570-x. [9] J. L. de Menezes Neto and H. E. Cabral, Parametric stability of a pendulum with variable length in an elliptic orbit, Regul. Chaotic Dyn., 25 (2020), 323-329.  doi: 10.1134/S1560354720040012. [10] A. D. Guerman, Spatial equilibria of multibody chain in a circular orbit, Acta Astronautica, 58 (2006), 1-14.  doi: 10.1016/j.actaastro.2005.05.002. [11] O. V. Kholostova, On the motions of a double pendulum with vibrating suspension point, Mechanics of Solids, 44 (2009), 184-197.  doi: 10.3103/S0025654409020034. [12] M. Lavagna and A. E. Finzi, Large multi-hinged space systems: A parametric stability analysis, Acta Astronautica, 54 (2004), 295-305.  doi: 10.1016/S0094-5765(02)00304-1. [13] A. P. Markeev, Linear Hamiltonian systems and some applications to the problem of stability of motion of satellites relative to the center of mass, $\mathcal{R}$ & $\mathcal{C}$ $\mathcal{D}$ynamics, Izhevsk-Moscow, 2009. [14] A. P. Markeev, On one special case of parametric resonance in problems of celestial mechanics, Astronomy Letters, 31 (2005), 350-356.  doi: 10.1134/1.1922534. [15] A. K. Misra, Z. Amier and V. J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronautica, 17 (1988), 1059-1068. [16] A. V. Sarychev, Equilibria of a double pendulum in a circular orbit, Acta Astronautica, 44 (1999), 63-65.  doi: 10.1016/S0094-5765(99)00015-6. [17] J. L. Synge, On the behavior, according to Newtonian theory of a plumb line or pendulum attached to an artificial satellite, Proc. Roy. Irish Acad. Sect. A, 60 (1959), 6 pp. [18] L. R. Valeriano, Parametric stability in Robe's problem, Regul. Chaotic Dyn., 21 (2016), 126-135.  doi: 10.1134/S156035471601007X.

show all references

##### References:
 [1] V. S. Aslanov, Orbital oscillations of an elastic vertically-tethered satellite, Mech. Solids, 46 (2011), 657-668.  doi: 10.3103/S0025654411050013. [2] L. Blitzer, Equilibrium and stability of a pendulum in an orbiting spaceship, Am. J. Phys., 47 (1979), 241-246.  doi: 10.1119/1.11561. [3] A. A. Burov, Oscillations of a vibrating dumbbell on an elliptic orbit, Dokl. Akad. Nauk, 437 (2011), 186-189. [4] A. A. Burov, A. D. Guerman and I. I. Kosenko, On plane oscillations of a pendulum with variable length suspended on the surface of a planet's satellite, Cosmic Research, 52 (2014), 289-294.  doi: 10.1134/S0010952514040029. [5] A. A. Burov and I. I. Kosenko, Planar vibrations of a solid with variable mass distribution in an elliptic orbit, Dokl. Phys., 56 (2011), 760-764. [6] A. A. Burov and I. I. Kosenko, Planar oscillations of a dumb-bell of a variable length in a central field of newtonian attraction, exact approach, Int. J. of Non-Linear Mech., 72 (2015), 1-5.  doi: 10.1016/j.ijnonlinmec.2015.01.011. [7] A. Burov, I. Kosenko and A. Guerman, Dynamics of a moon-anchored tether with variable lenght, Adv. Astronautical Sci., 142 (2012), 3495-3507. [8] G. Cruz Araujo and H. E. Cabral, Parametric stability in a $P +2$-body problem, J. Dyn. Diff. Equat., 30 (2018), 719-742.  doi: 10.1007/s10884-017-9570-x. [9] J. L. de Menezes Neto and H. E. Cabral, Parametric stability of a pendulum with variable length in an elliptic orbit, Regul. Chaotic Dyn., 25 (2020), 323-329.  doi: 10.1134/S1560354720040012. [10] A. D. Guerman, Spatial equilibria of multibody chain in a circular orbit, Acta Astronautica, 58 (2006), 1-14.  doi: 10.1016/j.actaastro.2005.05.002. [11] O. V. Kholostova, On the motions of a double pendulum with vibrating suspension point, Mechanics of Solids, 44 (2009), 184-197.  doi: 10.3103/S0025654409020034. [12] M. Lavagna and A. E. Finzi, Large multi-hinged space systems: A parametric stability analysis, Acta Astronautica, 54 (2004), 295-305.  doi: 10.1016/S0094-5765(02)00304-1. [13] A. P. Markeev, Linear Hamiltonian systems and some applications to the problem of stability of motion of satellites relative to the center of mass, $\mathcal{R}$ & $\mathcal{C}$ $\mathcal{D}$ynamics, Izhevsk-Moscow, 2009. [14] A. P. Markeev, On one special case of parametric resonance in problems of celestial mechanics, Astronomy Letters, 31 (2005), 350-356.  doi: 10.1134/1.1922534. [15] A. K. Misra, Z. Amier and V. J. Modi, Attitude dynamics of three-body tethered systems, Acta Astronautica, 17 (1988), 1059-1068. [16] A. V. Sarychev, Equilibria of a double pendulum in a circular orbit, Acta Astronautica, 44 (1999), 63-65.  doi: 10.1016/S0094-5765(99)00015-6. [17] J. L. Synge, On the behavior, according to Newtonian theory of a plumb line or pendulum attached to an artificial satellite, Proc. Roy. Irish Acad. Sect. A, 60 (1959), 6 pp. [18] L. R. Valeriano, Parametric stability in Robe's problem, Regul. Chaotic Dyn., 21 (2016), 126-135.  doi: 10.1134/S156035471601007X.
Double pendulum in an elliptical orbit. Orbital plane
Graphics of: (1) $\beta^{(2)}(e)$, (2) $\beta^{(3)}(e)$, (3) $\beta^{(5}(e)$, (4) $\beta^{(6)}(e)$, (5) $\beta^{(7)}(e)$, (6) $\beta^{(8)}(e)$ in the plane $e\times \beta$
Graphics of: (1) $\beta^{(1)}(e)$ and (2) $\beta^{(4)}(e)$ in the plane $e\times \beta$
Regions of stability and instability for the equilibrium point $E_2$. Graphics of (1): $\beta^{(1)}(e)$, (2): $\beta^{(2)}(e)$, (3): $\beta^{(3)}(e)$, (4): $\beta^{(4)}(e)$, (5): $\beta^{(5)}(e)$, (6): $\beta^{(6)}(e)$, (7): $\beta^{(7)}(e)$, (8): $\beta^{(8)}(e)$
Regions of stability and instability for the equilibrium point $E_3$. Graphics of (1): $\beta^{(1)}(e)$, (2): $\beta^{(2)}(e)$, (3): $\beta^{(3)}(e)$, (4): $\beta^{(4)}(e)$, (5): $\beta^{(5)}(e)$, (6): $\beta^{(6)}(e)$, (7): $\beta^{(7)}_+(e)$, (8): $\beta^{(7}_-(e)$, (9): $\beta^{(8)}(e)$, (10): $\beta^{(9)}(e)$
Regions of stability and instability for the equilibrium point $E_3$. Graphics of $\beta^{(7)}_+(e)$ and $\beta^{(7)}_-(e)$. The shaded region is where the equilibrium point is unstable
 [1] Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137 [2] Antonio Pumariño, Claudia Valls. On the double pendulum: An example of double resonant situations. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 413-448. doi: 10.3934/dcds.2004.11.413 [3] Qilin Wang, Shengji Li. Lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Journal of Industrial and Management Optimization, 2014, 10 (4) : 1225-1234. doi: 10.3934/jimo.2014.10.1225 [4] Nguyen Huy Chieu, Jen-Chih Yao. Subgradients of the optimal value function in a parametric discrete optimal control problem. Journal of Industrial and Management Optimization, 2010, 6 (2) : 401-410. doi: 10.3934/jimo.2010.6.401 [5] Bettina Klaus, Frédéric Payot. Paths to stability in the assignment problem. Journal of Dynamics and Games, 2015, 2 (3&4) : 257-287. doi: 10.3934/jdg.2015004 [6] Yarui Duan, Pengcheng Wu, Yuying Zhou. Penalty approximation method for a double obstacle quasilinear parabolic variational inequality problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022017 [7] Shengji Li, Chunmei Liao, Minghua Li. Stability analysis of parametric variational systems. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 317-331. doi: 10.3934/naco.2011.1.317 [8] Nguyen Thi Toan. Generalized Clarke epiderivatives of the extremum multifunction to a multi-objective parametric discrete optimal control problem. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021088 [9] Daniel Morales-Silva, David Yang Gao. Complete solutions and triality theory to a nonconvex optimization problem with double-well potential in $\mathbb{R}^n$. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 271-282. doi: 10.3934/naco.2013.3.271 [10] Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323 [11] Chunhua Wang, Jing Yang. Infinitely many solutions for an elliptic problem with double critical Hardy-Sobolev-Maz'ya terms. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1603-1628. doi: 10.3934/dcds.2016.36.1603 [12] Yacheng Liu, Runzhang Xu. Potential well method for initial boundary value problem of the generalized double dispersion equations. Communications on Pure and Applied Analysis, 2008, 7 (1) : 63-81. doi: 10.3934/cpaa.2008.7.63 [13] Li Cai, Fubao Zhang. The Brezis-Nirenberg type double critical problem for a class of Schrödinger-Poisson equations. Electronic Research Archive, 2021, 29 (3) : 2475-2488. doi: 10.3934/era.2020125 [14] Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413 [15] Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic and Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673 [16] Pedro Caro. On an inverse problem in electromagnetism with local data: stability and uniqueness. Inverse Problems and Imaging, 2011, 5 (2) : 297-322. doi: 10.3934/ipi.2011.5.297 [17] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control and Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [18] S. J. Li, Z. M. Fang. On the stability of a dual weak vector variational inequality problem. Journal of Industrial and Management Optimization, 2008, 4 (1) : 155-165. doi: 10.3934/jimo.2008.4.155 [19] Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems and Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551 [20] M. Zuhair Nashed, Alexandru Tamasan. Structural stability in a minimization problem and applications to conductivity imaging. Inverse Problems and Imaging, 2011, 5 (1) : 219-236. doi: 10.3934/ipi.2011.5.219

2020 Impact Factor: 0.857

## Tools

Article outline

Figures and Tables