Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

Lyudmila Grigoryeva; Juan-Pablo Ortega; Stanislav S. Zub

doi:10.3934/jgm.2014.6.373

Previous Article
Poisson structures for two nonholonomic systems with partially reduced symmetries
JGM Home
This Issue
Next Article
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

September 2014, 6(3): 373-415. doi: 10.3934/jgm.2014.6.373

Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

Lyudmila Grigoryeva ^1, , Juan-Pablo Ortega ^2, and Stanislav S. Zub ^3,

1.	Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16, route de Gray, F-25030 Besançon cedex, France
2.	Centre National de la Recherche Scientifique, Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, UFR des Sciences et Techniques, 16, route de Gray, F-25030 Besançon cedex
3.	Taras Shevchenko National University of Kyiv, 64, Volodymyrs'ka, 01601 Kyiv, Ukraine

Received November 2013 Revised June 2014 Published September 2014

Abstract
Related Papers

This work studies the symmetries, the associated momentum map, and relative equilibria of a mechanical system consisting of a small axisymmetric magnetic body-dipole in an also axisymmetric external magnetic field that additionally exhibits a mirror symmetry; we call this system the ``orbitron". We study the nonlinear stability of a branch of equatorial relative equilibria using the energy-momentum method and we provide sufficient conditions for their $\mathbb{T}^2$--stability that complete partial stability relations already existing in the literature. These stability prescriptions are explicitly written down in terms of some of the field parameters, which can be used in the design of stable solutions. We propose new linear methods to determine instability regions in the context of relative equilibria that allow us to conclude the sharpness of some of the nonlinear stability conditions obtained.

Keywords: Hamiltonian systems with symmetry, relative equilibrium, nonlinear stability/instability., momentum maps, orbitron, magnetic systems, generalized orbitron.

Mathematics Subject Classification: Primary: 37J15, 37J25; Secondary: 70E50, 70H14, 70H3.

Citation: Lyudmila Grigoryeva, Juan-Pablo Ortega, Stanislav S. Zub. Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies. Journal of Geometric Mechanics, 2014, 6 (3) : 373-415. doi: 10.3934/jgm.2014.6.373

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978). Google Scholar
[2]	J. Cees van der Meer, The Hamiltonian Hopf Bifurcation,, Springer-Verlag, (1985). doi: 10.1007/BFb0080357. Google Scholar
[3]	M. Dellnitz, I. Melbourne and J. E. Marsden, Generic bifurcation of Hamiltonian vector fields with symmetry,, Nonlinearity, 5 (1992), 979. doi: 10.1088/0951-7715/5/4/008. Google Scholar
[4]	H. R. Dullin, Poisson integrator for symmetric rigid bodies,, Regular and Chaotic Dynamics, 9 (2004), 255. doi: 10.1070/RD2004v009n03ABEH000279. Google Scholar
[5]	H. R. Dullin and R. W. Easton, Stability of levitrons,, Physica D, 126 (1999), 1. doi: 10.1016/S0167-2789(98)00251-6. Google Scholar
[6]	F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids,, Archive for Rational Mechanics and Analysis, 158 (2001), 259. doi: 10.1007/PL00004245. Google Scholar
[7]	V. Guillemin and S. Sternberg, A normal form for the momentum map,, in Differential Geometric Methods in Mathematical Physics* (ed. S. Sternberg)*, (1984), 161. Google Scholar
[8]	R. Harrigan, Levitation device,, US patent, (1983). Google Scholar
[9]	V. V. Kozorez, About a problem of two magnets,, Bull. of the Ac. of Sc. of USSR. Series: Mechanics of a Rigid Body (in Russian), 3 (1974), 29. Google Scholar
[10]	V. V. Kozorez, Dynamic Systems of Free Magnetically Interacting Bodies, (Russian), Naukova dumka, (1981). Google Scholar
[11]	N. E. Leonard, Stability of a bottom-heavy underwater vehicle,, Automatica, 33 (1997), 331. doi: 10.1016/S0005-1098(96)00176-8. Google Scholar
[12]	R. Krechetnikov and J. E. Marsden, On destabilizing effects of two fundamental non-conservative forces,, Physica D, 214 (2006), 25. doi: 10.1016/j.physd.2005.12.003. Google Scholar
[13]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities,, Nonlinearity, 15 (2002), 143. doi: 10.1088/0951-7715/15/1/307. Google Scholar
[14]	F. Laurent-Polz, J. Montaldi and M. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria,, Journal of Geometric Mechanics, 3 (2011), 439. doi: 10.3934/jgm.2011.3.439. Google Scholar
[15]	N. E. Leonard and J. E. Marsden, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry,, Physica D, 105 (1997), 130. doi: 10.1016/S0167-2789(97)83390-8. Google Scholar
[16]	E. Lerman and S. F. Singer, Stability and persistence of relative equilibria at singular values of the moment map,, Nonlinearity, 11 (1998), 1637. doi: 10.1088/0951-7715/11/6/012. Google Scholar
[17]	D. Lewis, Stacked Lagrange tops,, Journal of Nonlinear Science, 8 (1998), 63. doi: 10.1007/s003329900044. Google Scholar
[18]	D. Lewis, T. S. Ratiu, J. C. Simo and J. E. Marsden, The heavy top: A geometric treatment,, Nonlinearity, 5 (1992), 1. doi: 10.1088/0951-7715/5/1/001. Google Scholar
[19]	C. Lim, J. A. Montaldi and R. M. Roberts, Relative equilibria of point vortices on the sphere,, Physica D, 148 (2001), 97. doi: 10.1016/S0167-2789(00)00167-6. Google Scholar
[20]	C.-M. Marle, Le voisinage d'une orbite d'une action hamiltonienne d'un groupe de Lie,, Séminaire Sud-Rhodanien de Géométrie II, (1984), 19. Google Scholar
[21]	C.-M. Marle, Modéle d'action hamiltonienne d'un groupe the Lie sur une variété symplectique,, Rend. Sem. Mat. Univers. Politecn. Torino, 43 (1985), 227. Google Scholar
[22]	J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry,, Reports on Mathematical Physics, 5 (1974), 121. doi: 10.1016/0034-4877(74)90021-4. Google Scholar
[23]	K. R. Meyer, Symmetries and integrals in mechanics,, in Dynamical Systems* (ed. M. M. Peixoto)*, (1973), 259. Google Scholar
[24]	J. A. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques,, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553. doi: 10.1016/S0764-4442(99)80389-9. Google Scholar
[25]	J. A. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449. doi: 10.1088/0951-7715/10/2/009. Google Scholar
[26]	J. A. Montaldi and R. M. Roberts, Relative equilibria of molecules,, Journal of Nonlinear Science, 9 (1999), 53. doi: 10.1007/s003329900064. Google Scholar
[27]	J. A. Montaldi, R. M. Roberts and I. N. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems,, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237. doi: 10.1098/rsta.1988.0053. Google Scholar
[28]	J. Montaldi and M. Rodríguez-Olmos, On the stability of Hamiltonian relative equilibria with non-trivial isotropy,, Nonlinearity, 24 (2011), 2777. doi: 10.1088/0951-7715/24/10/007. Google Scholar
[29]	J.-P. Ortega, Symmetry, Reduction, and Stability in Hamiltonian Systems,, Ph.D thesis, (1998). Google Scholar
[30]	J.-P. Ortega and T. S. Ratiu, A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems,, Journal of Geometry and Physics, 32 (1999), 131. doi: 10.1016/S0393-0440(99)00025-X. Google Scholar
[31]	J.-P. Ortega and T. S. Ratiu, Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry,, Journal of Geometry and Physics, 32 (1999), 160. doi: 10.1016/S0393-0440(99)00024-8. Google Scholar
[32]	J.-P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria,, Nonlinearity, 12 (1999), 693. doi: 10.1088/0951-7715/12/3/315. Google Scholar
[33]	J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction,, Birkhauser Verlag, (2004). doi: 10.1007/978-1-4757-3811-7. Google Scholar
[34]	J.-P. Ortega and T. S. Ratiu, The reduced spaces of a symplectic Lie group action,, Annals of Global Analysis and Geometry, 30 (2006), 335. doi: 10.1007/s10455-006-9017-9. Google Scholar
[35]	J.-P. Ortega and T. S. Ratiu, The stratified spaces of a symplectic Lie group action,, Reports on Mathematical Physics, 58 (2006), 51. doi: 10.1016/S0034-4877(06)80040-6. Google Scholar
[36]	G. W. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space,, Journal of Geometry and Physics, 9 (1992), 111. doi: 10.1016/0393-0440(92)90015-S. Google Scholar
[37]	G. W. Patrick, Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift,, Journal of Nonlinear Science, 5 (1995), 373. doi: 10.1007/BF01212907. Google Scholar
[38]	G. W. Patrick, Two Axially Symmetric Coupled Rigid Bodies: Relative Equilibria, Stability, Bifurcations, and a Momentum Preserving Symplectic Integrator,, Ph.D thesis, (1995). Google Scholar
[39]	G. W. Patrick, M. Roberts and C. Wulff, Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods,, Archive for Rational Mechanics and Analysis, 174 (2004), 301. doi: 10.1007/s00205-004-0322-9. Google Scholar
[40]	S. Pekarsky and J. E. Marsden, Point vortices on a sphere: Stability of relative equilibria,, Journal of Mathematical Physics, 39 (1998), 5894. doi: 10.1063/1.532602. Google Scholar
[41]	M. Roberts, C. Wulff and J. S. W. Lamb, Hamiltonian systems near relative equilibria,, Journal of Differential Equations, 179 (2002), 562. doi: 10.1006/jdeq.2001.4045. Google Scholar
[42]	M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems,, Nonlinearity, 19 (2006), 853. doi: 10.1088/0951-7715/19/4/005. Google Scholar
[43]	M. Rodríguez-Olmos and M. E. Sousa-Dias, Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations,, Journal of Nonlinear Science, 19 (2009), 179. doi: 10.1007/s00332-008-9032-z. Google Scholar
[44]	J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method,, Archive for Rational Mechanics and Analysis, 115 (1991), 15. doi: 10.1007/BF01881678. Google Scholar
[45]	R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction,, Annals of Mathematics, 134 (1991), 375. doi: 10.2307/2944350. Google Scholar
[46]	W. R. Smythe, Static and Dynamic Electricity,, McGraw-Hill, (1939). Google Scholar
[47]	S. S. Zub, Research into orbital motion stability in system of two magnetically interacting bodies, preprint,, , (2012). Google Scholar
[48]	S. S. Zub, Stable orbital motion of magnetic dipole in the field of permanent magnets,, Physica D, 275 (2014), 67. doi: 10.1016/j.physd.2014.02.007. Google Scholar

show all references

References:

[1]	R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978). Google Scholar
[2]	J. Cees van der Meer, The Hamiltonian Hopf Bifurcation,, Springer-Verlag, (1985). doi: 10.1007/BFb0080357. Google Scholar
[3]	M. Dellnitz, I. Melbourne and J. E. Marsden, Generic bifurcation of Hamiltonian vector fields with symmetry,, Nonlinearity, 5 (1992), 979. doi: 10.1088/0951-7715/5/4/008. Google Scholar
[4]	H. R. Dullin, Poisson integrator for symmetric rigid bodies,, Regular and Chaotic Dynamics, 9 (2004), 255. doi: 10.1070/RD2004v009n03ABEH000279. Google Scholar
[5]	H. R. Dullin and R. W. Easton, Stability of levitrons,, Physica D, 126 (1999), 1. doi: 10.1016/S0167-2789(98)00251-6. Google Scholar
[6]	F. Fassò and D. Lewis, Stability properties of the Riemann ellipsoids,, Archive for Rational Mechanics and Analysis, 158 (2001), 259. doi: 10.1007/PL00004245. Google Scholar
[7]	V. Guillemin and S. Sternberg, A normal form for the momentum map,, in Differential Geometric Methods in Mathematical Physics* (ed. S. Sternberg)*, (1984), 161. Google Scholar
[8]	R. Harrigan, Levitation device,, US patent, (1983). Google Scholar
[9]	V. V. Kozorez, About a problem of two magnets,, Bull. of the Ac. of Sc. of USSR. Series: Mechanics of a Rigid Body (in Russian), 3 (1974), 29. Google Scholar
[10]	V. V. Kozorez, Dynamic Systems of Free Magnetically Interacting Bodies, (Russian), Naukova dumka, (1981). Google Scholar
[11]	N. E. Leonard, Stability of a bottom-heavy underwater vehicle,, Automatica, 33 (1997), 331. doi: 10.1016/S0005-1098(96)00176-8. Google Scholar
[12]	R. Krechetnikov and J. E. Marsden, On destabilizing effects of two fundamental non-conservative forces,, Physica D, 214 (2006), 25. doi: 10.1016/j.physd.2005.12.003. Google Scholar
[13]	F. Laurent-Polz, Point vortices on the sphere: A case with opposite vorticities,, Nonlinearity, 15 (2002), 143. doi: 10.1088/0951-7715/15/1/307. Google Scholar
[14]	F. Laurent-Polz, J. Montaldi and M. Roberts, Point vortices on the sphere: Stability of symmetric relative equilibria,, Journal of Geometric Mechanics, 3 (2011), 439. doi: 10.3934/jgm.2011.3.439. Google Scholar
[15]	N. E. Leonard and J. E. Marsden, Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry,, Physica D, 105 (1997), 130. doi: 10.1016/S0167-2789(97)83390-8. Google Scholar
[16]	E. Lerman and S. F. Singer, Stability and persistence of relative equilibria at singular values of the moment map,, Nonlinearity, 11 (1998), 1637. doi: 10.1088/0951-7715/11/6/012. Google Scholar
[17]	D. Lewis, Stacked Lagrange tops,, Journal of Nonlinear Science, 8 (1998), 63. doi: 10.1007/s003329900044. Google Scholar
[18]	D. Lewis, T. S. Ratiu, J. C. Simo and J. E. Marsden, The heavy top: A geometric treatment,, Nonlinearity, 5 (1992), 1. doi: 10.1088/0951-7715/5/1/001. Google Scholar
[19]	C. Lim, J. A. Montaldi and R. M. Roberts, Relative equilibria of point vortices on the sphere,, Physica D, 148 (2001), 97. doi: 10.1016/S0167-2789(00)00167-6. Google Scholar
[20]	C.-M. Marle, Le voisinage d'une orbite d'une action hamiltonienne d'un groupe de Lie,, Séminaire Sud-Rhodanien de Géométrie II, (1984), 19. Google Scholar
[21]	C.-M. Marle, Modéle d'action hamiltonienne d'un groupe the Lie sur une variété symplectique,, Rend. Sem. Mat. Univers. Politecn. Torino, 43 (1985), 227. Google Scholar
[22]	J. E. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry,, Reports on Mathematical Physics, 5 (1974), 121. doi: 10.1016/0034-4877(74)90021-4. Google Scholar
[23]	K. R. Meyer, Symmetries and integrals in mechanics,, in Dynamical Systems* (ed. M. M. Peixoto)*, (1973), 259. Google Scholar
[24]	J. A. Montaldi, Persistance d'orbites périodiques relatives dans les systèmes hamiltoniens symétriques,, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), 553. doi: 10.1016/S0764-4442(99)80389-9. Google Scholar
[25]	J. A. Montaldi, Persistence and stability of relative equilibria,, Nonlinearity, 10 (1997), 449. doi: 10.1088/0951-7715/10/2/009. Google Scholar
[26]	J. A. Montaldi and R. M. Roberts, Relative equilibria of molecules,, Journal of Nonlinear Science, 9 (1999), 53. doi: 10.1007/s003329900064. Google Scholar
[27]	J. A. Montaldi, R. M. Roberts and I. N. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems,, Phil. Trans. R. Soc. Lond. A, 325 (1988), 237. doi: 10.1098/rsta.1988.0053. Google Scholar
[28]	J. Montaldi and M. Rodríguez-Olmos, On the stability of Hamiltonian relative equilibria with non-trivial isotropy,, Nonlinearity, 24 (2011), 2777. doi: 10.1088/0951-7715/24/10/007. Google Scholar
[29]	J.-P. Ortega, Symmetry, Reduction, and Stability in Hamiltonian Systems,, Ph.D thesis, (1998). Google Scholar
[30]	J.-P. Ortega and T. S. Ratiu, A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems,, Journal of Geometry and Physics, 32 (1999), 131. doi: 10.1016/S0393-0440(99)00025-X. Google Scholar
[31]	J.-P. Ortega and T. S. Ratiu, Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry,, Journal of Geometry and Physics, 32 (1999), 160. doi: 10.1016/S0393-0440(99)00024-8. Google Scholar
[32]	J.-P. Ortega and T. S. Ratiu, Stability of Hamiltonian relative equilibria,, Nonlinearity, 12 (1999), 693. doi: 10.1088/0951-7715/12/3/315. Google Scholar
[33]	J.-P. Ortega and T. S. Ratiu, Momentum Maps and Hamiltonian Reduction,, Birkhauser Verlag, (2004). doi: 10.1007/978-1-4757-3811-7. Google Scholar
[34]	J.-P. Ortega and T. S. Ratiu, The reduced spaces of a symplectic Lie group action,, Annals of Global Analysis and Geometry, 30 (2006), 335. doi: 10.1007/s10455-006-9017-9. Google Scholar
[35]	J.-P. Ortega and T. S. Ratiu, The stratified spaces of a symplectic Lie group action,, Reports on Mathematical Physics, 58 (2006), 51. doi: 10.1016/S0034-4877(06)80040-6. Google Scholar
[36]	G. W. Patrick, Relative equilibria in Hamiltonian systems: The dynamic interpretation of nonlinear stability on a reduced phase space,, Journal of Geometry and Physics, 9 (1992), 111. doi: 10.1016/0393-0440(92)90015-S. Google Scholar
[37]	G. W. Patrick, Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift,, Journal of Nonlinear Science, 5 (1995), 373. doi: 10.1007/BF01212907. Google Scholar
[38]	G. W. Patrick, Two Axially Symmetric Coupled Rigid Bodies: Relative Equilibria, Stability, Bifurcations, and a Momentum Preserving Symplectic Integrator,, Ph.D thesis, (1995). Google Scholar
[39]	G. W. Patrick, M. Roberts and C. Wulff, Stability of Poisson equilibria and Hamiltonian relative equilibria by energy methods,, Archive for Rational Mechanics and Analysis, 174 (2004), 301. doi: 10.1007/s00205-004-0322-9. Google Scholar
[40]	S. Pekarsky and J. E. Marsden, Point vortices on a sphere: Stability of relative equilibria,, Journal of Mathematical Physics, 39 (1998), 5894. doi: 10.1063/1.532602. Google Scholar
[41]	M. Roberts, C. Wulff and J. S. W. Lamb, Hamiltonian systems near relative equilibria,, Journal of Differential Equations, 179 (2002), 562. doi: 10.1006/jdeq.2001.4045. Google Scholar
[42]	M. Rodríguez-Olmos, Stability of relative equilibria with singular momentum values in simple mechanical systems,, Nonlinearity, 19 (2006), 853. doi: 10.1088/0951-7715/19/4/005. Google Scholar
[43]	M. Rodríguez-Olmos and M. E. Sousa-Dias, Nonlinear Stability of Riemann Ellipsoids with Symmetric Configurations,, Journal of Nonlinear Science, 19 (2009), 179. doi: 10.1007/s00332-008-9032-z. Google Scholar
[44]	J. C. Simo, D. Lewis and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy-momentum method,, Archive for Rational Mechanics and Analysis, 115 (1991), 15. doi: 10.1007/BF01881678. Google Scholar
[45]	R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction,, Annals of Mathematics, 134 (1991), 375. doi: 10.2307/2944350. Google Scholar
[46]	W. R. Smythe, Static and Dynamic Electricity,, McGraw-Hill, (1939). Google Scholar
[47]	S. S. Zub, Research into orbital motion stability in system of two magnetically interacting bodies, preprint,, , (2012). Google Scholar
[48]	S. S. Zub, Stable orbital motion of magnetic dipole in the field of permanent magnets,, Physica D, 275 (2014), 67. doi: 10.1016/j.physd.2014.02.007. Google Scholar

[1]	Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems - A, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[2]	Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029
[3]	Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185
[4]	Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329
[5]	Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810
[6]	Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161
[7]	Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329
[8]	Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017
[9]	Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513
[10]	F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605
[11]	Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020027
[12]	Marcelo Messias. Periodic perturbation of quadratic systems with two infinite heteroclinic cycles. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1881-1899. doi: 10.3934/dcds.2012.32.1881
[13]	Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[14]	Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189
[15]	Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021004
[16]	Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[17]	Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1
[18]	Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006
[19]	Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[20]	Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

2019 Impact Factor: 0.649

American Institute of Mathematical Sciences

Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Stability of Hamiltonian relative equilibria in symmetric magnetically confined rigid bodies

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors