# American Institute of Mathematical Sciences

November  2019, 12(7): 1879-1903. doi: 10.3934/dcdss.2019124

## Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces

 1 Department of Mathematical Sciences, University of Texas at Dallas, 800 W Campbell Road, Richardson, Texas 75080-3021, USA 2 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China 3 University of Texas at Dallas, Department of Mathematical Sciences, 800 W Campbell Road, Richardson, Texas 75080-3021, USA 4 Department of Mathematics and Computer Science, Alcorn State University, 1000 ASU Drive, Lorman, MS, 39096, USA

* Corresponding author: Wieslaw Krawcewicz

Received  December 2017 Revised  July 2018 Published  December 2018

Fund Project: The first and second authors are supported by the Department of Mathematical Sciences University of Texas at Dallas. The third author is supported by the Center for Applied Mathematics at Guangzhou University, Guangzhou China and the Department of Mathematical Sciences University of Texas at Dallas.

In this paper, we investigate nonlinear periodic vibrations of a group of particles with a planar dihedral configuration governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide general formulae for the spectrum of the linearized system of equations describing the above configuration, which allows us to obtain the critical frequencies of the particles' motions. The obtained frequencies represent the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

Citation: Irina Berezovik, Wieslaw Krawcewicz, Qingwen Hu. Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 1879-1903. doi: 10.3934/dcdss.2019124
##### References:
 [1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. [2] Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2) × S1-symmetric variational problems: Equivariant gradient degree approach, Nonlinear Analysis and Optimization II. Optimization, 45-84, Contemp. Math., 514, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/514/10099. [3] Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory App., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9. [4] I. Berezovik, C. García-Azpeitia and W. Krawcewicz, Symmetries of nonlinear vibrations in tetrahedral molecular configurations, DCDS-B (accepted September 2018). [5] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985. doi: 10.1007/978-3-662-12918-0. [6] M. Dabkowski, W. Krawcewicz, Y. Lv and H-P. Wu, Multiple Periodic Solutions for Γ-symmetric Newtonian Systems, J. Diff. Eqns., 10 (2017), 6684-6730. doi: 10.1016/j.jde.2017.07.027. [7] T. tom Dieck, Transformation Groups., Walter de Gruyter, 1987. doi: 10.1515/9783110858372.312. [8] J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Eqns, 218 (2005), 216-252. doi: 10.1016/j.jde.2005.04.004. [9] C. Garcia-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators, J. Diff. Eqns, 251 (2011), 3202-3227.  doi: 10.1016/j.jde.2011.06.021. [10] C. Garcia-Azpeitia and M. Tejada-Wriedt, Molecular chains interacting by Lennard-Jones and Coulomb forces, Quali. Theory Dyn. Syst., 16 (2017), 591-608. doi: 10.1007/s12346-016-0221-0. [11] K. Geba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis Ⅱ (Frascati, 1995), Progr. Nonlinear Differential Equations App., 27, Birkhäuser, Boston, 1997,247-272. [12] A. Gołebiewska and S. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Analysis, TMA, 74 (2011), 1823-1834.  doi: 10.1016/j.na.2010.10.055. [13] E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'École Normale Supérieure, 6 (1889), 9-102. doi: 10.24033/asens.317. [14] J. Ize and A. Vignoli, Equivariant Degree Theory, vol. 8 of De Gruyter Series in Nonlinear Analysis and Applications, Berlin, Boston: De Gruyter., 2003. doi: 10.1515/9783110200027. [15] J. E. Lennard-Jones, On the determination of molecular fields, Proc. R. Soc. Lond. A, 106 (1924), 463-477. [16] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Math. Sciences, Vol. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [17] K. H. Mayer, G-invariante Morse-Funktionen, Manuscripta Math., 63 (1989), 99-114. doi: 10.1007/BF01173705. [18] H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal., 68 (2008), 1479-1516.  doi: 10.1016/j.na.2006.12.039. [19] S. Rybicki, Applications of degree for S1-equivariant gradient maps to variational nonlinear problems with S1-symmetries, Topol. Methods Nonlinear Anal., 9 (1997), 383-417.  doi: 10.12775/TMNA.1997.018. [20] E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto-Lomdon, 1966. [21] H.-P. Wu, GAP program for the computations of the Burnside ring A(Γ × O(2)), https://bitbucket.org/psistwu/gammao2, developed at University of Texas at Dallas, 2016. [22] , Symmetry Resources at Otterbein University, http://symmetry.otterbein.edu/gallery/.

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##### References:
 [1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006. [2] Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O(2) × S1-symmetric variational problems: Equivariant gradient degree approach, Nonlinear Analysis and Optimization II. Optimization, 45-84, Contemp. Math., 514, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/514/10099. [3] Z. Balanov, W. Krawcewicz, S. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory App., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9. [4] I. Berezovik, C. García-Azpeitia and W. Krawcewicz, Symmetries of nonlinear vibrations in tetrahedral molecular configurations, DCDS-B (accepted September 2018). [5] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985. doi: 10.1007/978-3-662-12918-0. [6] M. Dabkowski, W. Krawcewicz, Y. Lv and H-P. Wu, Multiple Periodic Solutions for Γ-symmetric Newtonian Systems, J. Diff. Eqns., 10 (2017), 6684-6730. doi: 10.1016/j.jde.2017.07.027. [7] T. tom Dieck, Transformation Groups., Walter de Gruyter, 1987. doi: 10.1515/9783110858372.312. [8] J. Fura, A. Ratajczak and S. Rybicki, Existence and continuation of periodic solutions of autonomous Newtonian systems, J. Diff. Eqns, 218 (2005), 216-252. doi: 10.1016/j.jde.2005.04.004. [9] C. Garcia-Azpeitia and J. Ize, Global bifurcation of polygonal relative equilibria for masses, vortices and dNLS oscillators, J. Diff. Eqns, 251 (2011), 3202-3227.  doi: 10.1016/j.jde.2011.06.021. [10] C. Garcia-Azpeitia and M. Tejada-Wriedt, Molecular chains interacting by Lennard-Jones and Coulomb forces, Quali. Theory Dyn. Syst., 16 (2017), 591-608. doi: 10.1007/s12346-016-0221-0. [11] K. Geba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis Ⅱ (Frascati, 1995), Progr. Nonlinear Differential Equations App., 27, Birkhäuser, Boston, 1997,247-272. [12] A. Gołebiewska and S. Rybicki, Global bifurcations of critical orbits of G-invariant strongly indefinite functionals, Nonlinear Analysis, TMA, 74 (2011), 1823-1834.  doi: 10.1016/j.na.2010.10.055. [13] E. Goursat, Sur les substitutions orthogonales et les divisions régulières de l'espace, Annales scientifiques de l'École Normale Supérieure, 6 (1889), 9-102. doi: 10.24033/asens.317. [14] J. Ize and A. Vignoli, Equivariant Degree Theory, vol. 8 of De Gruyter Series in Nonlinear Analysis and Applications, Berlin, Boston: De Gruyter., 2003. doi: 10.1515/9783110200027. [15] J. E. Lennard-Jones, On the determination of molecular fields, Proc. R. Soc. Lond. A, 106 (1924), 463-477. [16] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Math. Sciences, Vol. 74, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2061-7. [17] K. H. Mayer, G-invariante Morse-Funktionen, Manuscripta Math., 63 (1989), 99-114. doi: 10.1007/BF01173705. [18] H. Ruan and S. Rybicki, Applications of equivariant degree for gradient maps to symmetric Newtonian systems, Nonlinear Anal., 68 (2008), 1479-1516.  doi: 10.1016/j.na.2006.12.039. [19] S. Rybicki, Applications of degree for S1-equivariant gradient maps to variational nonlinear problems with S1-symmetries, Topol. Methods Nonlinear Anal., 9 (1997), 383-417.  doi: 10.12775/TMNA.1997.018. [20] E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto-Lomdon, 1966. [21] H.-P. Wu, GAP program for the computations of the Burnside ring A(Γ × O(2)), https://bitbucket.org/psistwu/gammao2, developed at University of Texas at Dallas, 2016. [22] , Symmetry Resources at Otterbein University, http://symmetry.otterbein.edu/gallery/.
Stationary solution to equation (7) with dihedral symmetries
Relative motions of all 6 particles with $\lambda_0^2 = \frac{l^2}{\mu}$, $l = 1$ and $\mu$ near the eigenvalue $\mu_0 = 10.10496819$ of $\nabla^2 V(u^o)$
Relative motions of all particles with $\lambda_0^2 = \frac{l^2}{\mu}$, $l = 1$ and $\mu$ near the eigenvalue $\mu = 6.442637681$ of $\nabla^2 V(u^o)$
Relative motions of all particles with $\lambda_0^2 = \frac{l^2}{\mu}$, $l = 1$ and $\mu$ near the eigenvalue $\mu = 8.469351217$ of $\nabla^2 V(u^o)$
Relative motions of all particles with Relative motions of all particles with $\lambda_0^2=\frac{l^2}{\mu}$, $l=1$ and $\mu$ near the eigenvalue $\mu=3.854423919$ of $\nabla^2 V(u^o)$
The values $\lambda_{j, l}$ in the critical set $\Lambda$
 $j$ $\mu_j$ $\lambda_{j, 1}$ $\lambda_{j, 2}$ $\lambda_{j, 3}$ $\lambda_{j, 4}$ 0 10.10496819 0.31458103 0.62916205 0.94374308 1.25832410 1 8.469351217 0.34361723 0.68723445 1.03085168 1.37446891 3 3.854423919 0.50935463 1.01870927 1.52806390 2.03741854 $2^+$ 6.442637681 0.62767390 1.25534781 1.88302171 2.51069561 $2^-$ 0.007288929 11.7130006 23.4260011 35.1390017 46.8520023
 $j$ $\mu_j$ $\lambda_{j, 1}$ $\lambda_{j, 2}$ $\lambda_{j, 3}$ $\lambda_{j, 4}$ 0 10.10496819 0.31458103 0.62916205 0.94374308 1.25832410 1 8.469351217 0.34361723 0.68723445 1.03085168 1.37446891 3 3.854423919 0.50935463 1.01870927 1.52806390 2.03741854 $2^+$ 6.442637681 0.62767390 1.25534781 1.88302171 2.51069561 $2^-$ 0.007288929 11.7130006 23.4260011 35.1390017 46.8520023
Maximal orbit types in $\mathscr W_{j,l}$
 $\mathscr W_{j,l}$, $l\ge 1$ maximal orbit types $\mathscr W_{0,l}$ $(D_6\times D_l)$ $\mathscr W_{1l}$ $({D_6}^{{\mathbb{Z}_1}}{ \times _{{D_6}}}{D_{6l}}) - ({D_2}^{{D_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2}^{{{\tilde D}_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$ $\mathscr W_{2,l}$ $({D_6}^{{\mathbb{Z}_2}}{ \times _{{D_3}}}{D_{3l}}) - ({D_2}^{{\mathbb{Z}_2}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2} \times {D_l})$ $\mathscr W_{3,l}$ $({D_6}^{{{\tilde D}_3}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
 $\mathscr W_{j,l}$, $l\ge 1$ maximal orbit types $\mathscr W_{0,l}$ $(D_6\times D_l)$ $\mathscr W_{1l}$ $({D_6}^{{\mathbb{Z}_1}}{ \times _{{D_6}}}{D_{6l}}) - ({D_2}^{{D_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2}^{{{\tilde D}_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$ $\mathscr W_{2,l}$ $({D_6}^{{\mathbb{Z}_2}}{ \times _{{D_3}}}{D_{3l}}) - ({D_2}^{{\mathbb{Z}_2}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2} \times {D_l})$ $\mathscr W_{3,l}$ $({D_6}^{{{\tilde D}_3}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
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