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June  2019, 24(6): 2473-2491. doi: 10.3934/dcdsb.2018261

Symmetries of nonlinear vibrations in tetrahedral molecular configurations

1. 

Department of Mathematical Sciences University of Texas at Dallas Richardson, 75080 USA

2. 

Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510 México DF, México

3. 

Center for Applied Mathematics, Guangzhou University, Guangzhou, China

Received  September 2017 Revised  April 2018 Published  October 2018

We study nonlinear vibrational modes of oscillations for tetrahedral configurations of particles. In the case of tetraphosphorus, the interaction of atoms is given by bond stretching and van der Waals forces. Using the equivariant gradient degree, we present a topological classification of the spatio-temporal symmetries of the periodic solutions with finite Weyl's group. This procedure describes all the symmetries of the nonlinear vibrations for general force fields.

Citation: Irina Berezovik, Carlos García-Azpeitia, Wieslaw Krawcewicz. Symmetries of nonlinear vibrations in tetrahedral molecular configurations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2473-2491. doi: 10.3934/dcdsb.2018261
References:
[1]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.  Google Scholar

[2]

Z. BalanovW. KrawcewiczS. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory Appl., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9.  Google Scholar

[3]

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, Contemp. Math., Amer. Math. Soc., Providence, RI, 514 (2010), 45–84. doi: 10.1007/s11784-010-0033-9.  Google Scholar

[4]

I. Berezovik, Q. Hu and W. Krawcewicz, Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces, accepted in Discrete & Continuous Dynamical Systems - S, (2018) arXiv: 1702.04234. Google Scholar

[5]

G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York-London, 1972.  Google Scholar

[6]

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.  Google Scholar

[7]

F. G. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for non-linear mappings in a Banach space, J. Functional Anal., 3 (1969), 217-245.   Google Scholar

[8]

M. Dabkowski, W. Krawcewicz and Y. Lv, H-P. Wu, Multiple periodic solutions for $Γ$-symmetric Newtonian systems, J. Differential Equations, 263 (2017), 6684-6730. Google Scholar

[9]

T. tom Dieck, Transformation Groups, Walter de Gruyter, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[10]

K. EfstathiouD. A. Sadovskii and B. I. Zhilinskii, Analysis of rotation-vibration relative equilibria on the example of a tetrahedral four atom molecule, SIAM J. Appl. Dyn. Sys., 3 (2004), 261-351.  doi: 10.1137/030600015.  Google Scholar

[11]

J. FuraA. 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.  Google Scholar

[12]

C. García-Azpeitia and J. Ize, Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n-body problem, J. Differential Equations, 254 (2013), 2033-2075.  doi: 10.1016/j.jde.2012.08.022.  Google Scholar

[13]

C. García-Azpeitia and M. Tejada-Wriedt, Molecular chains interacting by Lennard-Jones and Coulomb forces, Qualitative Theory of Dynamical Systems, 16 (2017), 591-608.  doi: 10.1007/s12346-016-0221-0.  Google Scholar

[14]

K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis Ⅱ (Frascati, 1995), Progr. Nonlinear Differential Equations A, Birkhäuser, Boston, 27 (1997), 247–272.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

K. Kawakubo, The Theory of Transformation Groups, The Clarendon Press, Oxford University Press, 1991.  Google Scholar

[18]

W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, John Wiley & Sons, Inc., 1997.  Google Scholar

[19]

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.  Google Scholar

[20]

J. Montaldi, M. Roberts and I. Stewart, Nonlinear normal modes of symmetric Hamiltonian systems, The Physics of Structure Formation, 354–371, Springer Ser. Synergetics, 37, Springer, Berlin, 1987. doi: 10.1007/978-3-642-73001-6_28.  Google Scholar

[21]

J. MontaldiM. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Philos. Trans. Roy. Soc. London, Ser. A, 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053.  Google Scholar

[22]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.  doi: 10.1007/BF01405263.  Google Scholar

[23]

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. Google Scholar

[24]

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.  Google Scholar

[25]

S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries, Topol. Methods Nonlinear Anal., 9 (1997), 383-417.  doi: 10.12775/TMNA.1997.018.  Google Scholar

[26]

E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto-London, 1966.  Google Scholar

show all references

References:
[1]

Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree, AIMS Series on Differential Equations & Dynamical Systems, Vol. 1, 2006.  Google Scholar

[2]

Z. BalanovW. KrawcewiczS. Rybicki and H. Steinlein, A short treatise on the equivariant degree theory and its applications, J. Fixed Point Theory Appl., 8 (2010), 1-74.  doi: 10.1007/s11784-010-0033-9.  Google Scholar

[3]

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, Contemp. Math., Amer. Math. Soc., Providence, RI, 514 (2010), 45–84. doi: 10.1007/s11784-010-0033-9.  Google Scholar

[4]

I. Berezovik, Q. Hu and W. Krawcewicz, Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces, accepted in Discrete & Continuous Dynamical Systems - S, (2018) arXiv: 1702.04234. Google Scholar

[5]

G. E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York-London, 1972.  Google Scholar

[6]

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.  Google Scholar

[7]

F. G. Browder and W. V. Petryshyn, Approximation methods and the generalized topological degree for non-linear mappings in a Banach space, J. Functional Anal., 3 (1969), 217-245.   Google Scholar

[8]

M. Dabkowski, W. Krawcewicz and Y. Lv, H-P. Wu, Multiple periodic solutions for $Γ$-symmetric Newtonian systems, J. Differential Equations, 263 (2017), 6684-6730. Google Scholar

[9]

T. tom Dieck, Transformation Groups, Walter de Gruyter, 1987. doi: 10.1515/9783110858372.312.  Google Scholar

[10]

K. EfstathiouD. A. Sadovskii and B. I. Zhilinskii, Analysis of rotation-vibration relative equilibria on the example of a tetrahedral four atom molecule, SIAM J. Appl. Dyn. Sys., 3 (2004), 261-351.  doi: 10.1137/030600015.  Google Scholar

[11]

J. FuraA. 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.  Google Scholar

[12]

C. García-Azpeitia and J. Ize, Global bifurcation of planar and spatial periodic solutions from the polygonal relative equilibria for the n-body problem, J. Differential Equations, 254 (2013), 2033-2075.  doi: 10.1016/j.jde.2012.08.022.  Google Scholar

[13]

C. García-Azpeitia and M. Tejada-Wriedt, Molecular chains interacting by Lennard-Jones and Coulomb forces, Qualitative Theory of Dynamical Systems, 16 (2017), 591-608.  doi: 10.1007/s12346-016-0221-0.  Google Scholar

[14]

K. Gȩba, Degree for gradient equivariant maps and equivariant Conley index, in Topological Nonlinear Analysis Ⅱ (Frascati, 1995), Progr. Nonlinear Differential Equations A, Birkhäuser, Boston, 27 (1997), 247–272.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

K. Kawakubo, The Theory of Transformation Groups, The Clarendon Press, Oxford University Press, 1991.  Google Scholar

[18]

W. Krawcewicz and J. Wu, Theory of Degrees with Applications to Bifurcations and Differential Equations, John Wiley & Sons, Inc., 1997.  Google Scholar

[19]

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.  Google Scholar

[20]

J. Montaldi, M. Roberts and I. Stewart, Nonlinear normal modes of symmetric Hamiltonian systems, The Physics of Structure Formation, 354–371, Springer Ser. Synergetics, 37, Springer, Berlin, 1987. doi: 10.1007/978-3-642-73001-6_28.  Google Scholar

[21]

J. MontaldiM. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Philos. Trans. Roy. Soc. London, Ser. A, 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053.  Google Scholar

[22]

A. Weinstein, Normal modes for nonlinear Hamiltonian systems, Invent. Math., 20 (1973), 47-57.  doi: 10.1007/BF01405263.  Google Scholar

[23]

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. Google Scholar

[24]

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.  Google Scholar

[25]

S. Rybicki, Applications of degree for $S^1$-equivariant gradient maps to variational nonlinear problems with $S^1$-symmetries, Topol. Methods Nonlinear Anal., 9 (1997), 383-417.  doi: 10.12775/TMNA.1997.018.  Google Scholar

[26]

E. H. Spanier, Algebraic Topology, McGraw-Hill Book Co, New York-Toronto-London, 1966.  Google Scholar

Figure 1.  Stationary solution to equation (2) with tetrahedral symmetries
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