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, 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[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.

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[26]

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

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