Dihedral Molecular Configurations Interacting by Lennard-Jones and   Coulomb Forces

Irina Berezovik; Qingwen Hu; Wieslaw Krawcewicz

arXiv:1702.04234·math.DS·February 15, 2017

Dihedral Molecular Configurations Interacting by Lennard-Jones and Coulomb Forces

Irina Berezovik, Qingwen Hu, Wieslaw Krawcewicz

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TL;DR

This paper studies the periodic vibrations of particles arranged in dihedral configurations influenced by Lennard-Jones and Coulomb forces, classifying solutions with symmetries and identifying critical frequencies for small oscillations.

Contribution

It introduces a topological classification of symmetric periodic solutions using the gradient equivariant degree and provides formulas for the spectrum of the linearized system.

Findings

01

Classification of periodic solutions with symmetries

02

Formulas for the spectrum of the linearized system

03

Identification of critical frequencies for particle motions

Abstract

In this paper, we investigate periodic vibrations of a group of particles with a dihedral configuration in the plane 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 with general formulae for the spectrum of the linearized system which allows us to obtain the critical frequencies of the particle motions which indicate the set of all critical periods of small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

Tables2

Table 1. Table 1. Conjugacy Classes of Subgroups in D 6 × O ( 2 ) subscript 𝐷 6 𝑂 2 D_{6}\times O(2)

ID	$(S)$	$\| W (S) \|$	ID	$(S)$	$\| W (S) \|$	ID	$(S)$	$\| W (S) \|$
1	$(ℤ_{1} \times ℤ_{n})$	$\infty$	35	$(D_{6} {}^{D_{3}}\times_{D_{1}} D_{n})$	$4$	69	$(D_{6} {}^{{\tilde{D}}_{3}}\times_{ℤ_{2}} D_{2 n})$	$2$
2	$(D_{1} \times ℤ_{n})$	$\infty$	36	$(D_{6} {}^{ℤ_{6}}\times_{D_{1}} D_{n})$	$4$	70	$(ℤ_{1} \times S O (2))$	$24$
3	$(ℤ_{2} \times ℤ_{n})$	$\infty$	37	$(D_{6} {}^{{\tilde{D}}_{3}}\times_{D_{1}} D_{n})$	$4$	71	$(D_{1} \times S O (2))$	$4$
4	$({\tilde{D}}_{1} \times ℤ_{n})$	$\infty$	38	$(D_{2} {}^{ℤ_{1}}\times_{D_{2}}^{D_{1}} D_{2 n})$	$4$	72	$(ℤ_{2} \times S O (2))$	$12$
5	$(ℤ_{3} \times ℤ_{n})$	$\infty$	39	$(D_{2} {}^{ℤ_{1}}\times_{D_{2}}^{ℤ_{2}} D_{2 n})$	$4$	73	$({\tilde{D}}_{1} \times S O (2))$	$4$
6	$(D_{2} \times ℤ_{n})$	$\infty$	40	$(D_{2} {}^{ℤ_{1}}\times_{D_{2}}^{{\tilde{D}}_{1}} D_{2 n})$	$4$	74	$(ℤ_{3} \times S O (2))$	$8$
7	$(D_{3} \times ℤ_{n})$	$\infty$	41	$(D_{6} {}^{ℤ_{3}}\times_{D_{2}}^{D_{3}} D_{2 n})$	$4$	75	$(D_{2} \times S O (2))$	$2$
8	$(ℤ_{6} \times ℤ_{n})$	$\infty$	42	$(D_{6} {}^{ℤ_{3}}\times_{D_{2}}^{ℤ_{6}} D_{2 n})$	$4$	76	$(D_{3} \times S O (2))$	$4$
9	$({\tilde{D}}_{3} \times ℤ_{n})$	$\infty$	43	$(D_{6} {}^{ℤ_{3}}\times_{D_{2}}^{{\tilde{D}}_{3}} D_{2 n})$	$4$	77	$(ℤ_{6} \times S O (2))$	$4$
10	$(D_{6} \times ℤ_{n})$	$\infty$	44	$(D_{3} {}^{ℤ_{1}}\times_{D_{3}} D_{3 n})$	$4$	78	$({\tilde{D}}_{3} \times S O (2))$	$4$
11	$(D_{1} {}^{ℤ_{1}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	45	$({\tilde{D}}_{3} {}^{ℤ_{1}}\times_{D_{3}} D_{3 n})$	$4$	79	$(D_{6} \times S O (2))$	$2$
12	$(ℤ_{2} {}^{ℤ_{1}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	46	$(D_{6} {}^{ℤ_{2}}\times_{D_{3}} D_{3 n})$	$2$	80	$(D_{1} {}^{ℤ_{1}}\times_{D_{1}} O (2))$	$4$
13	$({\tilde{D}}_{1} {}^{ℤ_{1}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	47	$(D_{6} {}^{ℤ_{1}}\times_{D_{6}} D_{6 n})$	$2$	81	$(ℤ_{2} {}^{ℤ_{1}}\times_{D_{1}} O (2))$	$12$
14	$(D_{2} {}^{D_{1}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	48	$(ℤ_{1} \times D_{n})$	$24$	82	$({\tilde{D}}_{1} {}^{ℤ_{1}}\times_{D_{1}} O (2))$	$4$
15	$(D_{2} {}^{ℤ_{2}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	49	$(D_{1} \times D_{n})$	$4$	83	$(D_{2} {}^{D_{1}}\times_{D_{1}} O (2))$	$2$
16	$(D_{2} {}^{{\tilde{D}}_{1}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	50	$(ℤ_{2} \times D_{n})$	$12$	84	$(D_{2} {}^{ℤ_{2}}\times_{D_{1}} O (2))$	$2$
17	$(D_{3} {}^{ℤ_{3}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	51	$({\tilde{D}}_{1} \times D_{n})$	$4$	85	$(D_{2} {}^{{\tilde{D}}_{1}}\times_{D_{1}} O (2))$	$2$
18	$(ℤ_{6} {}^{ℤ_{3}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	52	$(ℤ_{3} \times D_{n})$	$8$	86	$(D_{3} {}^{ℤ_{3}}\times_{D_{1}} O (2))$	$4$
19	$({\tilde{D}}_{3} {}^{ℤ_{3}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	53	$(D_{2} \times D_{n})$	$2$	87	$(ℤ_{6} {}^{ℤ_{3}}\times_{D_{1}} O (2))$	$4$
20	$(D_{6} {}^{D_{3}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	54	$(D_{3} \times D_{n})$	$4$	88	$({\tilde{D}}_{3} {}^{ℤ_{3}}\times_{D_{1}} O (2))$	$4$
21	$(D_{6} {}^{ℤ_{6}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	55	$(ℤ_{6} \times D_{n})$	$4$	89	$(D_{6} {}^{D_{3}}\times_{D_{1}} O (2))$	$2$
22	$(D_{6} {}^{{\tilde{D}}_{3}}\times_{ℤ_{2}} ℤ_{2 n})$	$\infty$	56	$({\tilde{D}}_{3} \times D_{n})$	$4$	90	$(D_{6} {}^{ℤ_{6}}\times_{D_{1}} O (2))$	$2$
23	$(ℤ_{3} {}^{ℤ_{1}}\times_{ℤ_{3}} ℤ_{3 n})$	$\infty$	57	$(D_{6} \times D_{n})$	$2$	91	$(D_{6} {}^{{\tilde{D}}_{3}}\times_{D_{1}} O (2))$	$2$
24	$(ℤ_{6} {}^{ℤ_{2}}\times_{ℤ_{3}} ℤ_{3 n})$	$\infty$	58	$(D_{1} {}^{ℤ_{1}}\times_{ℤ_{2}} D_{2 n})$	$4$	92	$(ℤ_{1} \times O (2))$	$12$
25	$(ℤ_{6} {}^{ℤ_{1}}\times_{ℤ_{6}} ℤ_{6 n})$	$\infty$	59	$(ℤ_{2} {}^{ℤ_{1}}\times_{ℤ_{2}} D_{2 n})$	$12$	93	$(D_{1} \times O (2))$	$2$
26	$(D_{1} {}^{ℤ_{1}}\times_{D_{1}} D_{n})$	$8$	60	$({\tilde{D}}_{1} {}^{ℤ_{1}}\times_{ℤ_{2}} D_{2 n})$	$4$	94	$(ℤ_{2} \times O (2))$	$6$
27	$(ℤ_{2} {}^{ℤ_{1}}\times_{D_{1}} D_{n})$	$24$	61	$(D_{2} {}^{D_{1}}\times_{ℤ_{2}} D_{2 n})$	$2$	95	$({\tilde{D}}_{1} \times O (2))$	$2$
28	$({\tilde{D}}_{1} {}^{ℤ_{1}}\times_{D_{1}} D_{n})$	$8$	62	$(D_{2} {}^{ℤ_{2}}\times_{ℤ_{2}} D_{2 n})$	$2$	96	$(ℤ_{3} \times O (2))$	$4$
29	$(D_{2} {}^{D_{1}}\times_{D_{1}} D_{n})$	$4$	63	$(D_{2} {}^{{\tilde{D}}_{1}}\times_{ℤ_{2}} D_{2 n})$	$2$	97	$(D_{2} \times O (2))$	$1$
30	$(D_{2} {}^{ℤ_{2}}\times_{D_{1}} D_{n})$	$4$	64	$(D_{3} {}^{ℤ_{3}}\times_{ℤ_{2}} D_{2 n})$	$4$	98	$(D_{3} \times O (2))$	$2$
31	$(D_{2} {}^{{\tilde{D}}_{1}}\times_{D_{1}} D_{n})$	$4$	65	$(ℤ_{6} {}^{ℤ_{3}}\times_{ℤ_{2}} D_{2 n})$	$4$	99	$(ℤ_{6} \times O (2))$	$2$
32	$(D_{3} {}^{ℤ_{3}}\times_{D_{1}} D_{n})$	$8$	66	$({\tilde{D}}_{3} {}^{ℤ_{3}}\times_{ℤ_{2}} D_{2 n})$	$4$	100	$({\tilde{D}}_{3} \times O (2))$	$2$
33	$(ℤ_{6} {}^{ℤ_{3}}\times_{D_{1}} D_{n})$	$8$	67	$(D_{6} {}^{D_{3}}\times_{ℤ_{2}} D_{2 n})$	$2$	101	$(D_{6} \times O (2))$	$1$
34	$({\tilde{D}}_{3} {}^{ℤ_{3}}\times_{D_{1}} D_{n})$	$8$	68	$(D_{6} {}^{ℤ_{6}}\times_{ℤ_{2}} D_{2 n})$	$2$

Table 2. Table 2. Maximal orbit types in 𝒲 j l subscript 𝒲 𝑗 𝑙 \mathscr{W}_{jl}

$𝒲_{j l}$ , $l \geq 1$	maximal orbit types
$𝒲_{0 l}$	$(D_{6} \times D_{l})$
$𝒲_{1 l}$	$(D_{6} {}^{ℤ_{1}}\times_{D_{6}} D_{6 l})$ , $(D_{2} {}^{D_{1}}\times_{ℤ_{2}} D_{2 l})$ , $(D_{2} {}^{{\tilde{D}}_{1}}\times_{ℤ_{2}} D_{2 l})$
$𝒲_{2 l}$	$(D_{6} {}^{ℤ_{2}}\times_{D_{3}} D_{3 l})$ , $(D_{2} {}^{ℤ_{2}}\times_{ℤ_{2}} D_{2 l})$ , $(D_{2} \times D_{l})$
$𝒲_{3 l}$	$(D_{6} {}^{{\tilde{D}}_{3}}\times_{ℤ_{2}} D_{2 l})$

Equations374

V (u) := (k, j) \in B \sum n U (∣ u_{j} - u_{k} ∣^{2}) + 0 \leq j < k \leq n - 1 \sum W (∣ u_{j} - u_{k} ∣^{2}),

V (u) := (k, j) \in B \sum n U (∣ u_{j} - u_{k} ∣^{2}) + 0 \leq j < k \leq n - 1 \sum W (∣ u_{j} - u_{k} ∣^{2}),

U (t) = t - 2 t, W (t) = \frac{B}{t ^{6}} - \frac{A}{t ^{3}} + \frac{σ}{t}, t > 0.

U (t) = t - 2 t, W (t) = \frac{B}{t ^{6}} - \frac{A}{t ^{3}} + \frac{σ}{t}, t > 0.

\overset{u}{¨} (t) = - \nabla V (u (t)),

\overset{u}{¨} (t) = - \nabla V (u (t)),

{\overset{u}{¨} (t) = - λ^{2} \nabla^{2} V (u (t)), u (t) \in Ω_{o}^{'}, t \in R u (0) = u (2 π), \overset{u}{˙} (0) = \overset{u}{˙} (2 π),

{\overset{u}{¨} (t) = - λ^{2} \nabla^{2} V (u (t)), u (t) \in Ω_{o}^{'}, t \in R u (0) = u (2 π), \overset{u}{˙} (0) = \overset{u}{˙} (2 π),

Φ (G)

Φ (G)

Φ_{n} (G)

(H) \leq (K) ⟺ \exists_{g \in G} g H g^{- 1} \subset K .

(H) \leq (K) ⟺ \exists_{g \in G} g H g^{- 1} \subset K .

G_{x}

G_{x}

(G_{x})

G (x)

X_{H}

X_{H}

X^{H}

X_{(H)}

X^{(H)}

V = V_{0} \oplus V_{1} \oplus \dots \oplus V_{r},

V = V_{0} \oplus V_{1} \oplus \dots \oplus V_{r},

(U = U_{0} \oplus U_{1} \oplus \dots \oplus U_{s}, respectively),

(U = U_{0} \oplus U_{1} \oplus \dots \oplus U_{s}, respectively),

(H) * (K) = (L) \in Φ (G) \sum n_{L} (L),

(H) * (K) = (L) \in Φ (G) \sum n_{L} (L),

n_{L} := χ_{c} ((G / H \times G / K)_{L} / N (L)),

n_{L} := χ_{c} ((G / H \times G / K)_{L} / N (L)),

(H) \cdot (K) = (L) \sum n_{L} (L),

(H) \cdot (K) = (L) \sum n_{L} (L),

n_{L} = \frac{n ( L , K ) ∣ W ( K ) ∣ n ( L , H ) ∣ W ( H ) ∣ - \sum _{(L) > (L)} n ( L , L ) n _{L} W ( L )}{∣ W ( L ) ∣},

n_{L} = \frac{n ( L , K ) ∣ W ( K ) ∣ n ( L , H ) ∣ W ( H ) ∣ - \sum _{(L) > (L)} n ( L , L ) n _{L} W ( L )}{∣ W ( L ) ∣},

n (L, K) = \frac{N ( L , K )}{N ( K )}, N (L, K) := {g \in G : g L g^{- 1} \subset K},

n (L, K) = \frac{N ( L , K )}{N ( K )}, N (L, K) := {g \in G : g L g^{- 1} \subset K},

π_{0} ((H)) = {(H) 0 if (H) \in Φ_{0} (G), otherwise.

π_{0} ((H)) = {(H) 0 if (H) \in Φ_{0} (G), otherwise.

π_{0} ((H) * (K)) = π_{0} ((H)) \cdot π_{0} ((K)),

π_{0} ((H) * (K)) = π_{0} ((H)) \cdot π_{0} ((K)),

\nabla_{G} -deg (\nabla φ, Ω) = (H_{i}) \in Φ (Γ) \sum n_{H_{i}} (H_{i}) = n_{H_{1}} (H_{1}) + \dots + n_{H_{m}} (H_{m}),

\nabla_{G} -deg (\nabla φ, Ω) = (H_{i}) \in Φ (Γ) \sum n_{H_{i}} (H_{i}) = n_{H_{1}} (H_{1}) + \dots + n_{H_{m}} (H_{m}),

\nabla_{G} -deg (\nabla φ, Ω) = \nabla_{G} -deg (\nabla φ, Ω_{1}) + \nabla_{G} -deg (\nabla φ, Ω_{2}) .

\nabla_{G} -deg (\nabla φ, Ω) = \nabla_{G} -deg (\nabla φ, Ω_{1}) + \nabla_{G} -deg (\nabla φ, Ω_{2}) .

\nabla_{G} -deg (\nabla_{v} Ψ (t, \cdot), Ω) = a constant .

\nabla_{G} -deg (\nabla_{v} Ψ (t, \cdot), Ω) = a constant .

\nabla_{G} -deg (\nabla φ, Ω) = (- 1)^{m^{-} (\nabla^{2} φ (v_{0}))} \cdot (H),

\nabla_{G} -deg (\nabla φ, Ω) = (- 1)^{m^{-} (\nabla^{2} φ (v_{0}))} \cdot (H),

\nabla_{G} -deg (\nabla φ_{1} \times \nabla φ_{2}, Ω_{1} \times Ω_{2}) = \nabla_{G} -deg (\nabla φ_{1}, Ω_{1}) * \nabla_{G} -deg (\nabla φ_{2}, Ω_{2})

\nabla_{G} -deg (\nabla φ_{1} \times \nabla φ_{2}, Ω_{1} \times Ω_{2}) = \nabla_{G} -deg (\nabla φ_{1}, Ω_{1}) * \nabla_{G} -deg (\nabla φ_{2}, Ω_{2})

\nabla_{G} -deg (\nabla φ \times \mbox Id_{W}, Ω \times B) = \nabla_{G} -deg (\nabla φ, Ω) .

\nabla_{G} -deg (\nabla φ \times \mbox Id_{W}, Ω \times B) = \nabla_{G} -deg (\nabla φ, Ω) .

\nabla_{G} -deg (\nabla φ_{1}, B (V)) = \nabla_{G} -deg (\nabla φ_{2}, B (V)) .

\nabla_{G} -deg (\nabla φ_{1}, B (V)) = \nabla_{G} -deg (\nabla φ_{2}, B (V)) .

T_{i} := T ∣_{V_{i}} : V_{i} \to V_{i}, T_{j, l} := T ∣_{V_{j, l}} : V_{j, l} \to V_{j, l} .

T_{i} := T ∣_{V_{i}} : V_{i} \to V_{i}, T_{j, l} := T ∣_{V_{j, l}} : V_{j, l} \to V_{j, l} .

\nabla_{G} \mbox - d e g (T, B (V)) = i \prod r \nabla_{G} \mbox - d e g (T_{i}, B (V_{i})) * j, l \prod \nabla_{G} \mbox - d e g (T_{j, l}, B (V_{j, l})) .

\nabla_{G} \mbox - d e g (T, B (V)) = i \prod r \nabla_{G} \mbox - d e g (T_{i}, B (V_{i})) * j, l \prod \nabla_{G} \mbox - d e g (T_{j, l}, B (V_{j, l})) .

m_{i} (μ) := dim (E (μ) \cap V_{i}) / dim V_{i}, m_{j, l} (μ) := dim (E (μ) \cap V_{j, l}) / dim V_{j, l} .

m_{i} (μ) := dim (E (μ) \cap V_{i}) / dim V_{i}, m_{j, l} (μ) := dim (E (μ) \cap V_{j, l}) / dim V_{j, l} .

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TopicsQuantum chaos and dynamical systems · Nonlinear Waves and Solitons · Advanced Differential Equations and Dynamical Systems

Full text

Dihedral Molecular Configurations Interacting by Lennard-Jones and Coulomb Forces

Irina Berezovik

[email protected]

,

Qingwen Hu

[email protected]

and

Wieslaw Krawcewicz

[email protected]

Department of Mathematical Sciences, the University of Texas at Dallas, Richardson, TX, 75080-3021, U.S.A.

Abstract.

In this paper, we investigate periodic vibrations of a group of particles with a dihedral configuration in the plane 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 with general formulae for the spectrum of the linearized system which allows us to obtain the critical frequencies of the particle motions which indicate the set of all critical periods of small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

1. Introduction

Classical forces used in molecular mechanics associated with bonding between the adjacent particles, electrostatic interactions and van der Waals forces, are modeled using bonding by Lennard-Jones and Coulomb potentials. In a typical molecule an atom is bonded only to few of its neighbors but it also interacts with every other atom in the molecule. The famous 6-12–Lennard-Jones potential, which was proposed in 1924 (cf. [17]), was found experimentally and since then it is successfully used in molecular modeling. Certainly one can expect that other types of more accurate potentials may be introduced in the future. To be more precise, consider $n$ identical particles $u_{j}$ , $j\in\{0,1,2,\dots,n-1\}=:X$ , in the space $\mathbb{R}^{3}$ . A set $\mathscr{B}\subset X\times X$ satisfying the conditions (1) $(i,j)\in\mathscr{B}$ then $(j,i)\in\mathscr{B}$ , and (2) $(j,j)\notin\mathscr{B}$ , can be considered as bonding set for a specific configuration of atoms in the molecule, that means we suppose that the particles $u_{k}$ and $u_{j}$ are bonded if $(k,j)\in\mathscr{B}$ . The symmetries of a molecular bonding are reflected in the set $\mathscr{B}$ . There are many examples of symmetric atomic molecules, for example, octahedral compounds of sulfur hexafluoride SF6 and molybdenum hexacarbonyl Mo(CO)6, the etraphosphorus P4, a spherical fullerene molecule with the formula C60 with icosahedral symmetry or dihedral molecule with 2-D interactions. One can find multiple example of symmetric molecule clusters in [24]. We describe the molecular model considered in this paper as follows. Let $u=(u_{0},u_{1},\dots,u_{n-1})\in\mathbb{R}^{2n}$ and $\Omega^{\prime}_{o}:=\{u\in\mathbb{R}^{2n}:u_{k}\not=u_{j},\text{ if }k\not=j,\text{ with }\,k,\,j=0,\,1,\,2,\,\cdots,\,n-1\}$ . Define the following energy functional $\bm{\mathcal{V}}:\Omega^{\prime}_{o}\to\mathbb{R}$ by

[TABLE]

where

[TABLE]

The following Newtonian equation describes the interaction between these $n$ -particles,

[TABLE]

In this paper we develop a new method allowing an extraction from model (2) a topological equivariant classification of $p$ -periodic ( $p>0$ ) molecular vibrations for a symmetric molecule in 2-D polygonal symmetric configuration with dihedral symmetry group. The vibrational motions, which are characteristic of all molecules can be easily detect using infrared or Raman spectroscopy, depend on the vibrational structure of electronic transitions in molecules. The vibrational motions are closely connected to the symmetric properties of $2\pi$ -periodic solutions the system

[TABLE]

where $\lambda=\frac{p}{2\pi}$ , which are exactly $p$ -periodic solutions to (2).

An important feature of a molecular vibration is that in general it admits spatial-temporal symmetries (depending on the actual molecular symmetries), which called a mode of vibration (reflected in atomic motions such as stretching, bending, rocking, wagging and twisting). These modes and the corresponding vibrational frequencies are of great importance in molecular dynamics. It is thus desirable to distinguish periodic motions with distinct symmetric modes of vibrations.

The content of this paper can be described as follows. In section 2 we recall the basic definitions and properties related to the equivariant degree theory. In section 3, we discuss a molecular model with Lennard-Jones and Coulomb potentials for identical atoms bonded in a polygonal configuration. In subsection 3.1 we show the existence of the symmetric equilibrium $u^{o}$ and in subsection 3.2 we formulate the problem of finding periodic vibration as a bifurcation problem for (23) and in subsection 3.3 we identify the $D_{n}$ -isotypical decomposition of the phase space. In section 4, the problem (23) is reformulated as an $D_{n}\times O(2)$ -equivariant bifurcation variational problem. The equivariant invariant $\omega(\lambda_{o})$ is provided in Theorem 4.2. Section 5 is devoted to the symbolic computations of the spectrum of $\nabla^{2}\bm{\mathcal{V}}(u_{o})$ (for a general potential $\bm{\mathcal{V}}$ ). In section 6 we formulate the main existence results based on the values of the equivariant invariants $\omega(\lambda_{o})$ (Theorem 6.1). In section 7, we consider a concrete system (2) with $D_{6}$ -symmetries and compute several equivariant invariants iand how to extract the relevant equivariant information. Finally, we confirm the obtained existence results with several computer simulations (in subsection 7.2).

2. Preliminaries

2.1. Equivariant Jargon:

$G$ -Actions:

In what follows $G$ always stands for a compact Lie group and all subgroups of $G$ are assumed to be closed. For a subgroup $H\subset G$ , denote by $N\left(H\right)$ the normalizer of $H$ in $G$ , and by $W\left(H\right)=N\left(H\right)/H$ the Weyl group of $H$ in $G$ . In the case when we are dealing with different Lie groups, we also write $N_{G}\left(H\right)$ ( $W_{G}\left(H\right)$ , respectively) instead of $N\left(H\right)$ ( $W\left(H\right)$ , respectively). We denote by $\left(H\right)$ the conjugacy class of $H$ in $G$ and define the following notations:

[TABLE]

The set $\Phi\left(G\right)$ has a natural partial order defined by

[TABLE]

For a $G$ -space $X$ and $x\in X$ , we define

[TABLE]

Moreover, for a subgroup $H\subset G$ , we use the following notations:

[TABLE]

The orbit space for a $G$ -space $X$ will be denoted by $X/G$ and for the space $G$ by $G\backslash X$ .

Isotypical Decomposition of Finite-Dimensional Representations:

As any compact Lie group admits only countably many non-equivalent real (complex, respectively) irreducible representations. Given a compact Lie group $G$ , we assume that we have a complete list of its all real (complex, respectively) irreducible representations, denoted $\mathcal{V}_{i}$ , $i=0,$ $1,$ $\ldots$ ( $\mathcal{U}_{j}$ , $j=0,$ $1,$ $\ldots$ , respectively). We refer to [1] for examples of such lists and the related notations.

Let $V$ ( $U$ , respectively) be a finite-dimensional real (complex, respectively) $\Gamma$ -representation. Without loss of generality, $V$ ( $U$ , respectively) can be assumed to be orthogonal (unitary, respectively). Then, $V$ ( $U$ , respectively) decomposes into the direct sum of $G$ -invariant subspaces

[TABLE]

which is called the $G$ *-*isotypical decomposition of $V$ ( $U$ , respectively), where each isotypical component $V_{i}$ (resp. $U_{j}$ ) is modeled on the irreducible $G$ -representation $\mathcal{V}_{i}$ , $i=0,$ $1,$ $\dots,$ $r$ , ( $\mathcal{U}_{j}$ , $j=0,$ $1,$ $\dots,$ $s$ , respectively), i.e. $V_{i}$ ( $U_{j}$ , respectively) contains all the irreducible subrepresentations of $V$ ( $U$ , respectively) which are equivalent to $\mathcal{V}_{i}$ ( $\mathcal{U}_{j}$ , respectively).

2.2. Gradient $G$ -Equivariant Degree

Euler Ring and Burnside Ring:

Definition 2.1.

(cf. [8]) Let $U\left(G\right):={\mathbb{Z}}\left[\Phi\left(G\right)\right]$ denote the free $\mathbb{Z}$ -module generated by $\Phi(G)$ . Define a ring multiplication on generators $\left(H\right)$ , $\left(K\right)\in\Phi\left(G\right)$ as follows:

[TABLE]

where

[TABLE]

for $\chi_{c}$ being the Euler characteristic taken in Alexander-Spanier cohomology with compact support (cf. [22]). The $\mathbb{Z}$ -module $U\left(G\right)$ equipped with the multiplication (7), (8) is a ring called the Euler ring of the group $G$ (cf. [6])

The ${\mathbb{Z}}$ -module $A\left(G\right)=A_{0}\left(G\right):={\mathbb{Z}}\left[\Phi_{0}\left(G\right)\right]$ equipped with a similar multiplication as in $U\left(G\right)$ but restricted only to generators from $\Phi_{0}\left(G\right)$ , is called a Burnside ring. That is, for $\left(H\right)$ , $\left(K\right),\left(L\right)\in\Phi_{0}\left(G\right)$

[TABLE]

where $n_{L}=\chi\left(\left(G/H\times G/K\right)_{L}/N\left(L\right)\right)=\left|\left(G/H\times G/K\right)_{L}/N\left(L\right)\right|$ and $\chi$ stands for the usual Euler characteristic. In this case, we have

[TABLE]

where

[TABLE]

and $\left(H\right),$ $\left(K\right),$ $\left(L\right)$ , $\left(\widetilde{L}\right)$ are taken from $\Phi_{0}\left(G\right)$ .Notice that $A\left(G\right)$ is a ${\mathbb{Z}}$ -submodule of $U\left(G\right)$ , but not a subring. Define $\pi_{0}:U\left(G\right)\rightarrow A\left(G\right)$ on generators $\left(H\right)\in\Phi\left(G\right)$ by

[TABLE]

Then we have,

Lemma 2.2.

(cf. [3]) The map $\pi_{0}$ defined by $(\mathrm{\ref{eq:pi_0-homomorphism}})$ is a ring homomorphism, that is,

[TABLE]

where $\left(H\right),\text{ }\left(K\right)\in\Phi\left(G\right).$

Lemma 2.2 allows us to use Burnside ring multiplication structure in $A\left(G\right)$ to partially describe the Euler ring multiplication structure in $U\left(G\right)$ .

$G$ -equivariant Gradient Degree $\nabla_{G}\text{\rm-deg}$ :

Assume that $G$ is a compact Lie group. Denote by $\mathcal{M}^{G}_{\nabla}$ the set of all admissible pairs $(\nabla\varphi,\Omega)$ .

Definition 2.3.

A $G$ -gradient $\Omega$ -admissible map $f:=\nabla\varphi$ is called a *special * $\Omega$ -Morse function if

(i)

$f|_{\Omega}$ is of class $C^{1}$ ;

(ii)

$f^{-1}(0)\cap\Omega$ is composed of regular zero orbits;

(iii)

for each $(H)$ with $f^{-1}(0)\cap\Omega_{(H)}\neq\emptyset$ , there exists a tubular neighborhood $\mathcal{N}(U,\varepsilon)$ such that $f$ is $(H)$ -normal on $\mathcal{N}(U,\varepsilon)$ .

We have,

Theorem 2.4.

(cf. [12]) There exists a unique map $\nabla_{G}\text{\rm-deg\,}:\mathcal{M}_{\nabla}^{G}\to U(G)$ , which assigns to every $(\nabla\varphi,\Omega)\in\mathcal{M}^{G}_{\nabla}$ an element $\nabla_{G}\text{\rm-deg\,}(\nabla\varphi,\Omega)\in U(G)$ , called the $G$ -gradient degree of $\nabla\varphi$ on $\Omega$ ,

[TABLE]

satisfying the following properties:

( $\nabla$ 1)

(Existence)* If $\nabla_{G}\text{\rm-deg\,}(\nabla\varphi,\Omega)\not=0$ , that is, there is in (11) a non-zero coefficient $n_{H_{i}}$ , then there exists ${x\in\Omega}$ such that $\nabla\varphi(x)=0$ and $(G_{x})\geq(H_{i})$ .*

( $\nabla$ 2)

(Additivity)* Let $\Omega_{1}$ and $\Omega_{2}$ be two disjoint open $G$ -invariant subsets of $\Omega$ such that $(\nabla\varphi)^{-1}(0)\cap\Omega\subset\Omega_{1}\cup\Omega_{2}.$ Then,*

[TABLE]

( $\nabla$ 3)

(Homotopy)* If $\nabla_{v}\Psi:[0,1]\times V\to V$ is a $G$ -gradient $\Omega$ -admissible homotopy, then*

[TABLE]

( $\nabla$ 4)

(Normalization)* Let $\varphi\in C^{2}_{G}(V,\mathbb{R})$ be a special $\Omega$ -Morse function such that $(\nabla\varphi)^{-1}(0)\cap\Omega=G(v_{0})$ and $G_{v_{0}}=H$ . Then,*

[TABLE]

where “ $\mathrm{m}^{-}(\cdot)$ ” stands for the total dimension of eigenspaces for negative eigenvalues of a symmetric matrix.

( $\nabla$ 5)

(Multiplicativity)* For all $(\nabla\varphi_{1},\Omega_{1})$ , $(\nabla\varphi_{2},\Omega_{2})\in\mathcal{M}^{G}_{\nabla}$ ,*

[TABLE]

where the multiplication ‘ $\ast$ ’ is taken in the Euler ring $U(G)$ .

( $\nabla$ 6)

(Suspension)* If $W$ is an orthogonal $G$ -representation and $\mathcal{B}$ an open bounded invariant neighborhood of $0\in W$ , then*

[TABLE]

( $\nabla$ 7)

(Hopf Property)* Assume that $B(V)$ is the unit ball of an orthogonal $\Gamma$ -representation $V$ and for $(\nabla\varphi_{1},B(V)),(\nabla\varphi_{2},B(V))\in\mathcal{M}^{G}_{\nabla}$ , one has*

[TABLE]

Then $\nabla\varphi_{1}$ and $\nabla\varphi_{2}$ are $G$ -gradient $B(V)$ -admissible homotopic.

Computations of the Gradient $G$ -Equivariant Degree:

Consider a symmetric $G$ -equivariant linear isomorphism $T:V\to V$ , where $V$ is an orthogonal $G$ -representation, that is, $T=\nabla\varphi$ for $\varphi(v)=\frac{1}{2}(Tv\bullet T)$ , $v\in V$ , where “ $\bullet$ ” stands for the inner product. We will show how to compute $\nabla_{G}\text{-deg\,}(T,B(V))$ . Consider the $G$ -isotypical decomposition (5) of $V$ and put

[TABLE]

Then, by the Multiplicativity property ( $\nabla$ 5),

[TABLE]

Take $\mu\in\sigma_{-}(T)$ , where $\sigma_{-}(T)$ stands for the negative spectrum of $T$ , and consider the corresponding eigenspace $E(\mu):=\ker(T-\mu\mbox{Id})$ . Define the numbers $m_{i}(\mu)$ and $m_{j,l}(\mu)$ by

[TABLE]

We also define the basic gradient degrees by

[TABLE]

We have that,

(i)

$\mbox{\rm Deg}_{\mathcal{V}_{i}}=\deg_{\mathcal{V}_{i}}$ ;

(ii)

$\mbox{\rm Deg}_{\mathcal{V}_{j,l}}=(G)-\deg_{\mathcal{V}_{j,l}}$ ,

where $\deg_{\mathcal{V}_{i}}$ is the basic $G$ -equivariant degree without free parameter and $\deg_{\mathcal{V}_{j,l}}$ is the basic $G$ -equivariant basic twisted degree, which was introduced in [1]. The basic degree

[TABLE]

can be computed from the recurrence formula

[TABLE]

and the twisted degree

[TABLE]

can be computed from the recurrence formula

[TABLE]

One can also find in [1] complete lists of these basic degrees for several groups $G=\Gamma\times S^{1}$ . Then, by using the properties of gradient $G$ -equivariant degree, one can establish

[TABLE]

2.3. Gradient Degree on the Slice

Let $G$ be a compact Lie group and $\mathscr{H}$ be a smooth Hilbert $G$ -representation (that is, $T:G\to O(\mathscr{H})$ is a smooth map). Let $\varphi:\mathscr{H}\rightarrow\mathbb{R}$ be a continuously differentiable $G$ -invariant functional. Then the gradient

[TABLE]

is a well-defined $G$ -equivariant operator. Let $u_{o}\in\mathscr{H}$ and put $H:=G_{u_{o}}$ . Since the $G$ -action on $\mathscr{H}$ is smooth, the orbit $G(u_{o})$ is a smooth submanifold of $\mathscr{H}$ . Denote by $S_{o}\subset\mathscr{H}$ the slice to the orbit $G(u_{o})$ at $u_{o}$ . Denote by $V_{o}:=\tau_{u_{o}}G(u_{o})$ the tangent space to $G(u_{o})$ at $u_{o}$ . Then clearly, $S_{o}=V_{o}^{\perp}$ and $S_{o}$ is a smooth Hilbert $H$ -representation.

Theorem 2.5.

(Slice Principle)* Let $\mathscr{E}$ be an orthogonal $G$ -representation, $\varphi:\mathscr{H}\rightarrow\mathbb{R}$ be a continuously differentiable $G$ -invariant functional, $u_{o}\in\mathscr{H}$ and $G(u_{o})$ be an isolated critical orbit of $\varphi$ such that $H:=G_{u_{o}}$ . Let $S_{o}$ be the slice to the orbit $G(u_{o})$ at $u_{o}$ and $\mathcal{U}$ an isolated tubular neighborhood of $G(u_{o})$ . Define $\varphi_{o}:S_{o}\to\mathbb{R}$ by $\varphi_{o}(v)=\varphi(u_{o}+v)$ , $v\in S_{o}$ . Then*

[TABLE]

where $\Theta:U(H)\to U(G)$ is defined on generators $\Theta(K)=(K)$ , $(K)\in\Phi(H)$ .

Proof.

Notice that $\nabla\varphi_{o}(u)=P\nabla\varphi(u)$ , where $P:\mathscr{E}\to S_{o}$ is an orthogonal projection. Since one can always approximate $\varphi_{o}$ on $\mathcal{U}\cap S_{o}$ by a generic map, which can be extended equivariantly on $\mathcal{U}$ and, in such a case, this extension is also generic, formula (18) follows directly from the definition of the gradient degree for generic maps. ∎

2.4. Product Group $G_{1}\times G_{2}$

Given two groups $\mathscr{G}_{1}$ and $\mathscr{G}_{2}$ , consider the product group and $\mathscr{G}_{1}\times\mathscr{G}_{2}$ . The following well-known result (see [7, 13]) provides a description of subgroups $\mathscr{H}$ of the product group $\mathscr{G}_{1}\times\mathscr{G}_{2}$ .

Theorem 2.6.

Let $\mathscr{H}$ be a subgroup of the product group $\mathscr{G}_{1}\times\mathscr{G}_{2}$ . Put $H:=\pi_{1}(\mathscr{H})$ and $K:=\pi_{2}(\mathscr{H})$ . Then, there exist a group $L$ and two epimorphisms $\varphi:H\rightarrow L$ and $\psi:K\rightarrow L$ , such that

[TABLE]

In this case, we will use the notation

[TABLE]

The conjugacy classes of subgroups of $\mathscr{G}_{1}\times\mathscr{G}_{2}$ , one needs the following statement (see [7]).

Proposition 2.7.

Let $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ be two groups. Two subgroups $H{{}^{\varphi}\times_{L}^{\psi}}K,H^{\prime}{{}^{\varphi^{\prime}}\times_{L}^{\psi^{\prime}}}K^{\prime}$ of $\mathcal{G}_{1}\times\mathcal{G}_{2}$ are conjugate if and only if there exist $(a,b)\in\mathcal{G}_{1}\times\mathcal{G}_{2}$ and $\alpha\in\text{\rm Aut\,}(L)$ such that the inner automorphisms $\mu_{a}:\mathcal{G}_{1}\to\mathcal{G}_{1}$ and $\mu_{b}:\mathcal{G}_{2}\to\mathcal{G}_{2}$ given by

[TABLE]

satisfy that $H^{\prime}=\mu_{a}(H)$ , $K^{\prime}=\mu_{b}(K)$ and $\varphi=\alpha\circ\varphi^{\prime}\circ\mu_{a}$ , $\psi=\alpha\circ\psi^{\prime}\circ\mu_{b}$ .

Remark 2.8.

For two groups $\mathcal{G}$ and $\mathcal{O}(2)$ , the introduced notation $\mathcal{H}=H^{\varphi}\times_{L}^{\psi}K\subset\mathcal{G}\times\mathcal{O}(2)$ , where $H\subset\mathcal{G}$ , $K\subset\mathcal{O}(2)$ are two subgroups and $\varphi:H\rightarrow L$ and $\psi:H\rightarrow L$ are epimorphismns, needs to be simplified. It was proposed by Hao-Pin Wu (cf [23]) to denote the conjugacy classes $(\mathcal{H})$ in a more comprehensive way. To be more precise, in order to identify $L$ with $K/\text{Ker\,}(\psi)\subset O(2)$ and denote by $r$ the rotation generator in $L$ . Next we put

[TABLE]

and define

[TABLE]

Of course, in the case when all the epimorphisms $\varphi$ with the kernel $Z$ are conjugate, there is no need to use the symbol $Z$ in (21) and we will simply write $\mathcal{H}=H\prescript{Z}{}{\times}_{L}K$ . Moreover, in this case all epimorphisms $\varphi$ from $H$ to $L$ are conjugate, we can also omit the symbol $L$ , i.e. we will write $\mathcal{H}=H\prescript{}{}{\times}_{L}K$ . The conjugacy classes of subgroups in $D_{6}\times O(2)$ are listed Table 1, which were obtained in [23] using G.A.P. programming.

3. Model for Atomic Interaction

Consider $n$ identical particles $u_{j}$ , $j=0,1,2,\dots,n-1$ , in the plane $\mathbb{C}$ . Assume that each particle interacts with the adjacent particles $u_{j-1}$ and $u_{j+1}$ , where the indices $j-1$ and $j+1$ are taken mod $n$ .

Put $u:=(u_{0},u_{1},\dots,u_{n-1})^{T}\in\mathbb{C}^{n}$ and $\Omega^{\prime}_{o}:=\{u\in\mathbb{C}^{n}:u_{k}\not=u_{j},\text{ if }k\not=j,\text{ with }\,k,\,j=0,\,1,\,2,\,\cdots,\,n-1\}$ . We define the following energy functional $V:\Omega^{\prime}_{o}\to\mathbb{R}$ by

[TABLE]

where

[TABLE]

The following Newtonian equation describes the interaction between these $n$ -particles,

[TABLE]

3.1. Symmetric Equilibrium for (23)

We notice that the space $\mathbb{C}^{n}$ is a representation of the group $D_{n}\times O(2)$ , where $D_{n}$ stands for the dihedral group corresponding to the group of symmetries of a regular $n$ -gone, which can be described as a subgroup of the symmetric group $S_{n}$ of $n$ -elements $\{0,1,\dots,n-2,n-1\}$ . More precisely, $D_{n}$ is generated by the “rotation” $\xi:=(0,1,2,3,\dots,n-1)$ and the “reflection” $\kappa:=(1,n-1)(2,n-2)\dots(m,n-m)$ , where $m=\left\lfloor\frac{n-1}{2}\right\rfloor$ . Then the action of $\mathfrak{G}$ on $\mathbb{C}^{n}$ is given by

[TABLE]

where $A\in O(2)$ and $\sigma\in D_{n}\subset S_{n}$ is given as a permutation of the vertices $C_{n}\subset\mathbb{C}$ of the $n$ -gone. To be more precise we have

[TABLE]

Let us point out that the group $D_{n}$ can be also described as a subgroup of $O(2)$ , where $\kappa$ is identified to the complex conjugation $\left[\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right]$ and $\xi^{j}$ is identified to the complex number $\gamma^{j}$ , $\gamma:=e^{i\frac{2\pi}{n}}$ , representing the rotation

[TABLE]

It is evident that $\Omega^{\prime}_{o}\subset\mathbb{C}^{n}$ is $G$ -invariant. Notice that the function $\bm{\mathcal{V}}:\Omega^{\prime}_{o}\to\mathbb{R}$ is invariant with respect to the action of $\mathbb{C}$ on $\mathbb{C}^{n}$ by shifting, that is, for all $z\in\mathbb{C}$ and $(z_{0},z_{1},\dots,z_{n-1})\in\Omega^{\prime}_{o}$ we have

[TABLE]

Therefore in order to make System (23) reference point independent, we put

[TABLE]

and $\Omega_{o}=\Omega^{\prime}_{o}\cap\mathscr{V}$ . Then, we obtain that $\mathscr{V}$ and $\Omega_{o}$ are $G$ -invariant and in addition $\Omega_{o}$ is flow-invariant for (23). Consider the point $v^{o}:=(1,\gamma,\gamma^{2},\dots,\gamma^{n-1})\in\Omega_{o}$ , where $\gamma:=e^{i\frac{2\pi}{n}}$ . One can verify that the isotropy group $\mathfrak{G}_{v^{o}}$ , which for simplicity we denote by $\Gamma$ , is given as the following amalgamated subgroup of $D_{n}\times O(2)$

[TABLE]

where in order to consider $D_{n}$ as a subgroup of $O(2)$ . Then $\mathscr{V}^{\Gamma}$ is a one dimensional subspace of $\mathscr{V}$ and we have (see (30) for more details) that

[TABLE]

Then, by Symmetric Criticality Condition, a critical point of $\bm{\mathcal{V}}^{\Gamma}:\Omega_{o}^{\Gamma}\to\mathbb{R}$ is also a critical point of $\bm{\mathcal{V}}$ . Notice that $V$ satisfies the coercivity condition, that is, $V(u)\to\infty$ as $u$ approaches $\partial\Omega_{o}$ or $\|u\|\to\infty$ . Then there exists a global minimum point $u^{o}$ of $\bm{\mathcal{V}}^{\Gamma}$ in $\Omega_{o}^{\Gamma}$ , which implies that $\nabla\bm{\mathcal{V}}^{\Gamma}(u^{o})=0$ . Since $\mathscr{V}^{\Gamma}$ is two-dimensional, we denote its vectors by $v:=z(1,\gamma,\gamma^{2},\dots,\gamma^{n-1})$ , $z\in\mathbb{C}$ . One can look for the orbit of equilibria for (23) where $z=t$ , $t\in\mathbb{R}$ . Put

[TABLE]

where $0\leq j<k\leq n-1$ . Define

[TABLE]

Notice that $|\gamma^{j+1}-\gamma^{j}|=\left|2\sin\frac{\pi}{n}\right|$ and $|\gamma^{k}-\gamma^{j}|=2\left|\sin\frac{(k-j)\pi}{n}\right|$ , $\phi$ is exactly the restriction of $\bm{\mathcal{V}}$ to the fixed-point subspace $\mathscr{V}_{o}^{\Gamma}=\{t(1,\gamma,\gamma^{2},\dots,\gamma^{n-1}):t>0\}$ , thus in order to find an equilibrium for (23), by Symmetric Criticality Principle, it is sufficient to identify a critical point $r_{o}$ of $\phi(t)$ . Clearly,

[TABLE]

thus there exists a minimizer $r_{o}\in(0,\infty)$ , which is a critical point of $\varphi$ and consequently

[TABLE]

is the $\Gamma$ -symmetric equilibrium of $\bm{\mathcal{V}}$ , providing the configuration of particles, shown at Figure 1 being a stationary solution to (23).

In the following, we write $u^{o}=(u_{0}^{o},u_{1}^{o},\dots,u_{n-1}^{o})\in\mathbb{C}^{n}$ , where $u_{k}^{o}=r_{o}\gamma^{k}$ , $k=0,1,2,\dots,n-1$ with $\gamma=e^{i\frac{2\pi}{n}}$ .

3.2. $G$ -Equivariant Bifurcation Problem for (23)

In what follows, we are interested in finding non-trivial $p$ -periodic solutions to (23), bifurcating from the orbit equilibrium points $\mathfrak{G}(u^{o})$ . By normalizing the period, namely, by making the substitution $v(t)=u\left(\frac{pt}{2\pi}\right)$ in (23) we obtain the following system,

[TABLE]

where $\lambda=\frac{p}{2\pi}$ . Since system (29) is symmetric with respect to the group action $G=D_{n}\times O(2)$ , we have the orbit of equilibria $M=G(u^{o})$ , which is a one-dimensional submanifold in $\mathscr{V}$ , with the tangent vector at $u^{o}\in M$ being

[TABLE]

Notice that the slice $S_{o}$ to the orbit $M$ at $u^{o}$ is $S_{o}=\{z\in\mathbb{C}^{n}:z\bullet v^{o}=0\}$ , which implies that

[TABLE]

Notice that the $G$ -isotropy group of $u_{o}$ is given by,

[TABLE]

which can be identified with $D_{n}$ .

3.3. $\Gamma$ -Isotypical Decomposition of $S_{o}$

Let $\mu:\mathbb{C}^{n}\to\mathbb{R}$ be given by

[TABLE]

Then we have $\ker\mu=S_{o}$ . Denote also by $\nu:\mathbb{C}^{n}\to\mathbb{C}$ the complex linear functional

[TABLE]

Then we also have $\ker\nu\subset\ker\mu$ . Notice that for $j=0,1,\dots,n-2$ , and $z_{k}=\gamma^{-jk}$ , $k=0,1,2,\dots,n-1$ , we have

[TABLE]

which implies for every $z\in\mathbb{C}$ ,

[TABLE]

Put

[TABLE]

The action of $\Gamma$ on $W_{j}$ which we identify with $D_{n}$ can be described as follows. Put

[TABLE]

Then we have

[TABLE]

where ‘ $\cdot$ ’ denotes the usual complex multiplication. Therefore, the sub-representation $W_{j}$ is equivalent to the irreducible $D_{n}$ -representation $\mathcal{V}_{j+1}$ (see [1] for more details). On the other hand, if $j=n-1$ , then for $z_{k}=z\gamma^{-(n-1)k}$ , $k=0,1,2,\dots,n-1$ , $z\in\mathbb{C}$ , we have

[TABLE]

which implies that $(z_{0},z_{1},\dots,z_{n-1})\in\ker\mu$ if and only if $z$ is real. Therefore, we have

[TABLE]

which is an additional component of $S_{o}$ on which $(\gamma,\gamma)$ and $(\kappa,\kappa)$ act trivially, namely, it is equivalent to $\mathcal{V}_{0}$ . Comparing the dimension of the slice with the dimensions of the irreducible components of $S_{o}$ , we recognize that this decomposition is the complete decomposition of $S_{o}$ into a product of irreducible $D_{n}$ -sub-representations and, as a consequence, we are now able to identify the $D_{n}$ -isotypical components of $S_{o}$ .

The Case of $n$ being an odd number:

In this case, the space $S_{o}$ has the following isotypical components,

[TABLE]

where $\mathscr{V}_{0}=W_{n-1}$ , that is,

[TABLE]

$\mathscr{V}_{1}=W_{n-2}$ and for $\left\lfloor\frac{n}{2}\right\rfloor\geq j>1$ ,

[TABLE]

Put

[TABLE]

Then for $j=1,2,\dots,r$ , $\mathscr{V}_{j}$ is a complex subspace of $\mathbb{C}^{n}$ such that

[TABLE]

The Case of $n$ being an even number:

In this case, the space $S_{o}$ has the following isotypical components,

[TABLE]

with the similar components $\mathscr{V}_{j}$ for $j<\frac{n}{2}=r$ , and an additional isotypical component $\mathscr{V}_{r}$ , given by

[TABLE]

4. Variational Reformulation of (29)

Sobolev Space of $V$ -Valued Periodic Functions:

Since $\mathscr{V}$ is an orthogonal $\Gamma$ - representation, we can consider the the first Sobolev space of $2\pi$ -periodic functions from $\mathbb{R}$ to $\mathscr{V}$ , that is,

[TABLE]

equipped with the inner product

[TABLE]

Let $O(2)$ denote the group of $2\times 2$ -orthogonal matrices. Notice that $O(2)=SO(2)\cup SO(2)\kappa$ , where $\kappa=\left[\begin{array}[]{cc}1&0\\ 0&-1\end{array}\right]$ . It is convenient to identify a rotation

[TABLE]

with $e^{i\tau}\in S^{1}\subset\mathbb{C}$ . Notice that $\kappa e^{i\tau}=e^{-i\tau}\kappa$ , that is, $\kappa$ as a linear transformation of $\mathbb{C}$ into itself, acts as complex conjugation. The space $H^{1}_{2\pi}(\mathbb{R},\mathscr{V})$ is an orthogonal Hilbert representation of $\mathfrak{G}=(D_{n}\times O(2))\times O(2)$ . Indeed, we have for $z\in H^{1}_{2\pi}(\mathbb{R},\mathscr{V})$ , $(\xi,A)\in D_{n}\times O(2)$ and $e^{i\tau}\in S^{1}$ we have (see (24))

[TABLE]

We identify a $2\pi$ -periodic function $x:\mathbb{R}\to V$ with a function $\widetilde{x}:S^{1}\to\mathscr{V}$ via the following commuting diagram:

$\mathbb{R}$$S^{1}$$\mathscr{V}$$\mathfrak{e}$$x$$\widetilde{x}$$\mathfrak{e}(\tau)=e^{i\tau}$

Using this identification, we write $H^{1}(S^{1},\mathscr{V})$ in place of $H^{1}_{2\pi}(\mathbb{R},\mathscr{V})$ . Let

[TABLE]

Then system (29) can be written as the following variational equation

[TABLE]

where $J:\mathbb{R}\times\Omega\to\mathbb{R}$ is defined by

[TABLE]

Assume that $u^{o}=r_{o}(1,\gamma,\gamma^{2},\gamma^{n-1})\in\mathbb{C}^{n}$ is the equilibrium point of (23) described in subsection 3.1. Then $u^{o}$ is a critical point of $J$ . We are interested in finding non-stationary $2\pi$ -periodic solutions bifurcating from $u^{o}$ , that is, non-constant solutions to system (33). Notice that $\mathfrak{G}_{u_{o}}=D_{n}\times O(2)$ , where $D_{n}\simeq D_{n}\times_{D_{n}}D_{n}\subset D_{n}\times O(2)$ . We consider the $\mathfrak{G}$ -orbit $\mathfrak{G}(u^{o})$ in the space $H^{1}(S^{1},\mathscr{V})$ . We denote by $\mathscr{H}$ the slice to $\mathfrak{G}(u^{o})$ in $H^{1}(S^{1},\mathscr{V})$ . We will also denote by $\mathscr{J}:\mathbb{R}\times\widetilde{\Omega}\to\mathbb{R}$ the restriction of $J$ to the set $\mathbb{R}\times\widetilde{\Omega}$ , where $\widetilde{\Omega}=\mathscr{H}\cap\Omega$ . Put $\mathbb{G}=\mathfrak{G}_{u^{o}}$ . Then $\mathscr{J}$ is $\mathbb{G}$ -invariant. Then by the Slice Criticality Principle (see Theorem 2.5), critical points of $\mathscr{J}$ are critical points of $J$ and are solutions to system (33). Consider the operator $L:H^{2}(S^{1};\mathscr{V})\to L^{2}(S^{1};\mathscr{V})$ , given by $Lu=-\ddot{u}+u$ , $u\in H^{2}(S^{1},\mathscr{V})$ . Then the inverse operator $L^{-1}$ exists and is bounded. Let $j:H^{2}(S^{1};\mathscr{V})\to H^{1}(S^{1},\mathscr{V})$ be the natural embedding operator. Then $j$ is a compact operator and we have

[TABLE]

where

[TABLE]

Consequently, the bifurcation problem (33) can be written as

[TABLE]

Moreover, we have

[TABLE]

where

[TABLE]

Consider the operator $\mathscr{A}(\lambda):\mathscr{H}\to\mathscr{H}$ , given by

[TABLE]

Notice that

[TABLE]

thus, by implicit function theorem, $\mathfrak{G}(u^{o})$ is an isolated orbit of critical points of $J$ , whenever $\mathscr{A}(\lambda)$ is an isomorphism. Therefore, a point $(\lambda_{o},u^{o})$ is a bifurcation point for (33), then $\mathscr{A}(\lambda_{o})$ is not an isomorphism. In such a case we put $\Lambda=\{\lambda>0:\mathscr{A}(\lambda_{o})$ is not an isomorphism $\}$ , and call the set $\Lambda$ the critical set for the trivial solution $u^{o}$ .

4.1. Application of Equivariant Gradient Degree

Consider the $S^{1}$ -action on $H^{1}(S^{1},\mathscr{V})$ , where $S^{1}$ acts on functions by shifting the argument (see (31)). Then, $(H^{1}(S^{1},\mathscr{V}))^{S^{1}}$ is the space of constant functions, which can be identified with the space $\mathscr{V}$ , that is,

[TABLE]

Then, the slice $\mathscr{S}_{o}$ in $H^{1}(S^{1},\mathscr{V})$ to the orbit $\mathfrak{G}(u^{o})$ at $u^{o}$ is exactly

[TABLE]

Definition 4.1.

We say that $\lambda_{o}\in\Lambda$ satisfies condition (C) if $P_{o}\circ\mathscr{A}(\lambda_{o})|_{S_{o}}:S_{o}\to S_{o}$ , where $P_{o}:\mathscr{V}\to S_{o}$ is an orthogonal projection, is an isomorphism.

Theorem 4.2.

Consider the bifurcation system (33) and assume that $\lambda_{o}\in\Lambda$ satisfies condition (C) and is isolated in the critical critical set $\Lambda$ , i.e. there exists $\lambda_{-}<\lambda_{o}<\lambda_{+}$ such that $[\lambda_{-},\lambda_{+}]\cap\Lambda=\{\lambda_{o}\}$ . Define

[TABLE]

where $B_{1}(0)$ stands for the open unit ball in $\mathscr{H}$ . If

[TABLE]

is non-zero, i.e. $n_{j}\not=0$ for some $j=1,2,\dots,m$ , then there exists a bifurcating branch of nontrivial solutions to (33) from the orbit $\{\lambda_{o}\}\times\mathfrak{G}(u^{o})$ with symmetries at least $(H_{j})$ .

Consider the $S^{1}$ -isotypical decomposition of $\mathscr{W}$ , that is,

[TABLE]

In a standard way, the space $\mathscr{W}_{l}$ , $l=1,2,\dots$ , can be naturally identified with the space $\mathscr{V}^{c}$ on which $S^{1}$ acts by $l$ -folding. To be more precise,

[TABLE]

Notice that, since the operator $\mathscr{A}(\lambda)$ is $\mathbb{G}$ -equivariant where $\mathbb{G}=D_{n}\times O(2)$ , it is also $S^{1}$ -equivariant and thus $\mathscr{A}(\lambda)(\mathscr{W}_{l})\subset\mathscr{W}_{l}$ . On the other hand, we have

[TABLE]

which under condition (C) implies that $\lambda_{o}\in\Lambda$ if and only if $\lambda_{o}^{2}=\frac{l^{2}}{\mu}$ for some $l=1,\,2,\,3,\dots$ and $\mu\in\sigma(\nabla_{u}^{2}\bm{\mathcal{V}}(u^{o}))$ .

5. Computation of the Spectrum $\sigma(\nabla^{2}{\mathcal{V}}(u^{o}))$

Computation of $\nabla\bm{\mathcal{V}}(u)$ :

Since the potential $\bm{\mathcal{V}}$ is given by (22), we can write that $\bm{\mathcal{V}}(u)=\Phi(u)+\Psi(u)$ , $u\in\Omega$ , where

[TABLE]

Notice that

[TABLE]

and

[TABLE]

Computation of $\nabla^{2}\bm{\mathcal{V}}(u^{o})$ :

For a given complex number $z=x+iy$ , which we write in a vector form $z=(x,y)^{T}$ , we define the matrix $\mathfrak{m}_{z}:=zz^{T}$ , i.e.

[TABLE]

We will also apply the following notation

[TABLE]

and we put $\mathfrak{m}_{jk}:=\mathfrak{m}_{z_{jk}}$ . Noticing that

[TABLE]

we know that the $2\times 2$ matrix $\mathfrak{m}_{jk}$ can be described using the complex operators as

[TABLE]

Put

[TABLE]

Notice that we have

[TABLE]

By direct computations we have that

[TABLE]

where

[TABLE]

and

[TABLE]

where

[TABLE]

Next, by direct computations one can derive the following matrix form of $\nabla^{2}\Psi(u^{o})=2\mathcal{C}+4r_{o}^{2}\mathcal{D}$ , where

[TABLE]

and

[TABLE]

where

[TABLE]

and

[TABLE]

Case 1: $n$ being an odd number: Put $\mathcal{A}_{k}:=\mathcal{A}|_{\mathscr{V}_{k}}$ , then we have the following matrices:

[TABLE]

where $1<k<\left\lfloor\frac{n}{2}\right\rfloor$ . Next, put $\mathcal{B}_{k}:=\mathcal{B}|_{\mathscr{V}_{k}}$ . Notice that

[TABLE]

Then, by direct computations we get

[TABLE]

where $1<k<\left\lfloor\frac{n}{2}\right\rfloor$ . Notice that

[TABLE]

In addition (since $n$ is odd), we put

[TABLE]

Then, put $\mathcal{C}_{k}:=\mathcal{C}|_{\mathscr{V}_{k}}$ , $\widetilde{\mathcal{C}}_{k}:=\widetilde{\mathcal{C}}|_{\mathscr{V}_{k}}$ and

[TABLE]

where $1<k<\left\lfloor\frac{n}{2}\right\rfloor$ . Next, notice that

[TABLE]

Put

[TABLE]

Then

[TABLE]

Put

[TABLE]

Then we have

[TABLE]

where

[TABLE]

also

[TABLE]

where

[TABLE]

and finally

[TABLE]

where

[TABLE]

Then we have the following explicit formulae for the spectrum

[TABLE]

where

[TABLE]

Notice that, for each eigenvalue $\mu\in\sigma(\mathscr{A})$ we have that its ${\mathcal{V}_{k}}$ -isotypical multiplicity $m_{k}(\mu)$ is given by

[TABLE]

Case 2: $n$ being an even number: In this case we have an additional isotypical component $\mathscr{V}_{r}$ , $r=\frac{n}{2}$ , and the entries of the matrix $\mathscr{A}_{k}$ and are slightly different. More precisely, notice that in this case we have

[TABLE]

where $\delta(k):=\begin{cases}1&\text{ if }k\text{ is even}\\ 0&\text{ if }k\text{ is odd}\end{cases}$ . Therefore, we have

[TABLE]

where

[TABLE]

and for $1<k\leq r-1$

[TABLE]

Consequently, we have the following explicit formulae for the spectrum

[TABLE]

where

[TABLE]

Of course, in this case the formula (43) is still valid.

6. Formulation of Results

6.1. Computation of the Gradient $\mathbb{G}$ -Equivariant Degree

In order to describe the $\mathbb{G}$ -isotypical decomposition of the slice $\mathcal{S}_{o}$ , first, we identify the irreducible $\mathbb{G}$ -representations related to the isotypical decomposition of $\mathscr{W}$ . These representations are $\mathcal{W}_{j,l}:=\mathcal{V}_{j}\otimes\mathcal{U}_{j}$ , where $\mathcal{U}_{l}$ is the $l$ -th irreducible $O(2)$ -representation (listed according to the convention introduced in [1]), $j=0,1,\dots,\lfloor\frac{n}{2}\rfloor$ , $l=1,2,3,\dots$ . The corresponding to $\mathcal{W}_{jl}$ isotypical components of $\mathscr{W}$ are

[TABLE]

These irreducible $\mathbb{G}$ -representations can be easily described. The representation $\mathcal{W}_{jl}=\mathbb{C}\oplus\mathbb{C}$ is a $4$ -dimensional (real) representation of real type with the action of $\mathbb{G}=D_{n}\times O(2)$ given by the formulae

[TABLE]

where $\xi\in SO(2)$ , $O(2)=SO(2)\cup SO(2)\bm{\kappa}$ , and

[TABLE]

For each positive eigenvalue $\mu_{j}^{\pm}\in\sigma(\nabla^{2}V(ar_{o}^{2}))$ , $1<j<\lfloor\frac{n-1}{2}\rfloor$ , we define the number $\lambda_{j,l}^{\pm}:=\frac{l^{2}}{\mu_{j}^{\pm}}$ , and for other eigenvalues $\mu_{j}$ , we put $\lambda_{j,l}:=\frac{l^{2}}{\mu_{j}}$ , $l\in\mathbb{N}$ . Then the critical set $\Lambda$ is composed of exactly all these numbers $\lambda_{j,l}^{\pm}$ and $\lambda_{j,l}$ . Since each of the eigenvalues $\mu_{j}^{\pm}$ (for $1<j<\lfloor\frac{n-1}{2}\rfloor$ ) and $\mu_{j}$ (otherwise) is of $\mathcal{V}_{j}$ -isotypical multiplicity one, it follows that for $\lambda_{-}<\lambda_{o}:=\lambda_{j,l}^{\pm}<\lambda_{+}$ (respectively $\lambda_{-}<\lambda_{o}:=\lambda_{j,l}<\lambda_{+}$ ), where $[\lambda_{-},\lambda_{+}]\cap\Lambda=\{\lambda_{o}\}$ , thus $\sigma_{-}(\mathscr{A}(\lambda_{-}))=\sigma_{-}(\mathscr{A}(\lambda_{-}))\cup\{\lambda_{o}\}$ . Consequently, we obtain that for

[TABLE]

is given by

[TABLE]

Example 6.1.

In the case of the group $\bm{G}=D_{6}\times O(2)$ , we have the following basic degrees111Let us point out that for practical applications of the gradient $D_{n}\times O(2)$ -degree, it is fully justified (see [7]) to use is values truncated to the Burnside ring $A(D_{n}\times O(2))$ $\text{Deg}_{\mathcal{W}_{jk}}$ (which were obtain in [23] using GAP programming)

[TABLE]

6.2. Existence Result

Theorem 6.2.

Under the assumptions formulated in section 3, for every $\lambda_{jl}$ , $0<j\leq\lfloor\frac{n}{2}\rfloor$ , there exists an orbit of bifurcating branches of nontrivial periodic solutions to (33) from the orbit $\{\lambda_{jl}\}\times\mathfrak{G}(u^{o})$ . More precisely, for every orbit type $(H_{j,l})$ in $D_{jl}$ there exists an orbit of periodic solutions with symmetries at least $H_{j,l}$ .

Proof.

This result is a direct consequence of the Existence Property ( $\nabla$ 1) of the gradient equivariant degree formulated in Theorem 2.4. ∎

7. Computational Example

In this section we consider a dihedral configuration of molecules composed of $n=6$ particles and put $A=0.2$ , $B=350$ , $\sigma=0.25$ for the function $W$ at (22), with which we obtain that $\phi$ defined at (27) assumes minimum at $r_{0}=1.836545792.$ The distinct eigenvalues of the Hessian matrix $\nabla^{2}V(u^{o})$ are $\mu_{0}=-10.36657914,$ $\mu_{1}=43.00585474,$ $\mu_{3}=19.58406142,$ $\mu_{2}^{-}=7.633501334$ and $\mu_{2}^{+}=11.42339623$ , Then one can easily compute the critical set $\Lambda$ , namely

[TABLE]

The positive values of the eigenvalues $\xi_{jl}(\lambda)$ of the operator $\mathscr{A}(\lambda)$ and the critical set $\Lambda$ are illustrated on Figure 2.

7.1. Topological Invariants $\omega(\lambda_{o})$

In Table 2 we list the maximal orbit types in $\mathscr{W}_{l}\setminus\{0\}$ , $l\geq 1$ :

Next, we list the values of the equivariant invariants $\omega(\lambda_{jl})$ (given by (38)):

[TABLE]

These sequence of equivariant invariants $\omega(\lambda_{j,l})$ can be continued indefinitely due to the the fact that any $p$ -periodic solution is also $2p$ , $3p$ , $4p$ , etc. periodic solution as well. However, in order to get a clear picture of the emerging from the symmetric equilibrium vibrations, it is sufficient to exhaust all the critical values $\lambda_{j,1}$ . Let us also point out that the exact value of the equivariant invariants $\omega(\lambda_{j,l}$ can be symbolically computed either in its truncated to the Burnside ring $A(D_{n}\times O(2))$ (such programs are already available) or in $U(D_{n}\times O(2))$ (we have all the needed algorithms so the appropriate computer programs were already created). However, one should understand that as each equivariant invariant $\omega(\lambda_{j,l})$ carry the full equivariant topological information about the emerging from the equilibrium $u^{o}$ periodic vibrations with the limit period $p=2\pi\lambda_{j,l}$ , so they can be significantly long. For example, we have

[TABLE]

Nevertheless, for the purpose of making predictions about the actual emerging periodic vibration with this particular limit period, one can look in $\omega(\lambda_{j,l})$ for the maximal orbit types listed in Table 2. Therefore, we can list some types222In order to provide the full list of possible symmetries of the emerfing periodic vibrations one needs to use the full topological invariant $\omega(\lambda_{j,l})$ . (according to their symmetries) of the branches of periodic vibrations emerging from the equilibrium $u^{o}$ :

$\lambda_{1,1}$ :

For the limit period $0.95811155$ there exist at least the following three orbits of $p$ -periodic vibrations with spatio-temporal symmetries at least $(D_{6}\prescript{\mathbb{Z}_{1}}{}{\times}_{D_{6}}D_{6})$ , $(D_{2}\prescript{D_{1}}{}{\times}_{\mathbb{Z}_{2}}D_{2})$ , $(D_{2}\prescript{{\tilde{D}}_{1}}{}{\times}_{\mathbb{Z}_{2}}D_{2})$ .

$\lambda_{3,1}$ :

For the limit period $1.41980428$ there exists at least the following orbit of $p$ -periodic vibrations with spatio-temporal symmetries at lest $(D_{6}\prescript{{\tilde{D}}_{3}}{}{\times}_{\mathbb{Z}_{2}}D_{2})$ .

$\lambda^{+}_{2,1}$ :

For the limit period $1.85901226$ there exist at least the following three orbits of $p$ -periodic vibrations with spatio-temporal symmetries at least $(D_{6}\prescript{\mathbb{Z}_{2}}{}{\times}_{D_{3}}D_{3l})$ , $(D_{2}\prescript{\mathbb{Z}_{2}}{}{\times}_{\mathbb{Z}_{2}}D_{2l})$ , $(D_{2}\prescript{}{}{\times}D_{l})$ .

$\lambda_{1,2}$ :

For the limit period $1.91622311$ there exist at least the following three orbits of $p$ -periodic vibrations with spatio-temporal symmetries at least $(D_{6}\prescript{\mathbb{Z}_{1}}{}{\times}_{D_{6}}D_{12})$ , $(D_{2}\prescript{D_{1}}{}{\times}_{\mathbb{Z}_{2}}D_{4})$ , $(D_{2}\prescript{{\tilde{D}}_{1}}{}{\times}_{\mathbb{Z}_{2}}D_{4})$ .

$\lambda^{+}_{2,1}$ :

For the limit period $2.27414407$ there exist at least the following three orbits of $p$ -periodic vibrations with spatio-temporal symmetries at least $(D_{6}\prescript{\mathbb{Z}_{2}}{}{\times}_{D_{3}}D_{3l})$ , $(D_{2}\prescript{\mathbb{Z}_{2}}{}{\times}_{\mathbb{Z}_{2}}D_{2l})$ , $(D_{2}\prescript{}{}{\times}D_{l})$

Notice that due to the isotypical type of the critical values $\lambda_{j,l}$ there are similar orbit types of branches emerging from $u^{o}$ with different limit period. One can ask about the global behavior of such branches. For instance, is it possible that such a branch emerge from one $\lambda_{j,l}$ and then ‘disappear’ into another $\lambda_{j^{\prime},l^{\prime}}$ ? By comparing the values of the equivariant invariants $\omega(\lambda_{j,l})$ with $\omega(\lambda_{j^{\prime},l^{\prime}})$ one can easily say that such situation would be very unlikely possible.

7.2. Numerical Simulations

In this subsection, we present some simulations of the periodic solutions predicted by our theoretical. On Figures 3–10. we show the periodic solutions which were found found for $\lambda^{2}=\frac{1}{\mu}$ with $\mu$ taking values near the last four eigenvalues.

8. Concluding Remarks

In this paper, we analyzed a system (2) with $n$ particle in the plane $\mathbb{R}^{2}$ admitting dihedral spatial symmetries. More precisely, we use the method of gradient equivariant degree [12, 4, 9, 20] to investigate the existence of periodic solution to (2), where $\bm{\mathcal{V}}$ is the Lennard-Jones and Coulomb potential, around an equilibrium admitting dihedral $D_{n}$ symmetries. The dynamics of system (2) can be very complicated with a large number of different periodic solutions exhibiting various spatio-temporal symmetries. The equivariant degree provides equivariant invariants for system (2) allowing a complete symmetric topological classification of the emanating (or bifurcating) branches of periodic solutions from a given equilibrium state. First, the critical periods $p_{jl}>0$ , which are the limit periods for those bifurcation branches can be identified from the so called critical set $\Lambda:=\{\lambda_{jl}=\frac{l^{2}}{\mu_{j}},\;l\in\mathbb{N},\;\mu_{j}\in\sigma(\nabla^{2}\bm{\mathcal{V}}(u^{o}))\}$ , where $\sigma(\nabla^{2}\bm{\mathcal{V}}(u^{o}))$ denotes the set of eigenvalues of the Hessian $\nabla^{2}\bm{\mathcal{V}}(u^{o})$ , and the symmetries of topologically possible solutions to (2) can be identified from the equivariant invariants $\omega_{\bf G}(\lambda_{jl})$ . The explicitly computed Hessian $\nabla^{2}\bm{\mathcal{V}}(u^{o})$ facilitated the formulation of general results for dihedral molecular configurations.

We developed a method using the isotypical decomposition of the phase space combined with block decompositions and the usual complex operations in order to represent $\nabla^{2}\bm{\mathcal{V}}(a)$ as a product of simple $2\times 2$ -matrices. Therefore, the spectrum $\sigma(\nabla^{2}\bm{\mathcal{V}}(a))$ is explicitly computed and these computations do not depend on a particular form of the potential $\bm{\mathcal{V}}$ . In addition, we provided an exact formula for computation of the equivariant invariants $\omega_{\bf G}(\lambda_{jl})$ . We should also mention that for larger groups $D_{n}$ , the actual computations of $\omega_{\bf G}(\lambda_{jl})$ can be quite complicated but still possible with the use of computer software. Such software was already developed for several types of groups $G=\Gamma\times O(2)$ and it is available at [23].

We reamrk that elements $\lambda_{jl}$ of the critical set $\Lambda$ correspond to the values of transitional frequencies. The equivariant invariant $\omega_{\bf G}(\lambda_{jl})$ provides a full topological classification of symmetric modes corresponding to the branches of molecular vibrations emerging from the equilibrium state at the critical frequency $\lambda_{jl}$ . This method can be applied to create, for a molecule with dihedral symmetries, an atlas of topologically possible symmetric modes of vibrations, the collection of actual distinct molecular vibrations (related the maximal symmetric types) emerging from the equilibrium state and the corresponding limit frequencies.

Bibliography24

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] Z. Balanov, W. Krawcewicz and H. Steinlein, Applied Equivariant Degree. AIMS Series on Differential Equations & Dynamical Systems , Vol. 1, 2006 2006 2006 .
2[2] Z. Balanov, W. Krawcewicz, S. Rybicki, and H. Steinlein, A short treatise on the equivariant degree theory and its applications , J. Fixed Point Theory Appl., 8 (2010), pp. 1–74.
3[3] Z. Balanov, W. Krawcewicz and H. Ruan, Periodic solutions to O ( 2 ) × S 1 O 2 superscript S 1 \mathrm{O}(2)\times\mathrm{S}^{1} -symmetric variational problems: Equivariant gradient degree approach. Nonlinear analysis and optimization II. Optimization, 45-84, Contemp. Math., 514, pp 4279–4296, Amer. Math. Soc., Providence , RI, 2010.
4[4] 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.
5[5] G.E. Bredon, Introduction to Compact Transformation Groups, Academic Press, New York-London, 1972.
6[6] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer-Verlag, New York-Berlin, 1985.
7[7] M. Dabkowski, W. Krawcewicz, Y. Lv, and H-P. Wu, Multiple Periodic Solutions for ?-symmetric Newtonian Systems , ar Xiv.org http://arxiv.org/abs/1612.07876
8[8] T. tom Dieck, Transformation Groups. Walter de Gruyter, 1987.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

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Taxonomy

Dihedral Molecular Configurations Interacting by Lennard-Jones and Coulomb Forces

Abstract.

1. Introduction

2. Preliminaries

2.1. Equivariant Jargon:

GGG-Actions:

Isotypical Decomposition of Finite-Dimensional Representations:

2.2. Gradient GGG-Equivariant Degree

Euler Ring and Burnside Ring:

Definition 2.1**.**

Lemma 2.2**.**

GGG-equivariant Gradient Degree ∇G-deg\nabla_{G}\text{\rm-deg}∇G​-deg:

Definition 2.3**.**

Theorem 2.4**.**

Computations of the Gradient GGG-Equivariant Degree:

2.3. Gradient Degree on the Slice

Theorem 2.5**.**

Proof.

2.4. Product Group G1×G2G_{1}\times G_{2}G1​×G2​

Theorem 2.6**.**

Proposition 2.7**.**

Remark 2.8**.**

3. Model for Atomic Interaction

3.1. Symmetric Equilibrium for (23)

3.2. GGG-Equivariant Bifurcation Problem for (23)

3.3. Γ\GammaΓ-Isotypical Decomposition of SoS_{o}So​

The Case of nnn being an odd number:

The Case of nnn being an even number:

4. Variational Reformulation of (29)

Sobolev Space of VVV-Valued Periodic Functions:

4.1. Application of Equivariant Gradient Degree

Definition 4.1**.**

Theorem 4.2**.**

5. Computation of the Spectrum σ(∇2V(uo))\sigma(\nabla^{2}{\mathcal{V}}(u^{o}))σ(∇2V(uo))

Computation of ∇V(u)\nabla\bm{\mathcal{V}}(u)∇V(u):

Computation of ∇2V(uo)\nabla^{2}\bm{\mathcal{V}}(u^{o})∇2V(uo):

6. Formulation of Results

6.1. Computation of the Gradient G\mathbb{G}G-Equivariant Degree

Example 6.1**.**

6.2. Existence Result

Theorem 6.2**.**

Proof.

7. Computational Example

7.1. Topological Invariants ω(λo)\omega(\lambda_{o})ω(λo​)

7.2. Numerical Simulations

8. Concluding Remarks

$G$ -Actions:

2.2. Gradient $G$ -Equivariant Degree

Definition 2.1.

Lemma 2.2.

$G$ -equivariant Gradient Degree $\nabla_{G}\text{\rm-deg}$ :

Definition 2.3.

Theorem 2.4.

Computations of the Gradient $G$ -Equivariant Degree:

Theorem 2.5.

2.4. Product Group $G_{1}\times G_{2}$

Theorem 2.6.

Proposition 2.7.

Remark 2.8.

3.2. $G$ -Equivariant Bifurcation Problem for (23)

3.3. $\Gamma$ -Isotypical Decomposition of $S_{o}$

The Case of $n$ being an odd number:

The Case of $n$ being an even number:

Sobolev Space of $V$ -Valued Periodic Functions:

Definition 4.1.

Theorem 4.2.

5. Computation of the Spectrum $\sigma(\nabla^{2}{\mathcal{V}}(u^{o}))$

Computation of $\nabla\bm{\mathcal{V}}(u)$ :

Computation of $\nabla^{2}\bm{\mathcal{V}}(u^{o})$ :

6.1. Computation of the Gradient $\mathbb{G}$ -Equivariant Degree

Example 6.1.

Theorem 6.2.

7.1. Topological Invariants $\omega(\lambda_{o})$