# Subharmonic Solutions in Reversible Non-Autonomous Differential Equations

Izuchukwu Eze, Carlos García-Azpeitia, Wieslaw Krawcewicz and Yanli Lv
Department of Mathematical Sciences the University of Texas at Dallas
Richardson, 75080 USA<EMAIL_ADDRESS>Matemáticas y Mecánica
IIMAS, Universidad Nacional Autónoma de México, Apdo. Postal 20-726, 01000
Ciudad de México, México, and Department of Mathematics, Xiangnan University,
889 Chen Zhou Da Dao, Chenzhou, Hunan 423000, China.
<EMAIL_ADDRESS>Mathematics Center at Guangzhou University,
Guangzhou, 510006 China, and Department of Mathematical Sciences the
University of Texas at Dallas Richardson, 75080 USA.
<EMAIL_ADDRESS>of Mathematical Sciences the China Three Gorges
University, Yichang, China<EMAIL_ADDRESS>
###### Abstract

We study the existence of subharmonic solutions in the system
$\ddot{u}(t)=f(t,u(t))$, where $u(t)\in\mathbb{R}^{k}$ and $f$ is an even and
$p$-periodic function in time. Under some additional symmetry conditions on
the function $f$, the problem of finding $mp$-periodic solutions can be
reformulated in a functional space as a $\Gamma\times\mathbb{Z}_{2}\times
D_{m}$-equivariant equation, where the group $\Gamma\times\mathbb{Z}_{2}$ acts
on the space $\mathbb{R}^{k}$ and $D_{m}$ acts on $u(t)$ by time-shifts and
reflection. We apply Brouwer equivariant degree to prove the existence of an
infinite number of subharmonic solutions for the function $f$ that satisfies
additional hypothesis on linear behavior near zero and the Nagumo condition at
infinity. We also discuss the bifurcation of subharmonic solutions when the
system depends on an extra parameter.

##  1 Introduction

In this paper we study the existence of subharmonic solutions of the system

$\ddot{u}(t)=f(t,u(t)),\;\;u(t)\in\mathbb{R}^{k},$ (1)

where $f:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is a
continuous function satisfying the following conditions:

  * ($A_{1}$)

For all $t\in\mathbb{R}$ and $x\in\mathbb{R}^{k}$ we have
$f(t+2\pi,x)=f(t,x)$;

  * ($A_{2}$)

For all $t\in\mathbb{R}$ and $x\in\mathbb{R}^{k}$ we have $f(-t,x)=f(t,x)$;

  * ($A_{3}$)

For all $t\in\mathbb{R}$ and $x\in\mathbb{R}^{k}$ we have $f(t,-x)=-f(t,x)$.

Notice that in condition ($A_{1}$), one could consider $f(t,x)$ being
$p$-periodic with respect to $t$, however by rescaling the time $t$, one can
always arrive to a $2\pi$-periodic function. The problem of finding multiple
subharmonic solutions to (1), especially in the case of Hamiltonian systems of
the type

$\ddot{u}+\nabla F(t,u)=h(t)$ (2)

attracted a lot of attention. Let us mention several contributions, beginning
with the classical work [3] and followed by the works [7, 21], with numerous
other articles that were devoted to this topic (see [1, 6, 8, 10, 11, 12, 15,
17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29]). It should be pointed out that
variational structure of the system (2) seems to play crucial role for the
application of the topological and geometric methods. Regarding the degree
theory (cf. [13, 16]), it has been successfully applied to non-Hamiltonian
systems in [4, 9]) (see also [5, 14, 28, 30]). Conditions ($A_{1}$)–($A_{3}$)
express the symmetric properties of the equation (1). Indeed, finding $2\pi
m$-periodic solutions to (1) leads to an operator which is
$D_{m}\times{\mathbb{Z}}_{2}$-equivariant. Notice that the
${\mathbb{Z}}_{2}$-action allows us to make a distinction between constant and
non-constant solutions. We do not require that $f$ is of a gradient-type or
has any differentiability properties, except for the existence of the
linearization at $0$. Since problem (1) leads naturally to a
$D_{m}\times{\mathbb{Z}}_{2}$-equivariant equation in functional spaces, one
should ask a question: what would be the impact of additional (geometric)
symmetries of equation (1) on the existence and multiplicity of subharmonic
solutions? Therefore, it is natural to assume that the system (1) has
additional symmetries represented by a group $\Gamma$. In this paper, we
assume that $\Gamma$ is a finite group acting on vectors in $\mathbb{R}^{k}$
by permuting their coordinates (see assumption ($A_{4}$)), i.e. the functional
equation has the symmetries

$G:=\Gamma\times D_{m}\times{\mathbb{Z}}_{2}.$

We use Brouwer $G$-equivariant degree to establish the existence and
multiplicity of subharmonic ${2\pi m}$-periodic solutions to (1). We make some
additional assumptions in order to illustrate an application of the Brouwer
equivariant degree to this systems of differential equations. First we assume
(see the assumption ($A_{5}$)) that the linearization at $0$ exists and is
non-degenerate. We also impose on $f$ the Nagumo growth condition, which
implies the existence of a priori bounds on periodic solutions to (1). We
explore in detail two cases of systems of equation: (a) non-symmetric (with
$\Gamma$ being trivial), and (b) with additional symmetries $\Gamma=D_{3}$ and
$\Gamma=D_{5}$. The group $D_{3}$ is the simplest non-abelian group, but it
already makes a significant impact on the existence of multiple subharmonic
solutions. Since the computations of Brouwer $G$-equivariant degree can be
technically challenging, in order to overcome these difficulties we use the
equivariant degree package EquiDeg for GAP programming, which was created by
Hao-Pin Wu and is available from https://github.com/psistwu/GAP-equideg As the
assumption ($A_{3}$) implies that $f(t,0)=0$, so (1) admits the (trivial)
solution $u(t)=0$. It is interesting to study a parametrized by $\alpha$
modification of the system (1) (see system (28)) for which the existence of
non-constant branches of subharmonic $2\pi m$-periodic solutions bifurcating
from $0$ can be analyzed. We apply the Brouwer $G$-equivariant degree method
to study the symmetric bifurcation problem for (28). We establish the
existence of multiple branches of subharmonic solutions emerging from the
trivial solutions as the parameter $\alpha$ crosses a critical value.
Theoretical results are illustrated by an example involving concrete
symmetries for the system (28).

##  2 Reversible Non-Autonomous Differential Equations

We are interested in studying the existence of the so-called subharmonic
periodic solutions to (1), i.e. in finding non-constant solutions, which for
some integer $m\geq 3$ satisfy

$u(t)=u(t+{2\pi m}),\;\;\dot{u}(t)=\dot{u}(t+{2\pi m}).$ (3)

We also consider a subgroup $\Gamma\leq S_{k}$ acting on $V:=\mathbb{R}^{k}$
by permuting the coordinates of vectors
$x=(x_{1},x_{2},\dots,x_{k})^{T}\in\mathbb{R}^{k}$, i.e. for $\sigma\in S_{k}$

$\sigma
x=\sigma(x_{1},x_{2},\dots,x_{k})^{T}:=(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(k)})^{T}.$
(4)

Clearly, the space $V:=\mathbb{R}^{k}$ equipped with this $\Gamma$-action is
an orthogonal $\Gamma$-representation. As it is also of our interest to study
the impact of the symmetries $\Gamma$ on the existence of subharmonic
solutions to (1), we introduce the following condition:

  * ($A_{4}$)

For all $t\in\mathbb{R}$, $x\in\mathbb{R}^{k}$ and $\sigma\in\Gamma$, we have
$f(t,\sigma x)=\sigma f(t,x)$.

The condition ($A_{4}$) implies that $f$ is $\Gamma$-equivariant, i.e. the
system (1) admits $\Gamma$-symmetries.

###  2.1 Reformulation of (1) in Functional Spaces

Consider the Banach space $\mathbb{F}:=C_{2\pi m}(\mathbb{R},V)$ of all $2\pi
m$-periodic continuous $V$-valued functions with the usual sup-norm

$\|\varphi\|_{\infty}:=\max_{t\in\mathbb{R}}|\varphi(t)|,\;\varphi\in\mathbb{F},$

and denote by $\mathbb{E}:=C^{2}_{2\pi m}(\mathbb{R},V)$ the Banach space of
all $2\pi m$-periodic $C^{2}$-differentiable $V$-valued functions with the
norm $\|\cdot\|:=\|\cdot\|_{2,\infty}$ given by

$\|u\|=\|u\|_{2,\infty}:=\max\\{\|u\|_{\infty},\|\dot{u}\|_{\infty},\|\ddot{u}\|_{\infty}\\},\quad
u\in\mathbb{E}.$ (5)

Notice that the natural injection operator
$\mathfrak{j}:\mathbb{E}\rightarrow\mathbb{F}$, $(\mathfrak{j}(u))(t):=u(t)$,
$t\in\mathbb{R}$, is a compact linear operator. We define the operator
$L:\mathbb{E}\rightarrow\mathbb{F}$ by $L(u)(t):=\ddot{u}(t)-u(t)$,
$u\in\mathbb{E}$, and the continuous map
$N_{f}:\mathbb{F}\rightarrow\mathbb{F}$ by
$N_{f}(\varphi)(t)=f(t,\varphi(t))$, $\varphi\in\mathbb{F}$. Then, the system
(1) is equivalent to the following operator equation

$Lu=N_{f}(\mathfrak{j}(u))-\mathfrak{j}(u),\quad u\in\mathbb{E}.$ (6)

Since the operator $L$ is an isomorphism, we can rewrite (6) as

$u=L^{-1}\Big{(}N_{f}(\mathfrak{j}(u))-\mathfrak{j}(u)\Big{)},\quad
u\in\mathbb{E}.$

Define the map $\mathscr{F}:\mathbb{E}\rightarrow\mathbb{E}$, by

$\mathscr{F}(u):=u-L^{-1}\Big{(}N_{f}(\mathfrak{j}(u))-\mathfrak{j}(u)\Big{)},\quad
u\in\mathbb{E}.$ (7)

Then $u\in\mathbb{E}$ is a solution to (1) if and only if

$\mathscr{F}(u)=0.$ (8)

One can easily observe that, by compactness of $\mathfrak{j}$, the map
$\mathscr{F}$ is a completely continuous field on $\mathbb{E}$. Obviously, by
the condition ($A_{3}$), we have $f(t,0)=0$ for all $t\in\mathbb{R}$, thus
$\mathscr{F}(0)=0$, i.e. the zero function is the trivial solution to (1). In
what follows we are interested in finding non-trivial (i.e. non-constant)
$2\pi m$-periodic solutions to (1). The group $G:=\Gamma\times
D_{m}\times{\mathbb{Z}}_{2}$ acts on the space $\mathbb{E}$ by

$\displaystyle(\sigma,\gamma^{j},\pm 1)u(t)$ $\displaystyle:=\pm\sigma
u(t+2\pi j),\quad
j=0,1,\dots,m-1,\;\;\sigma\in\Gamma,\;\gamma=e^{\frac{i2\pi}{m}},$
$\displaystyle(\sigma,\kappa,\pm 1)u(t)$ $\displaystyle:=\pm\sigma u(-t),\quad
t\in\mathbb{R},\;\;u\in\mathbb{E},$

thus $\mathbb{E}$ is an isometric Banach $G$-representation. One can easily
verify that the properties ($A_{1}$)—($A_{4}$) imply that $\mathscr{F}$ is
$G$-equivariant.

###  2.2 $G$-Isotypic Decomposition of $\mathbb{E}$

Actually, $\mathbb{E}$ is an isometric Banach $\Gamma\times
O(2)\times{\mathbb{Z}}_{2}$-representation, with $O(2)$-action given by

$e^{i\theta}u(t)=u\left(t+\theta m\right),\;\;\kappa u(t)=u(-t),\quad
u\in\mathbb{E},$

and $\Gamma$-action given by $(\gamma u)(t)=\gamma u(t)$, $\gamma\in\Gamma$,
$t\in\mathbb{R}$. Using the $\Gamma\times O(2)\times{\mathbb{Z}}_{2}$-action
on $\mathbb{E}$, one can easily recognize the $\Gamma\times
D_{m}\times{\mathbb{Z}}_{2}$-isotypic decomposition of $\mathbb{E}$. Indeed,
by using the usual Fourier series expansions of functions $u\in\mathbb{E}$, we
have the following $\Gamma\times O(2)\times{\mathbb{Z}}_{2}$-isotypic
decomposition of $\mathbb{E}$

$\mathbb{E}=\overline{\bigoplus_{j=0}^{\infty}\bigoplus_{l=0}^{\mathfrak{r}}\mathbb{V}_{j,l}},$
(9)

where

$\mathbb{V}_{j,l}=\left\\{u\in\mathbb{E}:u(t)=\cos(jt/m)a+\sin(jt/m)b,\;a,\,b\in
V_{l}\right\\}.$

and

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

is a $\Gamma$-isotypic decomposition of $V$, with the component $V_{l}$ being
modeled on the $\Gamma$-irreducible representation $\mathcal{U}_{l}$, $0\leq
l\leq\mathfrak{r}$.

######  Proposition 2.1.

For $j>0$, the $\Gamma\times O(2)\times{\mathbb{Z}}_{2}$-invariant subspace
$\mathbb{V}_{j,l}$ can be identified with the complexification
$V_{l}^{c}:=V_{l}\oplus iV_{l}$ of $V_{l}$, on which $O(2)$ acts by

$e^{i\theta}(a+ib):=e^{-ij\theta}\cdot(a+ib),\;\;\kappa(a+ib)=a-ib,\quad
a,\,b\in V_{l},$

where ‘$\cdot$’ stands for complex multiplication.

###### Proof.

Define the real isomorphism $\psi_{j}:V_{l}^{c}\rightarrow\mathbb{V}_{j,l}$ by
$\psi(a+ib)(t)=\cos(jt/m)a+\sin(jt/m)b$, where $a$, $b\in V_{l}$. Then for
$\mathbf{z}:=a+ib$ we have

$\displaystyle\psi_{j}\big{(}e^{i\theta}(\mathbf{z}\big{)})$
$\displaystyle=\psi_{j}\big{(}e^{i\theta}(a+ib)\big{)}$
$\displaystyle=\psi_{j}\Big{(}\cos(j\theta)a+\sin(j\theta)b+i(-\sin(j\theta)a+\cos(j\theta)b)\Big{)}$
$\displaystyle=\cos(jt/m)(\cos(j\theta)a+\sin(j\theta)b)+\sin(jt/m)(-\sin(j\theta)a+\cos(j\theta)b)$
$\displaystyle=\cos(\tfrac{j}{m}(t+m\theta))a+\sin(\tfrac{j}{m}(t+m\theta))b$
$\displaystyle=e^{i\theta}(\cos(jt/m)a+\sin(jt/m)b)=e^{i\theta}\psi_{j}(a+ib)=e^{i\theta}\psi_{j}(\mathbf{z}).$

$\square$

Consider $j$-th irreducible $O(2)$-representation
$\mathcal{W}_{j}\simeq{\mathbb{C}}$, $j>0$, where for $e^{i\theta}\in SO(2)$,
$e^{i\theta}z:=e^{i\theta j}\cdot z$ and $\kappa z:=\overline{z}$,
$z\in{\mathbb{C}}$. Clearly, since $D_{m}\leq O(2)$, $\mathcal{W}_{j}$ is a
$D_{m}$-representation. Put
$\mathfrak{s}:=\left\lfloor\frac{m+1}{2}\right\rfloor$. The irreducible
$D_{m}$-representations $\mathcal{V}_{i}$ are:

  * •

if $i=0$, then $\mathcal{V}_{0}\simeq\mathbb{R}$ with the trivial
$D_{m}$-action;

  * •

if $0<i<m/2$, then $\mathcal{V}_{i}\simeq\mathbb{R}^{2}={\mathbb{C}}$, where
$\gamma z=\gamma^{i}\cdot z$, $\kappa z=\overline{z}$, $z\in{\mathbb{C}}$;

  * •

if $i=\mathfrak{s}$, then $\mathcal{V}_{\mathfrak{s}}\simeq\mathbb{R}$ with
the $D_{m}$-action $\gamma x=x$, $\kappa x=-x$, $x\in\mathbb{R}$;

  * •

if $m$ is even, then we have the irreducible $D_{m}$-representation
$\mathcal{V}_{\mathfrak{s}+1}\simeq\mathbb{R}$ with the $D_{m}$-action $\gamma
x=-x$, $\kappa x=x$, $x\in\mathbb{R}$;

  * •

if $m$ is even, then we have the representation
$\mathcal{V}_{\mathfrak{s}+2}\simeq\mathbb{R}$ with $D_{m}$-action $\gamma
x=-x$, $\kappa x=-x$, $x\in\mathbb{R}$.

For the group $D_{m}\times{\mathbb{Z}}_{2}$, the corresponding irreducible
representations (with non-trivial ${\mathbb{Z}}_{2}$-action) will be denoted
by $\mathcal{V}_{i}^{-}$.

######  Proposition 2.2.

The $D_{m}$-representation $\mathcal{W}_{j}$ has the following
$D_{m}$-isotypic decomposition

  * $\bullet$

$\mathcal{W}_{mj}\simeq\mathcal{V}_{0}\oplus\mathcal{V}_{\mathfrak{s}}$,

  * $\bullet$

for $0<i<\frac{m}{2}$,
$\mathcal{W}_{mj+i}\simeq\mathcal{W}_{mj-i}\simeq\mathcal{V}_{i}$,

  * $\bullet$

if $m$ is even,
$\mathcal{W}_{mj-\frac{m}{2}}\simeq\mathcal{V}_{\mathfrak{s}+1}\oplus\mathcal{V}_{\mathfrak{s}+2}$.

For $j>0$ and $0\leq l\leq\mathfrak{r}$, we put

$\displaystyle\mathbb{V}_{jm,l}^{+}$
$\displaystyle=\left\\{u\in\mathbb{E}:u(t)=\cos(jt)a,\;a\in V_{l}\right\\},$
$\displaystyle\mathbb{V}_{jm,l}^{-}$
$\displaystyle=\left\\{u\in\mathbb{E}:u(t)=\sin(jt)b,\;b\in V_{l}\right\\},$
$\displaystyle\mathbb{V}_{jm-\frac{m}{2},l}^{+}$
$\displaystyle=\left\\{u\in\mathbb{E}:u(t)=\cos((j-\tfrac{1}{2})t)a,\;a\in
V_{l}\right\\},$ $\displaystyle\mathbb{V}_{jm-\frac{m}{2},l}^{-}$
$\displaystyle=\left\\{u\in\mathbb{E}:u(t)=\sin((j-\tfrac{1}{2})t)b,\;b\in
V_{l}\right\\}.$

Therefore, we have the following $\Gamma\times
D_{m}\times{\mathbb{Z}}_{2}$-isotypic decomposition of the space $\mathbb{E}$:

$\mathbb{E}=\bigoplus_{l=0}^{\mathfrak{r}}\bigoplus_{i=0}^{\mathfrak{s}^{\ast}}\mathcal{E}_{i,l}^{-},\quad
s^{\ast}:=\begin{cases}\mathfrak{s}&\text{ if $m$ is odd}\\\
\mathfrak{s}+2&\text{ if $m$ is
even}\end{cases},\quad\mathfrak{s}=\left\lfloor\frac{m+1}{2}\right\rfloor,$

where

$\mathcal{E}_{0,l}^{-}=\mathbb{V}_{0,l}\oplus\overline{\bigoplus_{j=1}^{\infty}\mathbb{V}_{mj,l}^{+}},\quad\mathcal{E}_{\mathfrak{s},l}^{-}=\overline{\bigoplus_{j=1}^{\infty}\mathbb{V}_{mj,l}^{-}},$

for $0<i<\frac{m}{2}$

$\mathcal{E}_{i,l}^{-}=\overline{\bigoplus_{j=0}^{\infty}\mathbb{V}_{mj+i,l}}\oplus\overline{\bigoplus_{j=1}^{\infty}\mathbb{V}_{mj-i,l}},$

and if $m$ is even then

$\mathcal{E}_{\mathfrak{s}+1,l}^{-}=\overline{\bigoplus_{j=1}^{\infty}\mathbb{V}_{mj-\frac{m}{2},l}^{+}},\quad\mathcal{E}_{\mathfrak{s}+2,l}^{-}=\overline{\bigoplus_{j=1}^{\infty}\mathbb{V}_{mj-\frac{m}{2},l}^{-}}.$

The component $\mathcal{E}_{i,l}^{-}$ ($0\leq i\leq\mathfrak{s}^{\ast}$,
$0\leq l\leq\mathfrak{r}$) is modeled on the irreducible $\Gamma\times
D_{m}\times{\mathbb{Z}}_{2}$-representation

$\mathcal{V}_{i,l}^{-}:=\mathcal{V}_{i}^{-}\otimes\mathcal{U}_{l}.$

Since the operator $L$ is $O(2)\times{\mathbb{Z}}_{2}$-equivariant
isomorphism, thus $L(\mathbb{V}_{j})=\mathbb{V}_{j}$ and
$L|_{\mathbb{V}_{j}}=-(j^{2}/m^{2}+1)\operatorname{Id}_{\mathbb{V}_{j}}$.

###  2.3 Linearization of Equation (1) at $0$ :

We make the following additional assumption

  * ($A_{5}$)

There exists a symmetric matrix $A:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$
such that

$\lim_{x\rightarrow 0}\frac{f(t,x)-Ax}{|x|}=0$ (10)

uniformly with respect to $t\in\mathbb{R}$, and for all integers $j\geq 0$ and
$\mu\in\sigma(A)$,

$\frac{j^{2}}{m^{2}}+\mu\not=0.$ (11)

We define the linear operator $\mathscr{A}:\mathbb{E}\rightarrow\mathbb{E}$
(associated with $A:\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$) by

$\mathscr{A}u:=u-L^{-1}\Big{(}N_{A}(\mathfrak{j}(u)-\mathfrak{j}(u)\Big{)},\quad
u\in\mathbb{E},$ (12)

where $N_{A}(\varphi)(t):=A(\varphi(t))$, $t\in\mathbb{R}$, $\varphi\in
C_{2\pi m}(\mathbb{R};V)$. Under the assumption ($A_{5}$) the operator
$\mathscr{A}:\mathbb{E}\rightarrow\mathbb{E}$ given by (12) is an isomorphism
and $D\mathscr{F}(0)=\mathscr{A}$.

######  Lemma 2.3.

Assume that $f:\mathbb{R}\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ satisfies the
conditions ($A_{1}$)—($A_{5}$). Then there exists $\varepsilon>0$ such that
the $G$-map $\mathscr{F}:\mathbb{E}\to\mathbb{E}$ (given by (7)) and
$\mathscr{A}:\mathbb{E}\to\mathbb{E}$ (given by (12)) are
$\Omega_{\varepsilon}$-admissibly $G$-homotopic (here
$\Omega_{\varepsilon}:=B_{\varepsilon}(0)$ in $\mathbb{E}$).

###### Proof.

Define the linear homotopy $\mathfrak{H}:[0,1]\times\mathbb{E}\to\mathbb{E}$
as $\mathfrak{H}(\lambda,u):=(1-\lambda)\mathscr{A}u+\lambda\mathscr{F}(u)$,
$u\in\mathbb{E}$, and suppose for contradiction that there exists a sequence
$\\{\lambda_{n},u_{n}\\}$ such that $u_{n}\not=0$, $\lambda_{n}\to\lambda_{o}$
and $u_{n}\to 0$ as $n\to\infty$ , then we have

$\displaystyle 0$
$\displaystyle=\mathfrak{H}(\lambda_{n},u_{n})=(1-\lambda_{n})\mathscr{A}u_{n}+\lambda_{n}F(u_{n})$
$\displaystyle=\mathscr{A}(u_{n})+\lambda_{n}(\mathscr{F}(u_{n})-\mathscr{A}u_{n}).$

Put $v_{n}:=\frac{u_{n}}{\|u_{n}\|}$. Then

$0=\mathscr{A}v_{n}+\lambda_{n}\frac{\mathscr{F}(u_{n})-\mathscr{A}u_{n}}{\|u_{n}\|}.$

Since $\|u_{n}\|\to 0$ and $\lambda_{n}$ is bounded, thus

$\lim_{n\to\infty}\frac{\mathscr{F}(u_{n})-\mathscr{A}u_{n}}{\|u_{n}\|}=0,$

which implies

$0=\lim_{n\to\infty}(\mathscr{A}v_{n}).$

On the other hand, since $\mathscr{A}=\operatorname{Id}-\mathscr{K}$, where
$\mathscr{K}:=L^{-1}(A\mathfrak{j}-\mathfrak{j})$ is a compact operator, one
can assume (by passing to a subsequence) that $\mathscr{K}v_{n}\to v_{o}$,
which implies $v_{n}\to v_{0}$ and $\|v_{o}\|=1$, so
$v_{o}\in\text{{Ker\,}}\mathscr{A}$, but this is a contradiction with
($A_{5}$). $\square$

###  2.4 Nagumo Growth Condition:

The following condition is often referred to as the Nagumo growth condition:

  * ($A_{6}$)

There exists a constant $M>0$ such that

$\forall_{t\in\mathbb{R}}\;\;\forall_{x\in\mathbb{R}^{k}}\;\;|x|\geq
M\;\;\Rightarrow\;\;f(t,x)\bullet x>0.$

We consider the following parametrized (by $\lambda\in[0,1]$) modification of
system (1):

$\begin{cases}\ddot{u}(t)=\lambda f(t,u(t))+(1-\lambda)u(t),\quad
t\in\mathbb{R},\;u(t)\in V,\\\ u(t)=u(t+{2\pi m}),\;\dot{u}(t)=\dot{u}(t+{2\pi
m}).\end{cases}$ (13)

Then we have:

######  Lemma 2.4.

Assume that $f:\mathbb{R}\oplus\mathbb{R}^{k}\to\mathbb{R}^{k}$ is a
continuous function satisfying conditions ($A_{1}$)—($A_{4}$) and ($A_{6}$).
If $u(t)$ is a ${2\pi m}$-periodic function of class $C^{2}$ such that
$\max_{t\in\mathbb{R}}|u(t)|\geq M$ (where $M$ is given in ($A_{6}$)), then
$u(t)$ cannot be a solution of (13) for $\lambda\in[0,1]$.

###### Proof.

Assume for the contradiction that $u(t)$ is a solution while
$\max_{t\in\mathbb{R}}|u(t)|\geq M$. Consider the function
$\phi(t):=\frac{1}{2}|u(t)|^{2}$. Suppose that
$\phi(t_{0})=\max_{t\in\mathbb{R}}\phi(t)$, then
$\phi^{\prime}(t_{0})=u(t_{0})\bullet\dot{u}(t_{0})=0$ and
$\phi^{\prime\prime}(t_{0})=\dot{u}(t_{0})\bullet\dot{u}(t_{0})+\ddot{u}(t_{0})\bullet
u(t_{0})\leq 0$. However, by condition ($A_{6}$),
$\phi^{\prime\prime}(t_{0})=\dot{u}(t_{0})\bullet\dot{u}(t_{0})+\ddot{u}(t_{0})\bullet
u(t_{0})=(\lambda(f(u(t_{0}))-u(t_{0}))+u(t_{0}))\bullet
u(t_{0})+\dot{u}(t_{0})\bullet\dot{u}(t_{0})>(1-\lambda)u(t_{0})\bullet
u(t_{0})+\lambda f(u(t_{0}))\bullet u(t_{0})>0$, which leads to a
contradiction with condition ($A_{6}$). $\square$

######  Lemma 2.5.

Assume that $f:\mathbb{R}\oplus\mathbb{R}^{k}\to\mathbb{R}^{k}$ is a
continuous function satisfying conditions ($A_{1}$)—($A_{3}$) and ($A_{6}$).
Then there exists $R>0$ such that for every solution $u\in\mathbb{E}$ to
$\eqref{eq:non-aut-lambda}$, $\lambda\in[0,1]$, we have $\|u\|<R$. In
addition, for $\Omega_{R}:=B_{R}(0)$, the map
$\mathscr{F}:\mathbb{E}\to\mathbb{E}$ is $\Omega_{R}$-admissibly $G$-homotopic
to $\operatorname{Id}$.

###### Proof.

By Lemma 2.4, there exists a $M>0$ such that any ${2\pi m}$-periodic solution
$u(t)$ to (13) satisfies $|u(t)|<M$. Take $A_{R}:=\\{(t,x)\in[0,{2\pi
m}]\times\mathbb{R}^{k}:|x|\leq M\\}$. Since the function
$F:[0,1]\times\mathbb{R}\oplus\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ given
by

$F(\lambda,t,x)=\lambda
f(t,x)+(1-\lambda)x,\;\;\;\;x\in\mathbb{R}^{k},\;\lambda\in[0,1]$

is continuous, and the set $[0,1]\times A_{R}$ is compact, then for every
solution $u(t)$ to (13) we have

$|\ddot{u}(t)|=|F(\lambda,t,u)|\leq\sup\\{|F(\lambda,t,x)|:(t,x)\in
A_{R},\,\lambda\in[0,1]\\}=:M_{2}.$

Put
$\dot{u}(t)=(u_{1}^{\prime}(t),u_{2}^{\prime}(t),\dots,u_{k}^{\prime}(t))^{T}$.
Then for every $1\leq l\leq k$, since the function $\dot{u}_{l}(t)$ is
periodic, there exists $\tau_{o}\in[0,{2\pi m}]$ such that
$u_{l}^{\prime}(\tau_{o})=0$. Thus the identity

$u_{l}^{\prime}(t)=\int_{\tau_{o}}^{t}u_{l}^{\prime\prime}(s)ds,\quad
t\in\mathbb{R}$

implies $|u_{l}^{\prime}(t)|\leq{2\pi m}M_{2}$ for $t\in\mathbb{R}$ and
consequently

$\|\dot{u}\|_{\infty}=\max_{t\in\mathbb{R}}\sqrt{|u_{1}^{\prime}(t)|^{2}+|u_{2}^{\prime}(t)|^{2}+\dots+|u_{k}^{\prime}(t)|^{2}}\leq\sqrt{k}{2\pi
m}M_{2}=:M_{1}.$

Consequently,

$\|\dot{u}\|=\max\\{\|u\|_{\infty},\|\dot{u}\|_{\infty},\|\ddot{u}\|_{\infty}\\}\leq\max\\{M,M_{1},M_{2}\\}<\max\\{M,M_{1},M_{2}\\}+1=:R,$
(14)

and the conclusion follows. $\square$

###  2.5 Abstract Existence Result

Assume that $f:\mathbb{R}\oplus V\to V$ satisfies the assumptions
($A_{1}$)—($A_{6}$). We denote the set of negative eigenvalues of the operator
$\mathscr{A}$ by $\sigma_{-}(\mathscr{A})$. Then by Lemma 2.3, there exists a
sufficiently small $\varepsilon>0$ such that $\mathscr{F}$ is
$\Omega_{\varepsilon}$-admissibly $G$-homotopic to $\mathscr{A}$ (given by
(12)) and therefore

$G\text{{-deg}}(\mathscr{F},\Omega_{\varepsilon})=G\text{{-deg}}(\mathscr{A},B(\mathbb{E}))=\prod_{\lambda\in\sigma_{-}(\mathscr{A})}G\text{{-deg}}(-\operatorname{Id}|_{E(\lambda)},B(E(\lambda))),$
(15)

where $E(\lambda)$ denotes the eigenspace of $\mathscr{A}$ corresponding to
$\lambda$, and $B(E(\lambda))$ stands for an open unit ball in $E(\lambda)$.

In order to use the formula (15) we need to compute the negative spectrum
$\sigma_{-}(\mathscr{A})$. Since $\mathscr{A}$ is $\Gamma\times
O(2)\times{\mathbb{Z}}_{2}$-equivariant, one can use the isotypic
decomposition (9) in order to determine eigenvalues (and eigenspaces) of
$\mathscr{A}$:

$\sigma(\mathscr{A})=\left\\{\lambda_{j,\mu}:=1+\frac{m^{2}(\mu-1)}{j^{2}+m^{2}}:j=0,1,2,\dots,\;\mu\in\sigma(A)\right\\}.$
(16)

Clearly,

$\lambda_{j,\mu}=\frac{j^{2}+m^{2}\mu}{j^{2}+m^{2}}<0$

if and only if $\mu<-j^{2}/m^{2}$. Notice that, in such a case we also have

$\lambda_{0,\mu}<\lambda_{1,\mu}<\dots<\lambda_{j-1,\mu}<\lambda_{j,\mu}<\dots<\lambda_{\mathfrak{j}_{\mu},\mu}<0<\lambda_{\mathfrak{j}_{\mu}+1,\mu},$

where $\mathfrak{j}_{\mu}$ is the integer number (by condition ($A_{5}$))
satisfying

$-\frac{(\mathfrak{j}_{\mu}+1)^{2}}{m^{2}}<\mu<-\frac{\mathfrak{j}_{\mu}^{2}}{m^{2}}.$

On the other hand, by Lemma 2.5, there exists a sufficiently large $R>0$ such
that $\mathscr{F}$ is $\Omega_{R}$-admissibly $G$-homotopic to
$\operatorname{Id}$. Therefore,
$G\text{{-deg}}(\mathscr{F},\Omega_{R})=G\text{{-deg}}(\operatorname{Id},\Omega_{R})=(G)$.
Put $\Omega:=\Omega_{R}\setminus\overline{\Omega_{\varepsilon}}$. Then the
$G\text{{-deg}}(\mathscr{F},\Omega)$ is well defined and by additivity
property we have

$\displaystyle G\text{{-deg}}(\mathscr{F},\Omega)$
$\displaystyle=G\text{{-deg}}(\mathscr{F},\Omega_{R})-G\text{{-deg}}(\mathscr{F},\Omega_{\varepsilon})$
$\displaystyle=(G)-G\text{{-deg}}(\mathscr{A},B(\mathbb{E})).$

In this way we can formulate the following abstract existence result:

######  Theorem 2.6.

Assume that $f:\mathbb{R}\oplus V\to V$ satisfies the assumptions
($A_{1}$)—($A_{6}$), $R>0$ is a sufficiently large (given by Lemma 2.5),
$\varepsilon>0$ is sufficiently small (given by Lemma 2.3) and
$\Omega:=\Omega_{R}\setminus\overline{\Omega_{\varepsilon}}$. If the
$G$-equivariant degree

$G\text{{-deg}}(\mathscr{F},\Omega)=n_{1}(H_{1})+n_{2}(H_{2})+\dots+n_{s}(H_{s})\in
A(G)$

has a non-zero coefficient $n_{j}$, then there exists a ${2\pi m}$-periodic
solution $u\in\Omega$ to (1) such that $G_{u}\leq H_{j}$. In addition, if
$D_{m}\not\leq H_{j}$ then $u$ is non-constant, and if for some $g\in D_{m}$,
$g\not=1$, we have $(g,-1)\in H_{j}$, then the solution $u$ can not be
$2\pi$-periodic solution, i.e. its minimal period is not $2\pi$.

###### Proof.

The existence of a ${2\pi m}$-periodic solution $x$ is a direct consequence of
the existence property for $G$-equivariant degree. Moreover, if $u(t)$ is
constant, then clearly $u(t+l2\pi)=u(t)$ and $u(-t)=u(t)$, for
$t\in\mathbb{R}$ and $l\in{\mathbb{Z}}$, so $D_{m}\leq H_{j}$. Assume that
there exists an element $(g,-1)\in H_{j}$ for some $1\not=g\in D_{m}$, which
implies that for some $1\leq l\leq m-1$ we have $g=\gamma^{l}$ or
$g=\gamma^{l}\kappa$. Then we also have

$\forall_{t\in\mathbb{R}}\;\;((g,-1)u)(t)=u(t)\;\;\Rightarrow x(0)=-x(2\pi
l).$

Since $x\not\equiv 0$ it follows that $u(t)\not=u(t+l2\pi)$ and consequently
$u(t)\not=u(t+2\pi)$. $\square$

###  2.6 Subharmonic Solutions in Non-Equivariant Case

To illustrate our previous theorem, in this section we assume that
$\Gamma=\\{e\\}$, i.e. we consider the case of (1) without additional
symmetries and $G=D_{m}\times{\mathbb{Z}}_{2}$.

######  Definition 2.7.

We define

$i(j):=\begin{cases}\alpha(j)&\text{ if
}\;\alpha(j)\leq\lfloor\frac{m}{2}\rfloor,\\\ m-\alpha(j)&\text{ if
}\;\alpha(j)>\lfloor\frac{m}{2}\rfloor,\end{cases}$ (17)

where $\alpha(j)\in\\{0,1,\dots,m-1\\}$ satisfies $\alpha(j)\equiv j\text{
(mod $m$)}$, i.e.

$\alpha(j):=j-\left\lfloor\frac{j}{m}\right\rfloor m\in\\{0,1,2,\dots,m-1\\}.$

We use notation $\mathfrak{m}(\mu)$ for the algebraic multiplicity of $\mu$
belonging to the spectrum of $A$. The negative spectrum
$\sigma_{-}(\mathscr{A})$ can be represented as

$\sigma_{-}(\mathscr{A})=\bigcup_{\mu\in\sigma_{-}(A)}\\{\lambda_{0,\mu},\lambda_{1,\mu},\dots,\lambda_{\mathfrak{j}_{\mu}-1,\mu},\lambda_{\mathfrak{j}_{\mu},\mu}\\}$
(18)

Denote by $E(\lambda_{j,\mu})$ the eigenspace of $\lambda_{j,\mu}$. In order
to compute the degree, we introduce the following notation:

$\beta_{0}(\mu):=\left(\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor+1\right)\mathfrak{m}(\mu),\quad\beta_{\mathfrak{s}}(\mu):=\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor\mathfrak{m}(\mu),$

and for $i=\mathfrak{s}+1$, $\mathfrak{s}+2$ (in the case $m$ is even),

$\beta_{i}(\mu):=\begin{cases}\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor\mathfrak{m}(\mu),&\text{
if }\;\alpha(\mathfrak{j}_{\mu})<\frac{m}{2},\\\
\left(\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor+1\right)\mathfrak{m}(\mu),&\text{
if }\;\alpha(\mathfrak{j}_{\mu})\geq\frac{m}{2},\end{cases}$

and finally for $0<i<\frac{m}{2}$ ,

$\beta_{i}(\mu):=\begin{cases}2\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor\mathfrak{m}(\mu)&\text{
if }\alpha(\mathfrak{j}_{\mu})<i,\\\
\left(2\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor+1\right)\mathfrak{m}(\mu)&\text{
if }\;i\leq\alpha(\mathfrak{j}_{\mu})<m-i,\\\
2\left(\left\lfloor\frac{\mathfrak{j}_{\mu}}{m}\right\rfloor+1\right)\mathfrak{m}(\mu)&\text{
if }\;m-i\leq\alpha(\mathfrak{j}_{\mu}).\end{cases}$

######  Definition 2.8.

We define

$\eta_{i}:=\sum_{\mu\in\sigma_{-}(A)}\beta_{i}(\mu),$

for $i=0,1,\dots,\mathfrak{s},\mathfrak{s}+1,\mathfrak{s}+2$. The number
$\eta_{i}$ counts the ”total number of times” that the irreducible
representation $\mathcal{V}_{i}^{-}$ appears in $\sigma_{-}(\mathscr{A})$.

We have the following list of basic degrees for the irreducible
$G$-representations (see appendix):

  * •

for $0\leq i\leq\lfloor\frac{m}{2}\rfloor$, $h:=\text{gcd}(m,i)$, $m/h$ is odd
then

$\deg_{\mathcal{V}_{i}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{h})-(D_{h}^{z})+({\mathbb{Z}}_{h});$

  * •

if $m/h\equiv 2$ (mod $4$) then

$\deg_{\mathcal{V}_{i}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{2h}^{d})-(D_{2h}^{\hat{d}})+({\mathbb{Z}}_{2h}^{d});$

  * •

if $m/h\equiv 0$ (mod $4$) then

$\deg_{\mathcal{V}_{i}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{2h}^{d})-(\widetilde{D}_{2h}^{d})+({\mathbb{Z}}_{2h}^{d});$

  * •

if $i=\mathfrak{s}$ then

$\deg_{\mathcal{V}_{\mathfrak{s}}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{m}^{z});$

  * •

if $i=0$ then

$\deg_{\mathcal{V}_{0}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{m});$

  * •

if $m$ is even and $i=\mathfrak{s}+1$ then

$\deg_{\mathcal{V}_{\mathfrak{s}+1}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{m}^{d});$

  * •

if $m$ is even and $i=\mathfrak{s}+2$ then

$\deg_{\mathcal{V}_{\mathfrak{s}+2}^{-}}=(D_{m}\times{\mathbb{Z}}_{2})-(D_{m}^{\hat{d}}).$

Notice that $\deg_{\mathcal{V}_{i}^{-}}=\deg_{\mathcal{V}_{i^{\prime}}^{-}}$
if and only if gcd$(i,m)=\text{gcd}(i^{\prime},m)$. Therefore, we introduce
the numbers $\rho_{i}$, $0\leq i\leq\mathfrak{s}+2$, that will allow us to
determine how many times the basic degree $\deg_{\mathcal{V}_{i}^{-}}$ appears
in the degree of $G\text{\rm-deg}(\mathscr{A},B(\mathbb{E}))$.

######  Definition 2.9.

We define
$\rho_{0}:=\eta_{0},\;\rho_{\mathfrak{s}}:=\eta_{\mathfrak{s}},\;\rho_{\mathfrak{s}+1}:=\eta_{\mathfrak{s}+1},\;\rho_{\mathfrak{s}+2}:=\eta_{\mathfrak{s}+2},$
and

$\rho_{i}:=\sum_{\text{gcd}(i^{\prime},m)=\text{gcd}(i,m)}\eta_{i^{\prime}},\qquad
0<i<\frac{m}{2}.$ (19)

Before proving our main theorem, we need to analyze the maximal $G$-orbit
types in the space $\mathbb{E}\setminus\\{0\\}$.

######  Lemma 2.10.

Suppose $m=2^{\mathfrak{n}}m^{\prime}$, where $m^{\prime}$ is an odd integer.
Then the maximal orbit types in $\mathbb{E}\setminus\\{0\\}$ are:

  * (a)

$(D_{m}^{z})$, $(D_{m})$, and if $\mathfrak{n}>0$,

  * (b)

$(D_{m}^{d})$, $(\widetilde{D}_{m}^{d})$, $(D_{\frac{m}{2}}^{d})$,
$(\widetilde{D}_{\frac{m}{2}}^{d})$,…,
$(D_{\frac{m}{2^{\mathfrak{n}-1}}}^{d})$,
$(\widetilde{D}_{\frac{m}{2^{\mathfrak{n}-1}}}^{d})$,

###### Proof.

The maximal $G$-orbit types in $\mathbb{E}\setminus\\{0\\}$ are exactly the
same as the maximal $G$-orbit types which occur in the space
$\mathcal{V}^{*}\setminus\\{0\\}$, where

$\mathcal{V}^{*}:=\mathcal{V}_{0}^{-}\oplus\mathcal{V}_{1}^{-}\oplus\mathcal{V}_{2}^{-}\oplus\mathcal{V}^{-}_{\mathfrak{s}}\oplus\mathcal{V}^{-}_{\mathfrak{s}+1}\oplus\mathcal{V}^{-}_{\mathfrak{s}+2}.$

First we identify the maximal orbit types in
$\mathcal{V}_{i}^{-}\setminus\\{0\\}$, $i=0,1,\dots,\mathfrak{s}+2$,

  * •

for $0\leq i\leq\lfloor\frac{m}{2}\rfloor$, $h:=\text{gcd}(m,i)$,
$p_{h}:=\frac{m}{h}$ and if $p_{h}$ is odd, then the orbit types are:
$(D_{h})$, $(D_{h}^{z})$;

  * •

if $p_{h}\equiv 2$ (mod $4$), then the maximal orbit types are:
$((D_{2h}^{d})$, $(D_{2h}^{\hat{d}})$;

  * •

if $p_{h}\equiv 0$ (mod $4$), then the maximal orbit types are:
$(D_{2h}^{d})$, $(\widetilde{D}_{2h}^{d})$;

  * •

if $i=\mathfrak{s,}$ then the maximal orbit type is: $(D_{m}^{z})$;

  * •

if $i=0,$ then the maximal orbit type is: $(D_{m})$;

  * •

if $m$ is even and $i=\mathfrak{s}+1,$ then the maximal orbit type is:
$(D_{m}^{d})$;

  * •

if $m$ is even and $i=\mathfrak{s}+2,$ then the maximal orbit type is:
$(D_{m}^{\hat{d}})$.

Notice that $(D_{m})$ and $(D_{m}^{z})$ are the maximal orbit types which
occur in
$\mathcal{V}_{0}^{-}\oplus\mathcal{V}_{\mathfrak{s}}^{-}\setminus\\{0\\}$. On
the other hand for $m$ being an even integer, we have (see Table 5.3 in [2]):

  * •

$(D_{2n}^{d}\leq(D_{m}^{d})$ if and only if $n|\frac{m}{2}$ and $\frac{m}{2n}$
is odd;

  * •

$(\widetilde{D}_{2n}^{d}\leq(\widetilde{D}_{m}^{d})$ if and only if
$n|\frac{m}{2}$ and $\frac{m}{2n}$ is odd;

  * •

$(D_{2n}^{\hat{d}}\leq(D_{m}^{\hat{d}})$ if and only if $n|\frac{m}{2}$ and
$\frac{m}{4n}$ is odd;

and the maximality of the orbit types $(D_{m}^{d})$,
$(\widetilde{D}_{m}^{d})$, $(D_{\frac{m}{2}}^{d})$,
$(\widetilde{D}_{\frac{m}{2}}^{d})$,…,
$(D_{\frac{m}{2^{\mathfrak{n}-1}}}^{d})$,
$(\widetilde{D}_{\frac{m}{2^{\mathfrak{n}-1}}}^{d})$ follows from this result.
$\square$ We have the main theorem.

######  Theorem 2.11.

Let $m$ be a natural number and
$f:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ be a continuous
function satisfying the assumptions ($A_{1}$)—($A_{3}$) and
($A_{5}$)—($A_{6}$). Suppose
$m=2^{\varepsilon_{0}}p_{1}^{\varepsilon_{1}}p_{2}^{\varepsilon_{2}}\dots
p_{s}^{\varepsilon_{s}}$, where $\varepsilon_{0}\geq 0$, $\varepsilon_{l}>0$,
and $p_{l}$, $l=1,2,\dots,s$, are the prime numbers such that
$2<p_{1}<p_{2}<\dots<p_{s}$. For $l=1,2,\dots s$ put $m_{l}:=\frac{m}{p_{l}}$.
Then

  * (i)

if $\rho_{0}$ is odd, then the system (1) admits a $G$-orbit of ${2\pi
m}$-periodic solutions with symmetries $(D_{m})$, and

  * (ii)

if $\rho_{\mathfrak{s}}$ is odd, then the system (1) admits a $G$-orbit of
${2\pi m}$-periodic solutions with symmetries $(D^{z}_{m})$, and

  * (iii)

if for some $l=1,2,\dots,s$, $\rho_{m_{l}}$ is odd, then the system (1) admits
a $G$-orbit of ${2\pi m}$-periodic solutions with symmetries either
$(D_{m_{i}}^{z})$ or $(D_{m}^{z})$, and

  * (iv)

if $\varepsilon_{0}>0$, and $\rho_{\mathfrak{s}+1}$ is odd, then the system
(1) admits a $G$-orbit of ${2\pi m}$-periodic solutions with symmetries
exactly $(D_{m}^{d})$, and

  * (iv)

if $\varepsilon_{0}>0$, and $\rho_{\mathfrak{s}+2}$ is odd, then the system
(1) admits a $G$-orbit of ${2\pi m}$-periodic solutions with symmetries
exactly $(D_{m}^{\hat{d}})$, and

  * (iv)

if $\varepsilon_{0}=\mathfrak{k}>0$, and for some $2\leq k\leq\mathfrak{k}$,
$\rho_{\frac{m}{2^{k}}}$ is odd, then the system (1) admits a $G$-orbit of
${2\pi m}$-periodic solutions with symmetries exactly
$(D_{\frac{m}{2^{k-1}}}^{d})$ and $(D_{\frac{m}{2^{k-1}}}^{\hat{d}})$.

###### Proof.

Clearly for $\mu\in\sigma_{-}(A)$ we have the following formula for the
$\mathcal{V}_{i}^{-}$-isotypic multiplicity of the eigenvalue
$\lambda_{j,\mu}$

$m_{i}^{-}(\lambda_{j,\mu})=\begin{cases}\mathfrak{m}(\mu)&\text{ if
}\;i=i(j),\;0<i<\frac{m}{2},\\\ \mathfrak{m}(\mu)&\text{ if
}\;i=\mathfrak{s},\;j>0,\;i(j)=0,\\\ \mathfrak{m}(\mu)&\text{ if
}\;i=\mathfrak{s}+1,\;i(j)=\frac{m}{2},\\\ \mathfrak{m}(\mu)&\text{ if
}\;i=\mathfrak{s}+2,\;i(j)=\frac{m}{2},\\\ 0&\text{ otherwise}.\end{cases}$
(20)

Therefore (by (20)), we have

$G\text{{-deg}}(\mathscr{A},B(\mathbb{E}))=\prod_{\mu\in\sigma_{-}(A)}\prod_{j=0}^{\mathfrak{j}_{\mu}}G\text{{-deg}}(-\operatorname{Id},B(E(\lambda_{j,\mu})),$
(21)

where

$G\text{{-deg}}(-\operatorname{Id},B(E(\lambda_{j,\mu}))=\begin{cases}(\deg_{\mathcal{V}_{0}^{-}})^{\mathfrak{m}(\mu)}&\text{
if }\;j=0,\\\
(\deg_{\mathcal{V}_{0}^{-}})^{\mathfrak{m}(\mu)}\circ(\deg_{\mathcal{V}_{\mathfrak{s}}^{-}})^{\mathfrak{m}(\mu)}&\text{
if }\;j>0,\;i(j)=0,\\\ (\deg_{\mathcal{V}_{i}^{-}})^{\mathfrak{m}(\mu)}&\text{
if }\;0<i=i(j)<\frac{m}{2},\\\
(\deg_{\mathcal{V}_{\mathfrak{s}+1}^{-}})^{\mathfrak{m}(\mu)}\circ(\deg_{\mathcal{V}_{\mathfrak{s}+2}^{-}})^{\mathfrak{m}(\mu)}&\text{
if }\;i(j)=\frac{m}{2}.\end{cases}$ (22)

Consequently, we obtain the following formula:

$\displaystyle G\text{{-deg}}(\mathscr{F},\Omega)$
$\displaystyle=(G)-G\text{{-deg}}(\mathscr{A},B(\mathbb{E}))$
$\displaystyle=(G)-\prod_{\mu\in\sigma_{-}(A)}\prod_{i=0}^{\mathfrak{s}+2}(\deg_{\mathcal{V}_{i}^{-}})^{\beta_{i}(\mu)}.$

Since $\eta_{i}$ counts the ”total number of times” that the irreducible
representation $\mathcal{V}_{i}^{-}$ appears in $\sigma_{-}(\mathscr{A})$, we
obtain that

$G\text{{-deg}}(\mathscr{F},\Omega)=(G)-\prod_{i=0}^{\mathfrak{s}+2}(\deg_{\mathcal{V}_{i}^{-}})^{\eta_{i}}.$
(23)

Since

$\deg_{\mathcal{V}_{i}^{-}}=\deg_{\mathcal{V}_{i^{\prime}}^{-}}\text{\qquad}\Leftrightarrow\text{\qquad
gcd}(i,m)=\text{gcd}(i^{\prime},m),$

and the numbers $\rho_{i}$ indicate the number of occurrences of the basic
degree $\deg_{\mathcal{V}_{i}^{-}}$ in the product (23), the conclusion
follows from Lemma 2.10 and the fact that the square of any basic degree is
the unit element $(G)\in A(G)$, i.e. $(\deg_{\mathcal{V}_{i}^{-}})^{2}=(G).$
$\square$ In order to illustrate the applications of Theorem 2.12, we only
consider a simple case when the operator $A$ has one simple negative
eigenvalue $\mu$ satisfying

$-\frac{(\mathfrak{p}+1)^{2}}{m^{2}}<\mu<-\frac{\mathfrak{p}^{2}}{m^{2}},\quad\mathfrak{p}:=\left\lfloor\frac{m}{2}\right\rfloor.$
(24)

Then we get the following result:

######  Corollary 2.12.

Let $m$ be a natural number and
$f:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ be a continuous
function satisfying the assumptions ($A_{1}$)—($A_{3}$), ($A_{5}$)—($A_{6}$)
and $\sigma_{-}(A)$ consists a single simple eigenvalue $\mu$ satisfying (24).
Suppose
$m=2^{\varepsilon_{0}}p_{1}^{\varepsilon_{1}}p_{2}^{\varepsilon_{2}}\dots
p_{s}^{\varepsilon_{s}}$, where $p_{l}>2$ are distinct prime numbers,
$\varepsilon_{0}\geq 0$, $\varepsilon_{l}\geq 1$ for $l=1,2,\dots,s$. Then

  * (i)

the system (1) admits $G$-orbit of non-zero ${2\pi m}$-periodic solutions with
symmetries $(D_{m})$;

  * (ii)

if for some $l=1,2,\dots,s$, the number $\frac{p_{l}-1}{2}$ is odd, the system
(1) admits $G$-orbit of ${2\pi m}$-periodic solutions with symmetries
$(D_{m_{l}}^{z})$ or $(D_{m}^{z})$ (here $m_{l}=\frac{m}{p_{l}}$);

  * (iii)

if $\varepsilon_{0}>0$, the system (1) admits $G$-orbit of ${2\pi m}$-periodic
solutions with symmetries exactly $(D_{m}^{d})$, $(D_{m}^{\hat{d}})$;

  * (iv)

if $\varepsilon_{0}>1$, the system (1) admits $G$-orbit of ${2\pi m}$-periodic
solutions with symmetries exactly $(D_{\frac{m}{2}}^{d})$,
$(\widetilde{D}_{\frac{m}{2}}^{d})$;

###### Proof.

Notice that we have

$\beta_{0}(\mu)=1,\;\;\beta_{\mathfrak{s}}(\mu)=0,\;\;\beta_{i}(\mu)=1\;\text{
for }\;0<i<\frac{m}{2},$

and in the case $m$ is even

$\beta_{\mathfrak{s}}(\mu+1)=\beta_{\mathfrak{s}}(\mu+2)=1.$

Consequently,

  * •

Since $\rho_{0}=\beta_{0}=1$, thus there exist a non-zero $2\pi m$-periodic
solution to the system (1) with symmetries exactly $(D_{m})$;

  * •

Since $\rho_{\mathfrak{s}}=\beta_{\mathfrak{s}}=0$, we cannot conclude the
existence of a $2\pi m$-periodic solution to the system (1) with symmetries
exactly $(D_{m}^{z})$;

  * •

Since $\rho_{m_{l}}=\beta_{m_{l}}\cdot\frac{p_{l}-1}{2}=\frac{p_{l}-1}{2}$,
$l=1,2,\dots,s$, if $\frac{p_{l}-1}{2}$ is odd there exists a $2\pi
m$-periodic solution to the system (1) with symmetries either
$(D_{m_{l}}^{z})$ or $(D_{m}^{z})$ (notice that $(D_{m_{l}}^{z})$ is not a
maximal orbit type);

  * •

In the case $m$ is even, i.e. $\varepsilon_{0}>0$, since
$\rho_{\mathfrak{s}+1}=\rho_{\mathfrak{s}+2}=1$, it follows that the system
(1) has an orbit of $2\pi m$-periodic solutions with symmetries either
$(D_{m}^{d})$ or $(D_{m}^{\hat{d}})$;

  * •

In addition, if $\varepsilon_{0}>1$, notice that
$\rho_{\frac{m}{4}}=\beta_{\frac{m}{4}}=1$, thus the system (1) admits
$G$-orbit of ${2\pi m}$-periodic solutions with symmetries
$(D_{\frac{m}{2}}^{d})$, $(\widetilde{D}_{\frac{m}{2}}^{d})$;

$\square$

In order to illustrate that the other maximal orbit types in in
$\mathbb{E}\setminus\\{0\\}$ can also appear as symmetries of $2\pi
m$-periodic subharmonic solutions to (1), we assume that
$m=2^{n}p_{1}^{\varepsilon_{1}}p_{2}^{\varepsilon_{2}}\dots
p_{s}^{\varepsilon_{s}}$, $n>1$, and the operator $A$ has one simple negative
eigenvalue $\mu$ satisfying

$-\frac{(\mathfrak{p}+1)^{2}}{m^{2}}<\mu<-\frac{\mathfrak{p}^{2}}{m^{2}},\quad\mathfrak{p}:=\left\lfloor\frac{m}{2^{n}}\right\rfloor.$
(25)

Then we have:

######  Corollary 2.13.

Let $m$ be a natural number and
$f:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ be a continuous
function satisfying the assumptions ($A_{1}$)—($A_{3}$), ($A_{5}$)—($A_{6}$)
and $\sigma_{-}(A)$ consists a single simple eigenvalue $\mu$ satisfying (25).
Suppose $m=2^{n}p_{1}^{\varepsilon_{1}}p_{2}^{\varepsilon_{2}}\dots
p_{s}^{\varepsilon_{s}}$, where $n>2$, $p_{l}>2$ are distinct prime numbers,
$\varepsilon_{0}\geq 0$, $\varepsilon_{l}\geq 1$ for $l=1,2,\dots,s$. Then

  * (i)

The system (1) admits $G$-orbit of non-zero ${2\pi m}$-periodic solutions with
symmetries $(D_{m}),$

  * (ii)

The system (1) admits $G$-orbit of ${2\pi m}$-periodic solutions with
symmetries exactly $(D_{\frac{m}{2^{n-1}}}^{d})$,
$(\widetilde{D}_{\frac{m}{2^{n-1}}}^{d})$;

###  2.7 Examples of Symmetric Systems

In this section we assume that $k=3$,
$f:\mathbb{R}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ satisfies
assumptions ($A_{1}$)—($A_{5}$) with $\Gamma=D_{3}$ and

$A=\frac{1}{4}\left[\begin{array}[]{ccc}-4&-2&-2\\\ -2&-4&-2\\\
-2&-2&-4\end{array}\right].$ (26)

Then we have

$\sigma(A)=\left\\{\mu_{0}=-2,\mu_{1}=-\frac{1}{2}\right\\}.$

#### Case $m=3$:

In this case $G=D_{3}\times D_{3}\times{\mathbb{Z}}_{2}$ and we have

$\sigma(\mathscr{A})=\left\\{\lambda_{j,l}:=1+\frac{9(\mu_{l}-1)}{j^{2}+9}:l=0,1,\;j=0,1,2,\dots\right\\}$

Then

$\displaystyle\sigma_{-}(\mathscr{A})=$
$\displaystyle\Big{\\{}\lambda_{0,0}=-2,\;\lambda_{1,0}=-\frac{17}{10},\;\lambda_{2,0}=-\frac{14}{13},\;\lambda_{3,0}=-\frac{1}{2},\;\lambda_{4,0}=-\frac{2}{25},$
$\displaystyle\lambda_{0,1}=-\frac{1}{2},\;\lambda_{1,1}=-\frac{7}{20},\;\lambda_{2,1}=\allowbreak-\frac{1}{26}\Big{\\}},$

and each of the eigenvalues $\lambda_{l,i}$ are isotypicaly simple, i.e. the
eigenspaces $E(\lambda_{l,i})$ are irreducible $G$-representations. More
precisely, we have

$\displaystyle E(\lambda_{0,0})\simeq\mathcal{V}_{0,0}^{-},\quad
E(\lambda_{3,0})\simeq\mathcal{V}_{0,0}^{-}\oplus\mathcal{V}_{2,0}^{-},\quad
E(\lambda_{1,0})\simeq E(\lambda_{2,0})\simeq
E(\lambda_{4,0})\simeq\mathcal{V}_{1,0}^{-},$ $\displaystyle
E(\lambda_{0,1})\simeq\mathcal{V}_{1,0}^{-},\quad E(\lambda_{1,1})\simeq
E(\lambda_{2,1})\simeq\mathcal{V}_{1,1}^{-}.$

Therefore, we obtain

$\displaystyle G\text{{-deg}}(\mathscr{F},\Omega)$
$\displaystyle=(G)-G\text{{-deg}}(\mathscr{A},B_{1}(0))$
$\displaystyle=(G)-(\deg_{\mathcal{V}_{0,0}^{-}})^{2}\circ\deg_{\mathcal{V}_{2,0}^{-}}\circ(\deg_{\mathcal{V}_{1,0}^{-}})^{3}\circ\deg_{\mathcal{V}_{0,1}^{-}}\circ(\deg_{\mathcal{V}_{1,1}^{-}})^{2}$
$\displaystyle=(G)-\deg_{\mathcal{V}_{0,1}^{-}}\circ\deg_{\mathcal{V}_{1,0}^{-}}\circ\deg_{\mathcal{V}_{2,0}^{-}}$

GAP Code: We use GAP package EquiDeg to compute
$G\text{{-deg}}(\mathscr{F},\Omega)$ for the groups $G:=D_{3}\times
D_{3}\times{\mathbb{Z}}_{2}$. The GAP code f is given below.

⬇

LoadPackage( "EquiDeg" );

gr1 := pDihedralGroup( 3 );

gr2 := SymmetricGroup( 2 );

# create the product of D_3 and Z_2

gr3:= DirectProduct( gr1, gr2 );

# create group G

G := DirectProduct( gr1, gr3 );

# create and name CCSs of gr1 and gr3

ccs_g1:= ConjugacyClassesSubgroups( gr1 );

ccs_g1_names := [ "Z1", "D1", "Z3", "D3" ];

ccs_gr3:=ConjugacyClassesSubgroups( gr3 );

ccs_gr3_names:=["Z1","Z1p","D1","D1z", "Z3",

"D1p","Z3p","D3","D3z","D3p"];

SetCCSsAbbrv(gr1, ccs_g1_names);

SetCCSsAbbrv(gr3, ccs_gr3_names);

ccs := ConjugacyClassesSubgroups( G );

cc := ConjugacyClasses( G );

# create characters of irreducible G-representations

irr := Irr( G );

# compute the corresponding to irr[k[] basic degree degk

For a subgroup $K\leq D_{n}$, we use the symbol
$K^{p}:=K\times{\mathbb{Z}}_{2}$, but in the code we simply write Kp.  By
using the list of conjugacy classes cc, one can easily recognize the
irreducible $G$-representations
$\mathcal{V}_{i,l}^{\pm}:=\mathcal{V}_{i}^{\pm}\otimes\mathcal{U}_{l}$. For
example, the character

| $(\pm\bm{1})$ | $(\pm\mathbf{\kappa}_{2})$ | $(\pm\bm{\gamma}_{2})$ | $(\pm\mathbf{\kappa}_{1})$ | $(\pm\bm{\kappa}_{1}\mathbf{\kappa}_{2})$ | $(\pm\mathbf{\kappa}_{1}\mathbf{\gamma}_{2})$ | $(\pm\mathbf{\gamma}_{1})$ | $(\pm\mathbf{\gamma}_{1}\mathbf{\kappa}_{2})$ | $(\pm\mathbf{\gamma}_{1}\mathbf{\gamma}_{2})$
---|---|---|---|---|---|---|---|---|---
Irr(G)[9] | $\pm 2$ | $\mp 2$ | $\pm 2$ | $0$ | $0$ | $0$ | $\mp 1$ | $\pm 1$ | $\mp 1$

is the character of the $G$-irreducible representation
$\mathcal{V}^{-}_{2,1}$. To be more precise, we have the following
correspondence of the characters.

Irr(G)[1]$=\chi_{\mathcal{V}^{+}_{0,0}}$ |  Irr(G)[7]$=\chi_{\mathcal{V}^{+}_{2,0}}$ |  Irr(G)[13]$=\chi_{\mathcal{V}^{-}_{1,2}}$
---|---|---
Irr(G)[2]$=\chi_{\mathcal{V}^{-}_{2,2}}$ |  Irr(G)[8]$=\chi_{\mathcal{V}^{+}_{0,2}}$ |  Irr(G)[14]$=\chi_{\mathcal{V}^{-}_{1,0}}$
Irr(G)[3]$=\chi_{\mathcal{V}^{-}_{2,0}}$ |  Irr(G)[9]$=\chi_{\mathcal{V}^{-}_{2,1}}$ |  Irr(G)[15]$=\chi_{\mathcal{V}^{+}_{1,2}}$
Irr(G)[4]$=\chi_{\mathcal{V}^{-}_{0,2}}$ |  Irr(G)[10]$=\chi_{\mathcal{V}^{-}_{0,1}}$ |  Irr(G)[16]$=\chi_{\mathcal{V}^{+}_{1,0}}$
Irr(G)[5]$=\chi_{\mathcal{V}^{-}_{0,0}}$ |  Irr(G)[11]$=\chi_{\mathcal{V}^{+}_{2,1}}$ |  Irr(G)[17]$=\chi_{\mathcal{V}^{+}_{1,1}}$
Irr(G)[6]$=\chi_{\mathcal{V}^{+}_{2,2}}$ |  Irr(G)[12]$=\chi_{\mathcal{V}^{+}_{0,1}}$ |  Irr(G)[18]$=\chi_{\mathcal{V}^{-}_{1,1}}$

To conclude the computations in GAP (we continue to use the package EquiDeg).

⬇

# unit element in A(G)

u := -BasicDegree( Irr( G )[1] );

# basic degrees

deg01 := BasicDegree( Irr( G )[10] );

deg10 := BasicDegree( Irr( G )[14] );

deg20 := BasicDegree( Irr( G )[3] );

deg := u-deg01*deg10*deg20;

The list of conjugacy classes of $G=D_{3}\times D_{3}\times{\mathbb{Z}}_{2}$
generated by GAP is $\\{(H_{k}):1\leq k\leq 69\\}$, where $(G)=(H_{69})$. As a
result we obtain

$\displaystyle G\text{{-deg}}(\mathscr{F},\Omega)$
$\displaystyle=-(H_{1})+(H_{4})+(H_{6})+(H_{8})+(H_{11})-2(H_{18})-(H_{22})-(H_{27})$
$\displaystyle+(H_{35})+(H_{43})+(H_{44})+(H_{45})-(H_{51})+(H_{63})-(H_{67}),$
(27)

where the coefficient of $G\text{{-deg}}(\mathscr{F},\Omega)$ can be easily
described using amalgamated notation, for example

⬇

gap> Print( AmalgamationSymbol( ccs[45] ) );

In this way we get the following description of some orbit types represented
in $G\text{{-deg}}(\mathscr{F},\Omega)$:

$\displaystyle H_{4}$
$\displaystyle=-(D_{1}\times_{{\mathbb{Z}}_{2}}D_{1}^{z}),\quad H_{6}$
$\displaystyle=(D_{1}\times{\mathbb{Z}}_{1}),$ $\displaystyle H_{8}$
$\displaystyle=({\mathbb{Z}}_{1}\times D_{1}),\quad H_{18}$
$\displaystyle=(D_{1}\times D_{1}),$ $\displaystyle H_{22}$
$\displaystyle=({\mathbb{Z}}_{1}\times D_{3}),\quad H_{21}$
$\displaystyle=(D_{1}\times{\mathbb{Z}}_{3}),$ $\displaystyle H_{43}$
$\displaystyle=(D_{3}\times D_{1}),\quad H_{44}$ $\displaystyle=(D_{1}\times
D_{3}),$ $\displaystyle H_{45}$
$\displaystyle=(D_{1}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p}),\quad H_{51}$
$\displaystyle=(D_{3}\times D_{1}^{z}),$ $\displaystyle H_{63}$
$\displaystyle=(D_{3}\times D_{3}^{z}),\quad H_{67}$
$\displaystyle=(D_{3}\times D_{3}).$

Next we find the maximal orbit types in the representation $\mathbb{E}$:

⬇

# characters appearing in E

chi := Irr(G)[3]+ Irr(G)[5]+ Irr(G)[9]+ Irr(G)[10]

+ Irr(G)[14]+ Irr(G)[18];

# find orbit types in E

orbtyps := ShallowCopy( OrbitTypes( chi ) );

Remove( orbtyps );

# find maximal orbit types in H-0

max_orbtyps := MaximalElements( orbtyps );

Print( List( max_orbtyps, IdCCS ) );

The maximal orbit types in $\mathbb{E}\setminus\\{0\\}$ are

$\displaystyle(H_{45})=(D_{1}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p}),\;\;\;(H_{52})=(D_{1}\times_{{\mathbb{Z}}_{2}}^{D_{3}^{z}}D_{3}^{p}),$
$\displaystyle(H_{63})=(D_{3}\times D_{3}^{z}),\;\;\;(H_{67})=(D_{3}\times
D_{3}).$

Consequently, we obtain the following

######  Theorem 2.14.

Let $m=3$ and $\Gamma=D_{3}=\langle(1,2,3),(2,3)\rangle$ (acting on
$\mathbb{R}^{3}$ by permuting the coordinates). Assume that
$f:\mathbb{R}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ and $A$ (given by
(34)) satisfy the conditions ($A_{1}$)–($A_{6}$). Then there exist

  * (i)

at least one $(D_{1}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p})$-orbit of
(i.e. at least 6 different) $6\pi$-periodic solutions to the system (1),

  * (ii)

at least one $(D_{3}\times D_{3}^{z})$-orbit of (i.e. at least 2 different)
$6\pi$-periodic solutions to the system (1),

  * (iii)

at least one $(D_{3}\times D_{3})$-orbit of (i.e. at least 2 different)
$6\pi$-periodic solutions to the system (1),

Therefore, the system (1) admits at least 10 different $6\pi$-periodic
solutions.

#### Case $m=4$:

In this case $G=D_{4}\times D_{3}\times{\mathbb{Z}}_{2}$, we have

$\sigma(\mathscr{A})=\left\\{\lambda_{j,l}:=1+\frac{16(\mu_{l}-1)}{j^{2}+16}:l=0,1,\;j=0,1,2,\dots\right\\}$

Then

$\displaystyle\sigma_{-}(\mathscr{A})=$
$\displaystyle\Big{\\{}\lambda_{0,0}=-2,\;\lambda_{1,0}=-\frac{31}{17},\;\lambda_{2,0}=-\frac{7}{5},\;\lambda_{3,0}=-\frac{23}{25},\;\lambda_{4,0}=-\frac{1}{2},$
$\displaystyle\lambda_{5,0}=-\frac{7}{41},\;\lambda_{0,1}=-\frac{1}{2},\;\lambda_{1,1}=-\frac{7}{17},\;\lambda_{2,1}=-\frac{1}{5}\Big{\\}},$

each of the eigenvalues $\lambda_{l,i}$ are isotypicly simple, i.e. the
eigenspaces $E(\lambda_{l,i})$ are irreducible $G$-representations. More
precisely, we have

$\displaystyle E(\lambda_{0,0})\simeq\mathcal{V}_{0,0}^{-},\quad
E(\lambda_{4,0})\simeq\mathcal{V}_{0,0}^{-}\oplus\mathcal{V}_{2,0}^{-},\quad
E(\lambda_{1,0})\simeq E(\lambda_{3,0})\simeq
E(\lambda_{5,0})\simeq\mathcal{V}_{0,1}^{-},$ $\displaystyle
E(\lambda_{2,0})\simeq\mathcal{V}_{3,0}^{-}\oplus\mathcal{V}_{4,0}^{-},\;E(\lambda_{0,1})\simeq\mathcal{V}_{0,1}^{-},\quad
E(\lambda_{1,1})\simeq\mathcal{V}_{1,1}^{-},\;E(\lambda_{2,1})\simeq\mathcal{V}_{3,1}^{-}\oplus\mathcal{V}_{4,1}^{-}$

Therefore, we obtain

$\displaystyle G\text{{-deg}}(\mathscr{F},\Omega)$
$\displaystyle=(G)-G\text{{-deg}}(\mathscr{A},B_{1}(0))$
$\displaystyle=(G)-(\deg_{\mathcal{V}_{0,0}^{-}})^{2}\circ(\deg_{\mathcal{V}_{2,0}^{-}})\circ(\deg_{\mathcal{V}_{0,1}^{-}})^{3}\circ(\deg_{\mathcal{V}_{3,0}^{-}})\circ(\deg_{\mathcal{V}_{4,0}^{-}})$
$\displaystyle\hskip
19.91684pt\circ\deg_{\mathcal{V}_{0,2}^{-}}\circ\deg_{\mathcal{V}_{1,0}^{-}}\circ\deg_{\mathcal{V}_{1,1}^{-}}\circ\deg_{\mathcal{V}_{3,1}^{-}}\circ(\deg_{\mathcal{V}_{4,1}^{-}})$
$\displaystyle=(G)-(\deg_{\mathcal{V}_{2,0}^{-}})\circ(\deg_{\mathcal{V}_{0,1}^{-}})\circ(\deg_{\mathcal{V}_{3,0}^{-}})\circ(\deg_{\mathcal{V}_{4,0}^{-}})\circ\deg_{\mathcal{V}_{0,2}^{-}}$
$\displaystyle\hskip
19.91684pt\circ\deg_{\mathcal{V}_{1,0}^{-}}\circ\deg_{\mathcal{V}_{1,1}^{-}}\circ\deg_{\mathcal{V}_{3,1}^{-}}\circ(\deg_{\mathcal{V}_{4,1}^{-}})$

GAP Code: $G:=D_{4}\times D_{3}\times{\mathbb{Z}}_{2}$.

⬇

LoadPackage( "EquiDeg" );

gr1 := pDihedralGroup( 3 );

gr2 := SymmetricGroup( 2 );

# create the product of D_3 and Z_2

gr3:= DirectProduct( gr1, gr2 );

# create group G

gr4 := pDihedralGroup( 4 );

G := DirectProduct( gr4, gr3 );

# create and name CCSs of gr4 and gr3

ccs_g1:= ConjugacyClassesSubgroups( gr1 );

ccs_g4_names := [ "Z1", "Z2", "D1", "tD1", "D2",

"Z4", "tD2", "D4" ];

ccs_gr3:=ConjugacyClassesSubgroups( gr3 );

ccs_gr3_names:=["Z1","Z1p","D1","D1z", "Z3",

"D1p","Z3p","D3","D3z","D3p"];

SetCCSsAbbrv(gr4, ccs_g4_names);

SetCCSsAbbrv(gr3, ccs_gr3_names);

ccs := ConjugacyClassesSubgroups( G );

cc := ConjugacyClasses( G );

# create characters of irreducible G-representations

irr := Irr( G );

# compute the corresponding to irr[k[] basic degree degk

For a subgroup $K\leq D_{n}$, we use the symbol
$K^{p}:=K\times{\mathbb{Z}}_{2}$ and in the code we simply write Kp.  The
group $G=D_{4}\times D_{3}\times{\mathbb{Z}}_{2}$ has 236 conjugacy classes of
subgroups. The conjugacy classes are denoted $(H_{k})$, $k=1,2,\dots,236$, and
are according to the same order as it is generated by GAP. The group G has 30
irreducible representations which can be easily identified in GAP.

Irr(G)[2]$=\chi_{\mathcal{V}^{-}_{2,2}}$ |  Irr(G)[7]$=\chi_{\mathcal{V}^{-}_{3,0}}$ |  Irr(G)[19]$=\chi_{\mathcal{V}^{-}_{4,1}}$
---|---|---
Irr(G)[3]$=\chi_{\mathcal{V}^{-}_{3,2}}$ |  Irr(G)[8]$=\chi_{\mathcal{V}^{-}_{4,0}}$ |  Irr(G)[20]$=\chi_{\mathcal{V}^{-}_{0,1}}$
Irr(G)[4]$=\chi_{\mathcal{V}^{-}_{4,2}}$ |  Irr(G)[9]$=\chi_{\mathcal{V}^{-}_{0,0}}$ |  Irr(G)[25]$=\chi_{\mathcal{V}^{-}_{1,2}}$
Irr(G)[5]$=\chi_{\mathcal{V}^{-}_{0,2}}$ |  Irr(G)[17]$=\chi_{\mathcal{V}^{-}_{2,1}}$ |  Irr(G)[26]$=\chi_{\mathcal{V}^{-}_{1,0}}$
Irr(G)[6]$=\chi_{\mathcal{V}^{-}_{2,0}}$ |  Irr(G)[18]$=\chi_{\mathcal{V}^{-}_{3,1}}$ |  Irr(G)[30]$=\chi_{\mathcal{V}^{-}_{1,1}}$

Therefore, we are set up to compute the degree
$G\text{{-deg}}(\mathscr{F},\Omega)$:

⬇

# unit element in A(G)

u := -BasicDegree( Irr( G )[1] );

# basic degrees

deg20 := BasicDegree( Irr( G )[6] );

deg01 := BasicDegree( Irr( G )[20] );

deg30 := BasicDegree( Irr( G )[7] );

deg40 := BasicDegree( Irr( G )[8] );

deg02 := BasicDegree( Irr( G )[5] );

deg10 := BasicDegree( Irr( G )[26] );

deg11 := BasicDegree( Irr( G )[30] );

deg31 := BasicDegree( Irr( G )[18] );

deg41 := BasicDegree( Irr( G )[19] );

deg := u-deg20*deg01*deg30* deg40* deg02* deg10

* deg11* deg31* deg41;

We also use the list of all irreducible $G$-representations generated by GAP.
Using this list, the corresponding basic $G$-degrees are easily computed by
the GAP program, so the exact value of $G\text{{-deg}}(\mathscr{F},\Omega)$ is
given by

$\displaystyle G\text{{-deg}}(\mathscr{F},\Omega)$
$\displaystyle=-3(H_{1})+(H_{2})+(H_{3})+(H_{4})-(H_{6})+2(H_{8})+(H_{9})$
$\displaystyle+(H_{10})+2(H_{11})-(H_{12})+(H_{13})+(H_{14})-(H_{17})+(H_{18})-(H_{19})$
$\displaystyle-(H_{20})-(H_{23})-(H_{25})-(H_{26})+(H_{27})-(H_{28})-(H_{32})+(H_{33})$
$\displaystyle-(H_{34})+(H_{35})-(H_{36})-(H_{39})+(H_{40})-(H_{43})-(H_{55})+2(H_{57})$
$\displaystyle-(H_{60})+(H_{62})+(H_{68})+(H_{71})+(H_{72})-(H_{73})-(H_{76})+(H_{86})$
$\displaystyle+(H_{87})-(H_{88})-(H_{90})-(H_{92})-(H_{95})+(H_{99})+(H_{100})+(H_{102})$
$\displaystyle+(H_{103})-(H_{104})+(H_{105})+(H_{109})--(H_{116})+(H_{117})+(H_{119})-(H_{120})$
$\displaystyle-(H_{122})+(H_{130})-(H_{135})-(H_{138})-(H_{144})+(H_{148})+(H_{151})+(H_{167})$
$\displaystyle+(H_{169})+(H_{170})-m(H_{173})+(H_{174})+(H_{175})+(H_{177})+(H_{179})-(H_{192})$
$\displaystyle-(H_{203})-(H_{208})-(H_{212})-(H_{214})+(H_{223})+(H_{225})+(H_{229})+(H_{233}).$

⬇

# characters appearing in E

chi := Irr(G)[2]+Irr(G)[3]+ Irr(G)[4]+ Irr(G)[5]

+ Irr(G)[6]+ Irr(G)[7]+Irr(G)[8]+Irr(G)[9]

+ Irr(G)[17]+ Irr(G)[18]+ Irr(G)[19]+ Irr(G)[20]

+ Irr(G)[25]+ Irr(G)[26]+ Irr(G)[30];

# find orbit types in E

orbtyps := ShallowCopy( OrbitTypes( chi ) );

Remove( orbtyps );

# find maximal orbit types in H-0

max_orbtyps := MaximalElements( orbtyps );

Print( List( max_orbtyps, IdCCS ) );

Since the $G$-isotypic components in $\mathbb{E}$ are easily identified, the
GAP program also allows a quick computation of all maximal orbit types in
$\mathbb{E}\setminus\\{0\\}$, namely

$\displaystyle(H_{177}),\;\;(H_{178}),\;\;(H_{179}),\;\;(H_{180}),\;\;(H_{223}),\;\;(H_{224}),\;\;(H_{225}),$
$\displaystyle(H_{228}),\;\;(H_{229}),\;\;(H_{232}),\;\;(H_{233}).$

One can notice that $G\text{{-deg}}(\mathscr{F},\Omega)$ has non-zero
coefficients for the following maximal orbit types:

$\displaystyle(H_{177})$
$\displaystyle=(D_{2}^{D_{1}}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p}),\quad(H_{179})$
$\displaystyle=(\widetilde{D}_{2}^{\widetilde{D}_{1}}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p})$
$\displaystyle(H_{223})$ $\displaystyle=(D_{4}\times
D_{3}^{z}),\quad(H_{225})$
$\displaystyle=(D_{4}^{D_{2}}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p}),\quad$
$\displaystyle(H_{229})$
$\displaystyle=(D_{4}^{D_{2}}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p}),(H_{233})$
$\displaystyle=(D_{4}^{{\mathbb{Z}}_{4}}\times_{{\mathbb{Z}}_{2}}^{D_{3}}D_{3}^{p})$

Consequently, we obtain the following result.

######  Theorem 2.15.

Let $m=4$, $k=3$ and $\Gamma=D_{3}=\langle(1,2,3),(2,3)\rangle$ (acting on
$\mathbb{R}^{3}$ by permuting the coordinates). Assume that
$f:\mathbb{R}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ and $A$ (given by
(34)) satisfy the conditions ($A_{1}$)—($A_{6}$). Then there exist

  * (i)

at least one $(D_{2}^{D_{1}}\times^{D_{3}}_{{\mathbb{Z}}_{2}}D_{3}^{p})$-orbit
of (i.e. at least 4 different) $8\pi$-periodic solutions to the system (1),

  * (ii)

at least one
$(\widetilde{D}_{2}^{\widetilde{D}_{1}}\times^{D_{3}}_{{\mathbb{Z}}_{2}}D_{3}^{p})$-orbit
of (i.e. at least 4 different) $8\pi$-periodic solutions to the system (1),

  * (iii)

at least one $(D_{4}\times D_{3}^{z})$-orbit of (i.e. at least 2 different)
$8\pi$-periodic solutions to the system (1),

  * (iv)

at least one $(D_{4}^{D_{2}}\times^{D_{3}}_{{\mathbb{Z}}_{2}}D_{3}^{p})$-orbit
of (i.e. at least 2 different) $8\pi$-periodic solutions to the system (1),

  * (v)

at least one $(D_{4}^{D_{2}}\times^{D_{3}}_{{\mathbb{Z}}_{2}}D_{3}^{p})$-orbit
of (i.e. at least 2 different) $8\pi$-periodic solutions to the system (1),

  * (vi)

at least one
$(D_{4}^{{\mathbb{Z}}_{4}}\times^{D_{3}}_{{\mathbb{Z}}_{2}}D_{3}^{p})$-orbit
of (i.e. at least 2 different) $8\pi$-periodic solutions to the system (1).

Therefore, the system (1) admits at least 16 different $8\pi$-periodic
solutions.

##  3 Bifurcation in Reversible Non-Autonomous Second Order Differential
Equaitons

Consider the following parametrized system with $2\pi$-periodic coefficients

$\ddot{u}(t)=\big{(}-\alpha\operatorname{Id}+A\big{)}u(t)+f(t,u(t)),\;\;u(t)\in\mathbb{R}^{k},$
(28)

where $A$ is a non-singular $k\times k$-matrix and
$f:\mathbb{R}\times\mathbb{R}^{k}\rightarrow\mathbb{R}^{k}$ is a continuous
function satisfying the conditions ($A_{1}$)—($A_{3}$) and :

  * ($B_{4}$)

$\displaystyle\lim_{x\to 0}\frac{f(t,x)}{|x|}=0$ uniformly with respect to
$t$.

We are interested in studying the bifurcation of the subharmonic ${2\pi
m}$-periodic solutions (for some integer $m$) to (28) from the trivial
solution $(\alpha,0)$, i.e. the solutions

$u(t)=u(t+{2\pi m}),\;\;\dot{u}(t)=\dot{u}(t+{2\pi m}).$ (29)

We also consider a subgroup $\Gamma\leq S_{k}$ which acts on
$V:=\mathbb{R}^{k}$ by permuting the coordinates of vectors
$x=(x_{1},x_{2},\dots,x_{k})^{T}$ in $\mathbb{R}^{k}$ given by

$\sigma
x=\sigma(x_{1},x_{2},\dots,x_{k})^{T}:=(x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(k)})^{T}.$
(30)

Clearly, the space $V:=\mathbb{R}^{k}$ equipped with this $\Gamma$-action is
an orthogonal $\Gamma$-representation. We also introduce the following
conditions

  * ($B_{5}$)

For all $t\in\mathbb{R}$, $x\in V$ and $\sigma\in\Gamma$, we have $f(t,\sigma
x)=\sigma f(t,x)$ and $A\sigma x=\sigma Ax$;

The condition ($B_{5}$) implies that the system (28) is $\Gamma$-symmetric.
The bifurcation problem (28) with the boundary conditions (29) can be
expressed as the following equation

$\mathscr{F}(\alpha,u)=0,\quad(\alpha,u)\in\mathbb{R}\oplus\mathbb{E},$ (31)

where

$\mathscr{F}(\alpha,u):=u-L^{-1}\Big{(}N_{A+f}\big{(}\mathfrak{j}(u)\big{)}-(\alpha+1)\mathfrak{j}(u)\Big{)},\quad\alpha\in\mathbb{R},\;u\in\mathbb{E}.$
(32)

Put $G:=\Gamma\times D_{m}\times{\mathbb{Z}}_{2}$. Notice that under the
assumptions ($A_{1}$)–($A_{3}$) and ($B_{5}$), the map $\mathscr{F}$ is
$G$-equivariant completely continuous field such that
$\mathscr{F}(\alpha,0)=0$ for all $\alpha\in\mathbb{R}$. Moreover, the
assumption ($B_{4}$) implies that $\mathscr{F}$ is differentiable at
$(\alpha,0)$ with respect to $u\in\mathbb{E}$, and

$\mathscr{A}(\alpha):=D_{u}\mathscr{F}(\alpha,0)=\operatorname{Id}-L^{-1}\Big{(}N_{A}\circ\mathfrak{j}-(\alpha+1)\mathfrak{j}\Big{)}:\mathbb{E}\rightarrow\mathbb{E}.$

The necessary condition for the point $(\alpha_{o},0)$ to be a bifurcation
point for (31) is that
$\mathscr{A}(\alpha_{o}):\mathbb{E}\rightarrow\mathbb{E}$ is not an
isomorphism, i.e. $0\in\sigma(\mathscr{A}(\alpha_{o})$.

The point $\alpha_{o}$ is called a critical point for (31) and the set of all
such critical points $\alpha_{o}$ is denoted $\Lambda$. One can easily compute
the spectrum of the operator $\mathscr{A}(\alpha_{o})$:

$\sigma(\mathscr{A}(\alpha_{o})):=\left\\{1+\frac{m^{2}(\mu-\alpha_{o}-1)}{j^{2}+m^{2}}:j=0,1,2,\dots,\;\mu\in\sigma(A)\right\\},$

which implies that

$\Lambda=\left\\{\alpha_{j,\mu}:=\frac{j^{2}+m^{2}\mu}{m^{2}}:j=0,1,2,\dots,\;\mu\in\sigma(A)\right\\}.$

###  3.1 Bifurcation in System (28) without Symmetries

Assume that $\Gamma=\\{e\\}$, i.e. $G:=D_{m}\times{\mathbb{Z}}_{2}$, and that
$\alpha_{j,\mu}\not=\alpha_{j^{\prime},\mu^{\prime}}$ for
$(j,\mu)\not=(j^{\prime},\mu^{\prime})$. Let us put all the elements of
$\Lambda$ in increasing order, i.e.
$\dots<\alpha_{j_{k},\mu_{k}}<\alpha_{j_{k+1},\mu_{k+1}}<\dots$. Then for
every $\alpha_{j_{o},\mu_{o}}\in\Lambda$ we have

$\omega_{G}(\alpha_{j_{o},\mu_{o}})=\prod_{\alpha_{j_{k},\mu_{k}}<\alpha_{j_{o},\mu_{o}}}(\deg_{\mathcal{W}_{i(j_{k})}^{-}})^{m(\mu_{k})}\Big{(}(G)-(\deg_{\mathcal{W}_{i(j_{o})}^{-}})^{m(\mu_{o})}\Big{)}.$
(33)

######  Theorem 3.1.

Suppose $A:V\rightarrow V$ and $f:\mathbb{R}\times V\rightarrow V$ satisfies
the assumptions ($A_{1}$)—($A_{3}$) and ($B_{4}$) and let $\Lambda$ be the
critical set for (31). Assume that for all $\alpha_{j,\mu}$,
$\alpha_{j^{\prime},\mu^{\prime}}\in\Lambda$ we have
$\alpha_{j,\mu}\not=\alpha_{j^{\prime},\mu^{\prime}}$ if
$(j,\mu)\not=(j^{\prime},\mu^{\prime})$. Then for every
$\alpha_{j_{o},\mu_{o}}\in\Lambda$ such that $m(\mu_{o})$ is odd we have
$\omega(\alpha_{j_{o},\mu_{o}})\not=0$, i.e. the point
$(\alpha_{j_{o},\mu_{o}},0)$ is a bifurcation point of non-trivial ${2\pi
m}$-periodic solutions for (28).

###### Proof.

Notice that under the assumption that $m(\mu_{o})$ is odd, we have

$(G)-(\deg_{\mathcal{W}_{i(j_{o})}^{-}})^{m(\mu_{o})}=(G)-(\deg_{\mathcal{W}_{i(j_{o})}^{-}})\not=0.$

Since the product
$\displaystyle\prod_{\alpha_{j_{k},\mu_{k}}<\alpha_{j_{o},\mu_{o}}}(\deg_{\mathcal{W}_{i(j_{k})}^{-}})^{m(\mu_{k})}$
is an invertible element in $A(G)$, it follows from formula (33) that
$\omega_{G}(\alpha_{j_{o},\mu_{o}})$ is non-zero. Therefore, by Theorem B.1,
the point $(\alpha_{j_{o},\mu_{o}},0)$ is a bifurcation point of non-trivial
${2\pi m}$-periodic solutions for (28). $\square$

Consequently, we obtain the following:

######  Theorem 3.2.

Suppose $A:V\rightarrow V$ and $f:\mathbb{R}\times V\rightarrow V$ satisfies
the assumptions ($A_{1}$)—($A_{3}$) and ($B_{4}$). Assume that $k>0$ is odd
and $\sigma(A)$ consists of exactly $k$ different eigenvalues such that for
all $(j,\mu)\not=(j^{\prime},\mu^{\prime})$ the critical points
$\alpha_{j,\mu}$, $\alpha_{j^{\prime},\mu^{\prime}}\in\Lambda$ are also
different. Suppose
$m=2^{\varepsilon_{0}}p_{1}^{\varepsilon_{1}}p_{2}^{\varepsilon_{2}}\dots
p_{s}^{\varepsilon_{s}}$, where $\varepsilon_{0}\geq 0$, $\varepsilon_{l}>0$
and $p_{l}$ are the prime numbers such that $2\leq p_{1}<p_{2}<\dots<p_{s}$.
For $l=1,2,\dots,s$ put $m_{l}:=\frac{m}{p_{l}}$. Then

  * (a)

if $p_{l}>2$ and $\rho_{l}$ is odd, then the system (31) admits an unbounded
branch of ${2\pi m}$-periodic solutions with symmetries at least
$(D_{m_{l}}^{z})$, $m_{l}:=\frac{m}{p_{l}}$, and

  * (b)

if $p_{1}=2$, $\varepsilon_{1}=1$, and $\rho_{1}$ is odd, then the system (31)
admits an unbounded branch of ${2\pi m}$-periodic solutions with symmetries at
least $(D_{m}^{d})$, $(D_{m}^{\hat{d}})$, and

  * (c)

if $p_{1}=2$ and $\varepsilon_{1}>1$, then and $\rho_{l}$ is odd, then the
system (31) admits an unbounded branch of ${2\pi m}$-periodic solutions with
symmetries at least $(D_{\frac{m}{2}}^{d})$,
$(\widetilde{D}_{\frac{m}{2}}^{d})$.

###  3.2 Bifurcation in System (28) with Additional Symmetries $\Gamma$

In this section we assume that $k=3$,
$f:\mathbb{R}\times\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ satisfies
assumptions ($A_{1}$)—($A_{3}$) and ($B_{4}$)—($B_{5}$) with $\Gamma=D_{3}$
and

$A=\frac{1}{4}\left[\begin{array}[]{ccc}-4&-2&-2\\\ -2&-4&-2\\\
-2&-2&-4\end{array}\right].$ (34)

Then we have

$\sigma(A)=\\{\mu_{0}=-2,\mu_{1}=-\frac{1}{2}\\}.$

#### Case $m=3$:

In this case $G=D_{3}\times D_{3}\times{\mathbb{Z}}_{2}$ and we have

$\sigma(\mathscr{A}(\alpha))=\left\\{\lambda_{j,l}:=1+\frac{9(\mu_{l}-\alpha-1)}{j^{2}+9}:l=0,1,\;j=0,1,2,\dots\right\\}$

Then

$\displaystyle\Lambda=$
$\displaystyle\Big{\\{}\alpha_{0,0}=-2,\;\alpha_{0,1}=-1/2,\;\alpha_{1,0}=-\frac{17}{9},\;\alpha_{1,1}=-\frac{7}{18},\dots,$
$\displaystyle\alpha_{j,0}=\frac{j^{2}-18}{9},\;\alpha_{j,1}=\frac{j^{2}-9/2}{9},\;\dots\Big{\\}},$

and each of the critical values $\alpha_{j,i}$ is isotypicly simple, i.e. the
eigenspace $E(\lambda_{j,i})$ is an irreducible $G$-representation
$\mathcal{V}_{i(j),l}^{-}$. Maximal orbit types in
$\mathbb{E}\setminus\\{0\\}$ are listed in (27). Using the same GAP code as in
Example (for $D_{3}$) in subsection 2.7, we can compute the exact bifurcation
invariants. Indeed, we have the following critical points from $\Lambda$

$\alpha_{0,0}<\alpha_{1,0}<\alpha_{2,0}<\alpha_{0,1}<\alpha_{1,1}<\alpha_{2,1}<\alpha_{3,0}<\alpha_{3,1}<\alpha_{4,0}<\alpha_{4,1}<\alpha_{5,1}<\dots$

Thus

$\displaystyle\omega(\alpha_{0,0})$
$\displaystyle=(G)-\deg_{\mathcal{V}_{0,0}^{-}}=(H_{67})$
$\displaystyle\omega(\alpha_{1,0})$
$\displaystyle=\deg_{\mathcal{V}_{0,0}^{-}}\circ((G)-\deg_{\mathcal{V}_{1,0}^{-}})=-(H_{43})+(H_{51})$
$\displaystyle\omega(\alpha_{2,0})$
$\displaystyle=\deg_{\mathcal{V}_{0,0}^{-}}\circ\deg_{\mathcal{V}_{1,0}^{-}}\circ((G)-\deg_{\mathcal{V}_{1,0}^{-}})=(H_{43})-(H_{51})$
$\displaystyle\omega(\alpha_{0,1})$
$\displaystyle=\deg_{\mathcal{V}_{0,0}^{-}}\circ((G)-\deg_{\mathcal{V}_{0,0}^{-}}\circ\deg_{\mathcal{V}_{2,0}^{-}}=(H_{63})+(H_{67}))$
$\displaystyle\omega(\alpha_{1,1})$
$\displaystyle=\deg_{\mathcal{V}_{2,0}^{-}}\circ((G)-\deg_{\mathcal{V}_{1,1}^{-}})=(H_{1})-2(H_{3})+(H_{4})-(H_{5})-(H_{8})+(H_{13})$
$\displaystyle+(H_{14})-(H_{16})+(H_{18})-(H_{28})+(H_{31})$
$\displaystyle\omega(\alpha_{4,0})$
$\displaystyle=\deg_{\mathcal{V}_{2,0}^{-}}\circ\deg_{\mathcal{V}_{1,1}^{-}}\circ((G)-\deg_{\mathcal{V}_{1,0}^{-}})=(H_{3})-(H_{4})-(H_{7})$
$\displaystyle+(H_{8})+2(H_{16})-2(H_{18})+(H_{43})-(H_{51})$
$\displaystyle\omega(\alpha_{2,1})$
$\displaystyle=\deg_{\mathcal{V}_{2,0}^{-}}\circ\deg_{\mathcal{V}_{1,1}^{-}}\circ\deg_{\mathcal{V}_{1,0}^{-}}\circ((G)-\deg_{\mathcal{V}_{1,1}^{-}})=-(H_{1})+(H_{3})+(H_{5})$
$\displaystyle+(H_{7})-(H_{13})-(H_{14})-(H_{16})+(H_{18})+(H_{28})-(H_{31})$

##  Appendix A APPENDIX: $G$-Equivariant Brouwer Degree

#### Equivariant Notation:

For a subgroup $H$ of $G$, i.e. $H\leq G$, we denote the normalizer of $H$ in
$G$ by $N(H)$ and the Weyl group of $H$ by $W(H)=N(H)/H$. The symbol $(H)$
stands for the conjugacy class of $H$ in $G$. We put $\Phi(G):=\\{(H):H\leq
G\\}$, i.e. $\Phi(G)$ is the set of conjugacy classes of subgroups in $G$.
$\Phi(G)$ has a natural partial order defined by $(H)\leq(K)$ iff $\exists
g\in G\;\;gHg^{-1}\leq K$. For two subgroups $L$, $H\leq G$, we denote by
$n(L,H)$ the number of different subgroups $H^{\prime}$ conjugate to $H$ such
that $L\leq H^{\prime}$.

For a $G$-space $X$ and $x\in X$, we denote the isotropy group of $x$ by
$G_{x}:=\\{g\in G:gx=x\\}$, the orbit of $x$ by $G(x):=\\{gx:g\in G\\}$, and
we call $(G_{x})$ the orbit type of $x\in X$. For a subgroup $H\leq G$ the
subspace $X^{H}:=\\{x\in X:G_{x}\geq H\\}$ is called the $H$-fixed-point
subspace of $X$. Clearly, $W(H)$ acts on $X^{H}$.

Consider a complete list of all irreducible $G$-representations
$\mathcal{V}_{i}$, $i=0,$ $1,$ $\ldots$, $r$. Such list for concrete group $G$
can be established by using GAP. Let $V$ be a finite-dimensional $G$\-
representation and (without loss of generality) we may assume that $V$ is an
orthogonal representation. Then $V$ decomposes into a direct sum

$V=V_{0}\oplus V_{1}\oplus\dots\oplus V_{r},$ (35)

where each component $V_{i}$ is modeled on the irreducible $G$-representation
$\mathcal{V}_{i}$, $i=0,1,2,\dots,r$, that is $V_{i}$ contains all the
irreducible subrepresentations of $V$ equivalent to $\mathcal{V}_{i}$. The
decomposition (35) is called $G$-isotypic decomposition of  $V$. Denote the
$\mathbb{R}$-algebra (resp. group) of all $G$-equivariant linear (resp.
invertible) operators on $V$ by $\text{L}^{G}(V)$ (resp. $\text{GL}^{G}(V)$) .
Clearly, the isotypic decomposition (35) induces the following direct sum
decomposition of $\text{GL}^{G}(V):$

$\text{GL}^{G}(V)=\bigoplus_{i=0}^{r}\text{GL}^{G}(V_{i}),$ (36)

where for every isotypic component $V_{i}$

$\text{GL}^{G}(V_{i})\simeq\text{GL}(m_{i},\mathbb{F}),\quad
m_{i}=\text{{dim\,}}V_{i}/\text{{dim\,}}\mathcal{V}_{i}$

and depending on the type of the irreducible representation $\mathcal{V}_{i},$
$\mathbb{F}$ ($=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$) is a finite-
dimensional division algebra $\text{L}^{G}(\mathcal{V}_{i})$.

###  A.1 Burnside Ring

We assume that $G$ is a finite group. Denote the free abelian group generated
by $(H)\in\Phi(G)$ by $A(G):={\mathbb{Z}}[\Phi(G)]$, i.e., an element $a\in
A(G)$ can be written as a sum

$\displaystyle a=n_{1}(H_{1})+\dots+n_{m}(H_{m}),$

where $n_{i}\in\mathbb{Z}$ and $(H_{i})\in\Phi(G)$. There is a natural
multiplication operation $\circ:A(G)\times A(G)\to A(G)$ which is defined on
generators $(H)$, $(K)\in\Phi(G)$ by

$(H)\cdot(K)=\sum_{(L)\in\Phi(G)}m_{L}\,(L),$ (37)

where the integer $m_{L}$ represents the number of $(L)$-orbits contained in
the space $G/H\times G/K$. The numbers $m_{L}$ can be easily computed from the
following recurrence formula

$m_{L}=\frac{n(L,H)|W(H)|n(L,K)|W(K)|-\sum_{(\widetilde{L})>(L)}m_{\widetilde{L}}\,n(L,\widetilde{L})\,|W(\widetilde{L})|}{|W(L)|}.$
(38)

Together with the multiplication ‘$\circ$’, $A(G)$ becomes a ring with the
unity $(G)$, which is called the Burnside ring of $G$. For more details see
[2]

###  A.2 Axioms of Brouwer $G$-Equivariant Degree

Consider an orthogonal $G$-representation $V$, a continuous $G$-map $f:V\to
V$, and an open bounded $G$-invariant set $\Omega\subset V$ such that for all
$x\in\partial\Omega$, we have $f(x)\neq 0$. Then $f$ is called
$\Omega$-admissible and $(f,\Omega)$ is called a $G$-admissible pair (in $V$).
The set of all possible $G$-pairs will be denoted by $\mathcal{M}^{G}$.

The following result (cf [2]) can be considered as an axiomatic definition of
the $G$-equivariant Brouwer degree:

######  Theorem A.1.

There exists a unique map $G\mbox{\rm-}\deg:\mathcal{M}^{G}\to A(G)$, which
assigns to every admissible $G$-pair $(f,\Omega)$ an element
$G\text{{-deg}}(f,\Omega)\in A(G)$

$G\mbox{\rm-}\deg(f,\Omega)=\sum_{(H)}{n_{H}(H)}=n_{H_{1}}(H_{1})+\dots+n_{H_{m}}(H_{m}),$
(39)

satisfying the following properties:

  * •

(Existence) If $G\mbox{\rm-}\deg(f,\Omega)\neq 0$, i.e., $n_{H_{i}}\neq 0$ for
some $i$ in (39), then there exists $x\in\Omega$ such that $f(x)=0$ and
$(G_{x})\geq(H_{i})$.

  * •

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

$\displaystyle
G\mbox{\rm-}\deg(f,\Omega)=G\mbox{\rm-}\deg(f,\Omega_{1})+G\mbox{\rm-}\deg(f,\Omega_{2}).$

  * •

(Homotopy) If $h:[0,1]\times V\to V$ is an $\Omega$-admissible $G$-homotopy,
then

$\displaystyle G\mbox{\rm-}\deg(h_{t},\Omega)=\mathrm{constant}.$

  * •

(Normalization) Let $\Omega$ be a $G$-invariant open bounded neighborhood of
$0$ in $V$. Then

$\displaystyle G\mbox{\rm-}\deg(\operatorname{Id},\Omega)=(G).$

  * •

(Multiplicativity) For any
$(f_{1},\Omega_{1}),(f_{2},\Omega_{2})\in\mathcal{M}^{G}$,

$\displaystyle G\mbox{\rm-}\deg(f_{1}\times
f_{2},\Omega_{1}\times\Omega_{2})=G\mbox{\rm-}\deg(f_{1},\Omega_{1})\circ
G\mbox{\rm-}\deg(f_{2},\Omega_{2}),$

where the multiplication ‘$\circ$’ is taken in the Burnside ring $A(G)$.

  * •

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

$\displaystyle
G\mbox{\rm-}\deg(f\times\operatorname{Id}_{W},\Omega\times\mathscr{B})=G\mbox{\rm-}\deg(f,\Omega).$

  * •

(Recurrence Formula) For an admissible $G$-pair $(f,\Omega)$, the $G$-degree
(39) can be computed using the following recurrence formula

$n_{H}=\frac{\deg(f^{H},\Omega^{H})-\sum_{(K)>(H)}{n_{K}\,n(H,K)\,\left|W(K)\right|}}{\left|W(H)\right|},$
(40)

where $\left|X\right|$ stands for the number of elements in the set $X$ and
$\deg(f^{H},\Omega^{H})$ is the Brouwer degree of the map $f^{H}:=f|_{V^{H}}$
on the set $\Omega^{H}\subset V^{H}$.

The $G\text{{-deg}}(f,\Omega)$ is called the $G$-equivariant Brouwer degree
(or simply $G$-degree) of $f$ in $\Omega$.

###  A.3 Computation of $\Gamma$-Equivariant Degree

Put $B(V):=\left\\{x\in V:\left|x\right|<1\right\\}$. For each irreducible
$G$-representation $\mathcal{V}_{i}$, $i=0,1,2,\dots$, we define

$\displaystyle\deg_{\mathcal{V}_{i}}:=G\mbox{\rm-}\deg(-\operatorname{Id},B(\mathcal{V}_{i})),$

and will call $\deg_{\mathcal{V}_{i}}$ the _basic degree_.

Consider a $G$-equivariant linear isomorphism $T:V\to V$ and assume that $V$
has a $G$-isotypic decomposition (35). Then by the Multiplicativity property,
we have

$G\mbox{\rm-}\deg(T,B(V))=\prod_{i=0}^{r}G\mbox{\rm-}\deg(T_{i},B(V_{i}))=\prod_{i=0}^{r}\prod_{\mu\in\sigma_{-}(T)}\left(\deg_{\mathcal{V}_{i}}\right)^{m_{i}(\mu)}$
(41)

where $T_{i}=T|_{V_{i}}$ and
$\sigma_{-}(T):=\left\\{\mu\in\sigma(T):\mu<0\right\\}$ denotes the real
negative spectrum of $T$.

Notice that the basic degrees can be effectively computed from (40), i.e.
formula

$\displaystyle\deg_{\mathcal{V}_{i}}=\sum_{(H)}n_{H}(H),$

where

$n_{H}=\frac{(-1)^{\dim\mathcal{V}_{i}^{H}}-\sum_{H<K}{n_{K}\,n(H,K)\,\left|W(K)\right|}}{\left|W(H)\right|}.$
(42)

##  Appendix B Local and Global Bifurcation Problems

Assume that $a<b$, $V$ is an orthogonal $G$-representation and $\Omega\subset
V$ is an open bounded $G$-invariant subset. Let $f:\mathbb{R}\oplus V\to V$ be
a continuous $G$-equivariant map such that $(f_{a},\Omega)$,
$(f_{b},\Omega)\in\mathcal{M}^{G}(V)$, where $f_{t}(x):=f(t,x)$,
$t\in\mathbb{R}$, $x\in V$. Then a continuous $G$-invariant function
$\varphi:\mathbb{R}\oplus V\to\mathbb{R}$ will be called
$\Omega$-complementing function for $f_{t}$ at $t=a$, $b$, if

$\begin{cases}\varphi(t,x)<0&\;\text{ if }\;t=a,\,b,\;x\in\Omega\\\
\varphi(t,x)>0&\;\text{ if }\;t=(a,b),\;x\in\partial\Omega.\end{cases}$ (43)

In such a case we define the map $F_{\varphi}:\mathbb{R}\oplus
V\to\mathbb{R}\oplus V$ by

$F_{\varphi}(t,x)=(\varphi(t,x),f(t,x)),\quad t\in\mathbb{R},\;x\in V.$ (44)

The following result is well-known in non-equivariant case.

######  Theorem B.1.

Suppose that $f:\mathbb{R}\oplus V\to V$ is a $G$-equivariant map such that
$(f_{a},\Omega)$, $(f_{b},\Omega)\in\mathcal{M}^{G}(V)$, and
$\varphi:\mathbb{R}\oplus V\to V$ is an $\Omega$-complementing function for
$f_{t}$ at $t=a$, $b$. Then
$(F_{\varphi},(a,b)\times\Omega)\in\mathcal{M}^{G}$, the $G$-equivariant
degree $G\text{{-deg}}(F_{\varphi},(a,b)\times\Omega)$ doesn’t depend on the
choice of the $\Omega$-complementing function $\varphi$ and we have

$G\text{\rm-deg}(F_{\varphi},(a,b)\times\Omega)=G\text{\rm-
deg}(f_{a},\Omega)-G\text{\rm-deg}(f_{b},\Omega).$ (45)

Let $f:\mathbb{R}\times V\to V$ be a continuous $G$-equivariant map such that
for all $t\in\mathbb{R}$, $f(t,0)=0$. We are interested in solutions of the
equation

$f(t,x)=0.$ (46)

Clearly any pair $(t,0)$ satisfies (46), thus we will call them a trivial
solutions to (46). All other solutions to (46) will be called nontrivial. We
denote the set of all nontrivial solutions to (46) by $\mathscr{S}$, i.e.

$\mathscr{S}:=\\{(t,x)\in\mathbb{R}\times\mathbb{R}^{n}\oplus
V:f(t,x)=0\;\text{ and }x\not=0\\}.$

######  Definition B.2.

Let $\mathcal{C}\subset\mathscr{S}$ and $\mathscr{U}\subset\mathbb{R}\times
V\to V$ be a $G$-invariant open subset. The set
$\mathcal{C}\subset\mathscr{S}$ is called a branch of nontrivial solutions to
(46) in $\mathscr{U}$ if $\overline{\mathcal{C}}$ is a connected component of
$\overline{\mathscr{S}}\cap\overline{\mathscr{U}}$. Moreover, we say that the
branch $\mathcal{C}$ bifurcates from a trivial solution $(t_{o},0)$ if
$(t_{o},0)\in\overline{\mathcal{C}}$.

######  Theorem B.3.

Let $V$ be an orthogonal $G$-representation and $\Omega\subset V$ be a
$G$-invariant bounded open set. Assume that $f:[0,1]\times V\to V$ is a
continuous $G$-equivariant map such that for every $t\in[0,1]$,
$(f_{t},\Omega)$ is an admissible $G$-pair and $\deg_{G}(f_{0},\Omega)\not=0$.
Then there exists a compact connected set $K_{o}\subset
f^{-1}(0)\cap[0,1]\times\Omega$ such that

$K_{o}\cap(\\{0\\}\times\Omega)\not=\emptyset\not=K_{o}\cap(\\{1\\}\times\Omega).$

Let us discuss the global bifurcation problem for (46). Under the assumption
that the derivative $D_{x}f(t,0)$ exists for all $t\in\mathbb{R}$ and the map
$t\mapsto D_{x}f(t,0)$ is continuous, one can easily show that if $(t_{o},0)$
is a bifurcation point for (46), then $D_{x}f(t_{o},0)$ is not an isomorphism.
Denote the set of all $t\in\mathbb{R}$ such that $(t,0)$ is a bifurcation
point of (46) by $\mathscr{B}$ and put

$\Lambda:=\\{t\in\mathbb{R}:\det D_{x}f(t,0)=0\\}.$ (47)

$\Lambda$ is called the set of critical points for (46) and we clearly have
$\mathscr{B}\subset\Lambda$.

The following result is called Rabinowitz’s Alternative:

######  Theorem B.4.

Suppose that $f:\mathbb{R}\oplus V\to V$ is a continuous $G$-equivariant map
such that $f(t,0)=0$ for all $t\in\mathbb{R}$ and $D_{x}f(t,0)$ exists and is
continuous with respect to $t\in\mathbb{R}$. We also assume that the set of
critical points $\Lambda$ for (46) (given by (47)) is discrete, and consider
an open bounded $G$-invariant set $\mathcal{U}\subset\mathbb{R}\oplus V$ such
that $(t_{o},0)\in\mathcal{U}$ for some $t_{o}\in\Lambda$. For a connected
component $\mathscr{C}$ of the set
$\overline{\mathcal{U}}\cap\overline{\mathscr{S}}$ such that
$(t_{o},0)\in\mathscr{C}$ we have the following alternative.

  * (i)

either $\mathscr{C}\cap\partial U\not=\emptyset$,

  * (ii)

or
$\mathscr{C}\cap(\Lambda\times\\{0\\})=\\{(t_{1},0),(t_{2},0),\dots,(t_{n},0)\\}$
for some $n\in{\mathbb{N}}$ (here $t_{j}\not=t_{k}$ for $j\not=k$) and

$\sum_{k=1}^{n}\omega_{G}(t_{k})=0.$ (48)

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