3. (c)

In Theorem 3.4, words
“Assumption 1.1, let $u_{0}:[0,\tau]\to{\mathbb{R}}^{2n}$ be a solution of
(1.2)”
is replaced by
“Assumption 1.2, let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ be a solution of
(1.7)”,
“to $u_{0}$ in $C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$” by “to $v_{0}$ on any
compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology”,
and “(1.2) in $\Lambda\times C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$” by
“(1.7)”.

As done in Remark 3.5, Theorem 4.1 has a more general version, that is, the
trivial branch $\\{(\lambda,v_{0})\,|\,\lambda\in\Lambda\\}$ of the set of
solutions of (1.7) may be replaced by
$\\{(\lambda,v_{\lambda})\,|\,\lambda\in\Lambda\\}$, where $v_{\lambda}$
solves (1.7) and $\Lambda\times{\mathbb{R}}\ni(\lambda,t)\mapsto
v_{\lambda}(t)\in\mathbb{R}^{2n}$ is continuous.

Corresponding to Corollary 3.6, we have also

######  Corollary 4.2.

For $M\in{\rm Sp}(2n,\mathbb{R})$, let continuous functions
$H_{0},\hat{H}:{\mathbb{R}}\times{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ satisfy

$\displaystyle
H_{0}(t+\tau,Mx)=H_{0}(t,x)\quad\forall(t,x)\in\Lambda\times{\mathbb{R}}\times\mathbb{R}^{2n},$
$\displaystyle\hat{H}(t+\tau,Mx)=\hat{H}(t,x)\quad\forall(t,x)\in\Lambda\times{\mathbb{R}}\times\mathbb{R}^{2n},$

and be such that all
$H_{0}(t,\cdot),\hat{H}(t,\cdot):{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ are
$C^{2}$ and all their partial derivatives depend continuously on
$(t,z)\in{\mathbb{R}}\times\mathbb{R}^{2n}$. Let
$v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ satisfy

$\dot{v}(t)=J\nabla(H_{0})_{t}(v(t))$ and $v(t+\tau)=Mv(t)\;\forall t$.

Suppose that $d\hat{H}_{t}(v_{0}(t))=0$ for all $t\in{\mathbb{R}}$ and that

either $\hat{H}_{t}^{\prime\prime}(v_{0}(t))>0\;\forall t$ or
$\hat{H}_{t}^{\prime\prime}(v_{0}(t))<0\;\forall t$.

Take $H(\lambda,t,z)=H_{0}(t,z)+\lambda\hat{H}(t,z)$ in (1.6). Then
$\Sigma:=\\{\lambda\in{\mathbb{R}}\,|\,\nu_{\tau,M}(\gamma_{\lambda})>0\\}$ is
a discrete set in ${\mathbb{R}}$; $(\mu,v_{0})$ with $\mu\in{\mathbb{R}}$ is a
bifurcation point for (1.7) if and only if $\nu_{\tau,M}(\gamma_{\mu})>0$; for
each $\mu\in\Sigma$ and $\rho>0$ small enough (3.5) and (3.8) hold in two
cases, and one of the following alternatives occurs:

  1. (i)

Equation (1.7) with $\lambda=\mu$ has a sequence of solutions, $v_{n}\neq
v_{0}$, $n=1,2,\cdots$, which converges to $v_{0}$ on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (ii)

For every $\lambda\in{\mathbb{R}}\setminus\\{\mu\\}$ near $\mu$ there is a
solution $v_{\lambda}\neq v_{0}$ of (1.7) with parameter value $\lambda$,
which converges to $v_{0}$ on any compact interval $I\subset{\mathbb{R}}$ in
$C^{1}$-topology.

  3. (iii)

There is an one-sided neighborhood $\mathcal{N}$ of $\mu$ in ${\mathbb{R}}$
such that for any $\lambda\in\mathcal{N}\setminus\\{\mu\\}$, (1.7) with
parameter value $\lambda$ has at least two distinct solutions,
$v_{\lambda}^{1}\neq v_{0}$ and $v_{\lambda}^{2}\neq v_{0}$, which converge to
$v_{0}$ on any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as
$\lambda\to\mu$.

Moreover, if $v_{0}=0$ and all $H_{0}(t,\cdot),\hat{H}(t,\cdot)$ are even,
then either (i) holds or the following occurs:

  1. (iv)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq d$, where $d$ is the number of maximal linearly independent
solutions of

$\dot{v}(t)=J(H^{\prime\prime}_{0t}(0)+\mu\hat{H}^{\prime\prime}_{t}(0))v(t)\quad\hbox{and}\quad
v(t+\tau)=Mv(t)\;\forall t\in{\mathbb{R}},$ (4.1)

and for $\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), (1.7) with parameter value
$\lambda$ has at least $n^{-}$ (resp. $n^{+}$) distinct pairs of nontrivial
solutions, $\\{v_{\lambda}^{i},-v_{\lambda}^{i}\\}$, $i=1,\cdots,n^{-}$ (resp.
$n^{+}$), which converge to zero on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

##  5 Bifurcations for solutions of the Hamiltonian system (1.11)

###  5.1 Statements of main results

######  Theorem 5.1.

Under Assumption 1.3, suppose also that the orthogonal symplectic matrix
$M\in{\rm Sp}(2n,\mathbb{R})$ satisfies $M^{l}=I_{2n}$ for some integer $l\geq
1$. Let

$v_{0}\in{\rm
Ker}(M-I_{2n})\cap(\cap_{\lambda\in\Lambda}\\{dH_{\lambda}=0\\})\subset{\mathbb{R}}^{2n}$
(5.1)

and let $\gamma_{\lambda}(t)=\exp(tJH^{\prime\prime}_{\lambda}(v_{0}))$, i.e.,
the fundamental matrix solution of
$\dot{v}(t)=JH^{\prime\prime}_{\lambda}(v_{0})v(t)$. Suppose that for some
$\tau>0$ and $\mu\in\Lambda$

  1. (a)

the following problem

$\dot{v}(t)=JH^{\prime\prime}_{\mu}(v_{0})v(t)$ and $v(t+\tau)=Mv(t)\;\forall
t$ (5.2)

has no nonzero constant solutions, i.e., ${\rm Ker}(M-I_{2n})\cap{\rm
Ker}(H^{\prime\prime}_{\mu}(v_{0}))=\\{0\\}$;

  2. (b)

$\dim{\rm Ker}(\gamma_{\mu}(\tau)-M)\neq 0$, $\dim{\rm
Ker}(\gamma_{\lambda}(\tau)-M)=0$ for each
$\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$, and
$i_{\tau,M}(\gamma_{\lambda})$ take, respectively, values
$i_{\tau,M}(\gamma_{\mu})$ and
$i_{\tau,M}(\gamma_{\mu})+\nu_{\tau,M}(\gamma_{\mu})$ as $\lambda\in\Lambda$
varies in two deleted half neighborhoods of $\mu$.

Then one of the following alternatives occurs for (1.11):

  1. (i)

Equation (1.11) with $\lambda=\mu$ has a sequence of ${\mathbb{R}}$-distinct
solutions, $v_{n}$, $n=1,2,\cdots$, which are ${\mathbb{R}}$-distinct with
$v_{0}$ and converges to $v_{0}$ on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (ii)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\nu_{\tau,M}(\gamma_{\mu})/2$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), (1.11) with parameter value
$\lambda$ has at least $n^{-}$ (resp. $n^{+}$) ${\mathbb{R}}$-distinct
solutions, $v_{\lambda}^{i}$, $i=1,\cdots,n^{-}$ (resp. $n^{+}$), which are
${\mathbb{R}}$-distinct with $v_{0}$ and converge to $v_{0}$ on any compact
interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

In particular, $(\mu,v_{0})=(\mu,{\mathbb{R}}\cdot v_{0})$ is a bifurcation
${\mathbb{R}}$-orbit of (1.11).

By the proof of Corollary 3.6 we may derive from Theorem 5.1:

######  Corollary 5.2.

Let $M\in{\rm Sp}(2n,\mathbb{R})$ be as in Theorem 5.1, $v_{0}\in{\rm
Ker}(M-I_{2n})$, and $H_{0},\hat{H}:{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ be
$M$-invariant $C^{2}$-functions. Suppose that
$dH_{0}(v_{0})=d\hat{H}(v_{0})=0$ and that either
$\hat{H}^{\prime\prime}(v_{0})>0$ or $\hat{H}^{\prime\prime}(v_{0})<0$. Then
for $\gamma_{\lambda}(t)=\exp(tJH_{0}^{\prime\prime}(v_{0})+\lambda
tJ\hat{H}^{\prime\prime}(v_{0}))$ and any $\tau>0$,
$\Sigma:=\\{\lambda\in{\mathbb{R}}\,|\,\nu_{\tau,M}(\gamma_{\lambda})>0\\}$ is
a discrete set in ${\mathbb{R}}$. Moreover, for each $\mu\in\Sigma$, if (5.2)
with $H(\mu,x)=H_{0}(x)+\mu\hat{H}(x)$ has no nonzero constant solution, then
the conclusions of Theorem 5.1 holds for (1.11) with
$H(\lambda,x)=H_{0}(x)+\lambda\hat{H}(x)$.

As the case of periodic solutions with prescribed periods near an equilibruim
(cf. [39, §6.6] and [2, Chapter 9]) we can derive from Theorem 5.1:

######  Corollary 5.3.

Let $M\in{\rm Sp}(2n,\mathbb{R})$ be as in Theorem 5.1, $v_{0}\in{\rm
Ker}(M-I_{2n})$, and $H:{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ be a $M$-invariant
$C^{2}$-function satisfying $dH(v_{0})=0$. Suppose that either
${H}^{\prime\prime}(v_{0})>0$ or ${H}^{\prime\prime}(v_{0})<0$. Then
$\Gamma(H,v_{0}):=\\{t\in\mathbb{R}\setminus\\{0\\}\,|\,\dim{\rm
Ker}(\exp(tJ{H}^{\prime\prime}(v_{0}))-M)>0\\}$ is a discrete set.

If ${H}^{\prime\prime}(v_{0})>0$ and
$\Gamma(H,v_{0})\cap(0,\infty)\neq\emptyset$, then for each positive
$\mu\in\Gamma(H,v_{0})$ one of the following alternatives occurs:

  1. (i)

The problem

$\dot{v}(t)=J\nabla H(v(t))\quad\hbox{and}\quad v(t+\mu)=Mv(t),\;\forall
t\in{\mathbb{R}}$ (5.3)

has a sequence of ${\mathbb{R}}$-distinct solutions, $v_{k}$, $k=1,2,\cdots$,
which are ${\mathbb{R}}$-distinct with $v_{0}$ and converges to $v_{0}$ on any
compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (ii)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\mathbb{R}$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\frac{1}{2}\dim{\rm
Ker}(\exp(\mu{H}^{\prime\prime}(v_{0}))-M)$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$) the problem

$\dot{v}(t)=J\nabla H(v(t))\quad\hbox{and}\quad v(t+\lambda)=Mv(t),\;\forall
t\in{\mathbb{R}}$ (5.4)

has at least $n^{-}$ (resp. $n^{+}$) ${\mathbb{R}}$-distinct solutions
solutions, $v_{\lambda}^{i}$, $i=1,\cdots,n^{-}$ (resp. $n^{+}$), which are
${\mathbb{R}}$-distinct with $v_{0}$ and converge to $v_{0}$ on any compact
interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

If ${H}^{\prime\prime}(v_{0})<0$ and
$\Gamma(H,v_{0})\cap(-\infty,0)\neq\emptyset$, for each negative
$\mu\in\Gamma(H,v_{0})$ one of the above (i) and (ii) with this $\mu$ holds.

###### Proof.

The first claim follows from the first one of Proposition 2.5. Suppose
${H}^{\prime\prime}(v_{0})>0$ and
$\Gamma(H,v_{0})\cap(0,\infty)\neq\emptyset$. We may assume
$\Gamma(H,v_{0})\cap(0,\infty)=\\{\lambda_{1}<\cdots<\lambda_{k}<\cdots<\lambda_{\kappa}\\}$,
where $\kappa\in\mathbb{N}\cup\\{\infty\\}$. Let
$\gamma_{\lambda}(t)=\exp(\lambda tJ{H}^{\prime\prime}(v_{0}))$. If
$\mu\in\Gamma(H,v_{0})\cap(0,\infty)$, it follows from (2.37) that
$i_{1,M}(\gamma_{\lambda})=i_{1,M}(\gamma_{\mu})$ for $\lambda\leq\mu$ close
to $\mu$ and that
$i_{1,M}(\gamma_{\lambda})=i_{1,M}(\gamma_{\mu})+\nu_{1,M}(\gamma_{\mu})$ for
$\lambda>\mu$ close to $\mu$. By Corollary 5.2 with $H_{0}=0$ and $\hat{H}=H$
we obtain one of the following alternatives:

  1. (a)

The problem

$\dot{v}(t)=\mu J\nabla H(v(t))\quad\hbox{and}\quad v(t+1)=Mv(t),\;\forall
t\in{\mathbb{R}}$ (5.5)

has a sequence of ${\mathbb{R}}$-distinct solutions, $v_{k}$, $k=1,2,\cdots$,
which are ${\mathbb{R}}$-distinct with $v_{0}$ and converges to $v_{0}$ on any
compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (b)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\mathbb{R}$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\nu_{1,M}(\gamma_{\mu})/2$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), the problem

$\dot{v}(t)=\lambda J\nabla H(v(t))\quad\hbox{and}\quad v(t+1)=Mv(t),\;\forall
t\in{\mathbb{R}}$ (5.6)

has at least $n^{-}$ (resp. $n^{+}$) ${\mathbb{R}}$-distinct solutions
solutions, $v_{\lambda}^{i}$, $i=1,\cdots,n^{-}$ (resp. $n^{+}$), which are
${\mathbb{R}}$-distinct with $v_{0}$ and converge to $v_{0}$ on any compact
interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

Clearly, (5.5) and (5.6) are equivalent to (5.3) and (5.4), respectively.

If ${H}^{\prime\prime}(v_{0})<0$ and
$\Gamma(H,v_{0})\cap(-\infty,0)\neq\emptyset$, we may assume
$\Gamma(H,v_{0})\cap(-\infty,0)=\\{\lambda_{-\kappa}\cdots<\lambda_{-k}<\cdots<\lambda_{-1}\\}$,
where $\kappa\in\mathbb{N}\cup\\{\infty\\}$. For each
$\mu\in\Gamma(H,v_{0})\cap(-\infty,0)$, by (2.38) we deduce that
$i_{1,M}(\gamma_{\lambda})=i_{1,M}(\gamma_{\mu})$ for $\lambda\geq\mu$ close
to $\mu$ and that
$i_{1,M}(\gamma_{\lambda})=i_{1,M}(\gamma_{\mu})+\nu_{1,M}(\gamma_{\mu})$ for
$\lambda<\mu$ close to $\mu$. As above the desired claim follows from
Corollary 5.2 with $H_{0}=0$ and $\hat{H}=H$. ∎

######  Example 5.4.

Let $H:{\mathbb{R}}^{2n}\to{\mathbb{R}}$ be a $C^{2}$ function satisfying
$dH(0)=0$ and $H^{\prime\prime}(0)>0$. By Williamson theorem there exists a
symplectic matrix $S\in{\rm Sp}(2n,{\mathbb{R}})$ such that

$SH^{\prime\prime}(0)S^{T}=D_{H^{\prime\prime}(0)}={\rm
diag}(\varrho_{1},\cdots,\varrho_{n},\varrho_{1},\cdots,\varrho_{n}),$

where $0<\varrho_{j}\leq\varrho_{k}$ for $j\leq k$, and each $\varrho_{j}$ is
such that $\det(JH^{\prime\prime}(0)\pm i\varrho_{j}I_{2n})=0$ for
$j=1,\cdots,n$. Consider the Hamiltonian system

$\dot{x}=J\nabla H(x)=X_{H}(x).$ (5.7)

By the transformation $x=S^{T}y$, this system is transformed into the system

$\dot{y}=J\nabla K(y)=X_{K}(y),$ (5.8)

where $K(y)=H(S^{T}y)$. A direct computation shows that
$K^{\prime\prime}(y)=SH^{\prime\prime}(S^{T}y)S^{T}$ and thus
$K^{\prime\prime}(0)=SH^{\prime\prime}(0)S^{T}=D_{H^{\prime\prime}(0)}$. Let
$\gamma^{H}$ and $\gamma^{K}$ be the fundamental matrix solutions of
linearizations of (5.7) and (5.8) at zero solutions, respectively. It is easy
to show that $\gamma^{K}(t)=(S^{T})^{-1}\gamma^{H}(t)S^{T}$ for all $t$, and
so

$i_{\tau}(\gamma^{H})=i_{\tau}(\gamma^{K})\quad\hbox{and}\quad\nu_{\tau}(\gamma^{H})=\nu_{\tau}(\gamma^{K})$

by (2.17). From [28, page 146, Theorem 6], (2.21) and [17, page 31, Lemma 13]
we can derive

$i_{\tau}(\gamma^{K})=n+2\sum^{n}_{i=1}E\left[\frac{\varrho_{i}\tau}{2\pi}\right]=n-2\sum^{n}_{i=1}\left[1-\frac{\varrho_{i}\tau}{2\pi}\right],$

where $E[\alpha]$ is the largest integer less than real $\alpha$. This result
is not needed below. Since

$\gamma^{K}(t)=\exp(tJK^{\prime\prime}(0))=\left(\begin{array}[]{cc}B_{1}(t)&B_{2}(t)\\\
B_{3}(t)&B_{4}(t)\end{array}\right)$

where $B_{1}(t)=B_{4}(t)={\rm
diag}(\cos(\varrho_{1}t),\cdots,\cos(\varrho_{n}t))$ and

$B_{2}(t)=-{\rm
diag}(\sin(\varrho_{1}t),\cdots,\sin(\varrho_{n}t))=-B_{3}(t),$

it is easily proved that
$\nu_{\tau}(\gamma^{H})=2\sharp\\{i\in\\{1,\cdots,n\\}\,|\,\varrho_{i}\tau=0\;{\rm
mod}\;2\pi\\}$ and

$\displaystyle\Gamma(H,0)\cap(0,\infty)$ $\displaystyle=$
$\displaystyle\\{\lambda>0\,|\,\dim{\rm Ker}(\exp(\lambda
JH^{\prime\prime}(0))-I_{2n})>0\\}$ $\displaystyle=$
$\displaystyle\cup^{n}_{i=1}\\{\lambda>0\,|\,\lambda\varrho_{i}=0\;{\rm
mod}\;2\pi\\}.$

For each $\mu\in\Gamma(H,0)\cap(0,\infty)$, one of (i) and (ii) in Corollary
5.3 with $v_{0}=0$ and $M=I_{2n}$ holds.

Note that every solution $u_{0}$ of (1.11) is $C^{3}$. As a strengthen version
of Theorem 4.1(a) we have:

######  Theorem 5.5.

Under Assumption 1.3, let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ is a
nonconstant solution of (1.11) for each $\lambda\in\Lambda$. Suppose that for
some $\mu\in\Lambda$ there exists a sequence $(\lambda_{k})\subset\Lambda$
converging to $\mu$ such that (1.11) with $\lambda=\lambda_{k}$ has solutions
$v_{k}$, $k=1,2,\cdots$, which are ${\mathbb{R}}$-distinct each other and
satisfies $v_{k}|_{[0,\tau]}\to v_{0}|_{[0,\tau]}$ in
$W^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$. Then the following problem

$\dot{v}(t)=JH^{\prime\prime}_{\mu}(v_{0}(t))u(t)$ and
$v(t+\tau)=Mv(t)\;\forall t$ (5.9)

has at least two linearly independent nontrivial solutions, i.e., $\dim{\rm
Ker}(\gamma_{\mu}(\tau)-M)\geq 2$, where $\gamma_{\lambda}(t)$ is the
fundamental matrix solution of
$\dot{v}(t)=JH^{\prime\prime}_{\lambda}(v_{0}(t))v(t)$.

Corresponding to (b) and (c) of Theorem 4.1 and Corollary 4.2 respectively, we
have also:

######  Theorem 5.6.

Under Assumption 1.3, let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ is a
nonconstant solution of (1.11) for each $\lambda\in\Lambda$, and let
$\gamma_{\lambda}(t)$ be the fundamental matrix solution of
$\dot{v}(t)=JH^{\prime\prime}_{\lambda}(v_{0}(t))u(t)$. Suppose for some
$\mu\in\Lambda$ that there exist two points in any neighborhood of
$\mu\in\Lambda$, $\lambda_{-}<\mu<\lambda_{+}$, such that
$i_{\tau,M}(\gamma_{\lambda_{-}})\neq i_{\tau,M}(\gamma_{\lambda_{+}})$ and

$\dim{\rm Ker}(\gamma_{\lambda_{-}}(\tau)-M)=1=\dim{\rm
Ker}(\gamma_{\lambda_{+}}(\tau)-M).$

Then there exists a sequence $(\lambda_{k})\subset\Lambda$ converging to $\mu$
and solutions $v_{k}$ of (1.11) with $\lambda=\lambda_{k}$, $k=1,2,\cdots$,
such that any two of these $v_{k}$ are ${\mathbb{R}}$-distinct and that
$(v_{k})$ converges to $v_{0}$ on any compact interval $I\subset{\mathbb{R}}$
in $C^{1}$-topology as $k\to\infty$.

######  Theorem 5.7.

Under Assumption 1.3, let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ is a
nonconstant solution of (1.11) for each $\lambda\in\Lambda$, and let
$\gamma_{\lambda}(t)$ be the fundamental matrix solution of
$\dot{v}(t)=JH^{\prime\prime}_{\lambda}(v_{0}(t))v(t)$. Suppose for some
$\mu\in\Lambda$ that $\dim{\rm Ker}(\gamma_{\mu}(\tau)-M)>1$, $\dim{\rm
Ker}(\gamma_{\lambda}(\tau)-M)=1$ for each
$\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$, and that
$i_{\tau,M}(\gamma_{\lambda})$ take values $i_{\tau,M}(\gamma_{\mu})$ and
$i_{\tau,M}(\gamma_{\mu})+\nu_{\tau,M}(\gamma_{\mu})-1$ as $\lambda\in\Lambda$
varies in two deleted half neighborhoods of $\mu$. Then one of the following
alternatives occurs:

  1. (i)

Equation (1.11) with $\lambda=\mu$ has a sequence of ${\mathbb{R}}$-distinct
solutions, $v_{k}$, $k=1,2,\cdots$, such that $(v_{k})$ converges to $v_{0}$
on any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as
$k\to\infty$.

  2. (ii)

For every $\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$ there is a solution
$v_{\lambda}$ of (1.11) with parameter value $\lambda$, which is
${\mathbb{R}}$-distinct with $v_{0}$ and converges to $v_{0}$ on any compact
interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

  3. (iii)

There is an one-sided neighborhood $\Lambda^{0}$ of $\mu$ such that for any
$\lambda\in\Lambda^{0}\setminus\\{\mu\\}$, (1.11) with parameter value
$\lambda$ has at least two ${\mathbb{R}}$-distinct solutions,
$v_{\lambda}^{1}$ and $v_{\lambda}^{2}$, which are ${\mathbb{R}}$-distinct
with $v_{0}$ and converge to $v_{0}$ on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

######  Remark 5.8.

For a real $\tau>0$ and an orthogonal symplectic matrix $M\in{\rm
Sp}(2n,\mathbb{R})$, let $H_{0},\hat{H}:{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ be
$M$-invariant $C^{2}$ functions and let
$v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ be a nonconstant solution of

$\dot{v}(t)=J\nabla H_{0}(v(t))$ and $v(t+\tau)=Mv(t)\;\forall t$.

There are not corresponding results with Corollaries 5.2, 5.3. That is, there
is not a $M$-invariant $C^{2}$ function
$\hat{H}:{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ satisfying such that
$\nabla\hat{H}(v_{0}(t))=0$ for all $t\in{\mathbb{R}}$ and that

either $\hat{H}^{\prime\prime}(v_{0}(t))>0\;\forall t$ or
$\hat{H}^{\prime\prime}(v_{0}(t))<0\;\forall t$.

This is implied in [15, Lemma 3.18] (i.e., the inequality in (2.31)). In fact,
by contradiction suppose that such a $\hat{H}$ exists. Take
$H(\lambda,z)=H_{0}(z)+\lambda\hat{H}(z)$ in (1.11). As in the proof of
Corollary 3.6 there exist only finite many points $\lambda\in[\mu-1,\mu+1]$
such that $\nu_{\tau,M}(B_{\lambda}|A)=\nu_{\tau,M}(B_{\lambda})\neq 0$ with
$B_{\lambda}(t)=(H_{0})^{\prime\prime}(u_{0}(t))+\lambda(\hat{H}_{0})^{\prime\prime}(u_{0}(t))$
and $A(t)=KI_{2n}$. On the one hand, for any $\theta\in{\mathbb{R}}$,
$v_{0}(\theta+\cdot)$ satisfies (1.11). It follows that
$\ddot{v}_{0}(t)=J\nabla H^{\prime\prime}_{\lambda}(v_{0}(t))\dot{v}_{0}(t)$
and $\dot{v}_{0}(t+\tau)=M\dot{v}_{0}(t)$ for all $t\in{\mathbb{R}}$. Since
$\dot{v}_{0}\neq 0$, we get $\nu_{\tau,M}(\gamma_{\lambda})\geq 1$ for all
$\lambda\in\mathbb{R}$. This is a contradiction.

Consider the Hilbert spaces

$\displaystyle\textsf{L}_{\tau,M}$ $\displaystyle=$ $\displaystyle\\{u\in
L^{2}_{\rm loc}({\mathbb{R}},{\mathbb{R}}^{2n})\,|\,u(t+\tau)=Mu(t)\;\hbox{for
a.e.}\;t\in{\mathbb{R}}\\},$ $\displaystyle\textsf{W}_{\tau,M}$
$\displaystyle=$ $\displaystyle\\{u\in W^{1,2}_{\rm
loc}({\mathbb{R}},{\mathbb{R}}^{2n})\,|\,u(t+\tau)=Mu(t)\;\hbox{for
all}\;t\in{\mathbb{R}}\\}$

equipped with inner products (2.1) and (2.2) respectively. There exists a
natural (continuous) $\mathbb{R}$-action on $\textsf{L}_{\tau,M}$ and
$\textsf{W}_{\tau,M}$:

$(\theta,u)\mapsto\theta\cdot u$ (5.10)

given by $(\theta\cdot u)(t):=u(\theta+t)$. Clearly, this action is a Banach
isometric if $M$ is also an orthogonal matrix, and becomes a $S^{1}$-action
with $S^{1}=\mathbb{R}/(l\tau\mathbb{Z})$ if $M^{l}=I_{2n}$ for some integer
$l\geq 1$.

Let $\mathcal{S}$ be one of the spaces
$C^{i}_{M}([0,\tau];{\mathbb{R}}^{2n})$,
${W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$ and
$L^{2}([0,\tau];{\mathbb{R}}^{2n})$. For each $u\in\mathcal{S}$ we define
$u_{M}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ by (1.9). Clearly, if $u\in
C^{i}_{M}([0,\tau];{\mathbb{R}}^{2n})$, then $u_{M}\in
C^{i}({\mathbb{R}},{\mathbb{R}}^{2n})$ and satisfies
$u_{M}(t+\tau)=Mu_{M}(t)\;\forall t\in{\mathbb{R}}$. When $u\in
L^{2}([0,\tau];{\mathbb{R}}^{2n})$ we have $u_{M}(t+\tau)=Mu_{M}(t)\;\forall
t\in{\mathbb{R}}\setminus(\tau{\mathbb{Z}})$, and so $u_{M}(t+\tau)=M_{M}u(t)$
for a.e. $t\in{\mathbb{R}}$. Thus there exists a natural (continuous)
$\mathbb{R}$-action on $\mathcal{S}$,

$\mathbb{R}\times\mathcal{S}\to\mathcal{S},\,(\theta,u)\mapsto(\theta\ast
u)(\cdot):=u_{M}(\theta+\cdot)|_{[0,\tau]}.$ (5.11)

Clearly, the actions in (5.10) and (5.11) are Banach isometries if $M$ is also
an orthogonal matrix, and become $S^{1}$-actions with
$S^{1}=\mathbb{R}/(l\tau\mathbb{Z})$ if $M^{l}=I_{2n}$ for some integer $l\geq
1$. Moreover Hilbert space isomorphisms
${W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})\to\textsf{W}_{\tau,M}$ and
$L^{2}([0,\tau];{\mathbb{R}}^{2n})\to\textsf{L}_{\tau,M}$ given by $u\to
u_{M}$ are equivariant with respect to these two actions.

Because of these, variational spaces will be chosen as
${W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$ and
$L^{2}([0,\tau];{\mathbb{R}}^{2n})$ in the following proofs though natural
choices should be $\textsf{M}_{\tau,M}$ and $\textsf{L}_{\tau,M}$.

Let ${{\mathbb{R}}}_{u}$ denote the isotropy group at a point
$u\in\mathcal{S}$ of the action in (5.11). Then it is equal to either
${\mathbb{R}}$, or $\\{0\\}$ or $\varrho{\mathbb{Z}}$ for some real
$\varrho>0$. These mean that $u$ is constant in the first case, injective in
the second case, and $\varrho=k\tau$ for some $k\in{\mathbb{N}}$ in the final
case.

######  Proposition 5.9.

Let $H:\mathbb{R}^{2n}\to{\mathbb{R}}$ be $C^{1,1}$ and $M$-invariant. Suppose
that $u:[0,\tau]\to{\mathbb{R}}^{2n}$ satisfies $\dot{u}=J\nabla H(u)$ and
$u(\tau)=Mu(0)$. Then $u$ is $C^{2}$ and

  1. (i)

if ${\mathbb{R}}_{u}={\mathbb{R}}$, $u$ is equal to a constant vector $u_{0}$
in ${\rm Ker}(M-I_{2n})$;

  2. (ii)

if ${\mathbb{R}}_{u}=\\{0\\}$, $\theta\mapsto\theta\ast u$ is an one-to-one
$C^{1}$ immersion from $\mathbb{R}$ to
$C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$;

  3. (iii)

if ${\mathbb{R}}_{u}=\varrho{\mathbb{Z}}$ with $\varrho>0$, the
${\mathbb{R}}$-orbit $\\{\theta\ast u\,|\,\theta\in{\mathbb{R}}\\}$ is a
$C^{1}$ embedded $S^{1}={\mathbb{R}}/(\varrho{\mathbb{Z}})$ in
$C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$.

Moreover, if $M=I_{2n}$ then the case (ii) does not occur.

(ii) comes from the existence and uniqueness theorem for solutions of
$\dot{u}=J\nabla H(u)$ with initial values. (iii) uses the elementary
topological fact that a continuous one-to-one map from a compact space to a
Hausdorff space is a homeomorphism.

###  5.2 Proofs of Theorem 5.1 and Corollary 5.3

###### Proof of Theorem 5.1.

The vector $v_{0}$ in (5.1) is also understood as a constant value map from
${\mathbb{R}}$ to ${\mathbb{R}}^{n}$. When viewed as a constant value map from
$[0,\tau]$ to ${\mathbb{R}}^{n}$ we write it as $u_{0}$, i.e.,
$u_{0}=v_{0}|_{[0,\tau]}$. Then $Mu_{0}=u_{0}$ and $u_{0}$ is a fixed point
for the ${\mathbb{R}}$-action in (5.11). As before we assume that
$\Lambda\subset{\mathbb{R}}$ contains $0$ and $\mu=0$. Following the notations
in proofs of Theorems 3.1, 3.2, 3.4, as in the first paragraph of Section 3.2
we can modify $H$ as $\tilde{H}(\lambda,t,z)=\chi(z)H(\lambda,t,z)$ so that it
is also $M$-invariant. (Indeed, $\hat{U}_{0}:=\sum^{l}_{j=0}M^{j}U_{0}$ is
$M$-invariant and $Cl(\hat{U}_{0})$ may be contained in $U$ by shrinking
$U_{0}$. We only need to replace $\chi$ by $\hat{\chi}$, where
$\hat{\chi}(z):=\chi(\sum^{l}_{j=0}M^{j}z)$.) Then functionals
$\mathcal{L}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$ satisfy
conditions in [33, Theorem 4.6] with $H=L^{2}([0,\tau];{\mathbb{R}}^{2n})$ and
$X=C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})$, and are invariant for
$S^{1}$-action with $S^{1}=\mathbb{R}/(lT\mathbb{Z})$ given by (5.11).

In order to use [33, Theorem 5.12] we also need prove that the fixed point set
of the induced $S^{1}$-action on $H_{0}^{0}$ is $\\{0\\}$. Recall that
$w_{0}=-\Lambda_{M,\tau,K}u_{0}=-J\dot{u}_{0}-Ku_{0}=-Ku_{0}$. Hence
$Mw_{0}=w_{0}$. Let $\xi\in H^{0}_{0}$ be a fixed point for the above
$S^{1}$-action. Then it is constant and satisfies $M\xi=\xi$. Let
$B_{0}(w_{0})$ be given by (3.22). Note that $\xi\in H$ sits in $H_{0}^{0}$ if
and only if
$B_{0}(w_{0})\xi=A_{M,\tau,K}\xi+(H_{K}^{\ast})^{\prime\prime}_{z}(0;w_{0})\xi=0$,
that is,

$\xi=-\Lambda_{M,\tau,K}((H_{K}^{\ast})^{\prime\prime}_{z}(0;w_{0})\xi)=-K(H_{K}^{\ast})^{\prime\prime}_{z}(0;w_{0})\xi=-K[H^{\prime\prime}_{z}(0,u_{0})-KI_{2n}]^{-1}\xi$

because $w_{0}=-J\dot{u}_{0}-Ku_{0}=\nabla H(0,u_{0})-Ku_{0}=\nabla
H_{K}(0,u_{0})$, or equivalently $H^{\prime\prime}_{z}(0,u_{0})\xi=0$. By the
assumption (a), i.e., ${\rm Ker}(M-I_{2n})\cap{\rm
Ker}(H_{z}^{\prime\prime}(0,u_{0}))=\\{0\\}$, we deduce $\xi=0$.

Combing these with the assumption (b) we see that functionals
$\mathcal{L}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$ satisfy
conditions in [33, Theorem 5.12] with $H=L^{2}([0,\tau];{\mathbb{R}}^{2n})$
and $X=C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})$. Hence one of the following
alternatives occurs:

  1. (I)

$(0,w_{0})$ is not an isolated solution in $\\{0\\}\times
C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})$ of $\nabla_{w}\psi_{K}(0,w)=0$.

  2. (II)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$0$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\frac{1}{2}\dim H^{0}_{0}$, and that for
$\lambda\in\Lambda^{-}\setminus\\{0\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{0\\}$) the functional
$\psi_{K}(\lambda,\cdot)$ has at least $n^{-}$ (resp. $n^{+}$) distinct
critical $S^{1}$-orbits disjoint with $w_{0}$, which converge to $w_{0}$ in
$C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})$ as $\lambda\to 0$.

In case (I), we have a sequence $(w_{j})\subset
C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})\setminus\\{0\\}$ such that
$\|w_{j}-w_{0}\|_{C^{0}}\to 0$ as $j\to\infty$ and that
$\nabla_{w}\psi_{K}(0,w_{j})=0$ for each $j\in{\mathbb{N}}$. Because $w_{0}$
is a fixed point for the $S^{1}$-action, the $S^{1}$-orbits are compact and
different $S^{1}$-orbits are not intersecting, by passing to a subsequence we
can assume that any two of $w_{j}$, $j=0,1,\cdots$, do not belong the same
$S^{1}$-orbit. The same claim also holds for $u_{0}$ and
$u_{j}:=-(\Lambda_{M,\tau,K})^{-1}w_{j}\neq u_{0}$, $j=1,2,\cdots$. Moreover,
each $u_{j}$ satisfies

$\displaystyle\dot{u}(t)=J\nabla_{u}H(0,u(t))\;\forall
t\in[0,\tau]\quad\hbox{and}\quad u(\tau)=Mu(0)$

and therefore $u_{j}\to u_{0}$ in $C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$.
Since $H(0,\cdot)$ is $M$-invariant, each $u_{j}$ can be extended into a
solution $v_{j}:=(u_{j})_{M}$ of (5.2) via (1.9). Clearly, any two of $v_{j}$,
$j=0,1,\cdots$, are ${\mathbb{R}}$-distinct, and $(v_{j})$ converges $v_{0}$
on any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology.

In case (II), note firstly that $\dim H_{0}^{0}$ is equal to the number of
maximal linearly independent solutions of

$\displaystyle\dot{u}(t)=JH_{z}^{\prime\prime}(0,u_{0})u(t)\;\forall
t\in[0,\tau]\quad\hbox{and}\quad u(\tau)=Mu(0).$ (5.12)

Since $M^{T}H_{z}^{\prime\prime}(0,Mz)=H_{z}^{\prime\prime}(0,z)$, the
extension via (1.9) establishes an isomorphism between the solution space of
(5.12) and that of

$\displaystyle\dot{v}(t)=JH_{z}^{\prime\prime}(0,v_{0})u(t)\quad\hbox{and}\quad
v(t+\tau)=Mv(t),\;\forall t\in{\mathbb{R}}.$

Hence $\dim H_{0}^{0}$ is equal to the number of maximal linearly independent
solutions of (5.2).

For $\lambda\in\Lambda^{-}\setminus\\{0\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{0\\}$) let $S^{1}\cdot w_{\lambda}^{i}$,
$i=1,\cdots,n^{-}$ (resp. $n^{+}$) be distinct critical $S^{1}$-orbits of
$\psi_{K}(\lambda,\cdot)$ disjoint with $w_{0}$, which converge to $w_{0}$ in
$C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})$ as $\lambda\to 0$. Then

$\hbox{$u^{i}_{\lambda}:=-\Lambda_{M,\tau,K}^{-1}(w_{\lambda}^{i})\notin
S^{1}\cdot u_{0}=\\{u_{0}\\}$,\quad$i=1,\cdots,n^{-}$ (resp. $n^{+}$)},$

satisfy

$\displaystyle\dot{u}(t)=J\nabla_{u}H(\lambda,u(t))\;\forall
t\in[0,\tau]\quad\hbox{and}\quad u(\tau)=Mu(0)$

and therefore $S^{1}\cdot u^{i}_{\lambda}\to u_{0}$ in
$C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$ as $\lambda\to 0$. The extensions
$v^{i}_{\lambda}:=(u^{i}_{\lambda})_{M}$ of $u^{i}_{\lambda}$ via (1.9),
$i=1,\cdots,n^{-}$ (resp. $n^{+}$), are ${\mathbb{R}}$-distinct solutions of
(1.11), and also ${\mathbb{R}}$-distinct with $v_{0}$. This completes the
proof. ∎

###  5.3 Proofs of Theorems 5.5, 5.6, 5.7

Since the $\mathbb{R}$-action in (5.11) is not $C^{1}$, we cannot directly use
the theorems in [33, §5.2]. But we can use the proof ideas therein to complete
our proofs.

If $u\in C^{k}_{M}([0,\tau];{\mathbb{R}}^{2n})$ with $k\geq 1$ is nonconstant,
since $u_{M}=u_{M}^{(0)},\cdots,u_{M}^{(k)}$ are uniformly continuous over any
compact interval of ${\mathbb{R}}$ it is easy to prove that the map
$f:{\mathbb{R}}\to L^{2}([0,\tau];{\mathbb{R}}^{2n})$ given by
$f(\theta)=\theta\ast u$ is $C^{k}$, and
$f^{(i)}(\theta)=u^{(i)}_{M}(\theta+\cdot)|_{[0,\tau]}$, $i=1,\cdots,k$.

Suppose that $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ is a nonconstant
solution of (1.11) for each $\lambda\in\Lambda$. Then $v_{0}$ is $C^{3}$. As
in Section 4 we write $u_{0}:=v_{0}|_{[0,\tau]}$. It is nonconstant. Because
$H(\lambda,\cdot)$ is $C^{2}$ in ${\mathbb{R}}^{2n}$, by the existence and
uniqueness theorem for solutions of $\dot{v}=JH_{z}^{\prime}(\lambda,v)$ with
initial values we see:
$\bullet$ either the map $\theta\mapsto\theta\ast u_{0}$ is an one-to-one
$C^{2}$ immersion from ${\mathbb{R}}$ to
$W^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$, (thus the restriction of it to any
compact submanifold of ${\mathbb{R}}$ is a $C^{2}$-embedding by an elementary
result in general topology,) and $\\{\theta\ast
u_{0}\,|\,\theta\in{\mathbb{R}}\\}$ is a $C^{2}$-immersed submanifold in
${W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$ without self-intersection points;
$\bullet$ or $\\{\theta\ast u_{0}\,|\,\theta\in{\mathbb{R}}\\}$ is a
$C^{2}$-embedded circle.

Let $w_{0}:=-\Lambda_{M,\tau,K}(u_{0})$. Though it is only $C^{2}$, since
$\Lambda_{M,\tau,K}$ is a Banach isomorphism from
${W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$ to
$L^{2}([0,\tau];{\mathbb{R}}^{2n})$, and $\Lambda_{M,\tau,K}(\theta\ast
u)=\theta\ast(\Lambda_{M,\tau,K}(u))$ for any
$(\theta,u)\in{\mathbb{R}}\times{W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$,

$\mathcal{O}:=\\{\theta\ast w_{0}\,|\,\theta\in{\mathbb{R}}\\}$ (5.13)

is either a $C^{2}$-immersed submanifold in
$L^{2}([0,\tau];{\mathbb{R}}^{2n})$ without self-intersection points or a
$C^{2}$-embedded circle in $L^{2}([0,\tau];{\mathbb{R}}^{2n})$.

Let $\\{\dot{w}_{0}\\}^{\bot}$ denote the orthogonal complementary of
$\dot{w}_{0}$ in $L^{2}([0,\tau];{\mathbb{R}}^{2n})$. For $\varepsilon>0$,
consider the following $C^{\infty}$ submanifold of
$L^{2}([0,\tau];{\mathbb{R}}^{2n})$ containing $w_{0}$,

$\displaystyle\mathcal{S}_{w_{0}}:$ $\displaystyle=$
$\displaystyle\\{w_{0}+u\,|\,u\in
L^{2}([0,\tau];{\mathbb{R}}^{2n}),\;\|u\|_{2}<\varepsilon,\;(\dot{w}_{0},u)_{2}=0\\}$
$\displaystyle=$ $\displaystyle
w_{0}+\left\\{u\in\\{\dot{w}_{0}\\}^{\bot}\,|\,\|u\|_{2}<\varepsilon\right\\}.$

Clearly, it is invariant under the isotropy group ${{\mathbb{R}}}_{w_{0}}$ of
the ${\mathbb{R}}$-action at a point $w_{0}\in
L^{2}([0,\tau];{\mathbb{R}}^{2n})$. Note that ${{\mathbb{R}}}_{w_{0}}$ is
equal to the isotropy group ${{\mathbb{R}}}_{u_{0}}$ of the
${\mathbb{R}}$-action at a point
$u_{0}\in{W}^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$. The tangent space
$T_{w_{0}}\mathcal{S}_{w_{0}}=\\{\dot{w}_{0}\\}^{\bot}$ and
$L^{2}([0,\tau];{\mathbb{R}}^{2n})=T_{w_{0}}\mathcal{S}_{w_{0}}\oplus({\mathbb{R}}\ast\dot{w}_{0})$.

######  Proposition 5.10.

${\mathbb{R}}\ast\mathcal{S}_{w_{0}}$ is a neighborhood of the orbit
$\mathcal{O}={\mathbb{R}}\ast w_{0}$.

Because the ${\mathbb{R}}$-action is not $C^{1}$ it has no sense to require
that ${\mathbb{R}}\ast w$ has a transversal intersection with
$\mathcal{S}_{w_{0}}$ for the orbit of every $w\in
L^{2}([0,\tau];{\mathbb{R}}^{2n})$ sufficiently close to $w_{0}$. But
Proposition 5.10 can be proved with the following:

######  Proposition 5.11 ([5, Proposition 3.5]).

Let $M$ be a finite-dimensional manifold, $N$ a (possibly infinite
dimensional) Banach manifold, $Q\subset N$ a Banach submanifold, and $A$ a
topological space. Assume that $\chi:A\times M\to N$ is a continuous function
such that there exists $a_{0}\in A$ and $m_{0}\in M$ with:

  1. (a)

$\chi(a_{0},m_{0})\in Q$; (b) $\chi(a_{0},\cdot):M\to N$ is of class $C^{1}$;

  2. (c)

$\partial_{2}\chi(a_{0},m_{0})(T_{m_{0}}M)+T_{\chi(a_{0},m_{0})}Q=T_{\chi(a_{0},m_{0})}N$.

Then, for $a\in A$ near $a_{0}$, $\chi(a,M)\cap Q\neq\emptyset$.

###### Proof of Proposition 5.10.

Applying Proposition 5.11 to $A=N=L^{2}([0,\tau];{\mathbb{R}}^{2n})$,
$M={\mathbb{R}}$, $Q=\mathcal{S}_{w_{0}}$, $a_{0}=w_{0}$, $m_{0}=0$ and

$\chi:A\times M\to N:(w,\theta)\mapsto\theta\ast w,$

we get that $({\mathbb{R}}\ast w)\cap\mathcal{S}_{w_{0}}\neq\emptyset$ if
$w\in L^{2}([0,\tau];{\mathbb{R}}^{2n})$ is close to $w_{0}$. ∎

As before we assume that $\Lambda\subset{\mathbb{R}}$ contains $0$ and
$\mu=0$, and follow the notations in proof of Theorem 3.1. But we can require
that all modified $H(\lambda,\cdot)$ as done in the proof of Theorem 3.1 are
also $M$-invariant. (See the proof of Theorem 5.1 in Section 4.) Then we have
a family of $C^{1}$ and twice Gâteaux-differentiable functionals on
$H=L^{2}([0,\tau];{\mathbb{R}}^{2n})$,

$\psi_{K}(\lambda,u)=\int^{\tau}_{0}\left[\frac{1}{2}(u(t),(A_{M,\tau,K}u)(t))_{\mathbb{R}^{2n}}+H_{K}^{\ast}(\lambda,t;u(t))\right]dt,$
(5.14)

each of which is also invariant for the ${\mathbb{R}}$-action given by (5.11).
Moreover, we have

$\displaystyle\nabla_{v}\psi_{K}(\lambda,v)=A_{M,\tau,K}v+\nabla(H_{K}^{\ast})_{\lambda}(v(\cdot)),$
(5.15) $\displaystyle
B_{\lambda}(v):=D_{v}\nabla_{v}\psi_{K}(\lambda,v)=A_{M,\tau,K}+(H_{K}^{\ast})^{\prime\prime}_{\lambda}(v(\cdot))\in\mathscr{L}_{s}(L^{2}([0,\tau];{\mathbb{R}}^{2n})).$
(5.16)

Fortunately, we can directly prove the following result, which naturally holds
if the ${\mathbb{R}}$-action in (5.11) is $C^{1}$.

######  Proposition 5.12.

$u\in\\{\dot{w}_{0}\\}^{\bot}$ near $0$ is a critical point of
$\mathcal{L}_{\lambda}(\cdot)=\psi_{K,\lambda}(w_{0}+\cdot)$ if and only if it
is a critical point of the restriction of $\mathcal{L}_{\lambda}$ to
$\\{\dot{w}_{0}\\}^{\bot}$.

###### Proof.

We only need to prove sufficiency. Since $u\in\\{\dot{w}_{0}\\}^{\bot}$ is a
critical point of the restriction of $\mathcal{L}_{\lambda}$ to
$\\{\dot{w}_{0}\\}^{\bot}$, $w_{0}+u$ is a critical point of
$\psi_{K,\lambda}|_{\mathcal{S}_{w_{0}}}$, i.e.,

$d\psi_{K,\lambda}(w_{0}+u)[\xi]=0\quad\forall\xi\in
T_{w}\mathcal{S}_{w_{0}}=\\{\dot{w}_{0}\\}^{\bot}.$ (5.17)

Step 1 (Prove that $u$ is $C^{1}$). Since

$d\psi_{K,\lambda}(w_{0}+u)[\xi]=(A_{M,\tau,K}(w_{0}+u)+\nabla(H_{K}^{\ast})_{\lambda}(w_{0}+u),\xi)_{2}=0$

for all $\xi\in\\{\dot{w}_{0}\\}^{\bot}$, we have $\mu(u)\in{\mathbb{R}}$ such
that

$\nabla\psi_{K,\lambda}(w_{0}+u)=A_{M,\tau,K}(w_{0}+u)+\nabla(H_{K}^{\ast})_{\lambda}(w_{0}+u)=\mu(u)\dot{w}_{0}.$
(5.18)

Note that $\dot{w}_{0}$ is $C^{1}$, $A_{M,\tau,K}(w_{0}+u)\in
W^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$, and the continuous map

$\Gamma:[0,\tau]\times\mathbb{R}^{2n}\to{\mathbb{R}}^{2n},\;(t,z)\mapsto\nabla(H_{K}^{\ast})_{\lambda}(z)-\mu(u)\dot{w}_{0}(t)+(A_{M,\tau,K}(w_{0}+u))(t)$

is $C^{1}$ in $z$. Applying the implicit function theorem to $\Gamma$ we
deduce that $w_{0}+u$ and therefore $u$ is $C^{0}$. By (2.6),
$A_{M,\tau,K}(w_{0}+u)\in C^{1}_{M}([0,\tau];{\mathbb{R}}^{2n})$ and hence

$t\mapsto\mu(u)\dot{w}_{0}(t)-(A_{M,\tau,K}(w_{0}+u))(t)$

is $C^{1}$. Using the implicit function theorem again we obtain that $u$ is
$C^{1}$.

Step 2 (Prove that $\|u\|_{2}\to 0$ implies $\|u\|_{C^{1}}\to 0$). Since
$\dot{w}_{0}\neq 0$, $\nabla\psi_{K,\lambda}(w_{0})=0$ and
$\nabla\psi_{K,\lambda}$ is $C^{0}$, (5.18) implies that $\mu(u)\to 0$ as
$\|u\|_{2}\to 0$. By Proposition 2.16 or its proof, as $\|u\|_{2}\to 0$ we
have
$\|\nabla(H_{K}^{\ast})_{\lambda}(w_{0}+u)-\mu(u)\dot{w}_{0}-\nabla(H_{K}^{\ast})_{\lambda}(w_{0})\|_{2}\to
0$ and therefore

$\|A_{M,\tau,K}(w_{0}+u)-A_{M,\tau,K}(w_{0})\|_{2}\to 0$ (5.19)

because
$A_{M,\tau,K}(w_{0}+u)=\nabla(H_{K}^{\ast})_{\lambda}(w_{0}+u)-\mu(u)\dot{w}_{0}$.
But $A_{M,\tau,K}$ is the composition of the Banach space isomorphism

$(\Lambda_{M,\tau,K})^{-1}:L^{2}([0,\tau];\mathbb{R}^{2n})\to{W}^{1,2}_{M,K}([0,\tau];{\mathbb{R}}^{2n})$

and the inclusion $\iota:{W}^{1,2}_{M,K}([0,\tau];{\mathbb{R}}^{2n})\to
L^{2}([0,\tau];\mathbb{R}^{2n})$. Then (5.19) implies that $\|u\|_{1,2}\to 0$
and hence $\|u\|_{C^{0}}\to 0$. It follows that

$\|A_{M,\tau,K}(w_{0}+u)-A_{M,\tau,K}(w_{0})\|_{C^{0}}=\|\nabla(H_{K}^{\ast})_{\lambda}(w_{0}+u)-\mu(u)\dot{w}_{0}-\nabla(H_{K}^{\ast})_{\lambda}(w_{0})\|_{C^{0}}\to
0$

and so $\|u\|_{C^{1}}\to 0$ by (2.6).

Step 3 (Complete the final proof). Let $v:=-(\Lambda_{M,\tau,K})^{-1}(u)$.
Then $v$ is $C^{2}$. If $\|u\|_{2}$ is small enough, so is $\|v\|_{1,2}$
(actually $\|v\|_{C^{2}}$ by (2.6)). Since ${u}_{0}$ is nonconstant, so is
${u}_{0}+v$ (if $\|u\|_{2}$ is small enough). It follows that the map
$\theta\mapsto\theta\ast(u_{0}+v)$ is a $C^{1}$ immersion from ${\mathbb{R}}$
to $W^{1,2}_{M}([0,\tau];{\mathbb{R}}^{2n})$. As arguments near (5.13) the map
$\theta\mapsto\theta\ast(w_{0}+u)$ is a $C^{1}$ immersion from ${\mathbb{R}}$
to $L^{2}([0,\tau];{\mathbb{R}}^{2n})$. The latter implies that
$\Delta:=[-\nu,\nu]\ast w$ with $w:=w_{0}+u$ is a $C^{1}$ embedded submanifold
for $\nu>0$ small enough. (Specially, $w$ is a nonconstant $C^{1}$ map.) Since
$T_{w}\Delta={\mathbb{R}}\dot{w}$ and
$L^{2}([0,\tau];{\mathbb{R}}^{2n})=({\mathbb{R}}\dot{w}_{0})\oplus
T_{w_{0}}\mathcal{S}_{w_{0}}$ we can obtain a direct sum decomposition

$L^{2}([0,\tau];{\mathbb{R}}^{2n})=T_{w}\Delta\dot{+}T_{w}\mathcal{S}_{w_{0}}$
(5.20)

if $\|u\|_{2}$ is small enough. Note that
$d\psi_{K,\lambda}(w)[\xi]=0\;\forall\xi\in T_{w}\Delta$, From this, (5.17)
and (5.20) it follows that $d\psi_{K,\lambda}(w)=0$. ∎

Consider the following Hilbert space, Banach space and functional

$\displaystyle H^{\bot}:=\\{\dot{w}_{0}\\}^{\bot},\quad X^{\bot}:=\\{u\in
C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})\,|\,(\dot{w}_{0},u)_{2}=0\\},$ (5.21)
$\displaystyle\mathcal{L}^{\bot}_{\lambda}:H^{\bot}\to{\mathbb{R}},\;u\mapsto\mathcal{L}_{\lambda}(u)=\psi_{K}(\lambda,w_{0}+u).$
(5.22)

Then these are invariant under the isotropy group $\mathbb{R}_{w_{0}}$, and
$d\mathcal{L}^{\bot}_{\lambda}(0)=0\;\forall\lambda$. Moreover, by shrinking
$\varepsilon>0$ if necessary, Proposition 5.12 shows that for $u\in
B_{H^{\bot}}(0,\varepsilon)$,

$d\mathcal{L}^{\bot}_{\lambda}(u)=0\;\quad\Longleftrightarrow\quad
d\mathcal{L}_{\lambda}(u)=d\psi_{K,\lambda}(w_{0}+u)=0.$ (5.23)

Denote by $\Pi:H\to H^{\bot}$ the orthogonal projection. Then
$\Pi(u)=u-\frac{(u,\dot{w}_{0})_{2}}{\|\dot{w}_{0}\|_{2}}\dot{w}_{0}$ for
$u\in H$, and

$\nabla\mathcal{L}^{\bot}_{\lambda}(u)=\nabla\psi_{K,\lambda}(w_{0}+u)-\frac{(\nabla\psi_{K,\lambda}(w_{0}+u),\dot{w}_{0})_{2}}{\|\dot{w}_{0}\|_{2}}\dot{w}_{0}\quad\forall
u\in H^{\bot},$ (5.24)

where
$\nabla\psi_{K,\lambda}(w_{0}+u)=A_{M,\tau,K}(w_{0}+u)+\nabla(H_{K}^{\ast})_{\lambda}(w_{0}(\cdot)+v(\cdot))$
by (3.21). Since $\dot{w}_{0}$ is $C^{1}$, it is easy to see that
$\nabla\mathcal{L}^{\bot}_{\lambda}(u)\in X^{\bot}$ for any $u\in X^{\bot}$,
and

$\mathfrak{A}_{\lambda}:X^{\bot}\to
X^{\bot},\;u\mapsto\nabla\mathcal{L}^{\bot}_{\lambda}(u)$ (5.25)

is $C^{1}$. From (3.21) and (3.22) we may deduce that
$\nabla\mathcal{L}^{\bot}_{\lambda}$ has a Gâteaux derivative at $u\in
H^{\bot}$, $\mathfrak{B}_{\lambda}(u)\in\mathscr{L}_{s}(H^{\bot})$, given by

$\mathfrak{B}_{\lambda}(u)v=B_{\lambda}(u+w_{0})v-\frac{(B_{\lambda}(u+w_{0})v,\dot{w}_{0})_{2}}{\|\dot{w}_{0}\|_{2}}\dot{w}_{0}\quad\forall
v\in H^{\bot},$ (5.26)

where
$B_{\lambda}(u+w_{0})=A_{M,\tau,K}+(H_{K}^{\ast})^{\prime\prime}_{\lambda}(u(\cdot)+w_{0}(\cdot))$
by (3.22). Since
$P_{\lambda}(v)=(H_{K}^{\ast})^{\prime\prime}_{\lambda}(v(\cdot))$ and
$Q_{\lambda}(v)=A_{M,\tau,K}$ are positive definite and compact self-adjoint
operators on $H$, respectively,

$\mathfrak{P}_{\lambda}(u):=\Pi\circ
P_{\lambda}(u+w_{0})|_{H^{\bot}}\quad\hbox{and}\quad\mathfrak{Q}_{\lambda}(u):=\Pi\circ
Q_{\lambda}(u+w_{0})|_{H^{\bot}}$ (5.27)

are such operators on $H^{\bot}$, respectively. Moreover, it is clear that
$\mathfrak{B}_{\lambda}(u)=\mathfrak{P}_{\lambda}(u)+\mathfrak{Q}_{\lambda}(u)$.
Note that ${\mathbb{R}}\dot{w}_{0}\subset{\rm Ker}(B_{\lambda}(w_{0}))$. By
(3.31) and (3.32) we get

$\displaystyle m^{-}(\mathcal{L}^{\bot}_{\lambda},0)$ $\displaystyle=$
$\displaystyle
m^{-}({\psi}_{K,\lambda},w_{0})=i_{\tau,M}(\gamma_{\lambda})-i_{\tau,M}(\gamma_{K})-\nu_{\tau,M}(\gamma_{K}),$
(5.28) $\displaystyle m^{0}(\mathcal{L}^{\bot}_{\lambda},0)$ $\displaystyle=$
$\displaystyle m^{0}({\psi}_{K,\lambda},w_{0})-1=\dim{\rm
Ker}(\gamma_{\lambda}(\tau)-M)-1.$ (5.29)

Since $w_{0}$ is $C^{2}$, by the proof of Theorem 3.2 it is easily seen that

$\\{u\in H^{\bot}\,|\,\mathfrak{B}_{\lambda}(0)u=su,\,s\leq 0\\}\subset
X^{\bot}\quad\hbox{and}\quad\\{u\in H^{\bot}\,|\,\mathfrak{B}_{\lambda}(0)u\in
X^{\bot}\\}\subset X^{\bot}.$ (5.30)

###### Proof of Theorem 5.5.

In the proof of Theorem 3.1 we have proved that
$\mathcal{F}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$ satisfies the
conditions of [33, Theorem 3.1] with $H=X=L^{2}([0,\tau];{\mathbb{R}}^{2n})$.
Combing the arguments therein with (5.24)-(5.27) it is easy to check that [33,
Theorem 3.1] is applicable to
$\mathcal{F}_{\lambda}=\mathcal{L}^{\bot}_{\lambda}$ near $0\in
H=X=\\{\dot{w}_{0}\\}^{\bot}$.

Let $(\lambda_{k},v_{k})$ be as in Theorem 5.5. Then $\lambda_{k}\to\mu=0$ and
$w_{k}:=-\Lambda_{M,\tau,K}(v_{k}|_{[0,\tau]})$ are critical points of
$\psi_{K,\lambda_{k}}$ on $L^{2}([0,\tau];{\mathbb{R}}^{2n})$ and
$\|w_{k}-w_{0}\|_{2}\to 0$ as $k\to\infty$. Since
${\mathbb{R}}\cdot\mathcal{S}_{w_{0}}$ is a neighborhood of the orbit
$\mathcal{O}={\mathbb{R}}\cdot w_{0}$ by Proposition 5.10, we have
$\theta_{k}\in{\mathbb{R}}$, $\theta_{k}\to 0$, such that
$u_{k}:=\theta_{k}\ast w_{k}\in\mathcal{S}_{w_{0}}$ for all $k$. Note that
$\|u_{k}-w_{0}\|_{2}\to 0$ and that each $u_{k}$ is a critical point of
$\psi_{K,\lambda_{k}}$ on $L^{2}([0,\tau];{\mathbb{R}}^{2n})$. By Proposition
5.12 each $u_{k}$ is also a critical point of the restriction of
$\mathcal{L}_{\lambda_{k}}$ to $\\{\dot{w}_{0}\\}^{\bot}$. Since all $v_{k}$
are ${\mathbb{R}}$-distinct, any two of all $w_{k}$ belong to different
${\mathbb{R}}$-orbits. It follows that all $u_{k}$ are distinct each other. By
[33, Theorem 3.1] we get $m^{0}(\mathcal{L}^{\bot}_{0},0)>0$. From (5.29) it
follows that $\dim{\rm Ker}(\gamma_{\lambda}(\tau)-M)>1$. ∎

###### Proof of Theorem 5.6.

Following the notations above, in the proof of Theorem 3.2 we have proved that
$\mathcal{L}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$ with
$\lambda\in\Lambda$ satisfy the conditions of [33, Theorem A.3] on
$H=L^{2}([0,\tau];{\mathbb{R}}^{2n})$ and
$X=C^{0}_{M}([0,\tau];{\mathbb{R}}^{2n})$. From these, (5.30) and [32, Theorem
6.4] (or its proof) we derive that $\mathcal{L}^{\bot}_{\lambda}$ with
$\lambda\in\Lambda$ satisfy the conditions of [33, Theorem A.3] on $H^{\bot}$
and $X^{\bot}$ in (5.21) and (5.22).

Recall that we have assumed $\mu=0$. By the assumptions there exist sequences
$(\lambda_{k}^{-})\subset[-\rho,0)$ and $(\lambda_{k}^{+})\subset(0,\rho]$
converging to zero such that for all $k\in{\mathbb{N}}$,

$\dim{\rm Ker}(\gamma_{\lambda_{k}^{-}}(\tau)-M)=1=\dim{\rm
Ker}(\gamma_{\lambda_{k}^{+}}(\tau)-M)\quad\hbox{and}\quad
i_{\tau,M}(\gamma_{\lambda_{k}^{-}})\neq
i_{\tau,M}(\gamma_{\lambda_{k}^{+}}).$

If $\dim{\rm Ker}(\gamma_{\mu}(\tau)-M)=1$, the claim may be proved as
arguments in a paragraph above (3.24) in the proof of Theorem 3.2.

Suppose now $\dim{\rm Ker}(\gamma_{\mu}(\tau)-M)>1$. By a contradiction,
suppose that $(0,0)\in\Lambda\times X^{\bot}$ is not a bifurcation point of
$\nabla\mathcal{L}_{{\lambda}}^{\bot}(v)=0$ in $\Lambda\times X^{\bot}$. Then
by shrinking $\rho>0$ we have

$C_{\ast}(\mathcal{L}^{\bot}_{\lambda},0;{\bf K})$ are independent of
$\lambda\in[-\rho,\rho]$ for any Abel group ${\bf K}$. (5.31)

As in the proof of Theorem 3.2 we can also derive that

$C_{q}(\mathcal{L}^{\bot}_{{\lambda_{k}^{+}}},0;{\bf
K})=\delta^{q}_{m^{-}_{\lambda_{k}^{+}}}{\bf K}\quad\hbox{and}\quad
C_{q}(\mathcal{L}^{\bot}_{{\lambda_{k}^{-}}},0;{\bf
K})=\delta^{q}_{m^{-}_{\lambda_{k}^{-}}}{\bf K},\quad\forall
k\in{\mathbb{N}},$

and hence arrive at a contradiction. Here $\delta^{q}_{p}=1$ if $p=q$, and
$\delta^{q}_{p}=0$ if $p\neq q$. ∎

###### Proof of Theorem 5.7.

As before we assume that $\Lambda\subset{\mathbb{R}}$ contains $0$ and
$\mu=0$. Following the notations above and combing (5.28)-(5.29) with the
assumptions about Morse indexes we obtain that $\mathcal{L}^{\bot}_{\lambda}$
with $\lambda\in\Lambda$ satisfy the conditions of [33, Theorem 4.6] on
$H^{\bot}$ and $X^{\bot}$ in (5.21) and (5.22). Then one of the following
alternatives occurs:

  1. (i)

$(0,0)$ is not an isolated solution in $\\{0\\}\times X^{\bot}$ of
$\nabla\mathcal{L}^{\bot}_{\lambda}(w)=0$.

  2. (ii)

For each nonzero $\lambda\in\Lambda$ near $0$,
$\nabla\mathcal{L}^{\bot}_{\lambda}(w)=0$ has a solution $u_{\lambda}\in
X^{\bot}$ different from $0$, which converges to $0$ in $X^{\bot}$ as
$\lambda\to 0$.

  3. (iii)

There is an one-sided neighborhood $\Lambda^{0}$ of $0$ such that for any
$\lambda\in\Lambda^{0}\setminus\\{0\\}$,
$\nabla\mathcal{L}^{\bot}_{\lambda}(w)=0$ has at least two solutions in
$X^{\bot}$ different from $0$, which converge to $0$ in $X^{\bot}$ as
$\lambda\to 0$.

As in the proof of Theorem 3.4 the desired result may follow from these. ∎

##  6 Bifurcations for solutions of the Hamiltonian system (1.13)

Under Assumption 1.4 let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ solve (1.13)
for each $\lambda\in\Lambda$. The linearized problem of (1.13) along $v_{0}$
is

$\dot{v}(t)=JH^{\prime\prime}_{\lambda,t}(v_{0}(t))v(t),\quad
v(t+\tau)=v(t)\quad\hbox{and}\quad v(-t)=Nv(t)$ (6.1)

for all $(\lambda,t)\in{\mathbb{R}}\times\Lambda$. By (1.12), $N^{T}\nabla
H_{\lambda,-t}(Nz)=\nabla H_{\lambda,t}(z)$ and so $N^{T}(\nabla
H_{\lambda,-t})^{\prime}(Nz)N=(\nabla H_{\lambda,t})^{\prime}(z)$. Hence each
$B_{\lambda}(t):=H^{\prime\prime}_{\lambda,t}(v_{0}(t))=(\nabla
H_{\lambda,t})^{\prime}(v_{0}(t))$ satisfies Assumption 2.6. Let
$\gamma_{\lambda}:[0,\tau]\to{\rm Sp}(2n,\mathbb{R})$ be the fundamental
matrix solution of $\dot{Z}(t)=JB_{\lambda}(t)Z(t)$. It has the Maslov-type
index $(\mu_{1,\tau}(\gamma_{\lambda}),\nu_{1,\tau}(\gamma_{\lambda}))$
defined by (2.18) and (2.19).

Corresponding to Theorems 3.1, 3.2, 3.4 and Corollary 3.6 we have:

######  Theorem 6.1.

Under Assumption 1.4, if $(\mu,v_{0})$ is a bifurcation point of (1.13), then
(6.1) with $\lambda=\mu$ has nontrivial solutions, i.e.,
$\nu_{1,\tau}(\gamma_{\mu})\neq 0$.

######  Theorem 6.2.

Under Assumption 1.4 let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ solve (1.13)
for each $\lambda\in\Lambda$. Suppose for some $\mu\in\Lambda$ that there
exist two points in any neighborhood of $\mu\in\Lambda$,
$\lambda_{-}<\mu<\lambda_{+}$, such that

$\mu_{1,\tau}(\gamma_{\lambda_{-}})\neq\mu_{1,\tau}(\gamma_{\lambda_{+}})\quad\hbox{and}\quad\nu_{1,\tau}(\gamma_{\lambda_{-}})=\nu_{1,\tau}(\gamma_{\lambda_{+}})=0.$

Then $(\mu,v_{0})$ is a bifurcation point of (1.13).

Under slightly strong assumptions we can obtain better results.

######  Theorem 6.3.

Under Assumption 1.4 let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ solve (1.13)
for each $\lambda\in\Lambda$. Suppose for some $\mu\in\Lambda$ that
$\nu_{1,\tau}(\gamma_{\mu})\neq 0$ and $\nu_{1,\tau}(\gamma_{\lambda})=0$ for
each $\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$, and that
$\mu_{1,\tau}(\gamma_{\lambda})$ take, respectively, values
$\mu_{1,\tau}(\gamma_{\mu})$ and
$\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})$ as $\lambda\in\Lambda$
varies in two deleted half neighborhoods of $\mu$ Then one of the following
alternatives occurs:

  1. (i)

Equation (1.13) with $\lambda=\mu$ has a sequence of solutions, $v_{k}\neq
v_{0}$, $k=1,2,\cdots$, which converges to $v_{0}$ in
$C^{1}(S_{\tau};{\mathbb{R}}^{2n})$.

  2. (ii)

For every $\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$ there is a solution
$v_{\lambda}\neq v_{0}$ of (1.13) with parameter value $\lambda$, which
converges to $v_{0}$ in $C^{1}(S_{\tau};{\mathbb{R}}^{2n})$ as
$\lambda\to\mu$.

  3. (iii)

There is an one-sided neighborhood $\Lambda^{0}$ of $\mu$ such that for any
$\lambda\in\Lambda^{0}\setminus\\{\mu\\}$, (1.13) with parameter value
$\lambda$ has at least two distinct solutions, $v_{\lambda}^{1}\neq v_{0}$ and
$v_{\lambda}^{2}\neq v_{0}$, which converge to $v_{0}$ in
$C^{1}(S_{\tau};{\mathbb{R}}^{2n})$ as $\lambda\to\mu$.

Moreover, if $v_{0}=0$ and all $H(\lambda,t,\cdot)$ are even, then either (i)
holds or the following occurs:

  1. (iv)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\nu_{1,\tau}(\gamma_{\mu})$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), (1.13) with parameter value
$\lambda$ has at least $n^{-}$ (resp. $n^{+}$) distinct pairs of nontrivial
solutions, $\\{v_{\lambda}^{i},-v_{\lambda}^{i}\\}$, $i=1,\cdots,n^{-}$ (resp.
$n^{+}$), which converge to zero in $C^{1}(S_{\tau};{\mathbb{R}}^{2n})$ as
$\lambda\to\mu$.

Corresponding to Corollaries 3.6 and 4.2, we have also

######  Corollary 6.4.

Let continuous functions
$H_{0},\hat{H}:{\mathbb{R}}\times{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ satisfy

$\displaystyle
H_{0}(-t,Nz)=H_{0}(t,z)=H_{0}(t+\tau,z)\quad\forall(t,z)\in\Lambda\times{\mathbb{R}}\times\mathbb{R}^{2n},$
$\displaystyle\hat{H}(-t,Nz)=\hat{H}(t,z)=\hat{H}(t+\tau,z)\quad\forall(t,z)\in\Lambda\times{\mathbb{R}}\times\mathbb{R}^{2n},$

and be such that all
$H_{0}(t,\cdot),\hat{H}(t,\cdot):{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ are
$C^{2}$ and all their partial derivatives depend continuously on
$(t,z)\in{\mathbb{R}}\times\mathbb{R}^{2n}$. Let
$v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ satisfy

$\dot{v}(t)=J\nabla(H_{0})_{t}(v(t))$, $v(t+\tau)=v(t)$ and
$v(-t)=Nv(t)\;\forall t$.

Suppose that $d\hat{H}_{t}(v_{0}(t))=0$ for all $t\in{\mathbb{R}}$ and that

either $\hat{H}_{t}^{\prime\prime}(v_{0}(t))>0\;\forall t$ or
$\hat{H}_{t}^{\prime\prime}(v_{0}(t))<0\;\forall t$.

Take $H(\lambda,t,z)=H_{0}(t,z)+\lambda\hat{H}(t,z)$ in (1.13). Then
(I)
$\Sigma_{1}:=\\{\lambda\in{\mathbb{R}}\,|\,\nu_{1,\tau}(\gamma_{\lambda})>0\\}$
is a discrete set in ${\mathbb{R}}$.
(II) $(\mu,v_{0})$ with $\mu\in{\mathbb{R}}$ is a bifurcation point for (1.13)
if and only if $\nu_{1,\tau}(\gamma_{\mu})>0$.
(III) For each $\mu\in\Sigma_{1}$ and $\rho>0$ small enough it holds that

$\displaystyle\mu_{1,\tau}(\gamma_{\lambda})=\left\\{\begin{array}[]{ll}\mu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in[\mu-\rho,\mu),\\\
\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in(\mu,\mu+\rho]\end{array}\right.$

if $\hat{H}_{t}^{\prime\prime}(v_{0}(t))>0\;\forall t$, and

$\displaystyle\mu_{1,\tau}(\gamma_{\lambda})=\left\\{\begin{array}[]{ll}\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in[\mu-\rho,\mu),\\\
\mu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in(\mu,\mu+\rho]\end{array}\right.$

if $\hat{H}_{t}^{\prime\prime}(v_{0}(t))<0\;\forall t$; and one of the
following alternatives occurs:

  1. (i)

Equation (1.13) with $\lambda=\mu$ has a sequence of solutions, $v_{k}\neq
v_{0}$, $k=1,2,\cdots$, which converges to $v_{0}$ on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (ii)

For every $\lambda\in{\mathbb{R}}\setminus\\{\mu\\}$ near $\mu$ there is a
solution solution $v_{\lambda}\neq v_{0}$ of (1.13) with parameter value
$\lambda$, which converges to $v_{0}$ on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  3. (iii)

There is an one-sided neighborhood $\mathcal{N}$ of $\mu$ in ${\mathbb{R}}$
such that for any $\lambda\in\mathcal{N}\setminus\\{\mu\\}$, (1.13) with
parameter value $\lambda$ has at least two distinct solutions,
$v_{\lambda}^{1}\neq v_{0}$ and $v_{\lambda}^{2}\neq v_{0}$, which converge to
$v_{0}$ on any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as
$\lambda\to\mu$.

Moreover, if $v_{0}=0$ and all $H_{0}(t,\cdot),\hat{H}(t,\cdot)$ are even,
then either (i) holds or the following occurs:

  1. (iv)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq d$, where $d$ is the number of maximal linearly independent
solutions of

$\dot{v}(t)=J(H^{\prime\prime}_{0t}(0)+\mu\hat{H}^{\prime\prime}_{t}(0))v(t),\quad
v(t+\tau)=v(t)\quad\hbox{and}\quad v(-t)=Nv(t)\;\forall t\in{\mathbb{R}},$
(6.4)

and for $\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), (1.13) with parameter value
$\lambda$ has at least $n^{-}$ (resp. $n^{+}$) distinct pairs of nontrivial
solutions, $\\{v_{\lambda}^{i},-v_{\lambda}^{i}\\}$, $i=1,\cdots,n^{-}$ (resp.
$n^{+}$), which converge to zero on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology as $\lambda\to\mu$.

The operator $\Lambda_{I_{2n},\tau,0}$ on $L^{2}(S_{\tau};\mathbb{R}^{2n})$
defined by (2.3) with domain $W^{1,2}(S_{\tau};\mathbb{R}^{2n})$ restricts to
a closed linear and self-adjoint operator on $\textsf{L}_{\tau}$ with domain
$\textsf{W}_{\tau}$, denoted by $\tilde{\Lambda}_{I_{2n},\tau,0}$ for
clearness.
$\sigma(\tilde{\Lambda}_{I_{2n},\tau,0})=\frac{2\pi}{\tau}\mathbb{Z}$ consists
of only eigenvalues, and each eigenvalues has multiplicity $n$.

As before, from now on we assume that $\Lambda\subset{\mathbb{R}}$ contains
$0$ and $\mu=0$.

Solutions of (1.13) are critical points of the functional

$\Phi_{\lambda}(v)=\int^{\tau}_{0}\left[\frac{1}{2}(J\dot{v}(t),v(t))_{\mathbb{R}^{2n}}+H(\lambda,t,{v}(t))\right]dt$

on the Hilbert space $\textsf{W}_{\tau}$. As done in the first paragraph in
Section 3.2 we can assume that $H$ satisfies inequalities in (3.11) and (3.12)
for all $(\lambda,t,z)\in\Lambda\times\mathbb{R}\times\mathbb{R}^{2n}$.

Then we may choose $K\notin\frac{2\pi}{\tau}\mathbb{Z}$, $c_{i}>0$, $i=1,2,3$
such that each $H_{K}(\lambda,t,z):=H(\lambda,t,z)-\frac{K}{2}|z|^{2}$
satisfies (ii)-(iii) in the proof of Theorem 3.1. Consider the dual action
$\Psi_{K}(\lambda,\cdot):\textsf{W}_{\tau}\to{\mathbb{R}}$ defined by (3.17).
Since

$\tilde{\Lambda}_{I_{2n},\tau,K}:\textsf{W}_{\tau}\to\textsf{L}_{\tau},\;v\mapsto\tilde{\Lambda}_{I_{2n},\tau,0}v+Kv$

is a Banach isomorphism, we obtain a family of $C^{1}$ and twice Gâteaux-
differentiable functionals

$\psi_{K}(\lambda,\cdot)=\Psi_{K}(\lambda,\cdot)\circ(-\Lambda_{M,\tau,K})^{-1}:\textsf{L}_{\tau}\to{\mathbb{R}}$

given by

$\psi_{K}(\lambda,v)=\int^{\tau}_{0}\left[\frac{1}{2}(v(t),((\tilde{\Lambda}_{I_{2n},\tau,K})^{-1}v)(t))_{\mathbb{R}^{2n}}+H_{K}^{\ast}(\lambda,t,v(t))\right]dt,$
(6.5)

such that $v\in\textsf{W}_{\tau}$ is a critical point of
$\Psi_{K}(\lambda,\cdot)$ if and only if
$w:=-\tilde{\Lambda}_{I_{2n},\tau,K}v$ is a critical point of
$\psi_{K}(\lambda,\cdot)$. In particular,
$(0,v_{0})\in\Lambda\times\textsf{W}_{\tau}$ is a bifurcation point of the
problem (1.13) if and only if $(0,w_{0})\in\Lambda\times\textsf{L}_{\tau}$
with $w_{0}:=-\tilde{\Lambda}_{I_{2n},\tau,K}v_{0}$ is a bifurcation point of
$\nabla_{w}\psi_{K}(\lambda,w)=0$.

Let $\psi_{K,\lambda}(v)=\psi_{K}(\lambda,v)$. As in Remark 2.18 we can
compute

$\displaystyle{\psi}^{\prime\prime}_{K,\lambda}(w_{0})[\xi,\eta]$
$\displaystyle=$
$\displaystyle\int^{\tau}_{0}\left[((\tilde{\Lambda}_{I_{2n},\tau,K})^{-1}\xi)(t),\eta(t))_{\mathbb{R}^{2n}}+((H_{K,\lambda,t})^{\ast})^{\prime\prime}(w_{0}(t))\xi(t),\eta(t))_{\mathbb{R}^{2n}})\right]dt$
$\displaystyle=$
$\displaystyle\int^{\tau}_{0}\left[((\tilde{\Lambda}_{I_{2n},\tau,K})^{-1}\xi)(t)+[(H_{K,\lambda,t})^{\prime\prime}(v_{0}(t))]^{-1}\xi(t),\eta(t))_{\mathbb{R}^{2n}})\right]dt$
$\displaystyle=$
$\displaystyle\textsf{Q}_{B_{\lambda},K}(\xi,\eta),\quad\forall\xi,\eta\in\textsf{{L}}_{\tau},$

where $\textsf{Q}_{B_{\lambda},K}$ is given by (2.42) with
$B_{\lambda}(t)=(H_{\lambda,t})^{\prime\prime}(v_{0}(t))$. By Theorem 2.7 we
obtain

$\displaystyle m^{-}({\psi}_{K,\lambda},w_{0})$ $\displaystyle=$
$\displaystyle\mu_{1,\tau}(\gamma_{\lambda})-n[K\tau/(2\pi)],$ (6.6)
$\displaystyle m^{0}({\psi}_{K,\lambda},w_{0})$ $\displaystyle=$
$\displaystyle\nu_{1,\tau}(\gamma_{\lambda}).$ (6.7)

Let $\iota:\textsf{W}_{\tau}\to\textsf{L}_{\tau}$ be the inclusion. Then

$A_{M,\tau,K}:\textsf{L}_{\tau}\to\textsf{L}_{\tau},\;v\mapsto\iota\circ(\tilde{\Lambda}_{I_{2n},\tau,K})^{-1}v,$
(6.8)

is a compact self-adjoint operator, and $\psi_{K}(\lambda,\cdot)$ has the
gradient

$\nabla_{v}\psi_{K}(\lambda,v)=A_{M,\tau,K}v+\nabla_{z}H_{K}^{\ast}(\lambda,\cdot;v(\cdot)).$
(6.9)

Moreover, $\textsf{L}_{\tau}\ni
v\mapsto\nabla_{v}\psi_{K}(\lambda,v)\in\textsf{L}_{\tau}$ has a Gâteaux
derivative

$B_{\lambda}(v):=D_{v}\nabla_{v}\psi_{K}(\lambda,v)=A_{M,\tau,K}+(H_{K}^{\ast})^{\prime\prime}_{z}(\lambda,\cdot;v(\cdot))\in\mathscr{L}_{s}(\textsf{L}_{\tau}).$
(6.10)

Let $P_{\lambda}(v)=(H_{K}^{\ast})^{\prime\prime}_{v}(\lambda,\cdot,v(\cdot))$
and $Q_{\lambda}(v)=A_{M,\tau,K}$.

###### Proof of Theorem 6.1.

As in the proof of Theorem 3.1 we can show that [33, Theorem 3.1] is
applicable to $H=X=\textsf{L}_{\tau}$ and
$\mathcal{F}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$. ∎

Consider Banach subspaces of $C^{k}(S_{\tau};{\mathbb{R}}^{2n})$,
$\textsf{C}^{k}_{\tau}=C^{k}(S_{\tau};{\mathbb{R}}^{2n})\cap\textsf{L}_{\tau}$,
$k=0,1,\cdots$.

###### Proofs of Theorems 6.2, 6.3.

Note that $\tilde{\Lambda}_{I_{2n},\tau,K}$ gives rise to a Banach isomorphism
from $\textsf{C}^{1}_{\tau}$ to $\textsf{C}^{0}_{\tau}$, denoted by
$\tilde{\Lambda}^{c}_{I_{2n},\tau,K}$. As in the proof of Theorem 3.2 we can
prove that $\mathcal{L}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$ with
$\lambda\in\Lambda$ satisfies the conditions of [33, Theorem A.3] with
$H=\textsf{L}_{\tau}$ and $X=\textsf{C}^{0}_{\tau}$, and therefore complete
the proofs of Theorems 6.2, 6.3 by using (6.6) and (6.7). ∎

###### Proof of Corollary 6.4.

As in the proof of Corollary 3.6 this can be completed using Theorems 2.7, 2.8
and the above arguments. ∎

######  Theorem 6.5.

Under Assumption 1.4, suppose also that $H$ is independent of time $t$. For
some $\tau>0$ let $v_{0}:{\mathbb{R}}\to{\mathbb{R}}^{2n}$ be an even
$\tau$-periodic solution of $\dot{v}(t)=J\nabla H_{\lambda}(v(t))$ for each
$\lambda\in\Lambda$, and let $\gamma_{\lambda}(t)$ be the fundamental matrix
solution of $\dot{v}(t)=JH^{\prime\prime}_{\lambda}(v_{0})v(t)$. Suppose that
for some $\mu\in\Lambda$

  1. (a)

the following problem

$\dot{v}(t)=JH^{\prime\prime}_{\mu}(v_{0}(t))v(t)$, $v(t+\tau)=v(t)$ and
$v(-t)=Nv(t)\;\forall t$ (6.11)

has no nonzero even solutions;

  2. (b)

$\nu_{1,\tau}(\gamma_{\mu})\neq 0$, $\nu_{1,\tau}(\gamma_{\lambda})=0$ for
each $\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$, and
$\mu_{1,\tau}(\gamma_{\lambda})$ take, respectively, values
$\mu_{1,\tau}(\gamma_{\mu})$ and
$\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})$ as $\lambda\in\Lambda$
varies in two deleted half neighborhoods of $\mu$.

Then one of the following alternatives occurs:

  1. (i)

Equation (1.13) with $\lambda=\mu$ has a sequence of solution pairs,
$\\{v_{k},Nv_{k}\\}$, $k=1,2,\cdots$, such that $v_{k}$ and $Nv_{k}$ converge
to $v_{0}$ on any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (ii)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\nu_{1,\tau}(\gamma_{\mu})$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), (1.13) with parameter value
$\lambda$ has at least $n^{-}$ (resp. $n^{+}$) distinct solution pairs
$\\{v_{\lambda}^{i},Nv_{\lambda}^{i}\\}$, $i=1,\cdots,n^{-}$ (resp. $n^{+}$),
such that all $v_{\lambda}^{i}$ and $Nv_{\lambda}^{i}$ converge to $v_{0}$ on
any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as
$\lambda\to\mu$.

In particular, $(\mu,v_{0})$ is a bifurcation point of (1.13).

###### Proof.

The action of the group $G=\\{I_{2n},N\\}\cong\mathbb{Z}_{2}$ on
$\mathbb{R}^{2n}$ gives an orthogonal one on $\textsf{L}_{\tau}$ and
$\textsf{W}_{\tau}$, which induces an isometric action on each
$\textsf{C}_{\tau}^{k}$. The set of fixed points of this action, ${\rm
Fix}_{G}$, is equal to $\\{u\in\textsf{L}_{\tau}\,|\,u(-t)=u(t),{\rm
a.e.}\,t\in\mathbb{R}\\}$. Clearly, each functional $\psi_{K}(\lambda,\cdot)$
defined by (6.5) is invariant under this $G$-action. As in the proof of
Theorem 5.1 we can obtain that the functional
$\mathcal{L}_{\lambda}(\cdot)=\psi_{K}(\lambda,w_{0}+\cdot)$ satisfies
conditions in [33, Theorem 4.6] with $H=\textsf{L}_{\tau}$ and
$X=\textsf{C}_{\tau}^{0}$.

Note that $v_{0}$ and so $w_{0}:=-\tilde{\Lambda}_{I_{2n},\tau,K}v_{0}$
belongs to ${\rm Fix}_{G}$. The assumption (a) implies that the fixed point
set of the induced $G$-action on $H_{0}^{0}$ is $\\{0\\}$. Combing the
assumption (b) we can use [33, Theorem 5.12] to get the desired result. ∎

Similarly, since
$G=\\{I_{2n},-I_{2n}\\}\times\\{I_{2n},N\\}\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
orthogonally acts on $\textsf{L}_{\tau}$ and $\textsf{W}_{\tau}$ by

$(I_{2n},I_{2n})u=u,\quad(I_{2n},N)u=Nu,\quad(-I_{2n},I_{2n})u=-u,\quad(-I_{2n},N)u=-Nu,$

using [33, Theorem 5.12] and [33, Remark 5.14] we may directly prove:

######  Theorem 6.6.

Under Assumption 1.4, assume that $H$ is independent of time $t$ and that each
$H_{\lambda}$ is even and satisfies $dH_{\lambda}(0)=0$ for all $\lambda$. Let
$\gamma_{\lambda}(t)=\exp(tJH^{\prime\prime}_{\lambda}(0))$. Suppose for some
$\tau>0$ and $\mu\in\Lambda$ that

  1. (b)

$\nu_{1,\tau}(\gamma_{\mu})\neq 0$, $\nu_{1,\tau}(\gamma_{\lambda})=0$ for
each $\lambda\in\Lambda\setminus\\{\mu\\}$ near $\mu$, and
$\mu_{1,\tau}(\gamma_{\lambda})$ take, respectively, values
$\mu_{1,\tau}(\gamma_{\mu})$ and
$\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})$ as $\lambda\in\Lambda$
varies in two deleted half neighborhoods of $\mu$.

Then one of the following alternatives occurs:

  1. (i)

Equation (1.13) with $\lambda=\mu$ has a sequence of solution quadruples,
$\\{v_{k},-v_{k},Nv_{k},-Nv_{k}\\}$, $k=1,2,\cdots$, such that $v_{k}$
converges to $0$ on any compact interval $I\subset{\mathbb{R}}$ in
$C^{1}$-topology.

  2. (ii)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\nu_{1,\tau}(\gamma_{\mu})$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), (1.13) with parameter value
$\lambda$ has at least $n^{-}$ (resp. $n^{+}$) distinct solution quadruples
$\\{v_{\lambda}^{i},-v_{\lambda}^{i},Nv_{\lambda}^{i},-Nv_{\lambda}^{i}\\}$,
$i=1,\cdots,n^{-}$ (resp. $n^{+}$), such that $v_{\lambda}^{i}$ converges to
$0$ on any compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as
$\lambda\to\mu$.

In particular, $(\mu,0)$ is a bifurcation point of (1.13).

######  Corollary 6.7.

Let $H_{0},\hat{H}:{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ be $N$-invariant
$C^{2}$-functions. Suppose that $dH_{0}(0)=d\hat{H}(0)=0$ and that either
$\hat{H}^{\prime\prime}(0)>0$ or $\hat{H}^{\prime\prime}(0)<0$. Then for
$\gamma_{\lambda}(t)=\exp(tJH_{0}^{\prime\prime}(0)+\lambda
tJ\hat{H}^{\prime\prime}(0))$ and any $\tau>0$,
$\Sigma_{\tau}:=\\{\lambda\in{\mathbb{R}}\,|\,\nu_{1,\tau}(\gamma_{\lambda})>0\\}$
is a discrete set in ${\mathbb{R}}$. Moreover, for each $\mu\in\Sigma_{\tau}$
and $\rho>0$ small enough it holds that

$\displaystyle\mu_{1,\tau}(\gamma_{\lambda})=\left\\{\begin{array}[]{ll}\mu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in[\mu-\rho,\mu),\\\
\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in(\mu,\mu+\rho]\end{array}\right.$
(6.14)

if $\hat{H}^{\prime\prime}(0)>0\;\forall t\in[0,\tau]$, and

$\displaystyle\mu_{1,\tau}(\gamma_{\lambda})=\left\\{\begin{array}[]{ll}\mu_{1,\tau}(\gamma_{\mu})+\nu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in[\mu-\rho,\mu),\\\
\mu_{1,\tau}(\gamma_{\mu})&\forall\lambda\in(\mu,\mu+\rho]\end{array}\right.$
(6.17)

if $\hat{H}^{\prime\prime}(0)<0\;\forall t\in[0,\tau]$. Consequently, for each
$\mu\in\Sigma_{\tau}$, if (6.11) with $H(\mu,x)=H_{0}(x)+\mu\hat{H}(x)$ and
$v_{0}=0$ has no nonzero even solution, then the conclusions of Theorem 6.5
holds for (1.13) with $H(\lambda,x)=H_{0}(x)+\lambda\hat{H}(x)$. Suppose that
both $H_{0}$ and $\hat{H}$ are also even. Then the conclusions of Theorem 6.6
holds for (1.13) with $H(\lambda,x)=H_{0}(x)+\lambda\hat{H}(x)$.

###### Proof.

Suppose $\nu_{1,\tau}(\gamma_{\mu})>0$. As in the proof of Corollary 3.6, by
modifying values of $H_{0}$ and $\hat{H}$ outside a neighborhood $U$ of
$0\in\mathbb{R}^{2n}$ we can choose
$K\in\mathbb{R}\setminus\frac{2\pi}{\tau}\mathbb{Z}$ and $c_{i}>0$, $i=1,2,3$
such that $H_{K}(\lambda,z):=H_{0}(z)+\lambda\hat{H}(z)-\frac{K}{2}|z|^{2}$
satisfies

$\displaystyle
c_{1}I_{2n}\leq(H_{0})^{\prime\prime}(z)+\lambda\hat{H}^{\prime\prime}(z)-KI_{2n}\leq
c_{2}I_{2n}\quad\hbox{and}$ $\displaystyle c_{1}|z|^{2}-c_{3}\leq
H_{K}(\lambda,z)\leq c_{2}|z|^{2}+c_{3}$

for all $(\lambda,t,z)\in[\mu-1,\mu+1]\times[0,\tau]\times{\mathbb{R}}^{2n}$.
Let $\psi_{K,\lambda}(u):=\psi_{K}(\lambda,u)$ be given by (6.5). Then
${\psi}^{\prime\prime}_{K,\lambda}(0)[\xi,\eta]=\textsf{Q}_{B_{\lambda},K}(\xi,\eta)$
for all $\xi,\eta\in\textsf{{L}}_{\tau}$, where $\textsf{Q}_{B_{\lambda},K}$
is given by (2.42) with
$B_{\lambda}(t)=(H_{0})^{\prime\prime}(0)+\lambda\hat{H}^{\prime\prime}(0)$.
Therefore $m^{-}({\psi}_{K,\lambda},0)=m^{-}(\textsf{Q}_{B_{\lambda},K})$ and
$m^{0}({\psi}_{K,\lambda},0)=m^{0}(\textsf{Q}_{B,K})$.

If $\hat{H}^{\prime\prime}(0)>0$ (resp. $\hat{H}^{\prime\prime}(0)<0$), by
Theorem 2.8 we deduce that

$\displaystyle m^{-}(\textsf{Q}_{B_{\lambda_{2}},K})\geq
m^{-}(\textsf{Q}_{B_{\lambda_{1}},K})+m^{0}(\textsf{Q}_{B_{\lambda_{1}},K})\quad$
(resp. $m^{-}(\textsf{Q}_{B_{\lambda_{1}},K})\geq
m^{-}(\textsf{Q}_{B_{\lambda_{2}},K})+m^{0}(\textsf{Q}_{B_{\lambda_{2}},K})$)

for any $\mu-1\leq\lambda_{1}<\lambda_{2}\leq\mu+1$. These imply that
$\\{\lambda\in[\mu-1,\mu+1]\,|\,m^{0}(\textsf{Q}_{B_{\lambda},K})>0\\}$ is a
finite set. It follows from this and Theorem 2.7 that
$\\{\lambda\in{\mathbb{R}}\,|\,\nu_{1,\tau}(\gamma_{\lambda})>0\\}$ is a
discrete set in ${\mathbb{R}}$. The first claim is proved.

As in the proof of Corollary 3.6 we can use Theorem 2.8 to derive that

$\displaystyle
m^{-}(\textsf{Q}_{B_{\lambda},K})=m^{-}(\textsf{Q}_{B_{\mu},K})\;\forall\lambda\in[\mu-\rho,\mu),$
$\displaystyle
m^{-}(\textsf{Q}_{B_{\lambda},K})=m^{-}(\textsf{Q}_{B_{\mu},K})+m^{0}(\textsf{Q}_{B_{\mu},K})\;\forall\lambda\in(\mu,\mu+\rho]$

if $\hat{H}^{\prime\prime}(0)>0$ and $\rho>0$ is small enough, and

$\displaystyle
m^{-}(\textsf{Q}_{B_{\lambda},K})=m^{-}(\textsf{Q}_{B_{\mu},K})+m^{0}(\textsf{Q}_{B_{\mu},K})\;\forall\lambda\in[\mu-\rho,\mu),$
$\displaystyle
m^{-}(\textsf{Q}_{B_{\lambda},K})=m^{-}(\textsf{Q}_{B_{\mu},K})\;\forall\lambda\in(\mu,\mu+\rho]$

if $\hat{H}^{\prime\prime}(0)<0$ and $\rho>0$ is small enough. These and
Theorem 2.7 lead to (6.14) and (6.17). Other conclusions follow from Theorems
6.5, 6.6. ∎

Corresponding to Corollary 5.3 we have

######  Corollary 6.8.

Let $H:{{\mathbb{R}}}^{2n}\to{\mathbb{R}}$ be a $N$-invariant $C^{2}$-function
satisfying $dH(0)=0$. Suppose that either ${H}^{\prime\prime}(0)>0$ or
${H}^{\prime\prime}(0)<0$. Then
$\Delta(H):=\\{\lambda\in\mathbb{R}\setminus\\{0\\}\,|\,\nu_{1,1}(\gamma_{\lambda})>0\\}$
is a discrete set, where $\gamma_{\lambda}(t)=\exp(\lambda
tJ{H}^{\prime\prime}(0))$.

If ${H}^{\prime\prime}(0)>0$ and $\mu\in\Delta(H)\cap(0,\infty)$ is such that
the following problem

$\dot{v}(t)=\mu JH^{\prime\prime}(0)v(t)$, $v(t+1)=v(t)$ and
$v(-t)=Nv(t)\;\forall t$ (6.18)

has no nonzero even solutions, then one of the following alternatives occurs:

  1. (i)

The problem

$\dot{v}(t)=J\nabla
H(v(t)),\quad\hbox{$v(t+\mu)=v(t)$\quad\hbox{and}\quad$v(-t)=Nv(t)\;\forall
t$}$ (6.19)

has a sequence of solution pairs, $\\{v_{k},Nv_{k}\\}$, $k=1,2,\cdots$, such
that $v_{k}$ and $Nv_{k}$ converge to $0$ on any compact interval
$I\subset{\mathbb{R}}$ in $C^{1}$-topology.

  2. (ii)

There exist left and right neighborhoods $\Lambda^{-}$ and $\Lambda^{+}$ of
$\mu$ in $\Lambda$ and integers $n^{+},n^{-}\geq 0$, such that
$n^{+}+n^{-}\geq\nu_{1,1}(\gamma_{\mu})$, and for
$\lambda\in\Lambda^{-}\setminus\\{\mu\\}$ (resp.
$\lambda\in\Lambda^{+}\setminus\\{\mu\\}$), the problem

$\dot{v}(t)=J\nabla
H(v(t)),\quad\hbox{$v(t+\lambda)=v(t)$\quad\hbox{and}\quad$v(-t)=Nv(t)\;\forall
t$}$ (6.20)

has at least $n^{-}$ (resp. $n^{+}$) distinct solution pairs
$\\{v_{\lambda}^{i},Nv_{\lambda}^{i}\\}$, $i=1,\cdots,n^{-}$ (resp. $n^{+}$),
such that all $v_{\lambda}^{i}$ and $Nv_{\lambda}^{i}$ converge to $0$ on any
compact interval $I\subset{\mathbb{R}}$ in $C^{1}$-topology as
$\lambda\to\mu$.

If ${H}^{\prime\prime}(0)<0$ and $\mu\in\Delta(H)\cap(-\infty,0)$ is such that
(6.18) has no nonzero even solutions, then one of the above (i) and (ii) with
this $\mu$ holds.

Suppose further that $H$ is also even. Then “$\\{v_{k},Nv_{k}\\}$” in (i) and
“$\\{v_{\lambda}^{i},Nv_{\lambda}^{i}\\}$” in (ii) are replaced by
“$\\{v_{k},Nv_{k},-v_{k},-Nv_{k}\\}$” and
“$\\{v_{\lambda}^{i},Nv_{\lambda}^{i},-v_{\lambda}^{i},-Nv_{\lambda}^{i}\\}$”,
respectively.

###### Proof.

The first claim follows from Corollary 6.7. Suppose ${H}^{\prime\prime}(0)>0$
and $\mu\in\Delta(H)_{\tau}\cap(0,\infty)$. Let
$\gamma_{\lambda}(t)=\exp(\lambda tJ{H}^{\prime\prime}(0))$. It follows from
(6.14) that $\mu_{1,1}(\gamma_{\lambda})=\mu_{1,1}(\gamma_{\mu})$ for
$\lambda\leq\mu$ close to $\mu$ and that
$\mu_{1,1}(\gamma_{\lambda})=\mu_{1,1}(\gamma_{\mu})+\nu_{1,1}(\gamma_{\mu})$
for $\lambda>\mu$ close to $\mu$. By the third conclusion of Corollary 6.7
with $H_{0}=0$ and $\hat{H}=H$ we obtain the desired claims. The final result
may follow from the fourth conclusion of Corollary 6.7 with $H_{0}=0$ and
$\hat{H}=H$. ∎

######  Example 6.9.

Let $K:{\mathbb{R}}^{2n}\to{\mathbb{R}}$ be as in Example 5.4. It was proved
in [31, Example 3.2] that

$\mu_{1,\tau}(\gamma^{K}_{\lambda})=\mu_{2,\tau}(\gamma^{K}_{\lambda})=n-k+\sum^{n}_{i=1}\left[\frac{\lambda\varrho_{i}\tau}{\pi}\right],$
(6.21)

where $k=\sharp\\{i\in\\{1,\cdots,n\\}\,|\,\lambda\varrho_{i}\tau=0\;{\rm
mod}\;\pi\\}$. It is easily proved that

$\nu_{1,\tau}(\gamma^{K}_{\lambda})=\nu_{2,\tau}(\gamma^{K}_{\lambda})=\sharp\\{i\in\\{1,\cdots,n\\}\,|\,\lambda\varrho_{i}\tau=0\;{\rm
mod}\;2\pi\\}.$ (6.22)

Then
$\Gamma(K,0)\cap(0,\infty)=\cup^{n}_{i=1}\\{\lambda>0\,|\,\lambda\varrho_{i}=0\;{\rm
mod}\;2\pi\\}$. For $\mu\in\Gamma(K,0)\cap(0,\infty)$, all nontrivial
solutions of the problem

$\dot{v}(t)=\mu JK^{\prime\prime}(0)v(t)=\mu J{\rm
diag}(\varrho_{1},\cdots,\varrho_{n},\varrho_{1},\cdots,\varrho_{n})v(t)\quad\hbox{and}\quad
v(t+1)=v(t)$

have forms $v(t)=\gamma^{K}_{\mu}(t)c$, where components $c_{i}$ and $c_{n+i}$
in $c=(c_{1},\cdots,c_{2n})^{T}\in\mathbb{R}^{2n}$ are equal to zeros if
$\mu\varrho_{i}\notin 2\pi\mathbb{Z}$. Since $NJ=-JN$ and
$NK^{\prime\prime}(0)=K^{\prime\prime}(0)N$ we deduce that
$Nv(t)=N\gamma^{K}_{\mu}(t)c=\gamma^{K}_{\mu}(-t)Nc$. This implies that
$Nv(t)=v(-t)\;\forall t$ if and only $Nc=c$, i.e., $c_{1}=\cdots=c_{n}=0$.
Thus if the above $v(t)$ satisfies $Nv(t)=v(-t)\;\forall t$, then

$v(t)=\big{(}-c_{n+1}\sin(\lambda\varrho_{1}t),\cdots,-c_{2n}\sin(\lambda\varrho_{n}t),c_{n+1}\cos(\lambda\varrho_{1}t),\cdots,c_{2n}\cos(\lambda\varrho_{1}t)\big{)}^{T},$

where $c_{n+i}=0$ if $\mu\varrho_{i}\notin 2\pi\mathbb{Z}$. Clearly, if it is
also even then $v\equiv 0$. Hence the problem (6.18) with $H=K$ has no nonzero
even solutions. Suppose that $K$ is also $N$-invariant. By Corollary 6.8, for
each $\mu\in\Gamma(K,0)\cap(0,\infty)$ one of the statements (i) and (ii) in
Corollary 6.8 holds after $H$ and $\nu_{1,1}(\gamma_{\mu})$ are replaced by
$K$ and $\nu_{1,1}(\gamma^{K}_{\mu})$, respectively.

######  Question 6.10.

For a positive definite matrix $A$ of order $2n$, Williamson theorem gives
rise to a symplectic matrix $S\in{\rm Sp}(2n,{\mathbb{R}})$ such that
$S^{T}AS={\rm
diag}(\varrho_{1},\cdots,\varrho_{n},\varrho_{1},\cdots,\varrho_{n})$, where
$0<\varrho_{j}\leq\varrho_{k}$ for $j\leq k$. Suppose $AN=NA$. Can the above
symplectic matrix $S$ be chosen to satisfy $NS=SN$ ?

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School of Mathematical Sciences, Beijing Normal University
---
Laboratory of Mathematics and Complex Systems, Ministry of Education
Beijing 100875, The People’s Republic of China
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