\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{-6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1999/16\hfil PERSISTENCE OF INVARIANT MANIFOLDS \hfil\folio} \def\leftheadline{\folio\hfil Chongchun Zeng \hfil EJDE--1999/16} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations, Vol. {\eightbf 1998}(1998), No.~16, pp.~1--13.\hfil\break ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfill\break ftp ejde.math.swt.edu (login: ftp)\bigskip} } \topmatter \title PERSISTENCE OF INVARIANT MANIFOLDS FOR PERTURBATIONS OF SEMIFLOWS WITH SYMMETRY \endtitle \thanks {\it 1991 Mathematics Subject Classifications:} 58F15, 58F35, 58G30, 58G35, 34C35.\hfil\break\indent {\it Key words and phrases:} Semiflow, invariant manifold, symmetry. \hfil\break\indent \copyright 1999 Southwest Texas State University and University of North Texas. \hfil\break\indent Manuscript received in April 1995, revised version April 6, 1999. Published May 18, 1999. \hfil\break\indent The inordinate delay was due to an oversight by editor P.W. Bates, for which he offers \hfil\break\indent his apologies. \endthanks \author Chongchun Zeng \endauthor \address Department of Mathematics, Brigham Young University, Provo, UT 84602, USA \endaddress \email zengc\@math.byu.edu \endemail \abstract Consider a semiflow in a Banach space, which is invariant under the action of a compact Lie group. Any equilibrium generates a manifold of equilibria under the action of the group. We prove that, if the manifold of equilibria is normally hyperbolic, an invariant manifold persists in the neighborhood under any small perturbation which may break the symmetry. The Liapunov-Perron approach of integral equations is used. \endabstract \endtopmatter \document \head I. Introduction \endhead In the study of dynamical systems in finite-dimensional spaces, the theory of invariant manifolds has proved to be an important tool. Invariant manifolds, along with invariant foliations, can be used to construct coordinate systems in which the differential equations are partially decoupled. These coordinate systems are very useful in tracking the asymptotic behavior of orbits in neighborhoods of equilibria. In recent years, the theory of invariant manifolds has been generalized to semiflows in Banach spaces. See, for example, [BJ], [Ca], [CH], [CL1], [CL2], [H], [He], [Ke], [MS], [BLZ] and others. Here we extend some of these results to the case where an infinite-dimensional dynamical system is invariant under the action of a smooth Lie group in such a way that an equilibrium gives rise to a manifold of equilibria through the group action. The principal question addressed here is, what happens to this manifold when the system is perturbed, possibly breaking the symmetry in the system? Let $X$ be a Banach Space. Suppose $S(t)$ is a semiflow generated by a semilinear equation in $X$ and suppose that it is invariant under the action in X of a connected compact symmetry group $G$. If the origin $0$ is an equilibrium and the group $G$ acts at $0$ in a nondegenerate way, then the image of $0$ under the group action is a manifold of equilibria, diffeomorphic to $G$. Here we establish the persistence of this manifold under small perturbations of the system provided the manifold is normally hyperbolic. One can find many examples of systems of PDE's which have an inherent symmetry arising from an idealized model. One is interested in the structural stability of such systems and in the behavior of solutions to a perturbed system. An example may be found in the work by Bates [Ba] and Barrow \& Bates [BB1], [BB2], [BB3], where periodic traveling waves for a Ginzburg-Landau system are considered and the unperturbed system is invariant under the group $O(2)\times O(2)$. In order to prove the persistence, we require that the center subspace of the linearized equation at $0$ coincides with the tangent space of the manifold of equilibria, that is, the manifold is normally hyperbolic. For finite-dimensional dynamical systems, Fenichel [F1] and, independently, Hirsch, Pugh, and Shub [HPS] proved that compact normally hyperbolic invariant manifolds persist under small perturbations. Ma\~n\'e [Mn] proved that normal hyperbolicity is also a necessary condition. In infinite-dimensional spaces, Henry [He] proved the persistence of normally hyperbolic invariant manifolds which are graphs of maps from closed linear subspaces to their complementary subspaces. A more general result can be found in [BLZ]. Traditionally, there are two methods dealing with invariant manifolds. One dates back to Hadamard [Ha] and the other to Liapunov [Ly] and Perron [Pe]. The Hadamard approach, which is also called the graph-transform method, is more geometric, while the Liapunov-Perron method is more analytic and the strategy is to finds the manifolds as fixed points of some integral equations. In this work, we use the analytic method of Liapunov-Perron. Consider the equation $$u_t = F(u), \tag{1}$$ where $$F(u)=Au + f(u)$$ and $t>0$. The operator $A$, defined on a dense subspace $D(A)$ of $X$, is the generator of a strongly continuous semigroup $T(t)$ on $X$. Assume that $f$ is Lipschitz on $X$ and such that {\narrower\smallskip\noindent $f(0)=0$ and for any $\theta > 0$, there exists a neighborhood $U$ of $0$, such that Lip~$f|_U < \theta$.\smallskip} \noindent Thus, $A$ is the linear part of the right side of (1). Following a standard result, (See, for example, page 184, [Pa]) (1) determines a $C_0$ semiflow $S(t)$ on $X$, i.e. $S: [0, \infty)\times X\to X$ is continuous in both variables and $$S(t_1)\circ S(t_2)=S(t_1+t_2)$$ for all $t_1, t_2\in [0, \infty)$. Let $G$ be an $n$-dimensional connected compact Lie group and assume that $G$ acts smoothly ($C^2$) on $X$ and $D(A)$ is invariant under the action of $G$. Furthermore, assume that the semiflow generated by (1) is also $G$-invariant, i.e., $$S(t)(gu)=g(S(t)u)\tag 2$$ for all $t \ge 0, \, g\in G$ and $u\in X$. Throughout this paper, we shall use $u, v$ and so on to denote elements in $X$ and $g,h$ and so on to denote elements in $G$. With a slight abuse of notation, for an element $g\in G$, we also use $g$ to represent the transformations on $X$ defined by the group action and the left transformations $L_g$ on $G$ defined by $L_g h =gh$ for all $h \in G$. The invariance of the semiflow $S(t)$ under action of $G$ can be written in another form: for all $u \in D(A)$ and $g\in G$, $$F(gu)= Dg(u)F(u), $$ where $Dg(u)v \equiv lim_{h \to 0}\frac{g(u+hv)-g(u)} h$ is the derivative of the action of $g$ on $X$. Let $\bar{\phi}:G \times X \to X$ be the action, i.e. $\bar{\phi}(g,x) = gx$, which is $C^2$ on $G \times X$. Let $\phi_x = \bar{\phi}|_{G \times \{x\}}$ for $x \in X$ so that $\phi_x$ is a smooth map from $G$ to $X$. Assume that $\phi_0$ is one-to-one and $D\phi_0(e)$ is of rank $n$ (where $D\phi_x = D_G\bar{\phi}(g,x)$). \proclaim{Lemma 1} $G(0)=\phi_0(G)$ is a $C^2$ compact submanifold of $X$, which is composed of equilibria of the semiflow $S(t)$. \endproclaim \demo{Proof} Since $$\phi_x \circ g = g \circ \phi_x : G \to X,$$ then, $$D\phi_x(g) \circ Dg(e) = Dg(x) D\phi_x(e),$$ thus, $$D\phi_x(g) = Dg(x)D \phi_x(e)(Dg(e))^{-1},$$ therefore for all $ g \in G, D\phi_0(g)$ is of rank $n$. Combining this result with the fact that $\phi_0$ is one-to-one gives us the first conclusion. Since $0$ is an equilibrium of equation (1), (2) implies $\phi_0(G)$ is composed of equilibria, so it is invariant under the action of $S(t). \quad \blacksquare$ \enddemo Since $G$ is compact and $D\phi_x(g)$ is continuous in $G \times X$, there exists $\delta > 0$, such that $D\phi_x(g)$ is of rank $n$ for all $g \in G$ and $x \in B_\delta(0)$, the ball of radius $\delta$ in $X$. So, for all $h \in G, \, D\phi_{h(x)}(g)=D\phi_x(gh)\circ DR_h(g)$ is of rank n for the above $g$ and $x$, where $R_h$ is the right translation in $G$. Also, there exists $\bar M \text{ such that for all } g,h \in G$ and $ \, x \in B_\delta(h(0))$, we have $ \, 1/\bar M < \|Dg(x)\|< \bar M$. (In fact, we will use $\bar M$ as a universal upper bound.) Let $\sigma(A)$ be the spectrum of $A$. Let $\sigma_s = \{\lambda \in \sigma(A) \, | \, \text{Re } \lambda < 0\}, \ \sigma_c =\{\lambda \in \sigma(A) \, | \, \text{Re } \lambda = 0 \}, \ \sigma_u = \{\lambda \in \sigma(A) \, | \, \text{Re } \lambda > 0\}$. Assume $A$ satisfies: \roster \item $\sigma_c = \{0\}$, \item $\sigma_u$ is compact, \item There exists $\alpha > 0$ such that $\alpha < \inf \text{Re } \sigma_u$ and $-\alpha > \sup \text{Re }\sigma_s$. \endroster Therefore, we have closed subspaces $X_u, X_s, X_c$ corresponding to $\sigma_u, \sigma_s, \sigma_c$, invariant under $A$, and $X=X_u \oplus X_c \oplus X_s$, (see page 321 [TL]). Let $P_u, P_s, P_c$ be the corresponding projections. Let $P_z = I - P_c = P_u + P_s$. Assume $$X_c = D\phi_0(e) {\Cal T}_e (G),$$ where ${\Cal T}_e(G)$ is the tangent space of $G$ at $e$. Because of the previous assumption, $X_c$ is an $n$-dimensional subspace of $X$, which is the tangent space of the submanifold $\phi_0(G)$ at 0. Let $A_s = A|_{X_s}, A_u = A|_{X_u}$, and $ A_c = A|_{X_c}$. Since $G(0)$ consists of equilibria, then $F|_{G(0)}=0$. Note that $f$ is differentiable at $u=0$ and $f'(0)=0$, so that $A_c = 0$. Since $A_u$ has compact spectrum, so it is bounded and generates a group $e^{A_ut} = T_u(t) = T|_{X_u}$ satisfying $\|T_u(t)\| \le M_1e^{\alpha t}$ for $t < 0$, where $M_1 \ge 1$. Also, $A_s$ generates a $C_0$-semiflow $T_s(t) = T(t)|_{X_s}$ on $X_s$. Assume $\|T_s(t)\| \le M_2e^{-\alpha t}$ for $t > 0$, where $M_2 \ge 1$. By renorming, we may assume $M_1 = M_2 = 1$. Now we consider a perturbed equation: $$u_t = Au + f(u) + \epsilon H(u) \equiv F_\epsilon(u), \tag{3}$$ where $H(\cdot)$ is Lipschitz on $X$ with Lipschitz constant $L$. For the same reason as for equation (1), it is clear that (3) determines a $C_0$ semiflow $S_\epsilon(t)$ on $X$. Our main result is: \proclaim{Theorem} Under the above conditions, when $\epsilon$ is sufficiently small, there is a Lipschitz invariant manifold of the semiflow $S_\epsilon(t)$ near $G(0)$. \endproclaim \proclaim{Remark} The same result is true with $H(u,\epsilon)$ in place of $\epsilon H(u)$ provided that $H(u,\epsilon)$ is continuous, $H(u,0)=0$, and $H$ is Lipschitz in $u$ with the Lipschitz constant converging to $0$ as $\epsilon\to 0$. \endproclaim \proclaim{II. Proof of the Theorem} \endproclaim Define $\|\cdot\|$ on ${\Cal T}_g(G)$ as $\|v\| = \|D\phi_0(g) v\|$. We may use this norm to define a metric $d(\cdot,\cdot)$ on $G$ as the infimum of the length of the $C^1$ curves lying in $\phi_0(G)$ joining two image points under $\phi_0$ in $X$. Clearly $d(g,h) \ge \|\phi_0 g - \phi_0 h\|$. If $\|\phi_0 g_k - \phi_0 g_0\| \to 0$ as $ k \to +\infty$, there exists a neighborhood $U$ of $g_0$ and a local coordinate $\psi : U \to B_1^n(0)$, the unit ball in ${\Bbb R}^n$, such that $\psi(g_0) = 0$. Since $\phi_0(G)$ is a compact submanifold of $X$, and in particular, a proper submanifold of $X$, therefore $\phi_0 g_k \to \phi_0 g_0 \text{ in } X$, implies $g_k \to g_0 $ in $G$. Suppose $g_k \in \psi^{-1} (B_{\frac 12}^n(0))$. Since $\phi_0 \circ \psi^{-1}$ is a diffeomorphism and $\|D\phi_0 \circ \psi^{-1}\|$ on $B_{\frac 12}^n(0)$ is bounded, it follows that $d(g_k, g_0) \to 0$. So, $d(\cdot,\cdot)$ induces the same topology on $\phi_0(G)$ as that inherited from $X$ and $\phi_0(G)$ is diffeomorphic to $G$. From this, there exits a constant $C_0$ such that $\|\phi_0g - \phi_0h \| \leq d(g,h) \leq C_0\|\phi_0g - \phi_0h\|$ for all $g,h \in G$. Suppose the diameter of $G$ under $d(\cdot, \cdot)$ is $M > 0$. Define $Y = G \times (X_s \oplus X_u), \phi = \bar{\phi}|_Y$ for $(g, x) \in Y, \, v \in {\Cal T}_g(G), \, z \in X_s \oplus X_u$, $$\split D \phi(g, x)(v, z) &= D\phi(g, x)(v, 0) + D\phi(g, x)(0, z) \\ &= D\phi_x(g) v + Dg(x)(z)\\ &= Dg(x) D\phi_x(e) (Dg(e))^{-1} v + Dg(x)(z) \\ &= Dg(x) (D\phi_x(e)(D g(e))^{-1} v + z), \endsplit \tag 4$$ as in the proof of Lemma 1. Since $$\align X &= X_c \oplus X_s\oplus X_u \\ &= D\phi_0(e) {\Cal T}_e(G) \oplus X_s \oplus X_u \\ &= D\phi_0(e) (Dg(e))^{-1} {\Cal T}_g(G) \oplus X_s \oplus X_u, \endalign$$ so, by the argument following Lemma 1, there exists $ \delta > 0$ such that for $(g, x) \in Y_\delta =\{(g, x) \in Y : \, \|x\| \leq \delta\}, \, D\phi(g, x)$ is one-to-one and onto from ${\Cal T}_gG\times (X_u \oplus X_s)$ to $X$. ${\Cal T}_gG\times (X_u \oplus X_s)$ may be identified with ${\Cal T}_{(g,x)}Y$, the direct sum of ${\Cal T}_gG$ and $X_u \oplus X_s$, with norm given by the sum of the two norms on ${\Cal T}_gG$ and $X_u \oplus X_s$. Thus, $D\phi(g, x)$ is an isomorphism between ${\Cal T}_{(g,x)}Y$ and $X$. With this norm on ${\Cal T}_{(g, x)}Y$ we may extend the metric $d$ on $Y$ in the natural way. It is easy to verify that $d((g_1, x_1), (g_2,x_2)) = d(g_1, g_2) + \|x_1 - x_2\|$ is a metric on $Y$. Next we prove that when $\delta$ is small enough $\phi : Y_\delta \to X$ is one-to-one. Otherwise, there exists $ (g_k, x_k), (h_k, y_k) \in Y$ with $ x_k \to 0$ and $ y_k \to 0$ such that $g_k x_k = h_k y_k$, which implies $h_k^{-1}g_kx_k= y_k$. Since $G$ is compact, without loss of generality, suppose that $h_k^{-1}g_k$ converges to $g$. Let $k \to \infty$, we get $g(0) = 0$, so, $g = e $, which implies $h_k^{-1}g_k \to e, h_k^{-1}g_kx_k = y_k$. But $D\phi(e, 0)$ is an isomorphism so, by the Inverse Function Theorem, $\phi$ is a local diffeomorphism near $(e,0)$. So, for $k$ sufficiently large, $ h_k^{-1}g_k= e$ and $x_k = y_k$, which is a contradiction. Therefore, there exists $\delta > 0$ such that $\phi : Y_\delta \to X$ is one-to-one. By Inverse Function Theorem, $\phi$ is a diffeomorphism from $Y_\delta$ to $\phi(Y_\delta)$, an open subset containing $\phi_0(G)$. For $g \in G$ define $g : Y\to Y$ as $g(h, x) = (gh, x)$. Note that $g \circ \phi = \phi \circ g$, where the $g$ on the left side denotes the action on $X$ and the $g$ denotes the transformation on $Y$. Define $$\align &\pi_1 : Y \to G, \, \pi_1(g, x) = g, \\ & \pi_2^+ : Y \to X_u, \, \pi_2^+(g, x)= P_ux, \\ & \pi_2^- : Y \to X_s, \,\pi_s^-(g, x) = P_sx, \\ & \pi_2 = \pi_2^+ +\pi_2^-. \endalign$$ These projections are clearly smooth. Since $\phi$ is smooth and $G$ is compact, from (4), it is easy to find constants $a_1, \, \delta > 0$ such that for all $(g, x) \in Y_\delta$, $$ a_1 \ge \|D\phi(g, x)\|, \, \|D\phi(g, x)^{-1}\|. \tag 5$$ So, for $(g_1, x_1), (g_2, x_2) \in Y_\delta,$ $$ \|\phi (g_1, x_1) - \phi(g_2, x_2)\|/a_1 \le d((g_1, x_1), (g_2, x_2)) \le a_1 \|\phi(g_1, x_1) - \phi(g_2, x_2)\|.$$ Now we pull back (1) and (3) through $\phi$ on $Y_\delta$: $$\tilde F(g, x) = (D\phi)^{-1} F(\phi(g, x)), \tag 6$$ $$\tilde F_\epsilon(g, x) = (D\phi)^{-1} F_\epsilon(\phi(g, x)), \tag 7$$ for $x \in X_s \oplus X_u \cap D(A)$ and $g\in G$. Let $\eta_\delta$ be a Lipschitz cut-off function such that $\eta_\delta$: $[0, +\infty) \to [0,1], \ \eta_\delta = 1$ on $[0, \frac\delta 2], \ \eta_\delta = 0$ on $[\delta, +\infty)$, and $\ 0 \le Lip\eta \le \frac 4\delta$. In fact, we will consider $\eta_\delta(\|x\|)\tilde F $ and $\eta_\delta(\|x\|)\tilde F_\epsilon $ instead of $\tilde F$ and $\tilde F_\epsilon$, but for simplicity, we will just write $\tilde F$ and $ \tilde F_\epsilon$. With this notation, $\tilde F $ and $ \tilde F_\epsilon$ are defined on all of $Y$. By definition $\tilde F(g, 0) =0$. Since $g \circ \phi = \phi \circ g$, $$\split \tilde F(g, x) &= (D\phi)^{-1} F(gx) = (D\phi)^{-1} Dg(x) F(x)\\ &= D(\phi^{-1} \circ g)(x) F(x) = D(g \circ \phi^{-1})( x) F(x) \\ &= Dg(e, x) D\phi^{-1}(x) F(x) = Dg(e, x) \tilde F(e, x). \endsplit \tag 8$$ So, $\tilde F$ is invariant under $G$. Let $\tilde F_1 = D \pi_1 \tilde F, \ \tilde F_2 = D \pi_2 \tilde F, \ \tilde F_2^+ = D \pi_2^+ \tilde F $ and $ \tilde F_2^- = D\pi_2^- \tilde F$. Similarly, define $\tilde H_1, \tilde H_2, \tilde H_2^+, \tilde H_2^-, \tilde F_{\epsilon, 1}, \tilde F_{\epsilon, 2}, \tilde F_{\epsilon, 1}^+$ and $\tilde F_{\epsilon, 2}^+$. Identity (8) and $\pi_1 \circ g =g \circ\pi_1$ imply that $$\split \tilde F_1(g, x) &= D\pi_1 \tilde F(g, x) = D \pi_1 Dg \ \tilde F(e, x)\\ &= Dg D \pi_1 \tilde F(e, x) = Dg \tilde F_1(e, x),\endsplit \tag 9$$ and $\pi_2 \circ g = \pi_2$ implies that $$\tilde F_2(g, x) = D \pi_2 Dg \tilde F(e, x) = \tilde F_2(e, x). \tag 10$$ So $F_2$ is independent of the first component and we can write $\tilde F_2(x)$ for $x \in (X_s \oplus X_u) \cap D(A)$. Also, $$\split \tilde F_2(x) &= D \pi_2 (D \phi(e, x))^{-1} F(x) = D \pi_2 (D \phi(e,x))^{-1} (Ax+ f(x)) \\ &= Ax + D \pi_2 (D \phi(e,x))^{-1}f(x) = Ax + \tilde f_2(x), \endsplit \tag 11$$ where $\tilde f_2(x)= D\pi_2 (D \phi (e,x))^{-1}f(x)$. From (9), $$\split \tilde F_1(g, x) &= Dg(e) D \pi_1(e, x) (D \phi(e, x))^{-1} F(x)\\ &= Dg(e) D \pi_1(e, x) (D \phi(e,x))^{-1}(Ax + f(x))\\ &=Dg(e) D \pi_1(e, x) (0, Ax) + Dg(e) \tilde f_1(x) = Dg(e) \tilde f_1(x), \endsplit \tag 12$$ where $\tilde f_1(x)= D\pi_1 (D\phi(e,x))^{-1}f(x)$. Let $$\align Q_{\epsilon, 1}(g, x) &= \tilde F_1(g, x) + \epsilon \tilde H_1(g, x) = Dg(e) \tilde f_1(x) + \epsilon \tilde H_1(g, x), \tag 13\\ Q_{\epsilon, 2}(g, x) &= \tilde f_2 (x) + \epsilon \tilde H_2(g, x). \tag 14 \endalign$$ Similarly, define $Q_{\epsilon, 2}^+ \ Q_{\epsilon, 2}^-$. All this quantities are defined on $Y_\delta$, the image of a tubular neighborhood of the manifold $G(0)$ under $\phi$. In the rest of the paper, we shall only work in $Y_{\frac \delta2}$. Let $A_2=A_u \oplus A_s$. Consider $$\align g' &= \tilde F_1(g, x) + \epsilon \tilde H_1(g, x) = Q_{\epsilon, 1}(g, x), \tag 15 \\ x' &= A_2x + \tilde f_2(x) + \epsilon \tilde H_2(g, x) = A_2x + Q_{\epsilon, 2}(g, x). \tag 16 \endalign $$ This system is equivalent to (3) on $Y_{\delta/2}$. Equation (16) can be written as $$\align \dot x^+ &= A_ux^+ + Q_{\epsilon,2}^+(g, x^+, x^-), \tag 17\\ \dot x^- &= A_sx^- + Q_{\epsilon,2}^-(g, x^+, x^-), \tag 18 \endalign$$ where $x^+=P_ux$ and $x^-=P_sx$. Notice by (5) that $\tilde H$, $\tilde H_1$, $\tilde H_2$, $\tilde H_2^+$, and $\tilde H_2^-$, are still Lipschitz functions in $Y_{\frac \delta2}$ and the Lipschitz constants are independent of $\epsilon $ and $ \delta$. Let $\bar M$ be a universal upper bound of $\text{Lip}D\phi, \|Dg(e)\|, \|D\pi\|, \tilde H$, the norms of $\tilde H_1$, $\tilde H_2$, $\tilde H_2^+$, and $\tilde H_2^-$, and their Lipschitz constants on $Y_{\delta/2}$ and also bigger than $a_1$ in (5). In fact, we have used $\bar M$ as an upper bound of $\|Dg\|$ before. In the following $L(\delta)$ always will denote a quantity, which depends on $x$ and $g$ such that $L(\delta) \to 0$ as $\delta \to 0$ uniformly in $x$ and $g$. Then we have $$\align \|Q_{\epsilon, 1}(g, x)\| &\le \epsilon \bar M + L(\delta)\|x\|, \tag 19\\ \|Q_{\epsilon, 2}(g, x)\| &\le \epsilon \bar M + L(\delta)\|x\|. \tag 20 \endalign$$ Since $$\align \tilde f_2(x_1) &- \tilde f_2(x_2) = D \pi_2 D \phi^{-1}(x_1) (f(x_1) - f(x_2))\\ &+ (D \pi_2 D \phi^{-1}(x_1) - D \pi_2 D\phi^{-1}(x_2)) f(x_2), \endalign$$ we have $$\|\tilde f_2(x_1) - \tilde f_2(x_2)\| \le \|x_1 - x_2\| L(\delta),\tag 21$$ In the same way we get $$\align \|\tilde H_2(g_1, x_1) &- \tilde H_2(g_2, x_2)\| \le \bar M (d(g_1, g_2) + \|x_1 - x_2\|)\\ &= \bar M d((g_1, x_1), (g_2, x_2)). \tag 22 \endalign$$ So, $$\align \|Q_{\epsilon, 2}(g_1, x_1) &- Q_{\epsilon, 2}(g_2, x_2)\| \le (\epsilon \bar M + L(\delta)) \|x_1 - x_2\|\\ &+ \epsilon \bar M d(g_1, g_2), \tag 23 \endalign$$ where all the above conclusions hold in $Y_{\delta/2}$. Let $$\align Z^+ &= \left\{\gamma^+ : G \to X_u|\, \|\gamma^+(e)\| \le \frac \delta 8 \text{ and } \gamma^+ \text{ is Lipschitz with } Lip\gamma^+\le \frac {\delta}{8M}\right\},\\ Z^- &= \left\{\gamma^- : G \to X_s| \, \|\gamma^-(e)\| \le \frac \delta 8 \text{ and } \gamma^- \text{ is Lipschitz with } Lip\gamma^-\le \frac{\delta}{8M}\right\}. \endalign$$ Recall that $M$ is the diameter of $G$. Define $|\cdot|$ on $Z^+, Z^-$ as $|\gamma^\pm| = \max_{g \in G}\|\gamma^\pm(g)\|$. Define $|\cdot|$ on $Z^+\times Z^-$ as $|\gamma| = |\gamma^+| + |\gamma^-| $. It is not hard to verify that $ Z^+, Z^-, \ Z^+ \times Z^-$ are complete. We shall define a contraction mapping $E$ on $Z^+\times Z^-$ so that the graph of its fixed point is the unique invariant manifold in $Y_{\frac \delta2}$ for system (15) and (16). The transformation $E=(E^+, E^-)$ is defined in the following way. For any fixed $\gamma\in Z^+\times Z^-$, we substitute $\gamma(g)$ into equation (15) and obtain a vector field on $G$ which depends on $\gamma$. For any initial point $g_0\in G$, the trajectory $g(t)$ of this vector field exists for all $t\in (\infty, \infty)$. Next, substitute $g(t)$ and $\gamma(g(t))$ into the high order part of equations (17) and (18) and derive two unautonomous equations for $x^+$ and $x^-$, respectively. Solve equation (17) and (18) with zero value at $\infty$ and $-\infty$, respectively. Suppose $x^+(t)$ and $x^-(t)$ are solutions, then we define $E^+\gamma(g_0)=x^+(0)$ and $E^-\gamma(g_0)=x^-(0)$. Finally, we verify that $E$ is a contraction and the graph of its fixed point is an invariant manifold. Take any $\gamma \in Z^+ \times Z^-$. The argument before Lemma 4 depends on the choice of $\gamma$. Since $\gamma(g) \in Y_{\delta/2}$, $$\dot g = Q_{\epsilon, 1}(g, \gamma(g)) \tag 24$$ defines a vector field on $G$. \proclaim{Lemma 2}Suppose $g_1, g_2\in G$, and $g_1(t), g_2(t)$ are solutions of (24) with initial values $g_1$ and $g_2$, respectively. We have $$d(g_1(t), g_2(t)) \le e^{C_3(\epsilon,\delta)t} d(g_1, g_2),$$ where $$C_3(\epsilon,\delta)=\epsilon \bar M + L(\delta)\delta+ (\epsilon \bar M + L(\delta)) \frac{\delta}{4M}$$ is a constant independent of $\gamma$, $g_1$, and $g_2$. \endproclaim When (24) is vector field on $R^n$, this estimate follows directly from the Gronwall's Inequality. Here, the difficulty is that $G$ only has a Finsler structure on it. \demo{Proof} Let $g(t)$ be the solution of (24) with initial point $g_0$. Obviously, $g(t)$ depends on $\gamma$. By (19) $$\align d(g(t), e) &\le d(g_0, e) + \int_0^t \|Q_{\epsilon, 1}(g(s), \gamma(g(s))) \|ds\\ &\le d(g_0, e) + \int_0^t( \epsilon \bar M + L(\delta) \| \gamma(g(s))\|) \, ds\\ &\le d(g_0, e) + \epsilon \bar M t + \int_0^t L(\delta) \frac{\delta}{4M} d(g(s), e) + L(\delta) \|\gamma(e)\| \, ds\\ &\le d(g_0, e) + (\epsilon \bar M + L(\delta) \|\gamma(e)\|)t + \int_0^t L(\delta) \frac{\delta}{4M} d(g(s), e)ds. \endalign$$ Let $$C_1(\delta) = \frac{L(\delta)\delta}{4M}, \qquad \dsize C_2(\epsilon, \delta) = \epsilon \bar M + L(\delta)\frac{\delta}{4M}.$$ By Gronwall's inequality, $$ d(g(t), e) \le d(g_0, e) e^{C_1(\delta)t} + \frac{C_2(\epsilon, \delta)}{C_1(\delta)} (e^{C_1(\delta)t} - 1).\tag 25$$ For $t < 0$ we have the same estimate by changing $t$ to $|t|$. For $g_1, g_2 \in G$, let $g_1(t), g_2(t)$ be solutions of (24) with initial values $g_1$ and $g_2$. (We use $Q_{\epsilon, 1}(g)$ to denote $Q_{\epsilon,1}(g,\gamma(g))$.) Before we go further, we do some more estimates. For $(g_1, x_1), \ (g_2, x_2) \in Y_{\delta/2}$, we have $$\align \|D &\phi_0(g_1) \tilde H_1(g_1, x_1) - D \phi_0(g_2) \tilde H_1(g_2, x_2)\|\\ &= \|D \phi_0(g_1) D \pi_1(g_1, x_1) D \phi^{-1}(g_1 x_1) H(g_1x_1)\\ &- D \phi_0(g_2) D \pi_1(g_2, x_2) D \phi^{-1}(g_2 x_2) H(g_2x_2)\|\\ &\le \|D\phi_0(g_1) D \pi_1(g_1, x_1) D \phi^{-1} (g_1 x_1)\\ &\quad - D \phi_0(g_2) D \pi_1(g_2, x_2) D \phi^{-1}(g_2 x_2)\| \cdot \| H(g_1x_1)\|\\ &+ \|D \phi_0(g_2) D \pi_1(g_2, x_2) D \phi^{-1}(g_2 x_2)\| \ \|H(g_1x_1) - H(g_2x_2)\|\\ &\le \bar M(d(g_1, g_2) + \|x_1 - x_2\|), \tag 26 \endalign$$ and $$\align \|D \phi_0(g_1) &\tilde F_1(g_1, x_1) - D\phi_0(g_2) \tilde F_1(g_2, x_2)\|\\ &\le \|D \phi_0(g_1) Dg_1(e) D \pi_1(e, x_1) D \phi^{-1}(x_1)f(x_1)\\ &\qquad - D \phi_0(g_2) Dg_2(e) D \pi_1(e, x_2) D \phi^{-1}(x_2) f(x_2)\|\\ &\le \bar M \|x_1 - x_2\| L(\delta) + L(\delta) \|x_1\| \cdot \\ & \qquad \|D\phi_1(g_1) D g_1(e) D \pi_1(e, x_1) D \phi^{-1}(x_1)\\ &\qquad- D \phi_0(g_2) D g_0(e) D \pi_1(e, x_2) D \phi^{-1}(x_2)\|\\ &\le \|x_1 - x_2\| L(\delta) + \bar M L(\delta) \delta(d(g_1, g_2) + \|x_1 - x_2\|)\\ &\le L(\delta) \|x_1 - x_2\| + L(\delta) \delta d(g_1, g_2). \tag 27 \endalign$$ So, $$\align \|D \phi_0 Q_{\epsilon, 1}(g_1, x_1) -&D \phi_0 Q_{\epsilon, 1}(g_2, x_2)\| \le (\epsilon \bar M + L(\delta) \delta) d(g_1, g_2)\\ &+ (\epsilon \bar M + L(\delta)) \|x_1 - x_2\|. \tag 28 \endalign$$ For all $ \, a > 0$ take a smooth curve $c(r)$ on $G, r \in [0, 1]$, such that $c(0) = g_1, c(1) = g_2$ and $\dsize d(g_1, g_2) \ge \int_0^1 \|c'(r)\| dr - a$. Let $g(t, r)$ denote the solution of (24) with initial value $c(r)$. If $\gamma$ and $Q_{\epsilon,1}$ are smooth, by (28), we have $$\align \int_0^1 &\left|\left|\frac{\partial}{\partial r} \, g(t, r)\right|\right| dr \le \int_0^1 \|c'(r)\|dr + \int_0^1 \left|\left|\int_0^t \ \frac{\partial}{\partial s} \ \frac{\partial}{\partial r} \, g(s, r)ds\right|\right|dr\\ &\le \int_0^1 \|c'(r)\|dr + \int_0^1 \int_0^t \left|\left|\frac{\partial}{\partial r} \, Q_{\epsilon,1}(g(s,r))\right|\right| dsdr \\ &\le \int_0^1 \|c'(r)\|dr + \int_0^t \int_0^1 \left|\left|DQ_{\epsilon,1} \left(\frac{\partial}{\partial r} \, g(s, r), D\gamma (\frac{\partial}{\partial r} \, g(s, r))\right)\right|\right|drds\\ & \le \int_0^1 \|c'(r)\|dr+ \int_0^t \int_0^1 \left(\epsilon \bar M + L(\delta)\delta + (\epsilon \bar M + L(\delta)) \frac{\delta}{4M}\right) \left|\left|\frac{\partial}{\partial r} \, g(s, r)\right|\right|drds. \endalign$$ Let $\dsize C_3(\epsilon, \delta) = \epsilon \bar M + L(\delta)\delta + (\epsilon \bar M + L(\delta)) \frac{\delta}{4M}$. So, $$ \int_0^1 \left|\left|\frac{\partial}{\partial r} \, g(t, r)\right|\right| dr \le \int_0^1 \|c'(r)\|dr + C_3(\epsilon, \delta) \int_0^t \int_0^1 \left|\left|\frac{\partial}{\partial r} \, g(s, r)\right|\right| drds,$$ which implies, $$ \int_0^1 \left|\left|\frac{\partial}{\partial r} \, g(t,r)\right|\right| dr \le e^{C_3(\epsilon, \delta)t} \int_0^1 \|c'(r)\|dr \tag 29$$ Now not all of them are smooth but they are Lipschitz and that is enough. We still have the same estimate. Therefore, $$ d(g_1(t), g_2(t)) \le \int_0^1 \left|\left|\frac{\partial}{\partial r} \, g(t, r)\right|\right| dr \le e^{C_3(\epsilon, \delta)t} (d(g_1, g_2) + a),$$ and thus, $$d(g_1(t), g_2(t)) \le e^{C_3(\epsilon, \delta)t} d(g_1, g_2). \tag 30 $$ \enddemo We will define a map $E=(E^+,E^-)$ on $Z^+ \times Z^-$. For brevity, below we use $Q(g(s))$ to denote $Q(g(s),\gamma (g(s)))$ and define $$\align E^+ \gamma(g_0) &=-\int_0^{+\infty} T_u(-s) Q_{\epsilon, 2}^+ (g(s))ds,\tag 31\\ E^- \gamma(g_0) &= \int_{-\infty}^0 T_s(-s) Q_{\epsilon, 2}^- (g(s))ds, \tag 32 \endalign$$ where $g(s)$ is the solution of (24) with initial value $g_0$ and recall that $T_u, T_s$ are the semiflows generated by $A_u, A_s$. \proclaim{Lemma 3}$E$ maps $Z^+\times Z^-$ to itself if $$\frac{\epsilon \bar M + L(\delta) \delta}{\alpha} < \frac \delta 8 \tag C1$$ and $$\frac{C_4(\epsilon, \delta)}{\alpha - C_3(\epsilon, \delta)} \le \frac{\delta}{8M} \tag C2$$ hold. \endproclaim \demo{Proof} First we verify that $E^+$ and $E^-$ are well-defined. Since $\dsize |\gamma| \le \frac \delta 2$ and by (20), $\|Q_{\epsilon, 2}^\pm (g(s))\|$ is bounded, along with the condition on $T_s, T_u$, we see that $E$ is well-defined. Second we prove $E \gamma \in Z^+ \times Z^-$ under proper conditions. By (20), $$\|E^+ \gamma(e)\| \le \int_0^{+\infty} e^{-\alpha s} \left(\epsilon \bar M + L(\delta) \frac \delta 2\right) ds \le \frac{\epsilon \bar M + L(\delta) \frac \delta 2}{\alpha}.$$ Similarly $$\|E^- \gamma(e)\| \le \int_{-\infty}^0 e^{\alpha s} \left(\epsilon \bar M + L(\delta) \frac \delta 2\right) ds \le \frac{\epsilon \bar M + L(\delta) \frac \delta 2}{\alpha}.$$ So, if condition (C1) holds, then $\dsize \|E^+ \gamma(e)\| \le \frac \delta 8 $ and $ \ \|E^- \gamma(e)\| \le \frac \delta 8$. For $g_1, g_2 \in G$, let $g_1(t), g_2(t)$ denote the solution of (24) with initial data $g_1, g_2$, respectively. Then by (23), $$\align &\|E^+ \gamma(g_1) - E^+ \gamma(g_2)\| \le \int_0^{+\infty} e^{-\alpha s} \|Q_{\epsilon, 2}^+ (g_1 (s)) - Q_{\epsilon, 2}^+ (g_2(s))\| ds\\ &\le \int_0^{+\infty} e^{-\alpha s} (\epsilon \bar M(d(g_1(s), g_2(s)) + (\epsilon \bar M + L(\delta)) \frac{\delta}{4M} d(g_1(s), g_2(s)))ds\\ &= \int_0^{+\infty} e^{-\alpha s} \left(\epsilon \bar M + \epsilon \delta \frac{\bar M}{4M} \, + \frac {L(\delta)\delta}{4M}\right) d(g_1(s), g_2(s)) ds. \endalign$$ Let $\dsize C_4(\epsilon, \delta) = \epsilon \bar M + \epsilon \delta \, \frac{\bar M}{4M} + \frac{L(\delta)\delta}{4M}$. By (30) $$\align \|E^+\gamma(g_1)-E^+\gamma(g_2)\| &\\ &\le \int_0^{+\infty} C_4(\epsilon, \delta) e^{-(\alpha - C_3(\epsilon, \delta))s} d(g_1, g_2)ds\\ & = \frac{C_4(\epsilon, \delta)}{\alpha - C_3(\epsilon, 2)} d(g_1, g_2). \tag 33 \endalign$$ The same is true for $\|E^- \gamma(g_1) - E^- \gamma(g_2)\|$. Therefore, if condition (C2) holds then $E\gamma$ satisfies the condition on the $C_0$ norm and Lipschitz constant and $E \gamma \in Z^+ \times Z^-$. \enddemo Finally, we prove \proclaim{Lemma 4} $E$ is a contraction if conditions (C1), (C2), and $$\epsilon \bar M + L(\delta) \left( \frac 1\alpha + \frac{C_4(\epsilon, \delta)}{C_3(\epsilon, \delta)} \, \frac{1}{\alpha - C_3(\epsilon, \delta)} \right) < \frac 12 \tag C3 $$ hold. \endproclaim \demo{Proof} For $\gamma_1, \gamma_2 \in Z^+ \times Z^-$ let $g_1(t), g_2(t)$ be the solutions of (24), with initial value $g_0$, and with $\gamma$ replaced by $\gamma_1, \gamma_2$, respectively. Then by (23), $$\align \|&E^+ \gamma_1(g_0) - E^+ \gamma_2(g_0)\| \le \int_0^{+\infty} e^{-\alpha s} \|Q_{\epsilon, 2}^+ (g_1(s), \gamma_1(g_1(s)))\\ &\qquad \quad - Q_{\epsilon, 2}^+ (g_2(s), \gamma_2(g_2(s)))\| ds\\ &\le \int_0^{+\infty} e^{-\alpha s} (\epsilon \bar M d(g_1(s), g_2(s)) + (\epsilon \bar M + L(\delta))(\|\gamma_1(g_1(s)) - \gamma_1(g_2(s))\| \\ &\qquad \quad + \|\gamma_1(g_2(s)) - \gamma_2(g_2(s))\|))ds \\ &\le \int_0^{+\infty} e^{-\alpha s} ((\epsilon \bar M + (\epsilon \bar M + L(\delta)) \frac{\delta}{4M}) \, d(g_1(s), g_2(s)) \\ &\qquad \quad + (\epsilon \bar M + L(\delta)) |\gamma_1 - \gamma_2|)ds. \tag 34 \endalign $$ Let $\gamma_{(r)} = (2-r)\gamma_1 + (r-1)\gamma_2, \ r \in [1, 2]$, which is a homotopy between $\gamma_1$ and $\gamma_2$ and $\gamma_{(1)} = \gamma_1,$ and $\gamma_{(2)} = \gamma_2$. Let $g(t,r)$ denote the solution of (24) with $\gamma =\gamma_{(r)}$ and initial data $ g(0, r) = g_0$. Similar to the derivation of (30) we find $$\align \int_1^2 &\left|\left|\frac{\partial}{\partial r} \, g(t, r)\right|\right| dr \le \int_1^2 \int_0^t \left|\left|\frac{\partial}{\partial r} \frac {\partial}{\partial s} \, g(s, r)\right|\right| dsdr\\ &\le \int_0^t \int_1^2 \left|\left|\frac{\partial}{\partial r} \, Q_{\epsilon, 1}(g(s, r), \gamma_{(r)}(g(s, r)))\right|\right| drds\\ &\le \int_0^t \int_1^2 \left|\left| DQ_{\epsilon, 1} \left(\frac{\partial}{\partial r} \, g(s, r), \frac{\partial}{\partial r} \, ((2-r)\gamma_1 (g(s, r))\right.\right.\right.\\ &\qquad \qquad + (r-1) \gamma_2(g(s, r)))\biggr)\biggr|\biggr| drds\\ &\le \int_0^t \int_1^2 (\epsilon \bar M + L(\delta)\delta) \left|\left| \frac{\partial}{\partial r} \, g(s, r)\right|\right| + (\epsilon \bar M + L(\delta))\\ &\qquad \qquad \left( \frac{\delta}{4M} \left|\left| \frac{\partial}{\partial r} \, g(s, r)\right|\right| + \|\gamma_1(g(s, r)) - \gamma_2(g(s, r))\|\right) drds\\ &\le \int_0^t \int_1^2 (\epsilon \bar M + L(\delta))|\gamma_1 - \gamma_2| + C_3(\epsilon, \delta) \left|\left| \frac{\partial}{\partial r} \, g(s, r)\right|\right| drds\\ &\le (\epsilon \bar M + L(\delta)) |\gamma_1 - \gamma_2|t + \int_0^t\int_1^2 C_3(\epsilon, \delta) \left|\left| \frac{\partial}{\partial r} \, g(s, r)\right|\right| drds. \endalign$$ Therefore, $$\int_1^2 \left|\left| \frac{\partial}{\partial r} \, g(t, r)\right|\right| dr \le \frac{(\epsilon \bar M + L(\delta)) e^{C_3(\epsilon, \delta) t}}{C_3(\epsilon, \delta)} \, |\gamma_1 - \gamma_2|.\tag 35$$ Returning to (34), $$\align \|E^+ \gamma_1(g_0) &- E^+ \gamma_2(g_0)\| \le \int_0^{+\infty} e^{-\alpha s}(C_4(\epsilon, \delta)(\epsilon \bar M + L(\delta)) \frac {e^{C_3(\epsilon, \delta)s}}{C_3(\epsilon, \delta)}\\ & + (\epsilon \bar M + L(\delta))) |\gamma_1 - \gamma_2|ds\\ &\le (\epsilon \bar M + L(\delta)) |\gamma_1 - \gamma_2| \left( \frac 1 \alpha + \frac{C_4(\epsilon, \delta)}{C_3(\epsilon, \delta)} \, \frac{1}{\alpha - C_3(\epsilon, \delta)}\right). \tag 36 \endalign $$ Similarly $$\align \|E^- &\gamma_1(g_0) - E^- \gamma_2(g_0)\|\\ &\le (\epsilon \bar M + L(\delta)) |\gamma_1 - \gamma_2| \left( \frac 1\alpha + \frac{C_4(\epsilon, \delta)}{C_3(\epsilon, \delta)} \, \frac{1}{\alpha - C_3(\epsilon, \delta)}\right). \tag 37 \endalign $$ Therefore, $E$ is a contraction if condition (C3) holds. \enddemo Therefore there exists a unique fixed point $\gamma_0 \in Z^+ \times Z^-$. Next, we prove \proclaim{Lemma 5} $\{(g,\gamma_0(g))| \, g \in \, G \}$ is an invariant set of system (15), (17), (18). \endproclaim \demo{Proof} For $g_0 \in G$, let $g(t)$ be the solution of (24) with$\gamma = \gamma_0$. Writing $\gamma_0(g)=(\gamma_0^+(g)+\gamma_0^-(g))\in X_u \oplus X_s$, we have $$ \gamma_0^+(g_0) = - \int_0^{+\infty} T_u(-s) Q_{\epsilon, 2}^+ (g(s))ds, $$ which implies, for $\dsize t_0 > 0$, $$\split T_u(t_0) \gamma_0^+(g_0) & = -\int_0^{+\infty} T_u(t_0 - s) Q_{\epsilon, 2}^+(g(s))ds\\ & = -\int_0^{t_0} T_u(t_0- s) Q_{\epsilon, 2}^+ (g(s))ds - \int_0^{+\infty} T_u(-s) Q_{\epsilon, 2}^+(g(s + t_0)) ds\\ & = - \int_0^{t_0} T_u(t_0 - s) Q_{\epsilon, 2}^+ (g(s)) ds + \gamma_0^+ (g(t_0)). \endsplit$$ In the same way, we have $$ \gamma_0^-(g_0) = T_s(t_0) \gamma_0^-(g_0) + \int_0^{t_0} T_s(t_0 - s) Q_{\epsilon, 2}^- (g(s)) ds.$$ Therefore, $(g(t), \gamma_0(g(t)))$ is a solution of that system, which implies that $(g,\gamma_0(g))$ is an invariant manifold. \enddemo Finally, we consider the condition C1, C2, C3. $$\align \text{(C1) } &: \frac{\epsilon \bar M + L(\delta)\delta}{\alpha} < \frac \delta 8,\\ \text{(C2) } &: \frac{\delta}{8M} \ge \frac{C_4(\epsilon, \delta)}{\alpha - C_3(\epsilon, \delta)} = \frac{\epsilon \bar M + \epsilon \delta \frac{\bar M}{4M} + \frac{L(\delta)\delta}{4M}}{\alpha - \left(\epsilon \bar M + L(\delta)\delta + \left(\epsilon \bar M + L(\delta)\right) \frac{\delta}{4M}\right)},\\ \text{(C3) } &: \frac 1\alpha > \epsilon \bar M + L(\delta) \left( \frac1 \alpha + \frac{C_4(\epsilon, \delta)}{C_3(\epsilon, \delta)} \ \frac{1}{\alpha - C_3(\epsilon, \delta)}\right)\\ &\qquad = \epsilon \bar M + L(\delta) \left( \frac 1 \alpha + \frac{\epsilon \bar M + \epsilon \delta \frac{\bar M}{4M} + \frac{L(\delta)\delta}{4M}}{\epsilon \bar M + L(\delta)\delta + \epsilon \delta \frac{\bar M}{4M} + \frac{L(\delta)\delta}{4M}} \ \frac{1}{\alpha - C_3(\epsilon, \delta)}\right). \endalign$$ Note that $L(\delta)\to 0$ as $\delta \to 0$. 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