Center Manifold Theory and Computation Using a Forward Backward Approach

Transcript

1 College of William and Mary W&M Publish Undergraduae Honors Theses Theses, Disseraions, & Maser Projecs Cener Manifold Theory and Compuaion Using a Forward Backward Approach Emily E. Schaal College of William and Mary Follow his and addiional works a: hp://publish.wm.edu/honorsheses Par of he Mahemaics Commons, and he Non-linear Dynamics Commons Recommended Ciaion Schaal, Emily E., "Cener Manifold Theory and Compuaion Using a Forward Backward Approach" (217). Undergraduae Honors Theses. Paper hp://publish.wm.edu/honorsheses/1129 This Honors Thesis is brough o you for free and open access by he Theses, Disseraions, & Maser Projecs a W&M Publish. I has been acceped for inclusion in Undergraduae Honors Theses by an auhorized adminisraor of W&M Publish. For more informaion, please conac wmpublish@wm.edu.

2 Cener Manifold Theory and Compuaion Using a Forward-Backward Approach by Emily Schaal Submied o he Deparmen of Mahemaics in parial fulfillmen of he requiremens for Honors in Mahemaics a he College of William and Mary May 217 Acceped by Dr. Yu-Min Chung Dr. Sarah Day Dr. Rui Pereira

3 Acknowledgmens I wan o give many hanks o my hesis advisor, Dr. Yu-Min Chung, wihou whom none of his would have been possible, and who has gone ou of his way o encourage and assis me in my pursui of mahemaics. I d like o hank Professor Sarah Day and Professor Rui Pereira for serving on my defense commiee. I also wan o hank Professor Mike Jolly from Indiana Universiy, Bloomingon for coming up wih he idea for his projec and for coming o William and Mary o offer his advice and words of por. Finally, I d like o hank my friends here a he College and across he counry, as well as my dear family, for heir unfailing por during he ime I worked on my hesis. i

4 Absrac The cener manifold, an objec from he field of differenial equaions, is useful in describing he long ime behavior of he sysem. The mos common way of compuing he cener manifold is by using a Taylor approximaion. A differen approach is o use ieraive mehods, as presened in [12, [9, and [19. In paricular, [19 presens a mehod based on a discreizaion of he Lyapunov-Perron (L-P) operaor. One drawback is ha his discreizaion can be expensive o compue and have error erms ha are difficul o conrol. Using a similar framework o [19, we develop a forward-backward inegraion algorihm based on a boundary value problem derived from he operaor. We include he deails of he proofs ha por his formulaion; noably, we show ha he operaor is a conracion mapping wih a fixed poin ha is a soluion o he differenial equaion in our funcion space. We also show he firs sep in he inducion o prove he exisence of a C k cener manifold. We demonsrae he algorihm wih Runge-Kua (R-K) mehods of O(k) and O(k 2 ). Finally, we presen an applicaion of our algorihm o sudying a semilinear ellipic boundary value problem from [2. ii

5 Conens 1 Inroducion Background Framework Cener Manifold Theory Exisence of he Cener Manifold Regulariy of he Manifold Cener Manifold Compuaion Algorihm Applicaion Discussion Appendix Nonlinear Term Seady Sae Tables Julia Code Bibliography 69 iii

6 Chaper 1 Inroducion In his work we develop a echnique for sudying he cener manifold of a differenial sysem. When presen, knowledge of a unique cener manifold can be useful when analyzing he enire invarian manifold for a paricular sysem. Therefore, he moivaion for his projec is in is applicaion o he sudy of differenial equaions. Having an undersanding of hese equaions is inegral o he res of our work. 1.1 Background Differenial sysems model flow, or raes of change, and have imporan applicaions o fields such as biology, physics, economics, and many ohers. Informally, an ordinary differenial sysem a collecion of equaions relaing imporan quaniies and heir raes of change. Usually, his derivaive is aken wih respec o ime, represened as. Ordinary differenial equaions are ofen used o model populaion dynamics, he behavior of neurons, and oher biological processes. They also crop up in physics. For example, he Newonian equaions of moion are ordinary differenial equaions. These sysems, alhough he simpler subse of differenial equaions, can sill exhibi srange and unpredicable behavior. The Lorenz equaions [26, which were developed by Edward Lorenz o model a leaky waer wheel, is a sysem of only hree dimensions and ye exhibis chaoic behavior. Whereas ordinary differenial equaions are differeniaed wih respec o he same variable, parial differenial equaions are made up of derivaives of several variables. Consider he equaion 2 u(x, y) x u(x, y) y 2 + λu(x, y) u(x, y) 2 =, (1.1) which has second derivaives wih respec o x and y and is invesigaed in [2. Oher examples of parial differenial equaions are he wave equaion, he hea equaion, and Laplace s equaions. The complexiy of analyzing even apparenly simple parial differenial sysems makes heir sudy a mahemaical field uno iself. Any paricular differenial equaion is eiher auonomous or non-auonomous. An auonomous equaion, dy = f(y), is implicily dependen on he independen variable, meaning d ha if an ordinary differenial sysem was differeniaed wih respec o, he variable does 1

7 no appear in he equaion. On he oher hand, a non-auonomous equaion, dy = f(y, ), d is one ha is explicily dependen on he variable i was differeniaed wih respec o. For example, is a non-auonomous equaion, while dy d = y2 + (1.2) dy d = y2 is auonomous. We focus on auonomous equaions. The nex disincion is beween linear and nonlinear equaions. An equaion is linear if i is linear in he dependen variable and all of is derivaives. Oherwise, i is nonlinear. For example, dy = y + 2 d is linear, bu equaion (1.2) is nonlinear. Wheher or no an equaion is linear makes a big difference for how well we can solve i. Closed-form soluions exis for linear ordinary differenial equaions unique up o a consan. Some nonlinear sysems can be solved his way, bu mos canno. Alhough we are limied in he sysems for which we can obain full soluions, we can learn a lo by analyzing he behavior of he sysem wihou obaining is soluion. The firs sep in his process is idenifying he seady saes and he behavior of he sysem around each seady sae, which is a poin a which here is a consan soluion o he ordinary differenial equaion; y() = y. Inuiively, if a rajecory sars a a seady sae, i will say a ha seady sae forever. For example, he seady sae for he sysem ẋ = x + 1 x() = x ẏ = y y() = y is (1, ), which we find by seing ẋ = and ẏ =. If, when he sysem is perurbed off of he seady sae, i reurns o he seady sae, hen he seady sae is sable. If, on he oher hand, i moves away from and never reurns o is original posiion, he seady sae is unsable. We deermine sabiliy by finding he eigenvalues and eigenvecors of he linearized Jacobian marix for he sysem. In his case, J(1, ) = ( ( x+1) x (y) x ( x+1) y (y) y ) (1,) = ( ) 1, 1 which is already diagonal and will have eigenvecors ( 1 ) and ( 1 ). The negaive eigenvalue implies ha some rajecories will approach he seady sae, while he posiive eigenvalue implies ha some rajecories will move away from he seady sae. From his, we say ha his seady sae is a saddle node, where some rajecories ha sar in a ball of small enough radius will approach he seady sae, while oher rajecories will approach 2

8 Figure 1-1: Bifurcaion diagram of (1.3). Saddle Node Bifurcaion a µ c =. The blue line represens he evoluion of he sable seady sae s = µ and he red line represens he evoluion of he unsable seady sae x = µ. posiive or negaive infiniy. Because his sysem is linear, he behavior of he sysem around his poin describes he behavior of he enire sysem, so near enough could be anywhere in he phase space. The naure of he seady saes of a differenial sysem, and he naure of he behavior of he sysem, can change based on he variaion of a single parameer. This is a bifurcaion, and he parameer being varied is known as he bifurcaion parameer. Consider he sysem ẋ = µ x 2 x() = x (1.3) where x and µ R. By seing µ x 2 =, we can calculae he seady sae as x = ± µ, where µ is he bifurcaion parameer. When µ is negaive, here are no seady saes. When µ =, he poin x = is half-sable, which is a hybrid of he sable and unsable case. When µ is posiive, he poin x = µ is sable and he poin x = µ is unsable. We deermine he sabiliy by observing he change in sign of ẋ as we vary x around he seady sae. If, for a given µ, ẋ < when x < x and ẋ > when x > x, hen he soluion x() is increasing and he seady sae x is sable. If ẋ > when x < x and ẋ < when x > x, hen he seady sae x is unsable. If ẋ < when x < x and ẋ < when x > x, hen he seady sae x is half-sable. The evoluion of he fixed poins is described succincly in a bifurcaion diagram, which is a plo of he seady saes, x, as a funcion of µ. The bifurcaion diagram for (1.3) is in Figure 1-1. This paricular bifurcaion is called a saddle-node bifurcaion and describes when a sable and an unsable fixed poin collide and annihilae. A Hopf bifurcaion occurs when a pair of complex conjugae eigenvalues of he Jacobian marix for a sysem of a leas wo dimensions become purely imaginary. The sysem evolves from one wih an (un)sable spiral o one wih an (un)sable limi cycle. A simple example of a sysem which has a subcriical pichfork bifurcaion is ẏ 1 = βy 1 + y 3 1 y 5 1 (1.4) ẏ 2 = ω + by 2 1, (1.5) where β is he bifurcaion parameer; ω and b are oher parameers ha slighly modify he behavior of he sysem, from [33. An example bifurcaion diagram for a sysem like (1.4) is in Figure 1-2. We see in Figure 1-2 ha he sysem evolves from having an unsable spiral 3

9 o an unsable limi cycle. For a rigorous and exensive discussion of Hopf bifurcaions ha ies in wih our work, see [27. Figure 1-2: Hopf Bifurcaions, from [23 Seady saes are single poins on he invarian manifold associaed wih he differenial sysem. Informally, he invarian manifold is a se made up of all he poins such ha any rajecory in he flow ha sars a one of hese poins will only ravel o oher poins on he same se. Inuiively, if a rajecory (a skech of a paricular soluion in he phase plane) sars on he invarian manifold, i will say on i forever. There are hree disinc invarian manifolds: he sable, unsable, and cener. In a linear sysem in R n and C n, he manifolds are deermined by he eigendecomposiion of he Jacobian marix a he seady sae (for nonlinear sysems, we linearize abou he origin). For proofs of he exisence of he invarian and cener manifolds in R n for a nonlinear sysem using he Lyapunov-Perron mehod, see [6. We include a formal definiion of invarian manifold from [8: Definiion (Invarian Manifold) A se S E is called an invarian se for he differenial equaion ẋ = Ax + F (x, y, z) ẏ = By + G(x, y, z) ż = Cz + H(x, y, z) if, for each u = (x, y, z) S, he soluion u(x, y, z), defined on (, ) has is image in S. Alernaively, he rajecory passing hrough each u S lies in S. If S is a manifold, hen S is called an invarian manifold. A more in-deph discussion of invarian manifolds in Banach spaces can be found in [15. The global behavior of he sysem is simply he behavior of he invarian manifold for linear sysems and for nonlinear sysems wih sufficienly conrolled nonlinear erms. Oherwise, nonlinear erms can cause he sysem o behave badly - explosively or chaoically. We are ineresed in he invarian manifold because i ofen describes how he sysem behaves over long periods of ime. Seady saes appear on he manifold, and hus bifurcaions show up on he manifold as well. The inerial manifold is an invarian manifold ha capures all such long-erm behavior. I conains he global aracor, o which all soluions will converge a an exponenial rae. The aracor is wha guaranees ha he inerial manifold will capure he asympoic behavior of he sysem [11. A rigorous definiion and discussion of he manifold can be found in [34. 4

10 Figure 1-3: The phase porrai for (1.6), which has fixed poin (, ). The sable manifold is x =. The unsable manifold is y =. The rajecories ha sar on he sable manifold will move oward he seady sae a an exponenial rae, while he rajecories ha sar on he unsable manifold will blow up o posiive or negaive infiniy a an exponenial rae. The eigenvecors associaed wih negaive eigenvalues make up he sable subspace, and eigenvecors associaed wih posiive eigenvalues make up he unsable subspace. As an illusraion, consider he following example: ẋ = x (1.6) ẏ = y. The seady sae for he sysem is (, ), and he eigenvalue decomposiion of he Jacobian gives ( 1, ( 1 ) ) and (1, ( 1 ) ) as eigenvalue, eigenvecor pairs. Thus, he invarian manifold for his sysem is he se of poins {(, y), y R {(x, ), x R, where x = is he sable manifold and y = is he unsable manifold. This is shown in Figure 1-3. Noice ha he rajecories in Figure 1-3 move oward he origin near he y axis and oward posiive or negaive infiniy near he x axis. On he oher hand, he cener manifold can roughly be defined as he seady saes of he sysem around which he rajecories neiher grow nor decay exponenially over ime. In oher words, a seady sae is in he cener manifold if he behavior of he rajecories near enough o i will never be governed by eiher he unsable or he sable manifolds. In C n, he cener manifold is he subspace of he eigenvecor space of he Jacobian marix of he linearized sysem associaed wih eigenvalues wih zero real par. However, a sysem ha has a zero eigenvalue is no necessarily one ha has a cener manifold, and he cener manifold for a sysem is no necessarily unique. This is demonsraed in Figure 1-4, where he graph of he cener manifold is shown in red bu could also be described by he mirror image of he line across he x-axis. The cener manifold has a paricularly nice propery, which is ha once we know he map whose graph is he cener manifold, we can express he sysem only as a funcion of he cener subspace. For example, consider his wo-dimensional sysem: ẋ = xy ẏ = y + x 2 2y 2. 5

11 Figure 1-4: The phase porrai for he sysem ẋ = x 2, ẏ = y. The unsable manifold is in green and he cener manifold is in red. According o [32, i has exac cener manifold y = x 2. We subsiue in for y o ge ha he evoluion of he sysem is he same as he evoluion of ẋ = x 3. This is he cener manifold reducion, where x is he variable associaed wih he cener subspace. The reducion resuls in a one-dimensional sysem ha we can analyze for bifurcaions as opposed o a wo-dimensional sysem. In general, he reducion will always resul in a lower dimensional sysem, and in many cases he lower dimensionaliy makes he sysem much easier o analyze. While a number of mehods for cener manifold reducion already exis [35, 38, 14, 18, 25, 12, 9, 19, finding improved mehods can allow us o analyze he bifurcaions of (parial) differenial sysems wih a greaer degree of accuracy. One of he more common mehods for compuing he cener manifold numerically is o do a Taylor expansion o pu he sysem ino normal form [15, 31, eiher manually or using sofware, and hen implemen he resul as a funcion. This mehod is difficul even when using sofware because he risk of human error is quie high and he number of erms needed for sufficien accuracy can make he approximaion slow o compue. Anoher mehod is o divide he manifold ino subdivisions, as in [9. On he oher hand, [12, 19 presen ieraive algorihms for compuing he global cener manifold. The approach in [19 involves a discreizaion of he Lyapunov- Perron map, which hey ierae on piecewise consan funcions of ime. An advanage of heir approach is ha i can compue non-smooh manifolds. However, one of he drawbacks o heir algorihm is ha each sep in he ieraion is increasingly compuaionally expensive. For an overview of mehods for compuing he invarian manifold, see [28 and [22. The boundary value problem ha we presen serves as he basis for he formulaion of an algorihm ha uses simple numerical inegraion, such as he Runge-Kua ype schemes, o compue he global manifold. The limiaion o using cener manifold reducion in sudying 6

12 parial differenial equaions is ha he reducion iself ends o be very difficul o implemen and adap. The algorihm we presen has he advanage ha i is relaively simple and can be adoped using radiional numerical schemes. These schemes have he advanage ha heir behavior is already well undersood, and he accuracy of compuaion can be improved wih he implemenaion of several known updaes. In Secion 2.1, we prove he exisence and uniqueness of he cener manifold using he Lyapunov-Perron mehod. Firs, we show ha he Lyapunov-Perron map is well-defined and a conracion mapping, from [17: Definiion Le (E, d) be a meric space. A mapping T : E E is a conracion mapping if here exiss a consan c wih c < 1, such ha d(t (u 1 ), T (u 2 )) cd(u 1, u 2 ) for all u 1, u 2 E. Theorem (Banach Fixed Poin Theorem). Le (E, d) be a Banach space wih a conracion mapping T : E E. Then T admis a unique fixed-poin u E such ha T (u ) = u. Theorem 1.1.3, from [2, gives ha we can ierae he operaor o find a fixed poin, which we show represens a unique soluion in a cerain funcion space o he ordinary differenial sysem. These resuls lead us o he formulaion of he boundary value problem on which we base our algorihm. In Secion 2.2, we make some exensions o he framework and follow he same line of proof for he derivaive of he map. We prove ha he map whose graph gives he cener manifold is C 1, meaning ha he derivaive exiss and is coninuous. We also arrive a a boundary value problem for he derivaive of he cener manifold similar o he one we found in Secion 2.1. In Secion 3.1, we presen he forward/backward inegraion algorihm we develop based off of he boundary value problem and implemen i using Runge-Kua schemes of orders one and wo. We demonsrae he accuracy of hese schemes using an example from [19. We show he resuls when a simple (one dimensional in each componen) differenial sysem fis he framework compleely and compare hem o he resuls we ge in analyzing a simple sysem ha does no compleely fi he framework. We use his o help inform our analysis of (1.1), which does no compleely saisfy he framework, in Secion 3.2. We use radiional echniques o analyze he equaion and compare he resuls o wha we ge using our mehod. We demonsrae he usefulness of our algorihm in his seing and include a discussion of some ineresing behavior ha arose as well as furher research. 1.2 Framework We consider nonlinear ordinary differenial sysems in a Banach space E where u E akes he form u = x + y + z and has an associaed norm u = max{ x, y, z : ẋ = Ax + F (x, y, z) ẏ = By + G(x, y, z) ż = Cz + H(x, y, z). (1.7) 7

13 The Banach space can be decomposed such ha E = X Y Z. The linear erms are such ha A L(X, X), B L(Y, Y ), and C L(C, C), where L is he space of linear operaors. The nonlinear erms are such ha F (x, y, z) C(E, X), G(x, y, z) C(E, Y ), and H(x, y, z) C(E, Z). We assume ha F (,, ) = G(,, ) = H(,, ) =, which is a common assumpion made for convenience. The res of he assumpions are as follows. A1. Exponenial Trichoomy Condiion: We assume ha for α x, α y, β y, β z R such ha α x < β y α y < β z and consans K x, K y, and K z R e A K x e αx, e C K z e βz, e B K y e βy, e B K y e αy,. A2. Lipschiz Coninuiy of Nonlinear Terms: For all (x 1, y 1, z 1 ) and (x 2, y 2, z 2 ) E, here exis consans δ x, δ y, and δ z R such ha δ x, δ y, and δ z and F (x 1, y 1, z 1 ) F (x 2, y 2, z 2 ) δ x (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) G(x 1, y 1, z 1 ) G(x 2, y 2, z 2 ) δ y (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) H(x 1, y 1, z 1 ) H(x 2, y 2, z 2 ) δ z (x 1, y 1, z 1 ) (x 2, y 2, z 2 ). A3. Gap Condiion: Given he nonlinear Lipschiz consans δ x, δ y, and δ z and exponenial richoomy consans α x, α y, β y, β z R and K x, K y, and K z R, we he following inequaliies hold: β y α x > K x δ x + K y δ y β z α y > K y δ y + K z δ z. A1 defines bounds for he linear pars of each componen. The sable componen is bounded forward in ime, he unsable componen is bounded backward in ime, and he cener is bounded in boh direcions. This is a generalizaion of he exponenial dichoomy condiion, which perains o he (un)sable case of he invarian manifold. A developmen of he exponenial dichoomy condiion can be found in [3. To help undersand A1, consider an ordinary differenial equaion in C n where he real marix A is he linear operaor in he sable componen. A will have eigenvalues wih all negaive real par, and we pick λ x λ(a) o be he eigenvalue such ha R(λ x ) R(λ) for all λ λ(a), he se of all eigenvalues of A, and α x = R(λ x ), K x = R e I(λx). Nex, A2 gives us ha he behavior of he nonlinear erms will no be explosive or erraic. Then, we have A3, which allows a unique cener manifold o exis. For an invesigaion of he gap condiion ha we use, see [7. These hree assumpions allow us o sudy he behavior of he invarian manifold as he global behavior of he sysem. We define a parameer σ() such ha { σ p σ() = (1.8) σ n and define he following ordering condiions wih respec o he consans in A1 and A2. 8

14 C1. α x < σ n < β y α y < σ p < β z, C2. α x + K x δ x < σ n < β y K y δ y α y + K y δ y < σ p < β z K z δ z. This places σ n arbirarily beween he sable and cener componens and σ p arbirarily beween he cener and unsable componens. We define a funcion space F σ such ha each global rajecory φ of he differenial sysem where φ : Y E is found as a fixed poin in F σ = {φ C(R Y, E) : (e σ() φ() ) = φ σ <. R This is he space of all coninuous funcions from R Y o he space E ha are exponenially bounded, and i is hese funcions ha we wish o sudy. F σ is also a Banach space wih he σ norm. Finally, le y Y, φ(, y ) := φ() and define he Lyapunov-Perron operaor T : F σ Y F σ as T (φ(), y ) = e B y + e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds {{{{ I II + e ( s)a F (φ(s))ds, (1.9) {{ III where I is he Y componen, II is he Z componen, and III is he X componen. 9

15 Chaper 2 Cener Manifold Theory In his chaper, we use he framework and assumpions o sudy he properies and regulariy of he cener manifold. One of he resuls of his sudy is he boundary value problem on which we base our algorihm. 2.1 Exisence of he Cener Manifold We prove ha he cener manifold for a differenial sysem wih iniial condiion y exiss and can be defined as he graph of he X and Z componens of a map φ F σ. We begin by showing ha he T map is well defined. Noe ha while we show a complee proof for finding coninuiy in each case, we omi some of he deails of simplificaion in showing ha T F σ. These deails are shown in Proposiion Proposiion Assume A1, A2, and C1. Le y Y and φ F σ, hen T (φ(), y ) C(R Y, E) and T (φ, y ) σ <. Proof. Firs, we show ha T (φ, y ) σ <. Consider he case ha. If we ake he norm of (1.9) and apply assumpions A1 and A2, we obain T (φ(), y ) max {K y e αy y + K y δ y e αy e αys φ(s) ds, K z δ z e βz e βzs φ(s) ds, [ K x δ x e αx e αxs φ(s) ds + e φ(s) ds αxs. Nex, we muliply by one in he form of e σ(s)s e σ(s)s according o he sign of s in each inegral: T (φ(), y ) max {K y e αy y + K y δ y e αy e (σp αy)s e σps φ(s) ds, [ e (σp αx)s e σps φ(s) ds + K z δ z e βz e (σp βz)s e σps φ(s) ds, K x δ x e αx 1 e (σn αx)s e σns φ(s) ds.

16 We ake s s e σps φ(s) e σ(s)s φ(s) = φ σ <, s R e σns φ(s) e σ(s)s φ(s) = φ σ < s R in each inegral. We also muliply hrough by e σp : e σp T (φ(), y ) max {K y e (αy σp) y + K y δ y φ σ e (αy σp) e (σ α y)s ds, Briefly, simplifying from here gives e σp T (φ(), y ) max [ K x δ x φ σ e (αx σp) e (σp αx)s ds + { K y e (αy σp) y + K z δ z φ σ e (βz σp) e (σp βz)s ds, K yδ y σ p α y φ σ, K z δ z β z σ p φ σ, e (σn αx)s ds. K x δ x φ σ. σ n α x We have by C1 ha α y < σ p and hus e (αy σp) 1 when [, ), giving ha each erm is finie. Then, {e σp T (φ(), y ) <. The proof for proceeds in he same fashion, giving us ha { e σn T (φ(), y ) max K y e (βy σn) y + K yδ y β y σ n φ σ, K z δ z β z σ p φ σ, Because e (βz σn) 1 when (,, each erm is finie and we ge ha Boh cases ogeher give us ha ha {e σn T (φ(), y ) <. e σ() T (φ(), y ) < R K x δ x φ σ. σ n α x and T (φ, y ) σ <. Nex, we show ha T (φ(), y ) is coninuous in, or lim d (T (φ(), y ) T (φ(d), y )) =. 11

17 Firs, we calculae T (φ(), y ) T (φ(d), y ): T (φ(), y ) T (φ(d), y ) = (e B e db )y + e ( s)b G(φ(s))ds [ e ( s)c H(φ(s))ds + e ( s)a F (φ(s))ds d e (d s)b G(φ(s))ds e (d s)c H(φ(s)) e (d s)a F (φ(s))ds. We spli he proof ino six cases: <, >, and = as d + and d. In each case, we assume ha d sars in a small enough ball around ha i maches he sign of. Firs, we consider > and ake d + : T (φ(), y ) T (φ(d), y ) = (e B e db )y + (e ( s)b e (d s)b )G(φ(s))ds [ e ( s)c H(φ(s))ds + + (e ( s)a e (d s)a )F (φ(s))ds d e (d s)b G(φ(s))ds (e ( s)c e (d s)c )H(φ(s)) e (d s)a F (φ(s))ds = (e B e db )y + [I e (d )B e ( s)b G(φ(s))ds e (d s)b G(φ(s))ds [ e ( s)c H(φ(s))ds + [I e (d )C e ( s)c H(φ(s))ds + [I e (d )A e ( s)a F (φ(s))ds d e (d s)a F (φ(s))ds (e B e db )y + [I e (d )B e ( s)b G(φ(s))ds [ e ( s)c H(φ(s))ds + [I e (d )C +[I e (d )A e ( s)a F (φ(s))ds d e (d s)b G(φ(s))ds e ( s)c H(φ(s))ds e (d s)a F (φ(s))ds 12

18 = (e B e db ((d )B) n )y {{ e ( s)b G(φ(s))ds e (d s)b G(φ(s))ds n! n=1 (I) {{{{ (III) (II) [ e ( s)c ((d )C) n H(φ(s))ds e ( s)c H(φ(s))ds n! n=1 d {{ (V) {{ (IV) ((d )A) n e ( s)a F (φ(s))ds e (d s)a F (φ(s))ds. n! n=1 {{{{ (VII) (VI) As d +, e db e B and (I) will approach zero. In erms (III), (IV), and (VII), as d + he bounds on he inegrals conrac and each inegral approaches zero. In (II), he bounds on he inegral are finie and hus he inegral will remain bounded while ((d )B) n n=1 will n! approach zero as d +, forcing he erm o zero. The summaion erms in (V) and (VI) will also converge o zero. The indefinie inegrals are bounded by he boundedness of he norm of he T map esablished in he firs par of he proof, and hus he erms (V) and (VI) will approach zero as d + and he limi as d + of his expression will be zero. The same reasoning applies o he fuure cases. Nex, consider d for > : T (φ(), y ) T (φ(d), y ) = (e B e db )y + d e ( s)b G(φ(s))ds + [ (e ( s)c e (d s)c )H(φ(s))ds + (e ( s)a e (d s)a )F (φ(s))ds + (e ( s)b e (d s)b )G(φ(s))ds d d e (d s)c H(φ(s)) e (d s)a F (φ(s))ds = (e B e db )y + d e ( s)b G(φ(s))ds + [I e (d )B [ [I e (d )C +[I e (d )A e ( s)c H(φ(s))ds e ( s)a F (φ(s))ds + d d e ( s)b G(φ(s))ds e (d s)c H(φ(s)) e (d s)a F (φ(s))ds 13

19 = (e B e db )y d e ( s)b G(φ(s))ds + ((d )C) n n=1 n! ((d )A) n n=1 n! ((d )B) n n=1 n! e ( s)c H(φ(s))ds e ( s)a F (φ(s))ds + The limi of his expression as d is zero. Nex, we consider when < and d + : T (φ(), y ) T (φ(d), y ) = (e B e db )y + (e ( s)b e (d s)b )G(φ(s))ds + [ e ( s)c H(φ(s))ds + + (e ( s)a e (d s)a )F (φ(s))ds d d d d e ( s)b G(φ(s))ds e (d s)c H(φ(s)) e (d s)a F (φ(s))ds. e ( s)b G(φ(s))ds (e ( s)c e (d s)c )H(φ(s))ds e (d s)a F (φ(s))ds = (e B e db )y + (I e (d )B ) e ( s)b G(φ(s))ds + [ e ( s)c H(φ(s))ds + (I e (d )C ) +(I e (d )A ) e ( s)a F (φ(s))ds d d e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e (d s)a F (φ(s))ds = (e B e db ((d )B) n )y e ( s)b G(φ(s))ds + n! n=1 [ e ( s)c ((d )C) n H(φ(s))ds n! n=1 d ((d )A) n e ( s)a F (φ(s))ds n! n=1 The limi of his expression as d + is zero. d e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e (d s)a F (φ(s))ds. 14

20 We consider when < and d : T (φ(), y ) T (φ(d), y ) = (e B e db )y + (e ( s)b e (d s)b )G(φ(s))ds [ (e ( s)c e (d s)c )H(φ(s))ds + (e ( s)a e (d s)a )F (φ(s))ds d d e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e (d s)a F (φ(s))ds = (e B e db )y + (I e (d )B ) [ (I e (d )C ) +(I e (d )A ) e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e ( s)a F (φ(s))ds d d e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e (d s)a F (φ(s))ds = (e B e db )y ((d )B) n n=1 n! [ ((d )C) n n! n=1 ((d )A) n n! n=1 e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e ( s)a F (φ(s))ds The limi of his expression as d is zero. Le = and consider d + : T (φ(), y ) T (φ(d), y ) = (1 e db )y + = (1 e db )y + [ e (d s)b G(φ(s))ds e sc H(φ(s))ds + + d d (e sa e (d s)a )F (φ(s))ds [ e (d s)b G(φ(s))ds e sc H(φ(s))ds + (I e dc ) +(I e da ) d e sa F (φ(s))ds e ( s)b G(φ(s))ds e ( s)c H(φ(s))ds e (d s)a F (φ(s))ds. (e sc e (d s)c )H(φ(s))ds e (d s)a F (φ(s))ds e sc H(φ(s))ds e (d s)a F (φ(s))ds 15

21 = (1 e db )y + [ e (d s)b G(φ(s))ds e sc H(φ(s))ds (da) n n=1 The limi of his expression as d + is zero. Consider as d : T (φ(), y ) T (φ(d), y ) = (1 e db )y = (1 e db )y = (1 e db )y n! (dc) n n=1 n! e sa F (φ(s))ds [ e (d s)b G(φ(s))ds (e sc e (d s)c )H(φ(s))ds + e (d s)b G(φ(s))ds (e sa e (d s)a )F (φ(s))ds [ (I e dc ) +(I e da ) [ e (d s)b (dc) n G(φ(s))ds n! n=1 (da) n n! n=1 e sc H(φ(s))ds e sa F (φ(s))ds e sc H(φ(s))ds e sa F (φ(s))ds The limi of his expression as d is zero and T is coninuous in. d d d d d e sc H(φ(s))ds e (d s)a F (φ(s))ds. e (d s)c H(φ(s))ds e (d s)a F (φ(s))ds e sc H(φ(s))ds e (d s)a F (φ(s))ds d d e sc H(φ(s))ds e (d s)a F (φ(s))ds. Now we show ha he map is a conracion, meaning ha as we ierae i, he disance beween each successive ieraion will shrink and he ieraed map will converge o a fixed poin. Proposiion is given in [19. However, hey omi a rigorous proof, which we show here. In his proof, we show each deail of he simplificaion of he approximaion of he norm of T. These deails are referenced in laer proofs as well. Proposiion Given assumpions A1, A2, A3, C1, and C2, he equaion in (1.9) is a conracion mapping wih Lipschiz consan δ φ = max { K yδ y σ p α y, Kyδy β y σ n, Kxδx σ n α x, Kzδz β z σ p. Proof. Le φ 1, φ 2 F σ and denoe T (φ 1 (), y ) := T (φ 1 ()) for a fixed y Y. T (φ 1 ()) T (φ 2 ()) = + e ( s)b [G(φ 1 (s)) G(φ 2 (s))ds e ( s)a [F (φ 1 (s)) F (φ 2 (s))ds. e ( s)c [H(φ 1 (s)) H(φ 2 (s))ds 16

22 Taking norms over he expression gives T (φ 1 ()) T (φ 2 ()) { max e ( s)b [G(φ 1 (s)) G(φ 2 (s))ds, e ( s)a [F (φ 1 (s)) F (φ 2 (s))ds, e ( s)c [H(φ 1 (s)) H(φ 2 (s))ds { e ( s)b G(φ 1 (s)) G(φ 2 (s)) ds, e ( s)a F (φ 1 (s)) F (φ 2 (s)) ds, e ( s)c H(φ 1 (s)) H(φ 2 (s)) ds. The inerval (, ) can be separaed ino (, [, ), where (, or [, ). The proof coninues in wo cases. Case 1: Consider (,. In he X componen, s (,, and s for all s in he inerval. For he Y componen, s [, and s. In he Z componen, s [, ), and s. Applying A1 and A2 gives { T (φ 1 ()) T (φ 2 ()) max K x e αx( s) δ x φ 1 (s) φ 2 (s) ds, K y e βy( s) δ y φ 1 (s) φ 2 (s) ds, K z e βz( s) δ z φ 1 (s) φ 2 (s) ds. In his case, and we muliply hrough by e σn. In he X and Y componens, s because and we muliply by one in he form e σns e σns. The inerval considered in he Z componen is [, ), where s when s [, and s when s [, ). Spliing he inegral up according o hese inervals gives K z e βz( s) δ z φ 1 (s) φ 2 (s) ds = + K z e βz( s) δ z φ 1 (s) φ 2 (s) ds K z e βz( s) δ z φ 1 (s) φ 2 (s) ds. In he inegral from o zero, muliply by one in he form e σns e σns. In he inegral from zero o, s, muliply by one in he form e σps e σps. Then, e σn T (φ 1 ()) T (φ 2 ()) max {K y δ y e σn e βy( s) e σns e σns φ 1 (s) φ 2 (s) ds, K x δ x e σn ( K z δ z e σn e βz( s) e σns e σns φ 1 (s) φ 2 (s) ds + e αx( s) e σns e σns φ 1 (s) φ 2 (s) ds, e βz( s) e σps e σps φ 1 (s) φ 2 (s) ds). 17

23 Nex, we ake he remum of he e σns φ 1 (s) φ 2 (s) erms over he proper domain. e σn T (φ 1 ()) T (φ 2 ()) max {K y δ y e (βy σn) e (σn βy)s e σns φ 1 (s) φ 2 (s) ds, s [, K x δ x e (αx σn) e (σn αx)s e σns φ 1 (s) φ 2 (s) ds, K z δ z e (βz σn) ( s (, e (σn βz)s e σns φ 1 (s) φ 2 (s) ds, s [, e (σp βz)s s [, ) e σps φ 1 (s) φ 2 (s) ds), φ 1 φ 2 σ max {K y δ y e (βy σn) e (σn βy)s e σns ds, K x δ x e (αx σn) e (σn αx)s ds, ( K z δ z e (βz σn) e (σn βz)s ds + e ds) (σp βz)s. From here, we evaluae he inegral expressions using classical mehods. { e σn T (φ 1 ()) T (φ 2 ()) φ 1 φ 2 σ max K y δ y e (βy σn) 1 (1 e (σn βy) ), σ n β y ( K x δ x e (αx σn) 1 ) e (σn αx) lim σ n α x T e(σn αx)t, [ K z δ z e (βz σn) 1 (1 e (σn βz) 1 ( ) ) + lim σ n β z σ p β z T e(σp βz)t 1. Now, we know ha σ n α x > and σ p β z < by C1, and hus each limi evaluaes o zero and we simplify he expressions furher. { e σn Ky δ y T (φ 1 ()) T (φ 2 ()) max (1 e (βy σn) K x δ x ),, β y σ n σ n α x K z δ z e (βz σn) e (σn βz) φ 1 φ 2 σ. β z σ p We wan o make sure ha he remum of he lef hand side says bounded. We ake he remum over he given domain in each componen as well. { e σn K y δ y T (φ 1 ()) T (φ 2 ()) max (1 e (βy σn) ), (, (, β y σ n K x δ x σ n α x, K z δ z β z σ p φ 1 φ 2 σ. Because β y σ n > by C1, e (βy σn) 1 for (,, giving ha 1 e (βy σn) 1. 18

24 From his, we obain hree consans over he hree componens ha bound he norm when. { Ky δ y K x δ x K z δ z T (φ 1 ) T (φ 2 ) σ max, φ 1 φ 2 σ. β y σ n σ n α x β z σ p Case 2: Consider [, ). In he X componen, s (, and s. In he Z componen, as s [, ) and s. The richoomy condiions on X and Z apply he same way as in he firs case. For he Y componen, as s [, and s. The appropriae richoomy condiion is e B K y e αy. We have { T (φ 1 ()) T (φ 2 ()) max K x e αx( s) δ x φ 1 (s) φ 2 (s) ds, K y e αy( s) δ y φ 1 (s) φ 2 (s) ds, K z e βz( s) δ z φ 1 (s) φ 2 (s) ds. Because, muliply hrough by e σp. In he Y - and Z-componens, we use e σps because s in each inegral. In he X-componen, he inegral can be spli up accordingly: K x e αx( s) δ x φ 1 (s) φ 2 (s) ds = + K x e αx( s) δ x φ 1 (s) φ 2 (s) ds K x e αx( s) δ x φ 1 (s) φ 2 (s) ds, (2.1) where s in he firs inegral and s in he second. Applying (2.1) his gives e σp T (φ 1 ()) T (φ 2 ()) max {K y δ y e (αy σp) e (σp αy)s e σps φ 1 (s) φ 2 (s) ds, K x δ x e (αy σp) [ e (σn αx)s e σns φ 1 (s) φ 2 (s) ds + e (σp αx)s e σps φ 1 (s) φ 2 (s) ds, K z δ z e (βz σp) e (σp βz)s e σps φ 1 (s) φ 2 (s) ds, max {K y δ y e (αy σp) e (σp αy)s e σps φ 1 (s) φ 2 (s) ds, s [, [ K x δ x e (αy σp) e (σn αx)s e σns φ 1 (s) φ 2 (s) ds + K z δ z e (βz σp) s (, e (σp αx)s s [, e (σp βz)s e σps φ 1 (s) φ 2 (s) ds, e σps φ 1 (s) φ 2 (s) ds, s [, ) 19

25 [ max {K y δ y e (αy σp) e (σp αy)s ds, K x δ x e (αy σp) e (σn αx)s ds + K z δ z e (βz σp) e (σp βz)s ds e (σp αx)s ds, φ 1 φ 2 σ, { Ky δ [ y max e (αy σp) (e (σp αy) 1), K x δ x e (αy σp) (e (σp αx) 1), σ p α y σ n α x σ p α x K z δ z e (βz σp) e (σp βz) φ 1 φ 2 σ, β z σ p { e σp Ky δ y T (φ 1 ()) T (φ 2 ()) max (1 e (αy σp) ), [, ) σ p α y [, ) [ e (α y σ p) K x δ x + e(αy σp) 1 K z δ z, φ 1 φ 2 σ, σ n α x α x σ p β z σ p [, ) { Ky δ y K x δ x T (φ 1 ) T (φ 2 ) σ max,, σ p α y σ n α x Finally, we have ha for all R, T (φ 1 ) T (φ 2 ) σ max { Ky δ y σ p α y, K y δ y β y σ n, K z δ z β z σ p K x δ x σ n α x, φ 1 φ 2 σ. K z δ z β z σ p φ 1 φ 2 σ. From he condiions α x + K x δ x < σ n < β y K y δ y and α y + K y δ y < σ p < β z K z δ z, we have K x δ x < σ n α x and K y δ y < β y σ n as well as K y δ y < σ p α y and K z δ z < β z σ p. Therefore, δ φ := max { K yδ y σ p α y, Kyδy β y σ n, Kxδx σ n α x, Kzδz β z σ p < 1 and T (, y ) is a conracion mapping in F σ. By he conracion mapping principle [2, here exiss a unique φ F σ such ha φ (, y ) = T (φ (), y ). Now we show ha he φ is a unique soluion in F σ o he original sysem. Proposiion The fixed poin of T (, y ), denoed by φ (, y ), is characerized as he unique elemen in he funcion space ha is he soluion o (1.7) wih iniial condiion φ(, y ), and saisfies he boundary value problem ẋ = Ax + F (x, y, z) x() = ẏ = By + G(x, y, z) y() = y ż = Cz + H(x, y, z) z( ) =. (2.2) Proof. We show ha φ (, y ) is a soluion o he sysem, which we do by aking derivaives 2

26 wih respec o : ẋ = F (x, y, z) + A e ( s)a F (φ(s))ds ẏ = G(x, y, z) + Be B y + B ż = H(x, y, z) C e ( s)c H(φ(s))ds e ( s)b G(φ(s))ds From he map, x = e( s)a F (φ(s))ds, y = e B y + e( s)b G(φ(s))ds, and z = e ( s)c H(φ(s))ds. Subsiuing in yields he sysem in (1.7) wih he iniial condiion, which is unique given he choice of φ. To show he second par of he proposiion, simply noe ha for he X componen of he T map, plugging in = yields zero, in he Y componen of he T map, plugging in = yields y, and for he Z componen, plugging in = yields zero. So, we have he boundary condiions. Definiion (Cener Manifold) The cener manifold is M c := {u E : u(, u ) F σ. We firs show he invariance of he cener manifold. Proposiion M c is invarian: if u M c, hen u(, u ) M c for fixed. Proof. Take u M c such ha u(, u ) F σ. Fix. We need ha u 1 := u(, u ) M c. Since M c = {u E : u(, u ) F σ, i remains o show ha u(, u 1 ) F σ. Then, by he fac ha u is auonomous, u(, u 1 ) = u(, u(, u )) = u( +, u ) F σ. Nex, Φ is a map defined such ha Φ : Y X Z by Φ(y ) = φ(, y ) X Z. We show he following se equivalence. Proposiion Given Given A1, A2, A3, C1, and C2, we have M c = {u : Φ(y ) = x + z, or = graphφ. Proof. This is a direc resul of Proposiion We sudy he cener manifold in erms of he fixed poin of he T map. In he nex proof, we use he following inequaliy aribued o Gronwall, a proof of which is found in [13: Lemma (Gronwall s Inequaliy). If u() p() + q(s)u(s)ds for funcions u, p, and q such ha u and q are coninuous and p is non-decreasing, hen ( ) u() p() exp q(s)ds. (2.3) 21

27 Proposiion Given A1, A2 and C1, fix y 1 and y 2 Y. Then, M c is Lipschiz coninuous wih real Lipschiz consan δ Φ : Φ(y 1 ) Φ(y 1 ) δ Φ y 1 y 2. Proof. Le φ 1() = T (φ 1(), y 1 ) and φ 2() = T (φ 2(), y 2 ) and noe ha by Proposiion Φ(y 1 ) Φ(y 2 ) = φ 1() X Z φ 2() X Z φ 1 φ 2 σ. This implies ha we ge a bound for Φ(y 1 ) Φ(y 2 ) if we bound φ 1 φ 2 σ. We calculae he difference using he equivalence o he T map: φ 1() φ 2() = e B (y 1 y 2 ) + e ( s)c [H(φ 1(s)) H(φ 2(s))ds + e ( s)b [G(φ 1(s)) G(φ 2(s))ds e ( s)a [F (φ 1(s)) F (φ 2(s))ds. As a firs-pass esimaion, we use he calculaions from Proposiion 2.1.2, o show ha when : { φ 1 φ 2 σ (e (βy σn) Kx δ x K y δ y K z δ z y 1 y 2 ) + max,, φ 1 φ (, σ n α x β y σ n β z σ 2 σ p and when : φ 1 φ 2 σ (e (αy σp) y 1 y 2 ) + max (, { Kx δ x σ n α x, K y δ y σ p α y, K z δ z β z σ p φ 1 φ 2 σ. We have ha δ φ = max { K xδ x σ n α x, Kyδy β y σ n, Kyδy σ p α y, Kzδz β z σ p < 1 by Proposiion Because β y σ n > and α y σ p <, boh y 1 y 2 erms reach a remum a =, and each expression simplifies down o φ 1 φ 2 σ y 1 y 2 + δ φ φ 1 φ 2 σ and Φ(y 1 ) Φ(y 2 ) φ 1 φ 2 σ 1 1 δ φ y 1 y 2. where δ Φ = 1 1 δ φ. However, his is unsaisfacory because as δ φ 1, δ Φ and we obain a beer approximaion using a slighly differen mehod. The norm φ 1() φ 2() is aken as he 22

28 maximum over each componen. Firs noice ha φ 1() φ 2() = max { e B (y 1 y 2 ) + e ( s)c [H(φ 1(s)) H(φ 2(s))ds, e ( s)b [G(φ 1(s)) G(φ 2(s))ds, e ( s)a [F (φ 1(s)) F (φ 2(s))ds, { max e B (y 1 y 2 ) + e B (y 1 y 2 ) + e B (y 1 y 2 ) + e ( s)b [G(φ 1(s)) G(φ 2(s))ds, e ( s)c [H(φ 1(s)) H(φ 2(s))ds, e ( s)a [F (φ 1(s)) F (φ 2(s))ds. φ 1() φ 2() max { e B y 1 y 2 + δ y e ( s)b φ 1(s) φ 1(s) ds, e B y 1 y 2 + δ z e ( s)c φ 1(s) φ 2(s) ds, e B y 1 y 2 + δ x e ( s)a φ 1(s) φ 2(s) ds, e σ φ 1() φ 2() max {e σ e B y 1 y 2 + δ y e σ e ( s)b e σs e σs φ 1(s) φ 2(s) ds, e σ e B y 1 y 2 + δ z e σ e ( s)c e σs e σs φ 1(s) φ 2(s) ds e σ e B y 1 y 2 + δ x e σ e ( s)a e σs e σs φ 1(s) φ 2(s) ds. We use he Gronwall inequaliy as in Lemma 2.3 in each componen o ge e σ φ 1() φ 2() max {e σ e B y 1 y 2 exp(δ y e σ e ( s)b e σs ds), e σ e B y 1 y 2 exp(δ z e σ e ( s)c e σs ds) e σ e B y 1 y 2 exp(δ x e σ e ( s)a e σs ds). 23

29 Now consider he case ha : e σn φ 1() φ 2() max {K y e (βy σn) y 1 y 2 exp(k y δ y e (βy σn) e (σn βy)s ds), [ K y e (βy σn) y 1 y 2 exp(k z δ z e (βz σn) e (σn βz)s ds + K y e (βy σn) y 1 y 2 exp(k x δ x e (αx σn) which, using he same argumens as in Proposiion 2.1.2, simplifies o φ 1 φ 2 σ K y max { e Kxδx σn αx, e Kyδy βy σn, e Kzδz βz σp y1 y 2. e (σp βz)s ds ) e (σn αx)s ds) When : e σp φ 1() φ 2() max {K y e (βy σp) y 1 y 2 exp(k y δ y e (βy σp) e (σp βy)s ds), K y e (βy σp) y 1 y 2 exp(k z δ z e (βz σp) e (σp βz)s ds) [ K y e (βy σp) y 1 y 2 exp(k x δ x e (αx σp) e (σn αx)s ds + e (σp αx)s ds) ) { and his simplifies o φ 1 φ 2 σ K y max e Kzδx σn αx, e Kyδy Kzδz σp αy, e βz σp Finally, we have ha Φ 1 Φ 2 δ Φ y 1 y 2 for all R where y 1 y 2. { δ Φ = K y max e Kxδx σn αx, e Kyδy Kyδy Kzδz βy σn, e σp αy, e βz σp = K y e δ φ. Because each exponen exiss in he inerval [, 1), K y δ Φ < K y e. Now ha we have ha he manifold is Lipschiz, we have compleed he final sep in proving he following heorem: Theorem Given Given A1, A2, A3, C1, and C2 here exiss a unique Lipschiz map Φ : Y X Z such ha graph(φ) is he cener manifold of (1.7). 2.2 Regulariy of he Manifold The nex sep is o sudy he differeniabiliy of he manifold. We add an addiional assumpion: A4. Nonlinear Terms are C 1 : F (x, y, z) C 1 (E, X), G(x, y, z) C 1 (E, Y ), and H(x, y, z) C 1 (E, Z). 24

30 In oher words, we assume ha he derivaives of he nonlinear erms exis and are coninuous. Wih A2 and he following heorem from [1 we have ha he norm of he derivaive of each nonlinear erm is uniformly bounded. Theorem (Rademacher s Theorem). Given A2 and A4, he nonlinear erms are bounded such ha DF (x, y, z) δ x DG(x, y, z) δ y (2.4) DH(x, y, z) δ z. We know from Secion 2.1 ha for a fixed iniial condiion y we find φ (, y ) = T (φ (), y ). Define φ () := φ (, y ). For he mos par, we drop he sar noaion in his secion because all such φ ha we refer o will be he fixed poin of ieraing he T map given an iniial condiion. We are sudying he derivaive of he φ map. To his end, define he space F 1,σ : F 1,σ = { C(R Y, L(Y, E)) : e σ() () L(Y,E) = 1,σ < R where σ() is defined as before in (1.8). The map φ () is wrien in erms of he T map: φ () = e B y + e ( s)b G(φ (s))ds + e ( s)c H(φ (s))ds e ( s)a F (φ (s))ds. We differeniae wih respec o y o ge T 1 : F 1,σ Y F 1,σ such ha T 1 ( (), y ) = e B + e ( s)b DG(φ (s)) (s)ds e ( s)c DH(φ (s)) (s)ds {{{{ Y Z + e ( s)a DF (φ (s)) (s)ds. (2.5) {{ We will use his o sudy he map DΦ whose graph is he derivaive of he manifold. Noe ha in he X componen, e ( s)a DF (φ (s)) (s)ds where e ( s)a X X, DF (φ (s)) X E, (s) E Y. So e ( s)a DF (φ (s)) (s) is of dimension X Y. We go hrough similar analyses for he oher wo componens o ge ha T 1 ( (), y ) is of dimension E Y. In his secion, we will show ha he map Φ C 1 (Y, X Z). As in he previous secion, we firs need o show ha he T 1 map is well-defined. This proof is similar o he proof for T and hus we leave ou several deails. Proposiion Given A1, A2, C1, and (2.4), T 1 ( (), y ) is well-defined. Proof. Firs, we show T 1 : F 1,σ Y F 1,σ by showing ha T 1 (, y ) 1,σ < for any X 25

31 F 1,σ and y Y. When, e σn T 1 ( (), y ) max {K y e (βy σn) + K y δ y e (βy σn) [ K z δ z e (βz σn) e (σn βz)s e σns (s) ds + K x δ x e (αx σn) e (σn βy)s e σns (s) ds, e (σp βz)s e σps (s) ds, e (σn αx)s e σns (s) ds. (2.6) We see ha, based on he seps from Proposiion 2.1.2, (2.6) will simplify down o a se of consans less han infiniy. Showing ha T 1 ( (), y ) is coninuous in follows he same seps as in Proposiion The nex sep is o show ha T 1 (, y ) is a conracion mapping. This proof is also very similar o he proof for he T map. Proposiion Given Given A1, A2, A3, C1, C2, and (2.4), T 1 ( (), y ) is a conracion mapping wih rae δ φ. Proof. Take he difference T 1 ( 1 (), y ) T 1 ( 2 (), y ): T 1 ( 1 (), y ) T 1 ( 2 (), y ) = + e ( s)b DG(φ (s))( 1 (s) 2 (s))ds e ( s)c DH(φ (s))( 1 (s) 2 (s))ds e ( s)a DF (φ (s))( 1 (s) 2 (s))ds where 1, 2 are arbirary funcions in F 1,σ. Then, we ake norms and apply A1 and A2 in he case ha : { T 1 ( 1 (), y ) T 1 ( 2 (), y ) max K y e ( s)βy δ y 1 (s) 2 (s) ds, K z e ( s)βz δ z 1 (s) 2 (s) ds, K x e ( s)αy δ x 1 (s) 2 (s) ds. 26

32 e σns T 1 ( 1 (), y ) T 1 ( 2 (), y ) max {K y δ y e (βy σn) e (σn βy)s e σns 1 (s) 2 (s) ds K z δ z e (βz σn) e (σn βz)s e σns 1 (s) 2 (s) ds [ +K x δ x e (αy σn) e (σn αy)s e σns 1 (s) 2 (s) ds + The proof proceeds in he same way as in Proposiion e (σp αy)s e σps 1 (s) 2 (s) ds. Le () = T 1 ( (), y ) be he fixed poin of he T 1 map given y. Before we proceed we need he following lemma. Lemma Given A1, A2, A3, C1, C2, hen le φ 1 () := φ (, y 1 ) and φ 2 () := φ (, y 2 ). We have he following bound on φ 1 () φ 2 () : { K y ee (Kyδy+αy) y 1 y 2 when φ 1 () φ 2 () K y ee (βy Kyδy) y 1 y 2 when. Proof. From Proposiion 2.1.8, we have ha where in he case ha. So φ 1 () φ 2 () e σn δ Φ y 1 y 2 δ Φ = K y max { e Kxδx σn αx, e Kyδy βy σn, e Kzδz βz σp < Ky e φ 1 () φ 2 () K y ee σn y 1 y 2. From C2 on σ n, we have ha α x + K x δ x < σ n < β y K y δ y. If we muliply hrough by, we ge ha (β y K y δ y ) < σ n < (α x + K x δ x ). We ge he mos precise bound on φ(, y 1 ) φ(, y 2 ) by leing σ n β y K y δ y. When, and When, φ 1 () φ 2 () K y ee (βy Kyδy) y 1 y 2. φ 1 () φ 2 () e σp δ Φ y 1 y 2 φ 1 () φ 2 () δ Φ e σp y 1 y 2. As before, from C2, we ge ha (α y +K y δ y ) < σ p < (β z K z δ z ). Taking σ p (α y +K y δ y ) gives ha φ 1 () φ 2 () K y ee (Kyδy+αy) y 1 y 2. 27

33 We show nex ha = φ / y. This follows a similar idea o he proof presened in [7. Proposiion Given A1, A2, A3, C1, C2, and (2.4), we have ha φ(y )/ y =, where () := T 1 ( (), y ) and φ (, y ) = T (φ (), y ) for a given y Y. Proof. For clariy of noaion in his proof, we wrie ou φ (, y ). To ge ha φ / y =, we use he represenaion of φ (, y ) as a fixed poin of he T map and differeniae wih respec o y using Fréche differeniaion because we are in a Banach space. () C(R Y, L(Y, E)) and herefore is bounded and linear in Y. Then, if φ (y + h) φ (y ) h σ lim h h = (2.7) where h Y, we have ha φ (y ) is Fréche differeniable wih derivaive φ(y )/ y =. Firs, le ρ(, y, h) = φ (, y + h) φ (, y ) ()h E h where R and y, h Y. Consider his as where ρ(, y, h) = max{ρ X (, y, h), ρ Y (, y, h), ρ Z (, y, h) ρ X (, y, h) = φ (, y + h) x φ (, y ) x ()h x, h ρ Y (, y, h) = φ (, y + h) y φ (, y ) y ()h y, h ρ Z (, y, h) = φ (, y + h) z φ (, y ) z ()h z. h If we show ha R e σ() ρ(, y, h) as h, we will have he resul. Le ζ(s) := φ (s, y ) and w(s) := φ (s, y + h) φ (s, y ). We spli he proof ino seps and proceed accordingly. Sep 1: We esablish he following esimaes: for ρ Y (, y, h) Kyee 2 βy e Kyδys R Y (ζ(s), w(s))ds + K y δ y e ( s)βy ρ(s, y, h)ds, 28

34 ρ X (, y, h) K x K y ee αx e (βy Kyδy αx)s R X (ζ(s), w(s))ds + K x δ x e ( s)αx ρ(s, y, h)ds [ ρ Z (, y, h) K z K y ee βz e (βy Kyδy βz)s R Z (ζ(s), w(s))ds + for e (Kyδy+αy βz)s R Z (ζ(s), w(s))ds + K z δ z e ( s)βz ρ(s, y, h)ds; ρ Y (, y, h) Kyee 2 αy e Kyδys R Y (ζ(s), w(s))ds + K y δ y e ( s)αy ρ(s, y, h)ds [ ρ X (, y, h) K x K y ee αx e (βy Kyδy αx)s R X (ζ(s), w(s))ds + e (Kyδy+αy αx)s R X (ζ(s), w(s))ds + K x δ x e ( s)αx ρ(s, y, h)ds, ρ Z (, y, h) K z K y ee βz e (Kyδy+αy βz)s R Z (ζ(s), w(s))ds Sep 2: From Sep 1, we obain + K z δ z e ( s)βz ρ(s, y, h)ds. e σ() ρ(, y, h) (1 δ φ ) 1 R(ζ(), w()) R such ha R(ζ(), w()) = max{r n (ζ(), w()), R p (ζ(), w()), where { R n (ζ(), w()) = max Kyee 2 (βy σn) e Kyδys R Y (ζ(s), w(s))ds, K x K y ee (αx σn) e (βy Kyδy αx)s R X (ζ(s), w(s))ds, [ K z K y ee (βz σn) e (βy Kyδy βz)s R Z (ζ(s), w(s))ds + e (Kyδy+αy βz)s R Z (ζ(s), w(s))ds, 29

35 and { R p (ζ(), w()) = max Kyee 2 (αy σp) e Kyδys R Y (ζ(s), w(s))ds, [ K x K y ee (αx σp) e (βy Kyδy αx)s R X (ζ(s), w(s))ds + e (Kyδy+αy αx)s R X (ζ(s), w(s))ds, K z K y ee (βz σp) e (Kyδy+αy βz)s R Z (ζ(s), w(s))ds. Sep 3: We show ha lim h R(ζ(), w()) = max{lim h R n (ζ(), w()), lim h R p (ζ(), w()) =. Once we have his, i follows ha lim h R and we have shown (2.7). Proof for Sep 1: Consider ρ Y when : e σ() ρ(, y, h) lim h (1 δ φ ) 1 R(ζ(), w()) =, ρ Y (, y, h) = 1 e B (y + h) + h [e B y + [e B h + e ( s)b G(ζ(s))ds e ( s)b G(ζ(s) + w(s))ds e ( s)b DG(ζ(s)) (s)hds [[G(ζ(s) e( s)b + w(s)) G(ζ(s)) DG(ζ(s)) (s)h ds =. h We add and subrac DG(ζ(s))w(s) in he inegrand: e( s)b[ [G(ζ(s) + w(s)) G(ζ(s)) DG(ζ(s))w(s) + DG(ζ(s))w(s) DG(ζ(s)) (s)h ds =. h Then, we apply A1 and muliply by 1 = w(s) E / w(s) E o ge G(ζ(s) + w(s) G(ζ(s)) DG(ζ(s))w(s) ( s)βy K y e w(s) E + K y e ( s)βy DG(ζ(s)) w(s) (s)h E ds. h w(s) E ds h 3

36 A his poin, i is convenien o esablish some shor-hand noaion: R X (ζ(s), w(s)) = R Y (ζ(s), w(s)) = R Z (ζ(s), w(s)) = F (ζ(s) + w(s)) F (ζ(s)) DF (ζ(s))w(s) 2δ x w(s) E G(ζ(s) + w(s)) G(ζ(s)) DG(ζ(s))w(s) 2δ y w(s) E H(ζ(s) + w(s)) H(ζ(s)) DH(ζ(s))w(s) 2δ z w(s) E (2.8) where w(z) and ζ(z) are defined as before and each erm is he Fréche differeniaion of he respecive nonlinear erm wih respec o w(z). The bounds come from he argumen ha F (ζ(s) + w(s)) F (ζ(s)) DF (ζ(s))w(s) w(s) E applying A2 and (2.4) gives F (ζ(s) + w(s)) F (ζ(s)) + DF (ζ(s)) w(s) w(s) E, F (ζ(s) + w(s)) F (ζ(s)) DF (ζ(s))w(s) w(s) E δ xw(s) + δ x w(s) w(s) E = 2δ x. Each R X, R Y, and R Z is uniformly bounded. Noe ha we have w(s) (s)h E = ρ(s, y h, h). Then, ρ Y (, y, h) K y e ( s)βy R Y (ζ(s), w(s) w(s) E h ds + K y δ y e ( s)βy ρ(s, y, h)ds. By Lemma 2.2.4, we have w(s) E K y ee (βy Kyδy)s h when s and w(s) E K y ee (Kyδy+αy)s h when s, and ρ Y (, y, h) Kyee 2 βy e Kyδys R Y (ζ(s), w(s))ds + K y δ y e ( s)βy ρ(s, y, h)ds. We use he same seps o ge ha ρ X and ρ Z are bounded such ha ρ X (, y, h) K x K y ee αx e (βy Kyδy αx)s R X (ζ(s), w(s))ds + K x δ x e ( s)αx ρ(s, y, h)ds, [ ρ Z (, y, h) K z K y ee βz e (βy Kyδy βz)s R Z (ζ(s), w(s))ds + e (Kyδy+αy βz)s R Z (ζ(s), w(s))ds + K z δ z e ( s)βz ρ(s, y, h)ds. 31

37 For, we use he same process o ge ha [ ρ X (, y, h) K x K y ee αx e (βy Kyδy αx)s R X (ζ(s), w(s))ds + e (Kyδy+αy αx)s R X (ζ(s), w(s))ds + K x δ x e ( s)αx ρ(s, y, h)ds, and ρ Y (, y, h) Kyee 2 αy e Kyδys R Y (ζ(s), w(s))ds + K y δ y e ( s)αy ρ(s, y, h)ds, ρ Z (, y, h) K z K y ee βz e (Kyδy+αy βz)s R Z (ζ(s), w(s))ds + K z δ z e ( s)βz ρ(s, y, h)ds. Proof for Sep 2: Now ha we have all he componens of ρ for R, we evaluae R e σ() ρ(, y, h) = max{ R e σ() ρ X (, y, h), R e σ() ρ Y (, y, h), e σ() ρ Z (, y, h). R When we ake he remum over each of he erms, he las inegral in each expression will evaluae he same way as in Proposiion 2.1.2, giving ha R e σ() ρ(, y, h) R(ζ(), w()) + δ φ e σ() ρ(, y, h). R For example, consider e σn ρ X (, y, h): e σn ρ X (, y, h) e σn K x K y ee αx e (βy Kyδy αx)s R X (ζ(s), w(s))ds + e σn K x δ x e ( s)αx ρ(s, y, h)ds. For now, we ignore he erm dependen on R X and consider only e σn K x δ x e ( s)αx ρ(s, y, h)ds e (αx σn) K x δ x e (σn αx) s e σns ρ(s, y, h)ds. Because ρ(s, y, h) F 1,σ, s R e σns ρ(s, y, h) <. inegral o ge ha We move he erm ouside he e σn K x δ x e ( s)αx ρ(s, y, h)ds e σn ρ(, y, h) e (αx σn) K x δ x e (σn αx) ds. 32