Màster en Matemàtica Aplicada

Transcription

1 Màster en Matemàtica Aplicada Títol: Transfer of energy in the nonlinear Schrödinger fnajajiequation Autor: Adrià Simon Director: Amadeu Delshams Departament: Matemàtica Aplicada I Convocatòria: Juny 011

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3 Universitat Politècnica de Catalunya Facultat de Matemàtiques i Estadística Treball de Fi de Màster Transfer of energy in the nonlinear Schrödinger equation Adrià Simon Advisor: Amadeu Delshams Departament de Matemàtica Aplicada I

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5 Agraïments Vull agraïr a l Amadeu Delshams, el meu director, que m hagi proposat de fer aquest treball amb ell i que hagi dedicat tantes hores a ajudar-me a realitzar-lo, sobretot quan li he demanat de fer coses amb poc temps de maniobra. Considero que, a part de les matemàtiques, m ha ensenyat a gestionar un projecte i a enfrontar-me a un problema introduint-me a la tasca que desenvolupen els investigadors, fet que trobo molt oportú de cara els meus possibles futurs interessos. També vull agrair a la Tere Seara, al Joan Solà i al Marcel Guàrdia totes les converses que he tingut amb ells que em van orientar cap aquest treball. D altra banda vull agrair a l Uri tot el suport que m ha donat en cada moment, animant-me quan ho he necessitat i escoltant la infinitud de dubtes que m han sorgit a part de tot el suport tècnic i lingüístic. També vull agrair a la Berta i als meus pares tot allò que han fet perquè aquest treball hagi tirat endavant. Finalment, també m agradaria agrair a tots els meus amics de la fme i del Villena per estar al meu costat tot aquest temps.

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7 Abstract Key words: Nonlinear Schrödinger equation, Arnold Diffusion, Šilnikov coordinates MSC000: 5Q55,7J40,7K99 This work aims to use tools of Dynamical Systems to prove that the nonlinear Schrödinger equation presents a global instability. To do that, we first reduce this PDE to a system of ODE s of dimension N which we call the Toy Model System. Consequently our new purpose is to study the existence of instability in a system of ODE s. The way of proving it will consist in taking invariant objects and showing that there exists a solution that flows near all of them. This strategy resembles Arnold Diffusion. The contribution of this work is to use the so-called Šilnikov coordinates to prove this result when N =. However, we detect a problem when we flow around one of this invariant objects that, in suitable coordinates, can be seen as a saddle) that prevents us from completing the proof in the expected way.

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9 Contents Introduction 1 Chapter 1. Simplifications and proof of main theorem 7 1. NLS as an Infinite System of ODE s 8. Resonant truncation of FN LS 11. From resonant to finite system 1 4. First Ingredient: The Frequency Set Λ Second Ingredient: Diffusion in the Toy Model Third Ingredient: The Approximation Lemma The scaling argument and proof of Theorem 1 18 Chapter. Diffusion in the Toy Model System 1 1. Introduction 1. Initial remarks 1. Reduction to case N= Conclusions 4 References 45 i

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11 Introduction The purpose of this memoir is to apply typical techniques of Dynamical Systems in order to show that a particular PDE presents some global instability. The motivation is given by [4] where this instability is proved for the nonlinear Schrödinger equation. To do that, the authors reduce the initial PDE to a finite system of ODE s, where they prove the instability. We think that once they get this big simplification, they do not fully exploit the strong machinery developed in Dynamical Systems. The main idea in the work will be to suggest a new way of proving via the so-called Šilnikov coordinates) the instability in the finite dimensional system of ODE s. However, we will only deal with the case which its dimension is three. So, let us start considering the periodic defocusing cubic nonlinear Schrödinger NLS) equation { i t u + u = u 1) u u0, x) = u 0 x) where ut, x) is a complex valued function with the spatial variable lying in the torus T = R /πz ). The nonlinear Schrödinger equation i t u = u + g 0 u σ u, is one of the most important equation in mathematical physics. It arises in various physical contexts [16], [15], like semiconductor physics [1], nonlinear optics [10], Bose-Einstein condensate [5], quantum mechanics [1], plasma physics [8] or biomolecular dynamics [6]. For Bose-Einstein condensate application, one considers a particular case of NLS, Gross-Pitaevskii equation: i t u = u + V ext x)u + g 0 u σ u, where one introduces some external potential in order to confine the condensed. The factor g 0 = 4π a/m corresponds to a coupling constant, where a is the scattering length of the wave which value is positive if our atomic interaction is repulsive and negative for attractive interactions. For instance, taking 87 Rb or Li we have a > 0 and we get the defocusing nonlinear Schrödinger equation. Taking σ = 1 one obtains the cubic defocusing one, which is the equation that we will deal with. 1

12 INTRODUCTION As many evolution PDE s, we can see NLS as infinite dimensional hamiltonian system see [11] and []). In the real case, it means that we can rewrite our equation as: t u = J u h) u, t), where the hamiltonian function, h : H R R is defined on a infinite dimensional Hilbert space H and J is a non degenerated antisymmetric operator. It becomes slightly different when we have a complex valued function. To illustrate the changes, consider a finite dimensional hamiltonian system, with hamiltonian: h : V R n R q, p) hq, p) Then, the equations of motion are given by t q = h q, p) p t p = h q, p) q where we have used the Darboux coordinates, so our antisymmetric operator is ) 0 In J =. I n 0 Now, consider the change of variables z = q + ip, z = q ip. Then the equation becomes t z = i h z, z). z Note that we do not write the equation for z because since we have a real system, we obtain it just conjugating this equation. Since the multiplication for i is an antisymmetric nondegenerated form we say that an infinite dimensional hamiltonian system is given in complex form if the equations can be written as t u = J ūh) u, ū, t). In our case, we must take H = H s T 1, C) for s 1 J = ii hu, u) = T u u uu), where H s T, C) = W s, T, C) is the Sobolev space as a Hilbert space with the usual inner product f, g) = fx)gx) dx. T Indeed, define for ξ C and τ R I τ) = d dt Iτ) = hu, u + τξ) I 0) = ūhu, ū) ξ 1 T u u + τξ) uu + 1 τξ)) = T u ξ + 1 uu + τξ)u ξ.

13 INTRODUCTION Integrating by parts, I 1 0) = T u ξ + 1 u uξ = 1 T u + 1 ) u u ξ, so ūhu, ū) = 1 u + 1 u u. Now, we can write our PDE as and we obtain the original problem. t u = i ūhu, ū), As in finite dimensional hamiltonian systems, if the hamiltonian function does not depend on time, h is a conserved quantity, it means hux, t), ūx, t)), that we will call the energy, is constant along the solution. To prove it, take t u = i ūhu, ū) t ū = i u hu, ū) and derive d dt hux, t), ūx, t)) = uhu, ū) t u + ūhu, ū) t ū = i u hu, ū) ūhu, ū) i ūhu, ū) u hu, ū) = 0 Another interesting and useful conserved quantity in our problem is the so-called conservation of mass or L T ) norm. It is: Mu) = u. T As we will be using this result continuously we present its proof. dm dt u) = d u = t uu) = t u) u + u t u) dt T T T = i u + i u u ) u + u i u i u u ) T = i u u u u = i u u + u u) = 0 T T In conclusion, we say that our system presents conservation of energy and mass: ) 1 hu)t) = T u u uu) = hu)0) Mu)t) = u = Mu)0) T Taking u 0 H s T ) with s > 0, [] proves the local-in-time well-posededness of equation 1). The conservation laws ), together with this result, give the existence of a global smooth solution to 1) from smooth initial data. We are interested in solutions that initially oscillate only on scales comparable to the spatial period and eventually oscillate on arbitrarily short spatial scale. One can quantify such motion in terms of the growth in time of higher Sobolev norm ut) Hs T ), which is defined using the Fourier transform by, ) ut) Hs T ) := ut, ) Hs T ) := n Z n s ût, n) ) 1

14 4 INTRODUCTION where n := 1 + n ) 1 and 1 ût, n) = ut, x)e in x dx. T Once we have introduced the problem we are able to write the main result of this work, the construction of solutions of 1) with arbitrarily large growth in higher Sobolev norms, Theorem 1. Let 1 < s, K 1 and δ 1 be given parameters. Then there exists a global smooth solution ut, x) to 1) and a time T > 0 with and u0) H s δ ut ) H s K. Taking into account the conservations laws ), the growth constructed here must involve both movements of energy to higher frequencies, and movement of mass to lower frequencies. The mass associated to the higher and higher frequency energy must be decreasing by energy conservation. This must be balanced, by mass conservation, by more and more mass at low frequencies. After presenting the problem and announce the main theorem we are going to sketch the outline of its proof. First, we write 1) as a infinite dimensional system of ODE s in {a n t)} n Z, where a n t) are closely related to the Fourier mode ût, n) of the solution. Then we identify a related system, which we call the resonant system, that we use as an approximation to the full system. Taking the initial data supported in a finite set of frequencies Λ we will be able to show that this infinite system reduces to a finite one, the Toy Model System. There are two related ingredients which complete the proof of the main theorem. The first one consists on building the frequency set Λ, which is defined in terms of the desired Sobolev norm growth and according to a wish-list of geometric and combinatorial properties aimed at simplifying the resonant system. The second main ingredient is to show that the Toy Model System exhibits unstable orbits that travel from an arbitrarily small neighborhood of one invariant manifold to near a distant invariant manifold. It is this instability which is ultimately responsible for the support of the solution s energy moving to higher frequencies. This kind of instability reminds us of Arnold Diffusion. In order to show this similarity, we sketch the main idea in Arnold Diffusion. For more details, see [7]. Consider a hamiltonian system of N degrees of freedom given by a hamiltonian function H = H 0 +εf, where H 0 define a completely integrable hamiltonian system and εf is a perturbation. Thus, considering the non-perturbed system and assuming that it is written in action-angle variables I i, φ i ), we get that I i are constants of motion, so the hypersurfaces of constant I i are invariant objects tori) that foliate our phase space. When we consider ε 0 but sufficiently small) these tori might 1 In what follows, we omit the factors π as these play no role in our analysis.

15 INTRODUCTION 5 be destructed they are not invariant) or may survive these ones will be called primary tori) so we do not have the phase space foliated by invariant objects yet gap). To prove the existence of Arnold Diffusion consists on seeing that there exists solutions to the hamiltonian system given by H which starts arbitrarily near of one of those primary tori and ends up arbitrarily far. In terms of the action variables, this result can be written as: for all δ and K, there exists a solution and a time T > 0 such that I i 0) < δ and I i T ) > K for some i {1,..., N}. Although this expression is very similar to Theorem 1, this is not the similarity that we want to show. The resemblance is shown when we try to prove the existence of Arnold Diffusion. To do that, one shows that the unstable manifold of one of these tori intersects transversally the stable manifold of the next one so, by an obstruction argument, one can prove that there exists an orbit that follows this transition chain, giving rise to an unstable orbit. This is exactly what we will do in the Toy Model System. For this reason, we will call the chapter which contains the proof of this instability Diffusion in the Toy Model System. This memoir is organized as follows. In the first Chapter we provide a more detailed overview of the argument, giving the proof of Theorem 1 modulo some intermediate claims, like the construction of the frequency set Λ and the proof of the diffusion in the Toy Model. The purpose of the last Chapter is to show the instability presented in the Toy Model System for the particular case when N =, using the so-called Šilnikov coordinates. This result only holds when f satisfies some conditions. Assume that we take such a perturbation.

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17 Chapter 1 Simplifications and proof of main theorem In this chapter we will prove Theorem 1. To do that we will make first some important reductions that will allow us to consider a finite system of ODE s instead of a PDE. We now sketch the steps we must take. First, we write the solution of the PDE in a Fourier series in order to obtain an infinite system of ODE s corresponding to the Fourier coefficients that will only depend on time). Then we approximate this equation considering only the so-called resonant frequencies. We will justify later this approximation via a lemma. After doing that, our equation will have a very interesting property: if the initial data consists on a finite number of frequencies, then the solution at each time will not involve new ones. For this reason we will have obtained a finite system of ODE s equivalent via an approximation lemma) to our initial PDE. In addition we will reduce a little more our system of ODE s taking the initial condition in a very particulary set of frequencies. We will not include the construction of this set in this memoir because it involves a lot of combinatorial techniques that will not contribute to the understanding of our result. However this construction can be seen in [4]. Once we have obtained a finite system of ODE s, that we will call the Toy Model System, we will show that this system presents some instability that will correspond to the instability that we want to show for a solution of our initial PDE in terms of its Sobolev norm. The proof of this result will be given in the next chapter because is the most important part of the work. In this one we will include only the final result. To end the chapter we will prove Theorem 1 using all these ingredients. In order to make this chapter more complete, we write again our PDE and the main theorem: { i t u + u = u u u0, x) = u 0 x) 7

18 8 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM Theorem. Let 1 < s, K 1 and 0 < δ 1 be given parameters. Then there exists a global smooth solution ut, x) to 1) and a time T > 0 with and u0) H s δ ut ) H s K. 1. NLS as an Infinite System of ODE s In this section we will transform our PDE to an infinite system of ODE s. First, we realize that our equation has gauge freedom. Consider vt, x) defined by vt, x) = e igt ut, x), where ut, x) is a solution of 1) and G R is some constant. Let us compute which equation satisfies vt, x): i t v + v = i t e igt u ) + e igt u) = Ge igt u e igt i t u + e igt u = Gv + e igt i t u + u) = Gv + u v G + v ) v v = u = and, for t = 0, v0, x) = u0, x). So, vt, x) satisfies 4) i t v + v = G + v )v with the same initial data. Then we will choose the constant G to cancel some terms. Since the explicit solutions of the linear problem associated to NLS are e in x+ n t), for n Z, we assume that we can expand our function vt, x) in a Fourier-like series, so we write: vt, x) = n Z a n t)e in x+ n t). This will simplify our equation because the functional form in the spatial variable will be fixed, so we will get a system of ODE s for the Fourier coefficients instead of a PDE. Lemma 1. The Fourier coefficients a n t) satisfy the equations: 5) i t a n = Ga n + a n1 a n a n e iω4t where n 1,n,n Z n 1 n +n =n 6) ω 4 = n 1 n + n n.

19 1. NLS AS AN INFINITE SYSTEM OF ODE S 9 Proof. We impose that vt, x) is a solution of 4). For the right hand side, we have: ) ) i t v + v = i t a n e in x+ n t) + a n e in x+ n t) n Z n Z = i t a n )e in x+ n t) + a n n e in x+ n t) n Z n Z a n n e in x+ n t) n Z = n Z i t a n )e in x+ n t). On the other hand, we must compute: v v = a n1 a n a n e in1 n+n) x+ n1 n 1,n,n Z n + n )t). Then, using the completeness of {e in x }, we obtain: i t a n )e i n t = Ga n e i n t + a n1 a n a n e i n1 n + n )t. n 1,n,n Z n 1 n +n =n Finally, dividing all the terms by e i n t we get what we wanted to prove. Now, we are going to use the gauge parameter, G, to simplify a little bit our equation. Split the sum of the right hand side of 5) into the following terms: = + + n 1,n,n Z n 1 n +n =n n 1,n,n Z n 1 n +n =n n 1,n n n 1,n,n Z n 1 n +n =n n 1=n n 1,n,n Z n 1 n +n =n n =n := Term I + Term II + Term III + Term IV. n 1,n,n Z n 1 n +n =n n =n 1=n Term IV is not a sum at all: since n 1 = n = n, necessarily n = n and ω 4 = 0, and so Term IV = a n t) a n t). In Term II III) we have n 1 = n n = n) which implies m := n = n m := n = n 1 ) and ω 4 = 0 again, so Term II + Term III = m Z a n t)a m t)a m t) = a n t) m Z a m t) = a n t) ut) L T ), where we have used Plancherel s theorem. As we have seen in the previous section, one conserved quantity is the L norm, so we can put M = ut) L T ), and the equation 5) becomes: i t a n = Ga n a n a n + Ma n + a n1 a n a n e i n1 n + n )t. n 1,n,n Z n 1 n +n =n n 1,n n

20 10 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM Now, we can cancel two terms of this equation by choosing G = M. Equation 5) takes then the following useful form, which we denote FNLS, 7) i t a n = a n a n + a n1 a n a n e iω4t, where n 1,n,n Γn) 8) Γn) = {n 1, n, n ) Z ) : n 1 n + n = n, n 1 n, n n}. In conclusion, we have seen that we can transform our initial PDE to an infinite system of ODE s. One can show that FNLS is locally well-posed 1 in l 1 Z ), and for completeness we sketch the argument here. Define the trilinear operator: by N t) : l 1 Z ) l 1 Z ) l 1 Z ) l 1 Z ) 9) N t)a, b, c)) n = a n b n c n + n 1,n,n Γn) a n1 b n c n e iω4t. With this notation, we can express FNLS as i t a n = N t)a, a, a)) n. Lemma. 10) N t)a, b, c)) n l 1 Z ) k a l 1 Z ) b l 1 Z ) c l 1 Z ) Proof. N t)a, b, c)) n l 1 Z ) = n Z a nb n c n + a n1 b n c n e iω4t n 1,n,n Γn) a n b n c n + a n1 b n c n n Z n Z Since l 1 Z ) l Z ) there exists k such that, n 1,n,n Γn) a n a l Z ) k a l1 Z ) b n b l Z ) k b l1 Z ) so for the first term we have n Z a n b n c n k a l 1 Z ) b l 1 Z ) n Z c n = k a l 1 Z ) b l 1 Z ) c l 1 Z ). Using the definition of Γn), n = n + n n 1 and we can split the sum in the second term by a n1 b n c n a n1 b n c n n1+n n Z n 1,n,n Γn) n Z n 1 Z n Z = a n1 b n c m n 1 Z n Z m Z = a l1 Z ) b l1 Z ) c l1 Z ). Putting all together, we obtain the desired bound. 1 It means that there exists a unique solution which depends continuously on the data.

21 . RESONANT TRUNCATION OF FNLS 11 This lemma allows us to use standard Picard iteration arguments to show local well-posedness in l 1 Z ), which is valid on [0, T ] with T a0) l 1 Z ). The main used idea is that the equation i t a n = N t)a, a, a)) n behaves roughly like the ODE t a = a for the purposes of local existence theory.. Resonant truncation of FN LS The reduction from a PDE to a system of ODE s does not involve any approximation of the solution: if we solve the equation 7) for the Fourier coefficients, we can recover exactly the solution ut, x) of 1). In the present section we will take the first step to get a finite system of ODE s. The main idea will be to consider a subset of Γn) and take equation 7) extending the sum in this subset. We will look for a subset of frequencies with a closure property: if our initial data consists in a finite set of frequencies, then the solution of our approximate equation at every time will not involve new frequencies. The appropriate subset that we will take is: Γ res n) = {n 1, n, n ) Γn) : ω 4 = n 1 n + n n = 0} Γn), called the set of all resonant non-self interactions. Remark. Note that n 1, n, n ) Γ res n) precisely when n 1, n, n, n) form four corners of a nondegenerate rectangle. To see that, first we claim that: n 1, n, n ) Γ res n) n 1 n 0, n n 0, n n 0 ) Γ res n n 0 ) for any n 0 Z. Indeed, let n 0 Z and n 1, n, n ) Γ res n). Then, so Furthermore, n 1 n 0 ) n n 0 ) + n n 0 ) = n n 0 n 1 n 0 n n 0 n n 0 n n 0, n 1 n 0, n n 0, n n 0 ) Γn n 0 ). n 1 n 0 n n 0 + n n 0 = n 1 n + n + n 0 so we have proved the direct implication. n 0 = 0. + n 0 n 1 n + n ) = n + n 0 + n 0 n = n n 0, For the other implication, just take With this result, it suffices to prove the geometric interpretation for n = 0. n 1, n, n, 0) will be the corners of a rectangle if and only if n 1 n = 0, n 1 +n = n and n 1, n 0. The second and third condition are verified by definition of Γ0) and for the first one, we write: n 1 n = n 1 + n n 1 n = n n = 0.

22 1 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM On the other hand, if we take a rectangle in Z with one corner at the origin, n 1 n = 0, so n 1 + n = n 1 + n = n. Heuristically, the resonant interactions dominate in 7) because they do not contain the e iω4t factor that oscillates in time. So we approximate solutions of 7) by simply discarding the nonresonant interactions and we define the resonant truncation RFNLS of FNLS by 11) i t r n = r n r n + n 1,n,n ) Γ resn) r n1 r n r n. We prefer not to include now the reason why we can do this approximation because it involves many notation that we will introduce later.. From resonant to finite system Even after making the resonant approximation, we still have an infinite ODE to work with in 11), n can range freely over Z ). Our strategy is to choose initial data for which the system simplifies in several ways. Suppose we have some finite set of frequencies Λ that satisfies the following two properties: Property I Λ : Initial data) All the excited frequencies for t = 0 belong to Λ i.e. r n 0) = 0 whenever n / Λ). We say that the initial data r n 0) is entirely supported in Λ. Property II Λ : Closure) If n 1, n, n ) Γ res n) and n 1, n, n Λ, then n Λ. Then one can sow that r n t) stays supported in Λ for all time. The idea is that the non-linearity in 11) cannot excite any modes outside Λ if one only starts with modes inside Λ. Now we are going to prove that these conditions guarantee a finite dimensional model. Lemma. If Λ is a finite set satisfying Property I Λ, Property II Λ above, and r0) rt) solves RFNLS 11) on [0, T ] then for all t [0, T ] rt) is entirely supported in Λ. Proof. Define Bt) = n / Λ r n t).

23 . FROM RESONANT TO FINITE SYSTEM 1 Deriving, B t) = ) t r n t)r n t) = [ ] t r n t)r n t) + t r n t)r n t) n/ Λ n / Λ = { } Re t r n t)r n t) n/ Λ = Re i r nt) 4 + ir n t) n/ Λ = Im r n t) n/ Λ n 1,n,n ) Γ resn) r n1 t)r n t)r n t) r n1 t)r n t)r n t), n 1,n,n ) Γ resn) so B t) r n t) r n1 t) r n t) r n t). n/ Λ n 1,n,n ) Γ resn) By Property II Λ, since n / Λ and we consider n 1, n, n ) Γ res n), necessarily one of the frequencies does not lie in Λ. In addition using the boundedness of r n t) for 0 < t < T given in the local well-posedness argument we get B t) K r n t) r m t). Since n,m/ Λ n m n,m/ Λ n m n,m / Λ n m r n t) r m t) r n t) l 1, which is bounded by a constant independent of t when 0 t T, we can find a constant K such that r n t) r m t) K r n t), n / Λ so B t) CBt). Then, since B0) = 0 by Property I Λ, by Gronwall s inequality we get Bt) = 0 for all 0 t T. The strategy will consist in choosing initial data with Fourier support in such a set Λ, so that the resonant system 11) reduces to a finite dimensional system. We now place more conditions on the set Λ and on the initial data which bring about more simplifications. We demand that for some positive integer N to be specified later), the set Λ splits into N disjoint generations Λ = Λ 1... Λ N which satisfy certain properties that we will specify later, after first introducing necessary terminology. Definition 1. A nuclear family is a rectangle n 1, n, n, n 4 ), where the frequencies n 1, n known as the parents) live in a generation Λ j, and the frequencies n, n 4 known as the children) live in the next generation Λ j+1. Remark. Note that if n 1, n, n, n 4 ) is a nuclear family, then so is n 1, n 4, n, n ), n, n, n 1, n 4 ) and n, n 4, n 1, n ); we shall call these the trivial permutations of the nuclear family.

24 14 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM In addition to the described properties we require the following ones: Property III Λ : Existence and uniqueness of spouse and children) For any 1 j N and any n 1 Λ j there exists a unique nuclear family n 1, n, n, n 4 ) up to trivial permutations) such that n 1 is a parent of this family. In particular each n 1 Λ j has a unique spouse n Λ j and has two unique children n, n 4 Λ j+1 up to permutation). Property IV Λ : Existence and uniqueness of sibling and parents) For any 1 j N and any n Λ j+1 there exists a unique nuclear family n 1, n, n, n 4 ) up to trivial permutations) such that n is a child of this family. In particular each n Λ j+1 has a unique sibling n 4 Λ j+1 and has two unique parents n 1, n Λ j up to permutation). Property V Λ : Nondegeneracy) The sibling of a frequency n is never equal to its spouse. Property VI Λ : Faithfulness) Apart from the nuclear families, Λ contains no other rectangles. Indeed, from the closure hypothesis, it does not even contain any right-angled triangles which are not coming from a nuclear family.) Remark. Since every pair of parents in one generation corresponds to a exactly one pair of children in the next, each generation must have exactly the same number of frequencies. Assume, for the moment, that such a Λ exists for any given N. simplify equation 11). Now it becomes: 1) i t r n t) = r n t) r n t) + r nchild 1 r)r nchild t)r nspouse t) + r nparent 1 r)r nparent t)r nsibling t) Then, we can where for each n Λ j, n spouse is its spouse, n child 1, n child Λ j+1 are its two children, n sibling Λ j is its sibling, n parent 1, n parent Λ j 1 are its parents and the factor arises from trivial permutations of nuclear families. If n is in the last generation Λ N then we omit the term involving spouse and children; if n is in the first generation Λ 1 we omit the term involving siblings and parents. Thus, we get a finite system of ODE s. However, we now simplify this ODE by making yet another assumption, this time again involving the initial data: Property VII Λ : Intragenerational equality) The function n r n 0) is constant on each generation Λ j. Thus 1 j N and n, n Λ j imply r n 0) = r n 0). Similar to Lemma, we can show that if one has intragenerational equality at time 0 then one has intragenerational equality at all later times. Thus we may collapse the function n r n t), which is currently a function on Λ = Λ 1... Λ N, to a function j b j t) on {1,..., N}, where b j t) := r n t) whenever n Λ j. The ODE 1) now collapses to the following system that we call the Toy Model System: 1) i t b j t) = b j t) b j t) + b j 1 t) b j t) + b j+1 t) b j t), with the convention that b 0 t) = b N+1 t) = 0.

25 4. FIRST INGREDIENT: THE FREQUENCY SET Λ 15 In conclusion, assuming that there exists such a set Λ, and that we can approximate FNLS for RFNLS, we have transformed our initial PDE 1) into a finite system of ODE s 1) where we will prove Theorem 1. In the rest of the chapter we will take the following steps which will lead us to the proof of the main theorem. 1) First Ingredient: Construct the finite set of frequencies Λ. ) Second Ingredient: Prove that the Toy Model System exhibits a particular instability: we show that there exist solutions of the Toy Model System which thread through small neighborhoods of an arbitrary number of distinct invariant tori. More specifically, we show that we have a multi-hop solution to this ODE in which the mass is initially concentrated at b 1 but eventually ends up at b N. In terms of the resonant system 11), this instability corresponds to the growth of higher Sobolev norms. ) Third Ingredient: Write an Approximation Lemma which gives conditions under which solutions of RFN LS, and hence solutions corresponding to the Toy Model System, approximate an actual solution of the original N LS equation. When the conditions of this Approximation Lemma are satisfied, it is enough to construct a solution evolving according to the Toy Model System which exhibits the desired growth in H s. 4) Use a scaling argument that shows that these conditions can indeed be satisfied and glues the three ingredients together to complete the proof. We now detail the claims of the three ingredients and prove theorem 1) modulo these intermediate claims, but first we introduce some notation that will appear in what follows: Remark. If X, Y are nonnegative quantities, we use X Y or X = OY ) to denote the estimate X CY for some C, and X Y to denote the estimate X Y X. We use X Y to mean X cy for some small constant c. 4. First Ingredient: The Frequency Set Λ The proof that this set of frequencies exists can be seen in [4]. Here we will not include it, we only record the precise claim we make about the set. Proposition 1. Given parameters δ 1, K 1, we can find an N 1 and a set of frequencies Λ Z with, Λ = Λ 1 Λ... Λ N disjoint union which satisfies Property II Λ - Property VI Λ and also, n Λ 14) N n s K n Λ 1 n s δ. In addition, given any R CK, δ), we can ensure that Λ consists of N N 1 disjoint frequencies n satisfying n R. Note than Property IΛ and Property VII Λ will be easy satisfied when we choose our initial data.

26 16 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM Remark. Note that we are free to choose the parameter R, called inner radius, as large as we wish. 5. Second Ingredient: Diffusion in the Toy Model Our claim is that we can construct initial data for the Toy Model System 1) so that at time zero, b0) is concentrated in its first component b 1 and this concentration then propagates from b 1 to b, then to b etc. until at some time the solution is concentrated at b N. We will measure the extent to which the solution is concentrated with the parameter ɛ. More precisely, Theorem. Given N > 1, 0 < ɛ 1, there is initial data b0) = b 1 0), b 0),..., b N 0)) C N for 1) and there is a time T = T N, ɛ) so that b 1 0) 1 ɛ b j 0) ɛ j 1, b N T ) 1 ɛ b j T ) ɛ j N In addition, the corresponding solution satisfies bt) l 1 for all 0 t T. We will prove this theorem in the case N =. The proof for the general case can be seen in [4]. The reason of this reduction is that we will propose an alternative proof, using Šilnikov coordinates. Anyway, the proof of this theorem is the central work in this memoir so we will dedicate an entirely chapter. 6. Third Ingredient: The Approximation Lemma Now, we are going to see why we can do this resonant truncation of FNLS via the approximation lemma below: Lemma 4. Let 0 < σ < 1 be an absolute constant all implicit constants in this lemma may depend on σ). Let B 1, and let T B log B. Let be a solution to the equation gt) := {g n t)} n Z 15) i t gt) = N t)gt), gt), gt))) + Et) for times 0 t T, where the initial data g0) is compactly supported. Assume also that the solution gt) and the error term E obey the bounds of the form 16) gt) l1 Z ) B 1 17) for all 0 t T. t 0 Es) ds l 1 Z ) B 1 σ We conclude that if at) denotes the solution to FNLS 7) with initial data a0) = g0), then we have 18) at) gt) l1 Z ) B 1 σ/ for all 0 t T.

27 6. THIRD INGREDIENT: THE APPROXIMATION LEMMA 17 Remark. Note that we make some non-trivial assumptions 16) and 17)). In the next section, we will prove that in our case, these hypotheses are satisfied. Proof. First note that since a0) = g0) is assumed to be compactly supported, the solution at) to 7) exists globally in time, is smooth with respect to time and is in l 1 Z ) in space. Write Then, F t) = i t 0 Es) ds, and dt) := gt) + F t). i t d = i t g i t F = N t)g, g, g) + E E = N t)d F, d F, d F ). For 16) and 17), we get that g = O l 1B 1 ) and F = O l 1B 1 σ ), where we use O l 1X) to denote any quantity with an l 1 Z ) norm of OX). In particular, we have d = O l 1B 1 ). By trilinearity and the bound 10) we thus have Now write a := d + e. Then we have, i t d = N d, d, d) + O l 1B σ ). i t d + e) = N d + e, d + e, d + e), which when subtracted from the previous equation gives after more trilinearity and 10)) i t e = O l 1 B σ ) + O l 1 B ) e 1 + Ol 1 B 1 e ) ) 1 + Ol 1 e 1, and so by differential form of Minkowski s inequality t e 1 t e 1 ), we have t e 1 B σ + B e 1 + B 1 e 1 + e 1. If we assume temporarily that e 1 = OB 1 ) for all t [0, T ], then one can absorb the third and fourth terms on the right-hand side in the second and obtain: Applying Gronwall s inequality, e 1 e t 0 CB ds t e 1 CB e 1 + CB σ [ e0) 1 + t t = e CB t CB σ e CB s ds = e CB t CB σ CB 1 e CB t ) e CB t B 1 σ, 0 0 ] e s 0 CB dr CB σ ds where we have used that since a0) = g0) and F 0) = 0, d0) = a0) so e0) = 0. In conclusion, we have e 1 B 1 σ exp CB t ) for all t [0, T ]. Since we have T B log B, T cb log B for some very small c, so we have e 1 B 1 σ+cc B 1 σ/, and so we can remove the a priori

28 18 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM hypothesis e 1 = OB 1 ) by continuity arguments. write To conclude the proof we at) gt) l1 Z ) at) dt) l1 Z ) + dt) gt) l1 Z ) B 1 σ/ + B 1 σ B 1 σ/. Remark. The exponent σ/ can be in fact replaced by any exponent between 0 and σ, but we choose σ/ for concreteness. 7. The scaling argument and proof of Theorem 1 Finally, we present the argument that glues the three main components together to get Theorem 1. Before proving it, we recall the definition of the Sobolev norm and make some remarks: ut) H s T ) = n Z n s ût, n) ) 1 = 1 + n ) s ût, n) n Z We note a n t) to the solution of FNLS 7), g n t) to the solution of the resonant truncation, RFNLS 11) and b j t) to the solution of the Toy Model System 1). Given δ and K, construct Λ as in proposition 1). Note that we are free to specify R. Recall that this parameter measures the inner radius of the frequencies involved in Λ if n Λ then n R). We will take it very large so we will be able to approximate the H s norm by n Z n s ût, n) Its size will be given shortly, with R = Rδ, K). With the number N = Nδ, K) from the construction of Λ recall that N represents the number of generations in the set of frequencies), and a number ɛ = ɛk, δ) which we will specify shortly, we construct a traveling wave solution bt) to the toy model concentrated at scale ɛ according to Theorem. This theorem also gives us a time T 0 = T 0 K, δ) at which we have traversed the N generations of frequencies. Note that the toy model has the following scaling, ) t b λ) t) := λ 1 b λ. We choose the initial data for 1) by setting ) 1 19) a n 0) = b λ) j 0) for all n Λ j, and a n t) = 0 when n / Λ. The basic idea is to choose the parameter λ large enough to ensure the Approximation Lemma 4) applies, with g n the solution of the resonant system of ODE s 11) also evolving from the data 19), over the time interval [ ] 0, λ T 0 which is the time the rescaled solution b λ) takes to travel through. ) 1.

29 7. THE SCALING ARGUMENT AND PROOF OF THEOREM 1 19 all generations in Λ). In other words, we want to apply the Approximation Lemma 4 with a parameter B chosen large enough so that, 0) B log B λ T 0. With λ and B so chosen, we will be able to prove that ut) H s grows by a factor of K/δ on [ ] 0, λ T 0. We finally choose R to ensure this quantity starts at size approximate δ, rather than a much small smaller scale. We detail now these general remarks. The aim is to apply lemma 4) with gt) := {g n t)} n Z defined by, g n t) = b λ) j t), for n Λ j and g n t) = 0 when n / Λ. Hence, Et) is the non-resonant terms of 7). That is, 1) Et) := g n1 g n g n e iω4t. [Γn)\Γ resn)] Λ We include the set Λ in the description of the sum above to emphasize once more that the frequency support of gt) is always in this set. We choose B = CN)λ and then show that for large enough λ the required conditions 16) and 17) hold true. Observe that 0) holds true with this choice for large enough λ. Note first that simply considering its support, the fact that Λ = CN), and the fact that bt) l 1, we can be sure that bt) l 1 CN) and therefore ) b λ) t) l 1 Z ), gt) l 1 Z ) λ 1 CN). Thus, 16) holds with the choice B = CN)λ. For the second condition 17), we claim t ) Es) ds CN) λ + λ 5 T ). l 1 0 Note that this is sufficient with our choices B = CN)λ and T = λ T 0. It remains only to show ). Since ω 4 0 in the set Γn)\Γ res n), we can replace e iω4s by d ds [ ] e iω 4 s iω 4 and integrate by parts. Three terms arise: the boundary terms at s = 0 and s = T, and the integral term involving d ds [g n 1 s)g n s)g n s)]. For the boundary term, we use ) to obtain and upper bound of CN)λ. For the integral term, the s derivative falls on one of the g factors. We replace this differentiated term using the equation to get an expression that is 5-linear in g and bounded by CN)λ 5 T. Once λ has been chosen as above, we choose R sufficiently large so that the initial data g0) = a0) has the right size: ) 1 4) g n 0) n s δ. n Λ This is possible since the quantity on the left scales like λ 1 in λ, and R s in the parameter R. It remains to show that we can guarantee, ) 1 5) a n λ T 0 ) n s K, n Λ

30 0 1. SIMPLIFICATIONS AND PROOF OF MAIN THEOREM where at) is the evolution of the data g0) under the full system 7). We do this by first establishing, ) 1 6) g n λ T 0 ) n s K, and second that, 7) n Λ n Λ g n λ T 0 ) a n λ T 0 ) n s ) 1 1. As for 6), consider the ratio of this norm of the resonant evolution at time λ T 0 to the same norm at time 0, n Z Q := g nλ T 0 ) n s N i=1 n Λ n Z g n0) n s = i b λ) i λ T 0 ) n s N, i=1 n Λ i b λ) i 0) n s since g n = 0 when n / Λ. We now use the notation S j := n Λ j n s, Q = = = N i=1 bλ) i λ T 0 ) S i N i=1 bλ) i 0) S i S N 1 ɛ) 1 ɛ)s 1 + ɛs ɛs N S N 1 ɛ) S n [1 ɛ) S1 S N + ɛ S S N ɛ S N 1 S N + ɛ 1 ɛ) 1 ɛ) S1 S N + Oɛ) K δ, where the last inequality is ensured by Proposition 1 and by choosing ɛ CN, K, δ) sufficiently small. As for the second inequality 7), using the Approximation Lemma 4 we obtain that 8) g n λ T 0 ) a n λ T 0 ) n s λ σ n s 1, n Λ n Λ by possibly increasing λ and R, maintaining 4). Together, the inequalities 18), 8) give us immediately 7). ]

31 Chapter Diffusion in the Toy Model System 1. Introduction In this chapter we must prove Theorem above, which claims a certain kind of instability for the system which we call the Toy Model System, 9) t b j = i b j b j + ib j b j 1 + b j+1), for j = 1,..., N. with the assumption b 0 = b N+1 = 0. Again, the complete proof can be found in [4]; here we will show the result in the case N =. The reason is that we will propose a new kind of proof, more typical in dynamical systems, by using the so-called Šilnikov coordinates which were introduced by Šilnikov in [14] and are very useful in this kind of problems as we will see. This chapter is structured in two parts. In the first one we make some remarks about equation 9) in the general case, that will simplify our system. The second part will consist on proving Theorem when N =.. Initial remarks In this first section we make some remarks about the equation 9) in general dimension, N) and introduce some definitions and notation that we will use in the proof. We write bt) for the vector composed of the N modes, b 1 t),..., b N t)) and we note that there is a first conserved quantity, the vector norm of bt). Indeed, we have t b j = t b j b j ) = t b j )b j + b j t b j ) = t b j )b j + b j t b j ) = t b j )b j + t b j )b j = Re ) b j t b j 9) = Re i b j 4 ) ) + ib j b j 1 + b j+1 = 4Im b j b j 1 + bj+1) ) ) = 4Im b j b j 1 b j+1 b j and hence by telescoping series we obtain the vector norm conservation law or mass conservation) t N j=1 b j = 0. Note that this is a new property: although 1

32 . DIFFUSION IN THE TOY MODEL SYSTEM in previous chapters we have shown that our initial PDE has mass conservation law, we have approximated the equations to obtain this toy model system so there is not a clear reason to think that the new system will have it too. After showing that, we want to prove that for t = 0 the total mass is concentrated in the first mode and for a later time, the mass is concentrated in the last mode. For convenience, we shall take b0) = 1, so we will always assume that bt) = 1 and we will work in Σ = {x C N : x = 1} instead of working in C N. If we have a look at the form that we have written Theorem, we can observe that we have already taken this convention. This is because 9) enjoys scaling symmetry: if bt) is a solution to 9), so is b λ t) = λbλ t) for any λ > 0. For this reason, we can work in this region and then, in order to obey other conditions, scale with the parameter λ. Another interesting remark is that since the vector field that defining our system of ODE s does not depend on time, assigning initial data, bt 0 ) Σ, 9) generates a group St) : Σ Σ of smooth flows on the smooth N 1- dimensional compact manifold Σ defined by St)bt 0 ) := bt+t 0 ). An immediate consequence of this fact is that we can forget about the time t 0 where we have defined the initial data. Finally, the last things we need to define are some invariant objects: define the circles T 1,..., T N by T j := {b 1,..., b N ) Σ : b j = 1; b k = 0 for all k j}. Then it is easy to show that the flow St) leaves each circles T j invariant: St)T j = T j. Indeed, for each j we have the following explicit oscillator solutions to 9): 0) b j t) = e it+θ), b k t) = 0 for all k j, where θ is an arbitrary phase. Using this notation, we can rewrite the main theorem as: Theorem. Let N. Given any ε > 0, there exists a point x 1 within ε of T 1 using the usual metric on Σ), a point x N within ε of T N and a time t 0 such that St)x 1 = x N. In other words, there exists solutions to 9) of total mass 1 which are arbitrarily concentrated at the mode j = 1 at some time, and then arbitrarily concentrated at the mode j = N. We will not prove this theorem directly, flowing from T 1 to T N. Instead, we will take some steps: choosing some point x j depending on ε and the previous steps) near T j we will see that there exists a point x j+1 near T j+1 such that St)x j = x j+1. Then we will link all of this steps. To motivate this result let us consider N =. In this case, we prove the theorem giving the following explicit solution say the slider solution): e it ω 1) b 1 t) = 1 + e t e it ω b t) = 1 + e t

33 . REDUCTION TO CASE N= where ω = e πi/ is a cube root of unity. Indeed, this solution approaches to T 1 when t and approaches T when t. Although this can be translated in the j parameter and obtain solutions that slide from T j to T j+1, we can t use this solutions because they require an infinite amount of time for going from each circle to the following one. However, we will use some suitable perturbed version of these slider solutions. In other words, since we have N invariant manifolds T j ) connected by heteroclinic orbits slider solutions), we will find a solution that flows near them. To end this section and prepare the next one, we are going to write b j = x j + iy j and find the equation that satisfy the real part, x j, and the imaginary part, y j, of b j in 9): t x j + iy j ) = ir j x j + iy j ) + ix j iy j ) [ x j 1 y j 1 + ix j 1 y j 1 ] + ix j iy j ) [ x j+1 y j+1 + ix j+1 y j+1 ] = r j y j 4x j [x j 1 y j 1 + x j+1 y j+1 ] + y j [x j 1 y j 1 + x j+1 y j+1] ir j x j + ix j [x j 1 y j 1 + x j+1 y j+1] + 4iy j [x j 1 y j 1 + x j+1 y j+1 ], where rj = x j + y j. Now, taking real an imaginary parts, we get t x j = r j y j 4x j [x j 1 y j 1 + x j+1 y j+1 ] + y j [x j 1 y j 1 + x j+1 y j+1] t y j = r j x j + x j [x j 1 y j 1 + x j+1 y j+1] + 4y j [x j 1 y j 1 + x j+1 y j+1 ].. Reduction to case N= In this section we want to prove Theorem for the case N =, so Σ = {z C : z = 1}. and, writing b = b, b 0, b + ), the equations become { t x x, y ) : = r y 4x [x 0 y 0 ] + y [x 0 y0] t y = r x + x [x 0 y0] + 4y [x 0 y 0 ] { t x x 0, y 0 ) : 0 = r0y 0 4x 0 [x y + x + y + ] + y 0 [x y + x + y+] t y 0 = r0x 0 + x 0 [x y + x + y+] + 4y j [x y + x + y + ] { t x x +, y + ) : + = r+y + 4x + [x 0 y 0 ] + y + [x 0 y0] t y + = r+x + + x + [x 0 y0] + 4y + [x 0 y 0 ] with r + r 0 + r + = 1. Consider the invariant circles defined in the previous section, that, analogously, here will be called T, T 0 and T + instead of T 1, T and T. The idea of the proof will be the following: 1) Take the initial data at p near the unstable manifold of T ) Prove that q = St p )p is near the stable manifold of T 0, for some t p.

34 4. DIFFUSION IN THE TOY MODEL SYSTEM ) Prove that there exists a time t q and a point w near the unstable manifold of T 0 such that w = St q )q. 4) Finally, observe that for some time t w, St w )w is near the stable manifold of T +. The more difficult step is ) and it is where we will use the Šilnikov coordinates. We will see later that steps ) and 4) are equivalent due to the time reversal symmetry that exhibits our equation. For these two reasons, we will focus our argument near T 0, which means r0 1 and r, r+ 1. So it is reasonably to refer our equations to the coordinate b 0 = x 0 + iy 0 which we call the primary mode..1. Local coordinates near T 0. Setting x 0 = r 0 cos θ 0 and y 0 = r 0 sin θ 0, we get that the equations for the secondary modes become: { t x x, y ) : = r y 4r0x cos θ 0 sin θ 0 + r0y [cos θ 0 sin θ 0 ] t y = r x + r0x [cos θ 0 sin θ 0 ] + 4r0y cos θ 0 sin θ 0 { t x x +, y + ) : + = r+y + 4r0x + cos θ 0 sin θ 0 + r0y + [cos θ 0 sin θ 0 ] t y + = r+x + + r0x + [cos θ 0 sin θ 0 ] + 4r0y + cos θ 0 sin θ 0 Since we are near T 0 we must assume that r0 1 so r+, r 1. In order to separate the equations in dominant and small terms, we write r0 = 1 r+ r and grouping terms we get, ) t x ± = 4 [ cos θ 0 sin θ 0 x ± + cos θ 0 sin θ 0 )y ± + r± 4 cos θ0 sin θ 0 x ± + 1 cos θ 0 sin θ 0 ) ) ] [ y ± + r 4 cos θ0 sin θ 0 x ± cos θ 0 sin ) ] θ 0 y± x ±, y ± ) : t y ± = cos [ θ 0 sin θ 0 )x ± + 4 cos θ 0 sin θ 0 y ± + r ± 1 + cos θ 0 sin θ 0 ) ) ] [ x ± 4 cos θ 0 sin θ 0 y ± + r cos θ 0 sin ) ] θ 0 x± 4 cos θ 0 sin θ 0 y ±. On the other hand, the equations for the primary mode become t r 0 = 4 1 r+ r ) 1/ ) cos θ0 sin θ 0 x y + x + y+ 4 1 r+ r ) 1/ cos θ 0 sin ) θ 0 x y + x + y + ) ) t θ 0 = 1 + r+ + r + cos θ 0 sin ) ) θ 0 x y + x + y+ + 8 cos θ 0 sin θ 0 x y + x + y + )... The coordinate change given by Floquet theory. In this section, we will find a change of coordinates that will remove the explicit dependence on the primary modes in the equations for the secondary modes we will delete θ 0 in )). It supposes that we will have reduced the number of variables, from six to four. This change of variables will be given by Floquet theory as we will show. Note that if we neglect the quadratic terms in the equation for θ 0 in ), we get the trivial ODE t θ 0 = 1, so θ 0 = t taking the initial condition at time t = 0).

35 . REDUCTION TO CASE N= 5 Inserting this solution in ) without considering the quadratic terms, we obtain a linear system with periodic coefficients: ) ẋ 4 cos t sin t cos 4) = θ 0 sin ) ) θ 0 ) x ẏ cos θ 0 sin. θ 0 ) 4 cos t sin t y The main idea in this section will be to use the Floquet theory to obtain a change of variables that transform our periodic linear system 4) into a linear system with constants coefficients. Then, we will recover the original system ) and there we will do some change of variables closely inspired in that one. The problem is that Floquet theory is not constructive in the sense that to obtain this change we must know a solution for our problem. In our case, this is not a problem at all because we can integrate 4). Indeed, scaling the time variable τ = t, we get ) ) ẋ sin τ cos τ x 5) = ẏ cos τ sin τ y Now we write this system in a complex form, taking z = x + iy and z = x iy, we obtain: ż = ie iτ z z = ie iτ z. Solving this linear system, we obtain: { zτ) = C1 e iα+τ + C e iα τ zτ) = C 1 α + e iα+τ e iτ + C e iα τ e iτ, C 1, C C where α ± = 1 ± i. Now, taking x = 1 z + z), y = 1 i z z) and setting xτ), yτ) R we obtain that C 1, C may have the form C 1 = a + ia C = b + ib a, b R and 6) xτ) = ae τ ) cos τ/ + sin τ/ + be yτ) = ae τ cos τ/ + sin τ/ ) + be ). ) τ cos τ/ + sin τ/ cos τ τ/ ) sin τ/. Finally, a fundamental matrix for the system 4) is 7) Φt) = e ) cos t + sin t e ) cos t + sin t e cos t + ) sin t e cos t ) sin t Taking into account that our system is π-periodic, we compute the monodromy matrix ) M Φ := Φ0) 1 e Φπ) = π 0 0 e. π

36 6. DIFFUSION IN THE TOY MODEL SYSTEM Now, we have all the ingredients to use the Floquet theory. It says that taking C such that e πc = M Φ, then the change of variables defined by P t) = Φt)e Ct converts our non autonomous periodic linear system into a linear system with constant coefficients. In our case, we need C such that, ) e πc e = π 0 0 e π ) 0 C =. 0 So, the linear change of variables that we shall take is: 8) P t) = cos t + sin t cos t + sin t cos t + sin t cos t sin t ). As we have said, we will take the change: ) x± 9) = P θ 0 ) y ± u± v ± ) where P θ 0 ) = P θ 0 ) = cos θ 0 sin θ 0 cos θ 0 sin θ 0 cos θ 0 sin θ + 0 cos θ 0 sin θ 0 Lemma 5. After making the change on variables defined in 9) the system ) become { t u 40) ± = u ± + u ± [fu ±, v ± ) + gu, v )] + v ± hu, v ) t v ± = v ± v ± [fv ±, u ± ) + gu, v )] u ± hu, v ) where fa, b) = 4 ga, b) = 4 ha, b) = 4 [ a + ab + b ] [ 4 a + 7 ab + 4 ] b = gb, a) [ a + 8 ab + ] b = hb, a) and for the primary mode, 41) t r 0 = 4 r 0u v + u + v+) t θ 0 = 1 4u + v + + u v ). In addition, in these coordinates 4) r ± = 4 u ± + u ± v ± + v ±). ).

37 . REDUCTION TO CASE N= 7 Proof. Denoting B 1 := cos θ 0 sin θ 0 and B := sin θ 0 cos θ 0 we can rewrite the system ) as: ) ) ) ẋ± 4B B = 1 x± ẏ ± B 1 4B y ± }{{} Then 4) u± with v ± D 0 = ) = = E 1 ) + r± 4B 1 B B 1 ) 4B }{{} + r [ ] d dt P 1 x± θ 0 ) [ ] y ± d P 1 θ 0 ) P θ 0 ) dθ 0 }{{} D 0 + r ± P 1 θ 0 )E P θ 0 ) }{{} D ) E ) 4B B 1 B 1 4B }{{} E 1 D 1 = x± ) + P 1 ẋ± θ 0 ) ẏ ± y ± ). ) x± y ± ) θ 0 u± v ± ) + P 1 θ 0 )E 1 P θ 0 ) }{{} D 1 u± u± v ± ) + r P 1 θ 0 ) E 1 )P θ 0 ) }{{} D ) D = On the other hand, putting ) ) x ± = cos θ 0 sin θ 0 u ± + cos θ 0 sin θ 0 ) ) y ± = cos θ 0 sin θ 0 u ± + cos θ 0 + sin θ 0 v ± ) u± v ± ) 0 0 in ) we can easily obtain the equations 41). Then, inserting the equation for θ 0 in 4), we finally get ) ) ) u± 0 = u± v ± 0 v ± [ u ± + u ± v ± + v ±] 0 0 [ u + u ± v + v ] [u v + u + v + ] 4 Now, to obtain 40) we only have to rewrite these terms. ) u± v ± ) u± v ± v ± ) ) u± v ± ) v ± ) )