Some Quasi-Periodic and almost periodic solutions in coupled map lattices and flows E. Fontich, Y. Sire D. Blazevski R. de la Llave Georgia Institute of Technology http://www.ma.utexas.edu/mp arc-bin/mpa?yn=12-26 Hamiltonian PDE’s, Fields Inst. 2014 Closely related work with M. Jiang, E. Fontich, P. Mart´ ın, A. Haro
Basic set up. Formulation of final result We consider a lattice Z d (or a more general network) of • Hamiltonian systems. (analytic) Each of the single site systems contains: • A positive measure of KAM tori, nondegenerate. • One hyperbolic fixed point • We couple different sites by a local interaction which is also Hamiltonian (analytic)
Some examples Klein-Gordon model � W 1 ( q i − q j ) ¨ q j = −∇ [ V ( q j ) + ε | i − j | = 1 � W 2 ( q i − q j ) + · · · ] + ε | i − j | = 2 Very often V ( q ) = cos ( q ) , W 1 ( t ) = 1 2 t 2 , W 2 ( t ) = 1 4 t 2 . XY -Heisenberg models of spin waves W 1 ( η ) = sin ( η ) V ( q ) = B sin ( q ) . These models appear also as discretization of non-linear wave equations. They also appear in Statistical mechanics, neuroscience,
Dynamics of each of the sites • Positive Lebesgue measure of frequency ω ∈ Ω , nondegenerate. NOTE: KAM tori of the system may have different topology.
Dynamics of each of the sites � H 0 ( x i ) H 0 ( x ) = i ∈ Z d In the full systems we can intersperse tori with fixed points All of those are “whiskered” KAM tori.
We construct solutions of the form of cluster of sites oscillating with frequencies ω i but separated by positions separated by wide swatches of systems that are close to the hyperbolic fixed point. • We will also obtain that the tori produced have decay properties. That is, the effect of one site on another far appart is very small
We consider a regime in which the energy is not finite. The energy per site is finite Indeed the average energy is close to a critical value.
Roughly, we start by choosing ω (satisfying some mild conditions) then, we start placing the oscillations. If we place them far enough apart, we get approximate solutions of an invariance equation. We will show that they persist in the full system.
For simplicity we will formulate the theorem for maps. (There are very simple arguments that show that the results for maps imply the results for flows, Douady82,...) The proofs can be easily adapted.
Theorem A There is a set of frequencies Ω( ε ) , ε < ε ∗ | Ω( ε ) | / | Ω | → 1 as ε → 0 so that we can find one breather localized at one site. Theorem B Given any probability measure on Ω( ε ) equivalent to Lebesgue. In a set of full measure in Ω( ε ) N we can find a breather with frequency ω .
Theorem A does not need that the system is translation invariant but assumes that the local properties are uniformly smooth. (e.g. random media). Theorem B uses translation invariance as formulated here, but there is a version weakening it. No smallness condition, no loss of measure of in Theorem B. We can be significantly more precise — even if more technical — about the shape of the tori and the frequencies which appear. We will show that the tori are rougly products.
Both theorems will be deduced from a a more technical theorem. If we have an approximation solution of an invariance equation, which satisfies some appropriate non-degeneracy conditions = ⇒ There exists a true solution nearby. (Furthermore, it is the only solution in this neighborhood up to change of origin in the torus.) We will make precise the conditions when we have motivated them. The proof is based on an interative method in an appropriate function space.
This is, in particular, a theorem on persistence of whiskered KAM tori. • It has an a posteriori format. Approximate solutions lead to true solutions. • The method does not rely on transformation theory. • The proof can be supplemented by algorithmic details to become very efficient numerical algorithms. (Developed and implemented with G. Huguet and Y. Sire). • A finite dimensional version of the proof has been published by (E. Fontich, R.L. and Y. Sire in Jour. Diff. Eq. (2009), ERA (2009) ). In this lecture, we will emphasize more the infinite dimensional aspectes.
• There are theorems (Soffer-Weinstein, Pyke, Sigal, Comech-Komech) showing that solutions with this form cannot happen in hyperbolic PDES (“damping by radiation”). We think that this damping by radiation mechanism also applies in other regions of phase space. These systems share properties with PDE’s.
• There are other invariant objects which also can be studied by similar formalisms. E. Fontich, R. L. P. Mart´ ın (Jour. Diff. Eq. (2010) and MP ARC 10-75, 10-76 ) consider also hyperbolic sets and their manifolds. Show decay properties and develop a structural stability. Invariant measures in some of these sets were already considered in the literature (M. Jiang, Y. Pesin, etc. ).
• The main idea is that the couplings are smooth and decay “fast enough” with the distance. One of the key ideas is how to quantify the decay of the couplings. We will obtain also that the invariant objects produced also can be written as local objects and some non-local corrections which decay fast.
� H 0 ( x i ) + ε H 1 ( x ) H ε ( x ) = i � ∂ j ∂ i H 1 ( x ) � ρ ≤ C Γ( i − j ) � � ρ an analytic norm Γ a decay function Γ ≥ 0 � i Γ( i ) = 1 � j Γ( i − j )Γ( j − k ) ≤ Γ( i − k ) Even if H ε is a formal sum, the derivatives are bona-fide functions.
Example 1 Γ( i ) = ce − β | i | not a decay function. Γ( i ) = c da | i | − d + θ e − α | i | is a decay function for θ > 0, α ≥ 0. Physical interpretation ∂ j ∂ i H 1 ≡ effect of particle j on particle i � ∂ j ∂ i H 1 � ≡ bound on the effect of j on i � j ∂ k ∂ j H 1 ∂ j ∂ i H 1 ≡ effect of k on i through affecting j . The bounds are decay functions, imply that we can bound the effect of indirect interaction by the direct interaction. No cluster expansions needed!
For matrices (may be ∞ matrices), define | A ij | Γ( i − j ) − 1 � A � Γ = sup i , j Hence, | A ij | ≤ � A � Γ( i − j ) � | ( A B ) ij | ≤ A i k B k j k � ≤ � A � Γ � B � Γ Γ( i − k )Γ( k − j ) k ≤ � A � Γ � B � Γ Γ( i − k ) Banach algebra.
Decay functions were introduced in M. Jiang, R.L. Comm. Math. Phys. (2000) to study dependence on parameters of SRB measures. Note that since sup-norms are used. We get pathologies inherited from ℓ ∞ . For example, there are “observables at infinity” , i.e. nontrivial functionals that have zero partial derivatives, non-duality, no smooth functions with bounded support, etc.
• We can choose any frequency which satisfies an ∞ -dimensional Diophantine condition � � � − τ R � | ω i · k i | ≥ c R | k i | | i |≤ R | i |≤ R The theorem will be deduced from another theorem on persistence of whiskered tori with decay properties.
Consider mappings on P ≡ ( R n × T n ) Z d (endowed with ℓ ∞ ) f such that • f analytic � ∂ f i • ( Df ( x ) η ) i = η j ∂ x j (The derivative is given by the Jacobian matrix) � � � � ∂ f i � � ≤ c Γ( i − j ) � � ∂ x j ρ � � ρ = sup in a complex neighborhood of size 0
In ℓ ∞ , it is non-trivial to assume that the derivative is given by the Jacobian matrix. Example 2 A η = lim i →∞ η i is a non-trivial linear operator (Hann-Banach) whose matrix is zero.
We consider embeddings K : ( T n ) N → P which are “centered around { c i } N i = 1 ” � � � � ∂ K i � � ≤ C Γ( i − c j ) � � ∂θ j ρ � K i � ρ ≤ C max (Γ( i − c k )) k
Symplectic geometry in infinite dimensional spaces has, in general many pathologies. Turns out one can develop a nice theory for maps with decay and embeddings with decay. The pull back of a formal symplectic form under embeddings with decay, is a well defined finite dimensional form.
These K give an invariant torus when f ◦ K ( θ ) = K ( θ + ω ) ⋆ K F K T d T d K( )
Theorem f analytic, decay Γ exact symplectic K : an embedding centered at c i , decay Γ ∗ Df ◦ K ( θ ) admits an approximate invariant splitting which is hyperbolic � � [ Df ◦ K ( θ )] E s , u , c ( θ ) , E s , u , c d Γ ≤ ε ( θ + ω )
� � � � � Df ◦ K ( θ ) | E s Γ ≤ λ < 1 � θ � � � Df − 1 ◦ K ( θ ) | E u � Γ ≤ λ < 1 θ � � � Df ◦ K ( θ ) | E c � Γ ≤ η θ � � � � � Df − 1 ◦ K ( θ ) | E c � Γ ≤ η ( θ ) λη < 1 [Actually what is used in the paper is a bit more general]
∗ The splittings have decay properties. Denote by π the projections. � π s , u , c � ρ, Γ < ∞ The embeddings of the model spaces are also centered around c
∗ The center direction is 2 Nd dimensional ∗ Df ◦ K is non-degenerate in the center direction (explicit expression to be given later) ∗ ω is σ − τ Diophantine ω ∈ ( R n ) u ∗ � f ◦ K − K ◦ T ω � c , Γ ,ρ ≤ ε ∗ ε ≤ ε ∗ ( σ, τ, Nd non-degeneracy, ρ − ρ ′ )
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