A Parameterization Method for Computing Normally Hyperbolic - PowerPoint PPT Presentation

A Parameterization Method for Computing Normally Hyperbolic Invariant Tori Some Numerical Examples Marta Canadell SIMBa - Universtitat de Barcelona 10 abril 2012

Definitions and introducing the problem The method Implementation Outline Definitions and introducing the problem 1 The method 2 STEP 1: Solve F ( K ( θ ) − K ( f ( θ )) = 0 STEP 2: Solve DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 Improvements in the method The discretization of the system Implementation 3 Some information of 3D-FAF Numerical Results Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation Definitions and introducing the problem Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation The system model Consider a map in R n : F : R n → R n (a discrete dynamical system). � � Consider a d -manifold M d = T d , S d , . . . ⊂ R n . A parameterization of it will be an immersion K : M d → R n ( DK has maximum rank d ), d < n , K = K ( M d ) . Let f : M d → M d be the dynamics of F restricted over the manifold M d . Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation The system model z K = K ( M d ) F R n F ( K ( θ )) K ( θ ) y K x θ 1 R d f f ( θ ) M d θ θ 2 Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation The Invariance equation Definition We say that the manifold parameterized by K , K , is invariant under F with internal dynamics f if K and f meets the invariance equation: F ◦ K = K ◦ f ie: if for each θ ∈ M d we have F ( K ( θ )) = K ( f ( θ )) . We want to find these functions K and f . Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation Vector Bundles, hyperbolicity and Normal Hyperbolicity Consider M ⊂ R n a manifold. For one point of the manifold M � vector spaces : T p M , tangent space to M at p . Particulary, T p R n = R n the tangent space to R n at p . N p = { v ∈ R n | v ⊥ T p M } , normal space to M at p . T p R n = T p M + N p , as a vectorial spaces sum. Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation Vector Bundles, hyperbolicity and Normal Hyperbolicity For all points (together) of the manifold M � vector bundles : a manifold such which have a linear vector space associated to every point of it: TM = { ( p, v ) ∈ M × R n | v ∈ T p M } , tangent bundle of M . N = { ( p, v ) ∈ M × R n | v ⊥ T p M } , normal bundle of M . T R n | M = TM ⊕ N , as a Whitney sum of the vector bundles. = R d and N p ∼ For each p ∈ M we have T p M ∼ = R n − d , called the fibers of the vector space. Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation Suppose we have found K = K ( M d ) and f . DK generates the tangent space to each point of the invariant manifold, T K ( θ ) K : ( DK ) θ : T θ M d → T K ( θ ) K for each θ ∈ M d , DK ( θ ) is represented as a n × d matrix, a fiber. Considering all θ ∈ M d , we have a parameterization of the tangent bundle, T K . if N ( θ ) is a n × n − d matrix composed by n − d vectors linearly independent to the vectors of DK ( θ ) , then it generates the normal space N K ( θ ) , complementary to T K ( θ ) K . Considering all θ ∈ M d , we have a parameterization of the normal bundle, N ( K ) . Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation So, we could write: T K ( θ ) R n = T K ( θ ) K + N K ( θ ) ∼ = R n as a sums of vector spaces. T R n |K = T K ⊕ N ( K ) as a sums of vector bundles. We could define P ( θ ) := ( DK ( θ ) | N ( θ )) n × n , a vector bundle which generates the total space. So, it is an adapted frame of if P : M d → T K ( θ ) K + N K ( θ ) Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation We could also define the matrix map Λ : M d → M n × n as the dynamics over the tangent and normal bundles, the linearized dynamics on this frame. Then, Λ and P must satisfy the invariance of the splitting on the bundles: P ( f ( θ )) − 1 DF ( K ( θ )) P ( θ ) = Λ( θ ) Remark The dynamics of the bundles on the model manifold will be of the form � � Λ t ( θ ) B ( θ ) Λ( θ ) = Λ n ( θ ) O where Λ t is the linearized dynamics on the tangent space (a d × d matrix), Λ n is the linearized dynamics on the normal space (a n − d × n − d matrix) and B ( θ ) is a d × n − d matrix. Marta Canadell A Parameterization Method for computing NHIT

Definitions and introducing the problem The method Implementation As � � Λ t ( θ ) B ( θ ) DF ( K ( θ )) ( DK ( θ ) | N ( θ )) = ( DK ( f ( θ )) | N ( f ( θ ))) : Λ n ( θ ) O If B ( θ ) = 0 , then the normal bundle is invariant: the invariance equation is satisfied on the normal subspace DF ( K ( θ )) N ( θ ) = N ( f ( θ ))Λ n ( θ ) If Λ t ( θ ) = Df ( θ ) , then the tangent bundle is invariant: the invariance equation is satisfied on the tangent subspace DF ( K ( θ )) DK ( θ ) = DK ( f ( θ )) Df ( θ ) This is always true, as this equation is just the derivative of invariance equation. So, we will consider B ( θ ) = 0 and Λ t ( θ ) = Df ( θ ) to have last two conditions true and have both bundles invariant. In this way, the invariant manifold will be normally hyperbolic. Marta Canadell A Parameterization Method for computing NHIT

STEP 1: Solve F ( K ( θ ) − K ( f ( θ )) = 0 Definitions and introducing the problem STEP 2: Solve DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 The method Improvements in the method Implementation The discretization of the system The method Marta Canadell A Parameterization Method for computing NHIT

STEP 1: Solve F ( K ( θ ) − K ( f ( θ )) = 0 Definitions and introducing the problem STEP 2: Solve DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 The method Improvements in the method Implementation The discretization of the system The algorithm is inspired in the parameterization method (Cabré,Fontich, de la Llave) for finding a parameterization of the invariant manifold and a dynamics on it. The framework leads to solving invariance equations , for which one uses a Newton method adapted to the dynamics and the geometry of the (invariant) manifold, normally hyperbolic invariant tori. Marta Canadell A Parameterization Method for computing NHIT

STEP 1: Solve F ( K ( θ ) − K ( f ( θ )) = 0 Definitions and introducing the problem STEP 2: Solve DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 The method Improvements in the method Implementation The discretization of the system Summary of the problem If we want to find a normally hyperbolic invariant manifold, we are looking for K , f , P and Λ such that: F ( K ( θ ) − K ( f ( θ )) = 0 DF ( K ( θ ) P ( θ ) − P ( f ( θ ))Λ( θ ) = 0 DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 But as tangent bundle and its dynamics are well defined directly using derivatives of known values K and f , so we only have to solve the second equation for the normal part. Marta Canadell A Parameterization Method for computing NHIT

STEP 1: Solve F ( K ( θ ) − K ( f ( θ )) = 0 Definitions and introducing the problem STEP 2: Solve DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 The method Improvements in the method Implementation The discretization of the system Summary of the problem If we want to find a normally hyperbolic invariant manifold, we are looking for K , f , P and Λ such that: F ( K ( θ ) − K ( f ( θ )) = 0 DF ( K ( θ ) P ( θ ) − P ( f ( θ ))Λ( θ ) = 0 DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 But as tangent bundle and its dynamics are well defined directly using derivatives of known values K and f , so we only have to solve the second equation for the normal part. Marta Canadell A Parameterization Method for computing NHIT

STEP 1: Solve F ( K ( θ ) − K ( f ( θ )) = 0 Definitions and introducing the problem STEP 2: Solve DF ( K ( θ ) N ( θ ) − N ( f ( θ ))Λ n ( θ ) = 0 The method Improvements in the method Implementation The discretization of the system A newton method to compute K , f , P , Λ Given an approximate normally hyperbolic invariant torus ( K 0 , f 0 ) and its bundles ( P 0 , Λ 0 ) with error: F ( K 0 ( θ ) − K 0 ( f 0 ( θ )) = R ( θ ) n × 1 DF ( K 0 ( θ )) P 0 ( θ ) − P 0 ( f 0 ( θ ))Λ 0 ( θ ) = S ( θ ) n × n We look for ( H, h, Q, ∆) satisfying � � h ( θ ) − ˜ R ( θ ) = Λ( θ ) H ( θ ) − − H ( f ( θ )) 0 � �� � n × d − ˜ S n ( θ ) = Λ( θ ) Q ( θ ) − Q ( f ( θ ))Λ n ( θ ) − ∆( θ ) And obtain the improved torus: K ( θ ) = K 0 ( θ ) + P 0 ( θ ) H ( θ ) , f ( θ ) = f 0 ( θ ) + h ( θ ) , Λ n ( θ ) = Λ n N ( θ ) = N 0 ( θ ) + P 0 ( θ ) Q ( θ ) , 0 ( θ ) + ∆( θ ) Marta Canadell A Parameterization Method for computing NHIT