Numerical Computation of Invariant Objects with Wavelets David Romero i S` anchez Director Dr. Ll. Alsed` a Departament de Matem` atiques Universitat Aut` onoma de Barcelona de novembre de
Outline Motivation A Primer on Wavelets and Regularity The construction of the wavelets Regularity with wavelet coefficients Numerical Computation of Invariant Objects with Wavelets Using the Fast Wavelet Transform Solving the Invariance Equation by means of Haar Solving the Invariance Equation by means of Daubechies
Motivation Wavelets in Theory Wavelets in Practice Motivation We are interested in approximate, via expansions of a truncated base of wavelets, complicated objects semianalitically. From such approximation, we want to predict and understand changes in the geometry or dynamical properties (among others) of such objects. As a testing ground of our developed techniques, we will be focused on skew products of the form � � θ n � θ n +1 = R ω ( θ n ) = θ n + ω (mod 1) , = F σ,ε () x n x n +1 = F σ,ε ( θ n , x n ) , here x ∈ R + , θ ∈ S 1 = R / Z , ω ∈ R \ Q . D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice The [GOPY]-Keller model: a testing ground In the System (), we take F σ,ε ( θ, x ) = f σ ( x ) g ε ( θ ) ( multiplicative forcing ) with → [0 , ∞ ) ∈ C 1 , bounded, strictly increasing, f σ : [0 , ∞ ) − strictly concave and verifying f (0) = 0 . g ε : S 1 − → [0 , ∞ ) bounded and continuous. √ 5 − 1 Fixing ideas, we will use ω = and the following 2 one-parameter family of skew products (with x ≡ 0 invariant) θ n +1 = θ n + ω (mod 1) , � θ n � = F σ,ε ( σ ) () ε ( σ ) x n ���� x n +1 = 2 σ tanh( x n )( ε + | cos(2 πθ n ) | ) , � where ( σ − 1 . 5) 2 when 1 . 5 < σ ≤ 2 , ε ( σ ) = 0 when 1 < σ ≤ 1 . 5 . The toy model is similar to the [GOPY] model. [GOPY] Grebogi, Celso et al. , Strange attractors that are not chaotic , Phys. D – –. D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice The [GOPY]-Keller model: a testing ground In this testing ground we want to approximate the attractor, ϕ , of the above system (if it exists). Pinching condition ⇒ SNA’s creation When g ε = 0 at some point it is called the pinched case , whereas if g ε is strictly positive it is called the non-pinched case . In the pinched case, any F σ,ε –invariant set has to be 0 on a point and, hence, on a dense set (in fact on a residual set ). This is because the circle x ≡ 0 is invariant and the θ -projection of every invariant object must be invariant under R ω . Our main goal: work with wavelet approximations Compute ϕ in terms of wavelet coefficients to recover the appearance of the residual set from such coefficients. D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice The [GOPY]-Keller model: a testing ground In the next slide will appear a theorem due to Keller [Kel] that makes the above informal ideas rigorous. Before stating it we need to introduce the constant σ : Since the line x = 0 is invariant, by Birkhoff Ergodic Theorem, the vertical Lyapunov exponent on the circle x ≡ 0 is the logarithm of �� � σ := f ′ (0) exp S 1 log g ε ( θ ) dθ < ∞ . A particular instance of the Keller-GOPY attractor The parameterization ε ( σ ) controls the Lyapunov Exponent and the pinched point at the same time . D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice Keller’s Theorem (shortened) There exists an upper semicontinuous map ϕ : S 1 − → [0 , ∞ ) whose graph is invariant under the Model (). Moreover, if σ > 1 and g ε ( θ 0 ) = 0 for some θ 0 then the set { θ : ϕ ( θ ) > 0 } has full Lebesgue measure and the set { θ : ϕ ( θ ) = 0 } is residual, if σ > 1 and g ε > 0 then ϕ is positive and continuous; if g ε is C 1 then so is ϕ , if σ � = 1 then | x n − ϕ ( θ n ) | → 0 exponentially fast for almost every θ and every x > 0 . [Kel] Keller, Gerhard, A note on strange nonchaotic attractors , Fund. Math. –. D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice On the use of wavelets Notice that the invariant objects that we want to compute are expressed as graphs of functions (from S 1 to R ). The standard approach to compute with such objects is to use finite Fourier approximations to get expansions as: N � ϕ ∼ a 0 + ( a n cos( nθ ) + b n sin( nθ )) . n =1 Since the topology and geometry of these objects is extremely complicate, the regularity and periodicity of the Fourier basis make this approach too costly. D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice On the use of wavelets In this case, it seems more natural to use wavelets (an orthonormal basis of L 2 ( R ) ) that adapt much better to oscillatory, irregular and highly discontinuous objects. 2 j − 1 N � � ϕ ∼ a 0 + d − j,n ψ PER − j,n ( θ ) , j =0 n =0 where ψ PER is a given wavelet. Summarizing : given a generic skew product we want to Derive properties of ϕ we need we do Massive approximations of ϕ we need Massive calculation of d − j,n and ψ PER − j,n ( θ ) D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice Outline Motivation A Primer on Wavelets and Regularity The construction of the wavelets Regularity with wavelet coefficients Numerical Computation of Invariant Objects with Wavelets Using the Fast Wavelet Transform Solving the Invariance Equation by means of Haar Solving the Invariance Equation by means of Daubechies D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice A primer on wavelets Let us start by the definition of Multi-resolution Analysis (MRA) Definition A sequence of closed subspaces of L 2 ( R ) , {V j } j ∈ Z , is a Multi-resolution Analysis if it satisfies: { 0 } ⊂ · · · ⊂ V 1 ⊂ V 0 ⊂ V − 1 ⊂ · · · ⊂ L 2 ( R ) . { 0 } = � j ∈ Z V j . �� � = L 2 ( R ) . clos j ∈ Z V j There exists a function φ ( x ) whose integer translates , φ ( x − n ) , form an orthonormal basis of V 0 . Such function is called the scaling function . For each j ∈ Z it follows that f ( x ) ∈ V j if and only if f ( x − 2 j n ) ∈ V j for each n ∈ Z . For each j ∈ Z it follows that f ( x ) ∈ V j if and only if f ( x/ 2) ∈ V j +1 . D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice A primer on wavelets Consider the bi-indexed family of maps obtained by dilations and translations of φ ( x ) : � x − 2 j n � 1 √ φ j,n ( x ) = 2 j φ . 2 j It is shown that { φ j,n } n ∈ Z is an orthonormal basis of V j for each j ∈ Z , and φ ( x ) characterizes the whole MRA (see [Mal]). [Mal] Mallat, St´ ephane, A wavelet tour of signal processing , Academic Press Inc., San Diego, CA, , xxiv+. D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice A primer on wavelets If we fix an MRA, we know that V j ⊂ V j − 1 . Then, we define the subspace W j as the orthogonal complement of V j on V j − 1 . That is V j − 1 = W j ⊕ V j . We are looking for an orthonormal basis of W j : the wavelets . This basis is given, from a function called the mother wavelet ψ ( x ) , by � x − 2 j n � 1 ψ j,n ( x ) = √ 2 j ψ . 2 j The integer translates , ψ ( x − n ) , of ψ ( x ) form an orthonormal basis of W 0 . Also, ψ ( x ) verifies a relation with φ ( x ) . Moreover, from [Mal]: Mallat and Meyer Theorem For every j ∈ Z the family { ψ j,n } n ∈ Z is an orthonormal basis of each W j , The wavelets { ψ j,n } ( j,n ) ∈ Z × Z are an orthonormal basis of L 2 ( R ) for all j, n ∈ Z . D. Romero Numerical Computation of Invariant Objects with Wavelets //
Motivation Wavelets in Theory Wavelets in Practice Summarizing � � W j := V j − 1 \V j L 2 ( R ) = clos L 2 ( R ) = clos V j W j j ∈ Z j ∈ Z V j = span { φ j,n ( x ) } n ∈ Z W j = span { ψ j,n ( x ) } n ∈ Z V 0 = span { φ ( x − n ) } n ∈ Z W 0 = span { ψ ( x − n ) } n ∈ Z � 1 2 e − iω � h ∗ ( ω + π ) � ψ ( ω ) := φ ( ω ) √ φ ( x ) ψ ( x ) ∞ h (2 − p ω ) � � � φ ( ω ) = √ 2 p =1 � � h [ n ] e − inω � � h ( ω ) = 1 2 φ ( x 1 2 ψ ( x √ 2 ) = h [ n ] φ ( x − n ) √ 2 ) = g [ n ] φ ( x − n ) n ∈ Z n ∈ Z n ∈ Z h [ n ] g [ n ] g [ n ] := ( − 1) 1 − n h [1 − n ] D. Romero Numerical Computation of Invariant Objects with Wavelets //
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