Dulac map and time in families of hyperbolic saddles David Mar n - PowerPoint PPT Presentation

Dulac map and time in families of hyperbolic saddles David Mar´ ın (UAB) joint work with Jordi Villadelprat (URV) Advances in Qualitative Theory of Differential Equations Castro Urdiales, 17–21 June 2019.

Motivation: Dulac map and time as building block Qualitative behavior (bifurcation) of the period function of a center at the outer boundary of its period annulus (polycycle). line at infinity (polar set) Difficulty: The regularity of the period function drops out at the polycycle.

Motivation: Dulac map and time as building block Qualitative behavior (bifurcation) of the period function of a center at the outer boundary of its period annulus (polycycle). line at infinity (polar set) Difficulty: The regularity of the period function drops out at the polycycle. Tool: Asymptotic expansion of the period function at the polycycle, uniform with respect to parameters.

Motivation: Dulac map and time as building block Qualitative behavior (bifurcation) of the period function of a center at the outer boundary of its period annulus (polycycle). Test Example: Loud family ( − y + xy ) ∂ x + ( x + Dx 2 + Fy 2 ) ∂ y , symmetric system with Darboux first integral (1 − x ) α ( y 2 − P 2 ( x )) for F ( F − 1)( F − 1 / 2) � = 0 (Liouville first integral in general). F = − D Symmetry implies half period is the Dulac time between transverse F = 1 sections located at the symmetry axis. F D D = − 1 D = 0

Motivation: Dulac map and time as building block Qualitative behavior (bifurcation) of the period function of a center at the outer boundary of its period annulus (polycycle). Test Example: Loud family ( − y + xy ) ∂ x + ( x + Dx 2 + Fy 2 ) ∂ y , symmetric system with Darboux first integral (1 − x ) α ( y 2 − P 2 ( x )) for F ( F − 1)( F − 1 / 2) � = 0 (Liouville first integral in general). F = − D Symmetry implies half period is the Dulac time between transverse F = 1 sections located at the symmetry axis. Increasing period function F outside the red line, where the polycycle’s topology changes. D D = − 1 D = 0

Dulac map and time of families of hyperbolic saddles Building block in hyperbolic monodromic polycycles: � � 1 P µ ( x , y ) x ∂ ∂ x + Q µ ( x , y ) y ∂ , λ = − Q µ (0 , 0) X µ = P µ (0 , 0) > 0 , x m y n ∂ y where P , Q are C ∞ functions on Ω × U ⊂ R 2 × R N amb m , n ∈ Z + . FLP: X µ is locally orbitally linearizable ( ⇐ Darboux integrable).

( L , K )-Flatness condition Definition: If W ⊂ R N +1 is an open neighborhood of { 0 } × U and f : W ∩ ((0 , + ∞ ) × U ) → R is of class C K we say that L ( µ 0 ) if ∀ ν = ( ν 0 , ν 1 , . . . , ν N ) ∈ Z N +1 f ( s ; µ ) ∈ F K , | ν | ≤ K , + ∃ V ∋ µ 0 , ∃ C , s 0 > 0 such that ∀ µ ∈ V and ∀ s ∈ (0 , s 0 ) | ∂ ν f ( s ; µ ) | ≤ Cs L − ν 0 , where ∂ ν = ∂ ν 0 s ∂ ν 1 µ 1 · · · ∂ ν N µ N and | ν | = ν 0 + ν 1 · · · + ν N .

( L , K )-Flatness condition Definition: If W ⊂ R N +1 is an open neighborhood of { 0 } × U and f : W ∩ ((0 , + ∞ ) × U ) → R is of class C K we say that L ( µ 0 ) if ∀ ν = ( ν 0 , ν 1 , . . . , ν N ) ∈ Z N +1 f ( s ; µ ) ∈ F K , | ν | ≤ K , + ∃ V ∋ µ 0 , ∃ C , s 0 > 0 such that ∀ µ ∈ V and ∀ s ∈ (0 , s 0 ) | ∂ ν f ( s ; µ ) | ≤ Cs L − ν 0 , where ∂ ν = ∂ ν 0 s ∂ ν 1 µ 1 · · · ∂ ν N µ N and | ν | = ν 0 + ν 1 · · · + ν N . Remark: s λ ◦ s L = s λ L is ( λ L , ∞ )-flat.

( L , K )-Flatness condition Definition: If W ⊂ R N +1 is an open neighborhood of { 0 } × U and f : W ∩ ((0 , + ∞ ) × U ) → R is of class C K we say that L ( µ 0 ) if ∀ ν = ( ν 0 , ν 1 , . . . , ν N ) ∈ Z N +1 f ( s ; µ ) ∈ F K , | ν | ≤ K , + ∃ V ∋ µ 0 , ∃ C , s 0 > 0 such that ∀ µ ∈ V and ∀ s ∈ (0 , s 0 ) | ∂ ν f ( s ; µ ) | ≤ Cs L − ν 0 , where ∂ ν = ∂ ν 0 s ∂ ν 1 µ 1 · · · ∂ ν N µ N and | ν | = ν 0 + ν 1 · · · + ν N . Remark: s λ ◦ s L = s λ L is ( λ L , ∞ )-flat. L ( µ 0 ) extends to a C K function ˜ Lemma: If L > K every f ∈ F K f in a neighborhood of (0 , µ 0 ) such that ∂ ν ˜ f (0; µ ) = 0 for | ν | ≤ K .

( L , K )-Flatness condition Definition: If W ⊂ R N +1 is an open neighborhood of { 0 } × U and f : W ∩ ((0 , + ∞ ) × U ) → R is of class C K we say that L ( µ 0 ) if ∀ ν = ( ν 0 , ν 1 , . . . , ν N ) ∈ Z N +1 f ( s ; µ ) ∈ F K , | ν | ≤ K , + ∃ V ∋ µ 0 , ∃ C , s 0 > 0 such that ∀ µ ∈ V and ∀ s ∈ (0 , s 0 ) | ∂ ν f ( s ; µ ) | ≤ Cs L − ν 0 , where ∂ ν = ∂ ν 0 s ∂ ν 1 µ 1 · · · ∂ ν N µ N and | ν | = ν 0 + ν 1 · · · + ν N . Remark: s λ ◦ s L = s λ L is ( λ L , ∞ )-flat. L ( µ 0 ) extends to a C K function ˜ Lemma: If L > K every f ∈ F K f in a neighborhood of (0 , µ 0 ) such that ∂ ν ˜ f (0; µ ) = 0 for | ν | ≤ K . Lemma: If L ′ > L and f ∈ F K L ′ ( µ 0 ) then f ∈ s L I K ( µ 0 ), i.e. for every n ≤ K there is a neighborhood V ∋ µ 0 such that D n ( f ( s ; µ ) / s L ) → 0 as s → 0 + uniformly on µ ∈ V , where D = s ∂ s is the Euler operator. ( I K are the Mourtada’s classes.)

Unifom asymptotic expansion (in the FLP case) Theorem: For every µ 0 ∈ U there exist a neighborhood V ∋ µ 0 and polynomials D ij , T ij ∈ C ∞ ( V )[ w ] such that ∀ L ∈ R � D ( s ; µ ) = s λ s i + λ j D ij ( ω ; µ ) + F ∞ L ( µ 0 ) , 0 ≤ i + λ 0 j ≤ L � s i + λ j T ij ( ω ; µ ) + F ∞ T ( s ; µ ) = τ 0 ( µ ) log s + L ( µ 0 ) , 0 ≤ i + λ 0 j ≤ L where λ 0 = λ ( µ 0 ).

Unifom asymptotic expansion (in the FLP case) Theorem: For every µ 0 ∈ U there exist a neighborhood V ∋ µ 0 and polynomials D ij , T ij ∈ C ∞ ( V )[ w ] such that ∀ L ∈ R � D ( s ; µ ) = s λ s i + λ j D ij ( ω ; µ ) + F ∞ L ( µ 0 ) , 0 ≤ i + λ 0 j ≤ L � s i + λ j T ij ( ω ; µ ) + F ∞ T ( s ; µ ) = τ 0 ( µ ) log s + L ( µ 0 ) , 0 ≤ i + λ 0 j ≤ L where λ 0 = λ ( µ 0 ). If λ 0 = p / q then ω is the Ecalle-Roussarie compensator (deformation of the logarithm − ln s = ω ( s ; 0)): � 1 x = s − α ( µ ) − 1 x − α ( µ ) dx ω ( s ; α ( µ )) := , α ( µ ) = p − λ ( µ ) q . α ( µ ) s Moreover deg D ij = deg T ij = 0 if λ 0 / ∈ ∆ ij ⊂ Q > 0 discrete subset and τ 0 ≡ 0 except for ( m , n ) = (0 , 0).

Unifom asymptotic expansion (in the FLP case) Theorem: For every µ 0 ∈ U there exist a neighborhood V ∋ µ 0 and polynomials D ij , T ij ∈ C ∞ ( V )[ w ] such that ∀ L ∈ R � D ( s ; µ ) = s λ s i + λ j D ij ( ω ; µ ) + F ∞ L ( µ 0 ) , 0 ≤ i + λ 0 j ≤ L � s i + λ j T ij ( ω ; µ ) + F ∞ T ( s ; µ ) = τ 0 ( µ ) log s + L ( µ 0 ) , 0 ≤ i + λ 0 j ≤ L where λ 0 = λ ( µ 0 ). If λ 0 = p / q then ω is the Ecalle-Roussarie compensator (deformation of the logarithm − ln s = ω ( s ; 0)): � 1 x = s − α ( µ ) − 1 x − α ( µ ) dx ω ( s ; α ( µ )) := , α ( µ ) = p − λ ( µ ) q . α ( µ ) s Moreover deg D ij = deg T ij = 0 if λ 0 / ∈ ∆ ij ⊂ Q > 0 discrete subset and τ 0 ≡ 0 except for ( m , n ) = (0 , 0). Work in progress: elimination of the FLP hypothesis.

Formulae for the first coefficients of the Dulac time (2018) Assume m = 0, n > 0 and define σ ij = σ ( j ) i (0), τ ij = τ ( j ) (0), i � 0 x n − 1 λ = − Q (0 , 0) T 00 = Q ( x , 0) dx , P (0 , 0) , σ 20 � u � P (0 , y ) � dy Q (0 , y ) + 1 L ( u )= exp y , λ 0 � u � Q ( x , 0) � dx M ( u )= exp P ( x , 0) + λ x . 0 ◮ If λ > 1 / n then T ( s ) = T 00 + T 10 s + s I 1 with � σ 20 Q (0 , σ 20 ) + σ 11 σ 1 /λ T 10 = − σ 21 σ n − 1 ∂ 1 Q (0 , y ) L ( y ) dy 20 20 Q (0 , y ) 2 L ( σ 20 ) y 1 /λ − n +1 0 ◮ If λ < 1 / n then T ( s ) = T 00 + T 0 n s λ n + s λ n I 1 with � M ( x ) n � � τ 10 L ( σ 20 ) λ n T 0 n τ − λ n P ( x , 0) − M (0) n dx 10 = nQ (0 , 0) + σ λ n 11 σ n x λ n +1 P (0 , 0) 0 20

Formulae for the first coefficients of the Dulac time (2018) Assume m = 0, n > 0 and define σ ij = σ ( j ) i (0), τ ij = τ ( j ) (0), i Theorem (Mardeˇ si´ c-M.-Villadelprat, 2003) ◮ If λ > 1 / n then T ( s ) = T 00 + T 10 s + s I 1 with � σ 20 Q (0 , σ 20 ) + σ 11 σ 1 /λ T 10 = − σ 21 σ n − 1 ∂ 1 Q (0 , y ) L ( y ) dy 20 20 Q (0 , y ) 2 y 1 /λ − n +1 L ( σ 20 ) 0 ◮ If λ < 1 / n then T ( s ) = T 00 + T 0 n s λ n + s λ n I 1 with � τ 10 � M ( x ) n � L ( σ 20 ) λ n T 0 n τ − λ n P ( x , 0) − M (0) n dx 10 = nQ (0 , 0) + σ λ n 11 σ n x λ n +1 P (0 , 0) 0 20 ◮ If λ ≈ 1 n then T ( s ) = T 00 + s [ T 100 + T 101 ω ( s ; 1 − λ n )] + s I 1 with T 101 = (1 − λ n ) T 0 n and T 100 = T 10 + T 0 n extending to λ = 1 n .

Modifying Mellin transform � ∞ 0 x α f ( x ) dx Mellin transform: f ( x ) �→ { M f } ( α ) = x . Example 0: The Gamma function  { M ( e − x ) } ( α ) if α > 0    { M ( e − x − 1) } ( α )  if α ∈ ( − 1 , 0) Γ( α ) = { M ( e − x − (1 − x )) } ( α ) if α ∈ ( − 2 , − 1)    . .  . . . . � r f ( i ) (0) x i and α / Definition-Proposition: If T r 0 f ( x ) = ∈ Z − then i ! i =0 � u k − 1 � f ( i ) (0) i !( i + α ) u i + u − α ˆ [ f ( x ) − T k − 1 f ( x )] x α − 1 dx f α ( u ) := 0 0 i =0 does not depend on k > − α .