Cyclicity of Nilpotent Centers Douglas S. Shafer April 21, 2015 - PowerPoint PPT Presentation

Cyclicity of Nilpotent Centers Douglas S. Shafer April 21, 2015

Collaborator and Computer Algebra Tools This research was done in collaboration with Isaac A. Garc´ ıa Lleida, Spain Focus quantities computed using: M ATHEMATICA 8.0—A General Purpose Computer Algebra System Computations with ideals done using: S INGULAR 3-1-6—A Computer Algebra System for Polynomial Computations

Polynomial Systems Contrasts Non-degenerate Nilpotent x = − y + R ( x , y ) ˙ x = y + R ( x , y ) ˙ y = ˙ x + S ( x , y ) y = ˙ S ( x , y ) monodromic conditionally monodromic a center iff there is a a center can exist without suitable first integral a formal or analytic first integral the Lyapunov quantities are polynomials in the the Lyapunov quantities coefficients are conditionally polynomials in the coefficients

Andreev’s Monodromy Theorem (1955, 1958) x = y + R ( x , y ) ˙ X : y = ˙ S ( x , y ) Let y = F ( X ) be the unique solution of y + R ( x , y ) = 0 and f ( x ) = S ( x , F ( x )) = ax α + · · · ϕ ( x ) = div X ( x , F ( x )) = bx β + · · · The origin is monodromic if and only if α = 2 n − 1 is an odd integer a < 0 ϕ ≡ 0 or β � n or β = n − 1 and b 2 + 4 an < 0.

Andreev Number The Andreev number for the system x = y + R ( x , y ) ˙ X : y = ˙ S ( x , y ) with f ( x ) = S ( x , F ( x )) = ax 2 n − 1 + · · · , a < 0 ϕ ( x ) = div X ( x , F ( x )) = bx β + · · · for which β = n − 1 and b 2 + 4 an < 0 ϕ ≡ 0 β � n or or is the number n .

Standard Form f ( x ) = S ( x , F ( x )) = ax α + · · · x = y + R ( x , y ) ˙ ϕ ( x ) = div X ( x , F ( x )) = bx β + · · · y = ˙ S ( x , y ) x = u ↓ y = v + F ( u ) f ( u ) = au α + · · · u = v + v � ˙ R ( u , v ) ϕ ( u ) = bu β + · · · v = f ( u ) + v ϕ ( u ) + v 2 � ˙ S ( u , v ) u = ξ x ↓ v = − ξ y f ( x ) = − ξ − 1 f ( ξ x ) = − a ξ α − 1 x α + · · · x = − y + y � � ˙ R ( x , y ) ϕ ( x ) = ϕ ( ξ x ) = b ξ β x β + · · · y = � ϕ ( x ) + y 2 � ˙ f ( x ) + y � S ( x , y ) �

Lyaunov’s Generalized Trigonometric Functions For n ∈ N let x = Cs θ y = Sn θ denote the unique solution of dx d θ = − y x (0) = 1 dy d θ = x 2 n − 1 y (0) = 0 Cs θ and Sn θ are periodic of least period � Γ( 1 2 n ) π T n = 2 n Γ( n +1 2 n ) and satisfy Cs 2 n θ + n Sn 2 θ = 1 .

Generalized Polar Coordinates For an analytic monodromic system with Andreev number n, x = − y + y � ˙ R ( x , y ) y = � ϕ ( x ) + y 2 � ˙ f ( x ) + y � S ( x , y ) define y = r n Sn θ x = r Cs θ, to obtain dr d θ = F [ n ]( r , θ ) for which F [ n ]( r , θ ) is defined and analytic on a neighborhood of r = 0, is T n -periodic, and satisfies F [ n ](0 , θ ) ≡ 0.

Generalized Lyapunov Quantities v j Let Ψ( r ; h ) solve dr d θ = F [ n ]( r , θ ), Ψ(0; h ) = h . � v j h j P ( h ) = Ψ( T n ; h ) d ( h ) = P ( h ) − h = j � 1 v 1 = Ψ 1 ( T n ) − 1 , v j = Ψ j ( T n ) , j � 2

Polynomial v j Given a monodromic polynomial family parametrized by the admissible coefficients, λ , x = − y + y � ˙ R ( x , y ) ϕ ( x ) + y 2 � y = � ˙ f ( x ) + y � S ( x , y ) with f ( x ) = a 2 n − 1 x 2 n − 1 + · · · , ϕ ( x ) = b β x β + · · · , � � the Poincar´ e-Lyapunov quantities v i are polynomials in the parameters if and only if 1. a 2 n − 1 is a fixed (positive) constant, not a parameter, which without loss of generality can be assumed to be 1; and 2. if � ϕ ( x ) �≡ 0 and β = n − 1 then b β is a fixed constant, not a parameter.

Polynomial v j : Proof x = − y + y � u = v + v � ˙ R ( x , y ) ˙ R ( u , v ) ( y 2 ) ( v 2 ) y = � ˙ f ( x ) + y � ϕ ( x ) + · · · u = f ( u ) + v ϕ ( u ) + ˙ · · · u = ξ x ← − − − − f = a 2 n − 1 x 2 n − 1 + · · · f = ax 2 n − 1 + · · · v = − ξ y � ϕ = bx n − 1 + · · · ϕ = b n − 1 x β + · · · � d θ = H 1 r + H 2 r 2 + · · · dr = H 1 r + H 2 J 0 − H 1 J 1 r 2 + · · · J 2 J 0 + J 1 r + · · · J 0 0 where each H i and J i is a polynomial in λ , Cs θ , and Sn θ , and a 2 n − 1 Cs 2 n θ + n Sn 2 θ ) + b n − 1 Cs n θ Sn θ J 0 = ( = ( − a ξ 2 n − 2 Cs 2 n θ + n Sn 2 θ ) + b ξ n − 1 Cs n θ Sn θ

Bautin Ideal Analytic Polynomial x = y + R ( x , y , λ ) ˙ x = y + yR ( x , y ) ˙ y = ˙ S ( x , y , λ ) y = ˙ S ( x , y ) parametrized by admissible coefficients � v j ( λ ) h j d ( h ) = P ( h ) − h = j � 1 B = � v 1 ( λ ) , v 2 ( λ ) , · · ·� ∈ R [ λ ] B = � v 1 ( λ ) , v 2 ( λ ) , · · ·� ∈ G λ ∗

Minimal Basis The minimal basis of a finitely generated ideal I with respect to an ordered basis B = { f 1 , f 2 , f 3 , . . . } is the basis M I defined by the following procedure: (a) initially set M I = { f p } , where f p is the first non-zero element of B ; (b) sequentially check successive elements f j , starting with j = p + 1, adjoining f j to M I if and only if f j / ∈ � M I � , the ideal generated by M I . Example For I = � x 3 , x 2 , x � in R [ x ], M I = { x , x 2 , x 3 }

Small Zeros of Analytic Functions (Bautin) Technical Lemma If { f j 1 , . . . , f j s } is the minimal basis for the ideal � f j : j ∈ N � with generators ordered by the indices, then the analytic function Z ( h , λ ) = � f j ( λ ) h j can be validly expressed as Z ( h , λ ) = f j 1 ( λ )[1 + ψ 1 ( h , λ )] h j 1 + · · · + f j s ( λ )[1 + ψ s ( h , λ )] h j s . Zeros Theorem Suppose ψ j (0 , λ ∗ ) = 0 for all j . Then there exist δ and ǫ such that for each λ satisfying | λ − λ ∗ | < δ the equation Z ( h , λ ) = 0 has at most s − 1 isolated solutions in the interval 0 < h < ǫ .

Cyclicity Bound Theorem � v j h j d ( h ) = P ( h ) − h = j � 1 If the minimal basis of the Bautin ideal B = � v 1 , v 2 , · · ·� is M B = { v j 1 , v j 2 , · · · , v j s } then the cyclicty of a center at the origin of any member of the family is at most s − 1. Problems: 1. The generalized Lyapunov quantities are difficult to compute. 2. Even if we know a collection { v j 1 , . . . , v v s } whose vanishing at λ ∗ implies that all v j vanish, we do not necessarily know a basis of B .

Odd Degree Homogeneous Nonlinearities Henceforth restrict to x = y + P 2 m +1 ( x , y ) ˙ y = ˙ Q 2 m +1 ( x , y ) Andreev’s Monodromy Theorem: (0 , 0) is monodromic iff Q 2 m +1 (1 , 0) < 0 Make the change x = u − p ( − q ) − 1 / 2 v y = ( − q ) 1 / 2 v t = ( − q ) − 1 / 2 τ where p = P 2 m +1 (1 , 0) and q = Q 2 m +1 (1 , 0)) .

Odd Degree Homogeneous Nonlinearities in Standard Form x = y + yP ( x , y ) ˙ − x 2 m +1 + bx 2 m y + · · · + cy 2 m +1 y = ˙ for which f ( x ) = − x 2 m +1 � bx 2 m if b � = 0 ϕ ( x ) = 0 if b = 0 Andreev number n = m + 1 v j ( λ ) ∈ R [ λ ], λ the admissible coefficients

Focus Quantities (Amel’kin, Lukashevich, Sadovskii, 1982) For x = y + yP ( x , y ) ˙ X : − x 2 m +1 + bx 2 m y + · · · + cy 2 m +1 y = ˙ there exists a formal series � W ( x , y ) = ( m + 1) y 2 + W 2( km +1) ( x , y ) , k � 1 W j homogeneous of degree j such that X W = x 2(2 m +1) � k � 1 g k x 2 km = � k � 1 g k x K ( k ) g k ∈ R [ λ ] (0 , 0) is a center for system λ ∗ iff g k ( λ ∗ ) = 0 for all k

The Focus Quantities and the Generalized Lyapunov Quantities Generalized Lyapunov quantities v j : control stability and cyclicity recursively computed via integrations Focus quantities g j : pick out centers recursively computed via algebra Theorem Let I k = � g 1 , g 2 , . . . , g k � . There exist positive constants w k that are independent of λ such that v 1 = · · · = v m = 0 and v m +1 = w 1 g 1 for k ∈ N , v (2 k − 1) m + j ∈ I k for j = 2 , . . . , 2 m v (2 k +1) m +1 − w k +1 g k +1 ∈ I k

The Lyapunov and Focus Quantities v 1 = 0 v m +2 ∈ � g 1 � v 3 m +2 ∈ � g 1 , g 2 � . . . . . . . . . . . . v m = 0 v 3 m ∈ � g 1 � v 5 m ∈ � g 1 , g 2 � v m +1 = w 1 g 1 v 3 m +1 − w 2 g 2 ∈ � g 1 � v 5 m +1 − w 3 g 3 ∈ � g 1 , g 2 �

The Lyapunov and Focus Quantities: Sketch of the Proof Truncate the formal series for W at sufficiently large N = 2( κ m + 1). Relate ∆ W and ∆ h in one turn about the origin. ∆ W ( h ; λ ) � τ ( h ) � � d = W ( x ( t ; h , λ ) , y ( t ; h , λ ) dt dt 0 � τ ( h ) κ � g k ( λ ) x K ( k ) ( t ; h , λ ) dt = 0 k =1 � T m +1 g k ( λ ) x K ( k ) ( t ( θ ); h , λ ) h − m � u j ( θ ; λ ) h j � κ � � = 1 + d θ 0 j � 1 k =1

The Lyapunov and Focus Quantities: Sketch of the Proof Apply � � Ψ i ( θ ; λ ) h i � � � x ( θ ( t ); h , λ ) = r ( θ ; h , λ ) Cs θ = Cs θ = h + · · · Cs θ. i � 1 ∆ W ( h ; λ ) � � T m +1 � � u j ( θ ; λ ) h j � κ � � = h − m x K ( k ) ( t ( θ ); h , λ ) 1 + d θ g k ( λ ) 0 k =1 j � 1 � � T � � u j ( θ ; λ ) h K ( k )+ j � κ � � Cs K ( k ) θ h K ( k ) + = h − m � d θ g k ( λ ) 0 k =1 j � 2 � � κ � w k h K ( k ) + g k , 1 ( λ ) h K ( k )+1 + g k , 2 ( λ ) h K ( k )+2 + · · · = h − m g k ( λ ) k =1 � T 0 Cs K ( k ) θ d θ > 0, independent of λ w k =