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. . E E S S P P . Escola Polit` ecnica Superior . Universitat de Lleida The center cyclicity of the Lorenz, Chen and L¨ u systems Isaac A. Garc´ ıa AQTDE2019 (Castro Urdiales, June 17-21, 2019) Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Co-authors Susanna Maza , Universitat de Lleida, Spain Douglas S. Shafer , University of North Carolina at Charlotte, USA Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The Hopf points in R 3 We consider real polynomial families of differential equations x = − y + F 1 ( x , y , z ; µ ) , ˙ y = x + F 2 ( x , y , z ; µ ) , ˙ (1) z = λ z + F 3 ( x , y , z ; µ ) , ˙ Parameter space E ⊆ { ( λ, µ ) ∈ R ∗ × R p } , where R ∗ = R \ { 0 } ; F j ( j = 1 , 2 , 3) contain only nonlinear terms in ( x , y , z ). Hopf singular point The origin is a Hopf singularity of all the family (1): it possesses the eigenvalues ± i ∈ C and λ ∈ R ∗ . Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Local dynamics of a member on a center manifold W c Let W c be a local center manifold at the origin of system (1) ( λ,µ )=( λ † ,µ † ) : The origin is a center of (1) ( λ,µ )=( λ † ,µ † ) if all the orbits on W c are periodic; Otherwise, the origin is a saddle-focus: a focus on each W c . The center problem for a Hopf singularity in R 3 To decide for which parameters ( λ, µ ) ∈ R ∗ × R p the origin of (1) is a center or not. Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The Lyapunov solution to the center problem Lyapunov Center Theorem The origin is a center for system (1) ( λ,µ )=( λ † ,µ † ) if and only if it admits a real analytic local first integral of the form H ( x , y , z ) = x 2 + y 2 + · · · in a neighborhood of the origin in R 3 . Remark When there is a center, the W c is unique and analytic. Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Formal Lyapunov function There is a formal series H ( x , y , z ) = x 2 + y 2 + · · · ∈ R [[ x , y , z ]] such that � η j ( λ, µ )( x 2 + y 2 ) j , X ( H ) = � (2) j � 2 where X is the vector field associated to family (1). Focus quantities: � η j ( λ, µ ) � � η j ( λ, µ ) = η j ( λ, µ ) � ∈ Q ( λ )[ µ ] , d j ( λ ) = 0 ⇒ λ ∈ i Q . d j ( λ ) Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Unfolding the Hopf points In order to capture the full range of perturbations of Hopf points starting with a member of family (1) we need to consider the larger family: ˙ = α x − y + F 1 ( x , y , z ; µ ) , x y ˙ = x + α y + F 2 ( x , y , z ; µ ) , (3) z ˙ = λ z + F 3 ( x , y , z ; µ ) , Parameter set: E ′ = E × R . X ( H ) = � η j ( λ, µ, α )( x 2 + y 2 ) j where X is the vector j � 1 � field associated to family (3). ˜ η 1 ( λ, µ, α ) = 2 α and � η j ( λ, µ, α ) are analytic functions. Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The cyclicity of the Hopf points Given a family of vector fields (1) ( λ,µ ) we have a local Poincar´ e return map at the Hopf point The cyclicity of O is the maximum number of limit cycles that can bifurcate from it under small perturbations within family (3) ( λ,µ,α ) . Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Main goal: to get the cyclicity (or a bound) in the center case Difficulties Lack of analyticity of W c when α = 0 (in general). There is no W c when α � = 0 (the singularity becomes hyperbolic). λ is a trouble parameter when (for further convenience) we allow λ ∈ i Q ⊂ C . Target The goal is to overcome these difficulties, presenting a method for bounding the cyclicity in the center case without any kind of reduction to W c . Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The displacement map around the Hopf singularity Polar-directional blow-up Φ : R 3 → S 1 × R × R ( x , y , z ) �→ ( θ, r , w ) x = r cos θ, y = r sin θ, z = r w . Blows up ( x , y , z ) = (0 , 0 , 0) to the set { ( θ, r , w ) : r = 0 } . Φ is a diffeomorphism outside the solid cone C τ = { ( x , y , z ) : z 2 � τ ( x 2 + y 2 ) } for any τ > 0. ——————————————————————————— Family (3) ( λ,µ,α ) is written as the analytic system dr dw d θ = R ( θ, r , w ; µ ) , d θ = ( λ − α ) w + W ( θ, r , w ; λ, µ ) (4) on some cylinder { ( θ, r , w ) : | r | ≪ 1 , w ∈ K} where K ⊂ R is an arbitrary compact neighborhood of 0 in R ( K = {| w | < √ τ } ). Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The displacement map around the Hopf singularity Small amplitude periodic solutions of X around the origin There is a one-to-one correspondence between 2 π -periodic solutions of (4) and small amplitude periodic orbits of (3) ( λ,µ,α ) . When α = 0: Any small periodic orbit is contained in W c . W c ∩ C τ = { (0 , 0 , 0) } . When α � = 0: we prove that under perturbation ( | α | ≪ 1) the normally hyperbolic W c is replaced by a normally hyperbolic invariant two-manifold M through (0 , 0 , 0) such that: Any small periodic orbit is contained in M . M ∩ C τ = { (0 , 0 , 0) } . Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The reduced displacement map δ ( r 0 ; λ, µ, α ) Let Ψ( θ ; r 0 , w 0 ; λ, µ, α ) be the unique solution of (4) with initial condition ( r 0 , w 0 ). e map : Π( r 0 , w 0 ; λ, µ, α ) = Ψ(2 π ; r 0 , w 0 ; λ, µ, α ). Poincar´ Displacement map : d ( r 0 , w 0 ; λ, µ, α ) = Π( r 0 , w 0 ; λ, µ, α ) − ( r 0 , w 0 ). d ( r 0 , w 0 ; λ, µ, α ) = ( d 1 ( r 0 , w 0 ; λ, µ, α ) , d 2 ( r 0 , w 0 ; λ, µ, α )) . Applying a Lyapunov-Schmidt reduction to the displacement map: The reduced displacement map � v j ( λ, µ, α ) r j δ ( r 0 ; λ, µ, α ) := d 1 ( r 0 , ¯ w ( r 0 , λ, µ, α ); λ, µ, α ) = 0 j � 1 v 1 ( λ, µ, α ) = e 2 πα − 1. X ( λ ∗ ,µ ∗ , 0) has a center at the origin ⇔ δ ( r 0 ; λ ∗ , µ ∗ , 0) ≡ 0. Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Some ideals in the Noetherian ring R ( λ )[ µ ] Poincar´ e-Lyapunov quantities (when α = 0) � � v j ( λ, µ, 0) = V j ( λ,µ ) D j ( λ ) ∈ R ( λ )[ µ ] , D j ( λ ) = 0 ⇒ λ ∈ i Q . Bautin ideal (when α = 0) B = � v j : j ∈ N � B k = � v 3 , . . . , v k � . I = � � η j : j ∈ N � I k = � � η 2 , . . . , � η k � Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Relation between the ideals B and I Theorem The following holds: B = � v 2 k +1 : k � 1 � = I ; The minimal bases of B and I have the same finite cardinality; The Bautin depth: the cardinality of the minimal basis of B # min B Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

A uniform bound on the cyclicity of centers Using the classical strategy : # min B � v j k ( λ, µ ) [1 + ψ k ( r 0 ; λ, µ )] r j k δ ( r 0 ; λ, µ ) = (5) 0 k =1 where ψ k ( r 0 ; λ, µ ) are analytic functions at r 0 = 0, ψ k (0; λ, µ ) = 0 and � � v j 1 , . . . , v j # min B is a minimal basis of B . Theorem The cyclicity of any center at the origin perturbing in E ′ is at most # min B . But, how to compute # min B ? Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Main tool (inspired by the work of Colin Christopher) def = V ( � η j : j ∈ N � ) ⊂ R p +1 Center variety V C An element of the family (1) corresponding to ( λ, µ ) has a center at the origin if and only if ( λ, µ ) ∈ V C ∩ E . The mapping F κ : R ∗ × R p → R κ For any κ ≤ # min B , define F κ ( λ, µ ) = ( � η j 1 ( λ, µ ) , . . . , � η j κ ( λ, µ )) , (6) where { � η j 1 ( λ, µ ) , . . . , � η j κ ( λ, µ ) } is the minimal basis of the ideal I j κ in R ( λ )[ µ ]. Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

Main tool (inspired by the work of Colin Christopher) Let Cyc ( X ( λ ∗ ,µ ∗ ) , 0) be the cyclicity of a point ( λ ∗ , µ ∗ ) ∈ V C ∩ E under perturbations in E ′ . Theorem Let κ and F κ be as before and C be an irreducible component of V C . Suppose: κ � p + 1; rank ( d P F κ ) = κ at some point P = ( λ ∗ , µ ∗ ) ∈ C ∩ E . Then the following holds: (i) Cyc ( X ( λ ∗ ,µ ∗ ) , 0) ≥ κ ; (ii) If moreover codim { ( λ, µ, 0) ∈ R p +2 : ( λ, µ ) ∈ C } = κ + 1 then Cyc ( X ( λ ∗ ,µ ∗ ) , 0) = κ . Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems

The Lorenz family x = σ ( y − x ) , ˙ y = ρ x − y − xz , ˙ z = − bz + xy , ˙ (7) with parameters ( ρ, σ, b ) ∈ R 3 such that b σ � = 0 else no singularity is isolated. The origin is always a singularity; When b ( ρ − 1) > 0 there also exists the symmetric singularities � � E ± := ( ± b ( ρ − 1) , ± b ( ρ − 1) , ρ − 1) . Note: System (7) is invariant under the involution ( x , y , z ) �→ ( − x , − y , z ). Isaac A. Garc´ ıa The center cyclicity of the Lorenz, Chen and L¨ u systems