Local invariant sets of analytic vector fields Niclas Kruff RWTH - PowerPoint PPT Presentation

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Generalization to invariant ideals Application to polynomial vector fields Local invariant sets of analytic vector fields Niclas Kruff RWTH Aachen University August 3, 2016 Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Generalization to invariant ideals Application to polynomial vector fields Table of contents 1 Introduction 2 Semi-invariants and some of their properties 3 Poincar´ e-Dulac Normal Forms 4 Generalization to invariant ideals 5 Application to polynomial vector fields Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Introduction Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Autonomous differential equations Consider the autonomous ordinary differential equation ˙ x = f ( x ) , t ∈ R , on an open subset U ⊆ K n , where K ∈ { R , C } . Furthermore let 0 ∈ U be a stationary point of f . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Autonomous differential equations Consider the autonomous ordinary differential equation ˙ x = f ( x ) , t ∈ R , on an open subset U ⊆ K n , where K ∈ { R , C } . Furthermore let 0 ∈ U be a stationary point of f . Components of vector field f = ( f 1 , . . . , f n ): i) K [ x ] n , K [ x ] the polynomial ring over K . ii) K { x } n , K { x } the ring of convergent power series over K . Later on, we will also need formal power series. In the following: R ∈ { K [ x ] , K [[ x ]] , K { x }} . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Invariant sets Definition A subset V ⊆ U is called an invariant set for x = f ( x ) ˙ if for every x 0 ∈ V the whole trajectory through x 0 is a subset of V . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Invariant sets Definition A subset V ⊆ U is called an invariant set for x = f ( x ) ˙ if for every x 0 ∈ V the whole trajectory through x 0 is a subset of V . Invariant sets are useful for qualitative analysis, and special solutions of a differential equation. Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Invariant sets: Example Consider the differential equation x = − y + x (1 − x 2 − y 2 ) ˙ y = x + y (1 − x 2 − y 2 ) . ˙ The set C := { ( x , y ) ∈ R 2 | x 2 + y 2 = 1 } is invariant for this equation. Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Poincar´ e-Dulac Normal Forms Invariant sets Generalization to invariant ideals Application to polynomial vector fields Invariant sets: Example Consider the differential equation x = − y + x (1 − x 2 − y 2 ) ˙ y = x + y (1 − x 2 − y 2 ) . ˙ The set C := { ( x , y ) ∈ R 2 | x 2 + y 2 = 1 } is invariant for this equation. Restriction of differential equation to C yields: x = − y ˙ y = x ˙ Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Semi-invariants and some of their properties Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields The Lie derivative Let f ∈ R n . The map: L f : R − → R , ψ �→ L f ( ψ ) := D ( ψ )( x ) · f ( x ) , is called Lie derivative along f . Lie derivative plays an important role in study of invariant sets. Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields The Lie derivative Let f ∈ R n . The map: L f : R − → R , ψ �→ L f ( ψ ) := D ( ψ )( x ) · f ( x ) , is called Lie derivative along f . Lie derivative plays an important role in study of invariant sets. Properties: i) L f is linear. ii) Product rule: L f ( ψ 1 ψ 2 ) = ψ 1 L f ( ψ 2 ) + ψ 2 L f ( ψ 1 ) . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Lie brackets The K -vector space R n becomes a Lie algebra with the following map: [ · , · ] : R n × R n − → R n , ( f , g ) �→ [ f , g ] := Dg · f − Df · g . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Lie brackets The K -vector space R n becomes a Lie algebra with the following map: [ · , · ] : R n × R n − → R n , ( f , g ) �→ [ f , g ] := Dg · f − Df · g . Useful property: Let f , g ∈ R n . If φ ∈ R one has L f ( L g ( φ )) − L g ( L f ( φ )) = L [ f , g ] ( φ ) . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Semi-invariants and invariant sets Definition Let φ ∈ R . If there exists λ ∈ R such that L f ( φ ) = λ · φ holds, then φ is called a semi-invariant of f . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Semi-invariants and invariant sets Definition Let φ ∈ R . If there exists λ ∈ R such that L f ( φ ) = λ · φ holds, then φ is called a semi-invariant of f . Consequently, φ is a semi-invariant iff L f ( � φ � ) ⊆ � φ � , for ideal generated by φ . Semi-invariants are useful on the study of invariant sets. Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Semi-invariants and invariant sets Lemma Let R = K [ x ] or R = K { x } and φ be a semi-invariant of f . Then, the set V ( φ ) := { x ∈ U | φ ( x ) = 0 } is an invariant set of f . Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Important operators Poincar´ e-Dulac Normal Forms Semi-invariants Generalization to invariant ideals Application to polynomial vector fields Previous example Check invariance for C := { ( x , y ) ∈ R 2 | x 2 + y 2 = 1 } . Example x = − y + x (1 − x 2 − y 2 ) ˙ y = x + y (1 − x 2 − y 2 ) , ˙ Let φ := x 2 + y 2 − 1. Then L f ( φ ) = − (2 x 2 + 2 y 2 ) φ. Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields

Introduction Semi-invariants and some of their properties Normal Forms Poincar´ e-Dulac Normal Forms Normal Forms and semi-invariants Generalization to invariant ideals Application to polynomial vector fields Poincar´ e-Dulac Normal Forms Niclas Kruff RWTH Aachen University Local invariant sets of analytic vector fields