A Dynamical Systems Approach to Singularities of Ordinary Differential Equations Matthias Seiß and Werner M. Seiler Institut f¨ ur Mathematik Werner M. Seiler (Kassel) Singularities of ODEs 1 / 8
Singularities of Differential Equations Many forms of singular behaviour in the context of differential equations • (derivatives of) solutions become singular “blow-up”, “shock” � • stationary points of vector fields • bifurcations in parameter dependent systems • singular integrals (solutions not contained in the “general integral”) • multi-valued solutions (like “breaking waves”) • . . . here: geometric modelling of differential equations critical points � of natural projection map geometric singularities � Werner M. Seiler (Kassel) Singularities of ODEs 2 / 8
Singularities of Differential Equations • differential topology • definition of singularities (of smooth maps) • main emphasis on classifications and local normal forms • hardly any works on (general) systems • classical analysis • mainly quasi-linear systems (including DAEs) • rich literature on scalar equations • existence , uniqueness and regularity of solutions through singularity • main techniques: fixed point theorems , sub- and supersolutions • differential algebra • main emphasis on singular integrals • motivating problem for differential ideal theory • (geometric) singularities eliminated • useful for algorithmic approaches • singularities related to differential Galois theory Werner M. Seiler (Kassel) Singularities of ODEs 2 / 8
Geometric Setting • fibred manifold: π : E → T with dim T = 1 • trivial case: E = T × U , π = pr 1 • adapted local coordinates: ( t , u ) ( independent variable t , dependent variables u ) • section: smooth map σ : T → E with π ◦ σ = id (locally: σ ( t ) = ( t , s ( t )) with function s : T → U ) • q-jet [ σ ] ( q ) : class of all sections with same Taylor polynomial of t degree q around expansion point t set of all q -jets [ σ ] ( q ) • jet bundle J q π : t ( t , u ( q ) ) • local coordinates: (derivatives up to order q ) • natural hierarchy with projections π q r : J q π − → J r π 0 ≤ r < q π q : J q π − → T Werner M. Seiler (Kassel) Singularities of ODEs 3 / 8
Geometric Setting Definition ordinary differential equation of order q � submanifold R q ⊆ J q π such that im π q | R q dense in T • more general definition than usual in geometric theory • no conditions on independent variable allowed • no distinction scalar equation or system • basic assumption: equation formally integrable (no “hidden” integrability conditions) Werner M. Seiler (Kassel) Singularities of ODEs 3 / 8
Geometric Setting prolongation of section σ : T → E section j q σ : T → J q π � s ( t ) , . . . , s ( q ) ( t ) � � j q σ ( t ) = t , s ( t ) , ˙ Definition classical solution section σ : T → E such that im( j q σ ) ⊆ R q � Werner M. Seiler (Kassel) Singularities of ODEs 3 / 8
Vessiot Distribution Definition contact distribution C q ⊂ T ( J q π ) generated by vector fields q − 1 m C ( q ) � � trans = ∂ t + u α j +1 ∂ u α j α =1 j =0 C ( q ) 1 ≤ α ≤ m = ∂ u α α q Proposition section γ : T → J q π of the form γ = j q σ ⇐ ⇒ T im ( γ ) ⊂ C q Proof. chain rule! Werner M. Seiler (Kassel) Singularities of ODEs 4 / 8
Vessiot Distribution Consider prolonged solution j q σ of equation R q ⊆ J q π : • integral elements � � T ρ im ( j q σ ) for ρ ∈ im ( j q σ ) � � � • solution ⇒ ⊆ T ρ R q = T ρ im ( j q σ ) � � • prolonged section = ⇒ T ρ im ( j q σ ) ⊆ C q | ρ Definition Vessiot space at point ρ ∈ R q : V ρ [ R q ] = T ρ R q ∩ C q | ρ • generally: dim V ρ [ R q ] depends on ρ � regular distribution only on open subset of R q • computing Vessiot distribution V [ R q ] corresponds to “projective” form of prolonging from R q to R q +1 • computation requires only linear algebra Werner M. Seiler (Kassel) Singularities of ODEs 4 / 8
Vessiot Distribution Consider square first-order ordinary differential equation R 1 ⊂ J 1 π with u ) = 0 where Φ : J 1 π → ❘ m local representation Φ ( t , u , ˙ • define m × m matrix A and m -dimensional vector d A = C (1) Φ = ∂ Φ trans Φ = ∂ Φ ∂ t + ∂ Φ d = C (1) ∂ u · ˙ u ∂ ˙ u assume A almost everywhere non-singular • compute determinant δ = det A and adjugate C = adj A • V [ R 1 ] almost everywhere generated by single vector field X = δ C (1) trans − ( C d ) T C (1) ( X essentially lift of “evolutionary vector field” associated to given differential equation to J 1 π ) Werner M. Seiler (Kassel) Singularities of ODEs 4 / 8
Vessiot Distribution Definition ordinary differential equation R q ⊆ J q π • generalised solution integral curve N ⊆ R q of V [ R q ] � projection π q • geometric solution 0 ( N ) of generalised solution N � • geometric solution in general not image of a section (thus no interpretation as a function!) • geometric solution π q 0 ( N ) is classical solution ⇐ ⇒ N everywhere transversal to π q • geometric solutions allow for modelling of multi-valued solutions Werner M. Seiler (Kassel) Singularities of ODEs 4 / 8
Geometric Singularities Ordinary differential equation R q ⊂ J q π Φ ( t , u ( q ) ) = 0 • local description: (dim u = m ) (not necessarily square “DAEs” included ) � • Assumptions: • equation formally integrable • equation of finite type ∂ Φ /∂ u q has almost everywhere rank m � • second assumption = ⇒ almost everywhere dim V ρ [ R q ] = 1 Werner M. Seiler (Kassel) Singularities of ODEs 5 / 8
Geometric Singularities Definition point ρ ∈ R q ⊂ J q π is V ρ [ R q ] 1-dimensional and transversal to π q • regular � • regular singular � V ρ [ R q ] 1-dimensional and not transversal to π q • irregular singular ( s-singular ) � dim V ρ [ R q ] = 1 + s with s > 0 critical points of π q 0 | R q ) (singular points � Proposition point ρ ∈ R q ⊂ J q π � C ( q ) Φ � • ρ regular ⇐ ⇒ rank ρ = m • ρ regular singular ⇐ ⇒ ρ not regular and C ( q ) Φ | C ( q ) � � rank trans Φ ρ = m Werner M. Seiler (Kassel) Singularities of ODEs 5 / 8
Regular Singularities Theorem R q ⊂ J q π equation without irregular singularities • ρ ∈ R q regular point = ⇒ (i) unique classical solution σ exists with ρ ∈ im j q σ (ii) solution σ can be extended in any direction until j q σ reaches either boundary of R q or a regular singularity • ρ ∈ R q regular singularity ⇒ = dichotomy (i) either two classical solutions σ 1 , σ 2 exist with ρ ∈ im j q σ i (both ending or both starting in ρ ) (ii) or one classical solution σ exists with ρ ∈ im j q σ whose derivative of order q + 1 blows up at t = π q ( ρ ) Werner M. Seiler (Kassel) Singularities of ODEs 6 / 8
Regular Singularities Proof. • V [ R q ] locally generated by vector field X X vertical wrt π q • ρ regular singularity = ⇒ • dichotomy ∂ t -component of vector field X does or does not � change sign at ρ Werner M. Seiler (Kassel) Singularities of ODEs 6 / 8
Regular Singularities u 3 + u ˙ Example: ˙ u − t = 0 (hyperbolic gather) singularity manifold ( criminant ): u 2 + u = 0 3 ˙ (visible part contains only regular singularities) Werner M. Seiler (Kassel) Singularities of ODEs 6 / 8
Regular Singularities u 3 + u ˙ Example: ˙ u − t = 0 (hyperbolic gather) second derivative of geo- metric solution touching “tip” of discriminant (projection of criminant) blows up below intersections of criminant and generalised solutions geometric solu- tions “change direction” Werner M. Seiler (Kassel) Singularities of ODEs 6 / 8
Irregular Singularities let ρ ∈ R q be an irregular singularity • consider simply connected open set U ⊂ R q without any irregular singularities such that ρ ∈ U • in U Vessiot distribution V [ R q ] generated by single vector field X Proposition Generically any smooth extension of X vanishes at ρ Proof. • elementary linear algebra of adjugate matrix • problem: do components of X possess common divisor? Conjecture: not true, if and only if ρ lies on singular integral Werner M. Seiler (Kassel) Singularities of ODEs 7 / 8
Irregular Singularities u 3 + u ˙ Example: ˙ u − t = 0 (hyperbolic gather) neighbourhood of an irregular singularity stable/unstable manifolds define intersecting generalised solutions! Werner M. Seiler (Kassel) Singularities of ODEs 7 / 8
Conclusions Comparison with dynamical systems theory • use of Vessiot distribution transforms implicit differential equation into an explicit (and autonomous) one • one-dimensional distribution defines only direction , not an arrow X and − 42 X define same distribution! � • absolute signs of (real parts of) eigenvalues meaningless; only relative signs matter • different (smooth) centre manifolds yield different generalised solutions • generalised solutions through irregular singularity are one-dimensional invariant manifolds of vector field X with discrete α and ω limit sets consist of orbits separated by isolated stationary points � Werner M. Seiler (Kassel) Singularities of ODEs 8 / 8
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