Blow-up of vector fields and dynamical systems of compactified Painleve equations
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1 Blow-up of vector fields and dynamical systems of compactified Painleve equations Institute of Mathematics for Industry Kyushu University Hayato Chiba Nov/29/2012
2 Introduction
3 The Painlevé equations are now related to many fields of mathematics such as differential geometry, foliations, probability theory, representation theory, algebraic geometry, number theory, and dynamical systems, etc. They are related to phase transition, catastrophe, bifurcations. from O s web page.
4 Purpose. Dynamical systems-approach to Painleve. 3-dim vector fields with a bifurcation parameter blow-up at a bifurcation point. Painleve equations as 3-dim vector fields on cpt mfd. on a weighted projective space.
5 Many special solutions are characterized as special invariant manifolds of dynamical systems. For example Airy function (for Airy eq.) Tri-tronquee sol. (for P1, P2 ) are center manifolds of certain fixed points. Pole-free sol. (cf. Hastings-McLeod sol. for P2) are heteroclinic orbits connecting two fixed points. Airy sol. for P2 with are center-stable mfd of a certain fixed point.
6 Theorem. Painleve I to VI are locally transformed into integrable equations near poles. More precisely for (P1) transformation defined on the region, there exists a local such that (P1) is transformed into Cor. Any solutions of (P1), (P2), (P4) are meromorphic.
7 Geometry of weighted blow-up
8 Blow-up method resolution of singularities in algebraic variety differential equation integration : divisor bijective
9 The weighted action: The quotient space is called the weighted projective space. A weighted projective space is an orbifold (algebraic variety) with several singularities. cf.
10 Weighted Blow-up. action : principle bundle. : the associated line bundle. cf.
11 Vector field on blow-up Suppose that the origin is a singularity ; non-hyperbolic fixed point. (phase transition). ODE on the line bundle projection ODE on scaling limit (fiber coordinate ). A new dynamical system on the divisor determines the local behavior of the given system on. In many applications, ODEs on weighted projective spaces are those define special functions; Airy, Hermite, Painleve
12 Applications of the blow-up to dynamical systems
13 Application to a bifurcation problem. : small parameter : no periodic orbits. : stable periodic orbit. The existence of a periodic orbit follows from the analysis of the Airy function! Applications: chemical reaction, neural network, biological populations
14 Known fact (classical). : small parameter Thm. For small, a stable periodic orbit whose position from the cubic curve is where is the first negative zero of the Airy function.
15 Application to a bifurcation problem. unperturbed Saddle-node singularity
16 Application to a bifurcation problem. Divisor is. Dynamics on the divisor is the Airy equation.
17 Local coordinates of the line bundle coordi. on the base space coordi. on the fiber
18 Change the coordinates, and take the scaling limit Equations on the base space are Airy eq. ODE on the divisor is the compactified Airy eq. For the real case, the base is a two disk = compactification of the Airy flow on by
19 Bifurcation type Saddle-node Transcritical Pitch-fork Bogdanov-Takens Bogdanov-Takens with Z2 symmetry Quadratic Equation on the divisor Airy Hermite First order linear equation Painleve (I) Painleve (II) Painleve (IV) Painleve property!!
20 Vector fields blow-up New dynamics on the divisor defining special functions. Knowledge of special functions is used to investigate behavior of vector fields. Purpose in this talk. Dynamical systems theory is applied to investigate properties of the Painleve equations.
21 Analysis of the Painleve equations (i) Painleve property
22 3-dim vector field with a Bogdanov-Takens singularity. The unperturbed system undergoes the BT-bifurcation at the origin. Invariance. Blow-up coordinates.
23 Blow-up at the origin, and scaling limit gives an ODE on the divisor as is covered by four coord. One of them is The first Painleve eq.
24 For the real case, the divisor is ODE on = compactification of the Painleve flow on by : inside : front hemisphere : right hemisphere : upper hemisphere
25 Suppose that a sol. of diverges at finite. The coordinate change from 4th coord to 2nd coord. A Any poles corresponds to the fixed point A : in the 2 nd coordinate. The eigenvalues of the Jacobi matrix at the point satisfy the condition for the Poincare linearization thm.
26 Thm.(Poincare) : holomorphic vec. field on. : fixed point of. If eigenvalues of the Jacobi matrix satisfy a certain algebraic condition, then local analytic coord. transformation near s.t. is transformed into the linear vec. field Thm. analytic transformation near infinity s.t. is transformed into. Cor. Any solutions of are meromorphic functions on. A similar argument is valid for P1 to P6.
27 Analysis of the Painleve equations (ii) Boutroux s tritronquee solutions
28 The coordinate change from 4th coord to 3rd coord. Fixed point : Eigenvalues: there exists 1-dim center mfd. Known facts. Center manifolds are not analytic, while they may be analytic on a certain sector. Center manifolds are not unique, while they are exponentially close to each other. The center manifold theory is regarded as a higher dimensional version of the theory of irregular singular points.
29 The center manifold theory proves center manifold analytic on a sector. By the transformation we obtain Thm.(Boutroux, 1913) five solutions of s.t. they are analytic on each sectors when they are exponentially close to each other on the common region. they have no poles on each sector when
30 Analysis of the Painleve equations (iii) Hamiltonian system
31 Equations on. The sets are 2-dim invariant manifolds, which reflect the cellular decomposition;
32 Equations on is given by a Hamiltonian system (Weirstrass s equation) Thm. For compactified (P1) to (P6) on certain weighted projective spaces, equations on (w.r.t. the cellular decomposition) are autonomous Hamiltonian systems.
33 It is possible to construct approximate solutions near by using exact solution of the Hamiltonians. Elliptic asymptotics.
34 Analysis of the Painleve equations (iv) unique characterization
35 All compactified (P1) to (P6) on the following properties. Rem. The Jacobi matrix at the fixed point for (P1) is where. Recall that (P1) is transformed into. have (i) ODEs written in local coordinates are meromorphic. (ii) Equations on (w.r.t. the cellular decomposition) are Hamiltonian systems. (iii) hyperbolic fixed point satisfying Poincare s linearization theorem s.t. (3,2) component of the Jacobi matrix is not zero. If were zero, (P1) is transformed into and the Painleve property may be violated.
36 Thm. Give an ODE on, where are holomorphic in, meromorphic in If this ODE satisfies (i),(ii),(iii), then this is (P1). (the same statement holds for (P2) and (P4)). (i) ODEs written in local coordinates are meromorphic. (ii) Equation on (w.r.t. the cellular decomposition) is a Hamiltonian system. (iii) hyperbolic fixed point satisfying Poincare s linearization theorem s.t. (3,2) component of the Jacobi matrix is not zero.
37 (i) ODEs written in local coordinates are meromorphic. (ii) Equation on (w.r.t. the cellular decomposition) is a Hamiltonian system. (iii) hyperbolic fixed point satisfying Poincare s linearization theorem s.t. (3,2) component of the Jacobi matrix is not zero. The condition (i) is a global one reflecting the geometry of. The conditions (ii) and (iii) are local conditions. Rem. For (P1), the condition (ii) can be removed. (geometry of ) + (behavior at a pole) = (P1)!
38 Thm. Give an ODE on, where are holomorphic in, If this ODE satisfies (i),(ii), then this is (P1). (the same statement holds for (P2) and (P4)). (i) ODEs written in local coordinates are meromorphic. (ii) Equation on (w.r.t. the cellular decomposition) is a Hamiltonian system with the Hamiltonian (geometry of ) + (dynamics on codim 1 submanifold) = (Painleve).
39 Summary, Future works. (P1) to (P6) are transformed into integrable systems near certain fixed points on. For example, (P1) In general, the transformation is analytic near a small neighborhood of the fixed point. + (behavior at a pole) = (P1) It seems that the transformation everything on (P1). knows
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