Bilinear discretization of integrable systems with quadratic vector fields Yuri B. Suris (Technische Universität München) “Geometry and Integrability”, Obergurgl, 19.12.2008 Yuri B. Suris Hirota-Kimura Discretizations
The problem of integrable discretization. Hamiltonian approach (Birkhäuser, 2003) Consider a completely integrable flow ˙ x = f ( x ) = { H , x } (1) with a Hamilton function H on a Poisson manifold P with a Poisson bracket {· , ·} . Thus, the flow (1) possesses many functionally independent integrals I k ( x ) in involution. The problem of integrable discretization : find a family of diffeomorphisms P → P , � x = Φ( x ; ǫ ) , (2) depending smoothly on a small parameter ǫ > 0, with the following properties: Yuri B. Suris Hirota-Kimura Discretizations
1. The maps (2) approximate the flow (1): Φ( x ; ǫ ) = x + ǫ f ( x ) + O ( ǫ 2 ) . 2. The maps (2) are Poisson w. r. t. the bracket {· , ·} or some its deformation {· , ·} ǫ = {· , ·} + O ( ǫ ) . 3. The maps (2) are integrable , i.e. possess the necessary number of independent integrals in involution, I k ( x ; ǫ ) = I k ( x ) + O ( ǫ ) . Yuri B. Suris Hirota-Kimura Discretizations
Missing in the book: Hirota-Kimura discretizations ◮ R.Hirota, K.Kimura. Discretization of the Euler top. J. Phys. Soc. Japan 69 (2000) 627–630, ◮ K.Kimura, R.Hirota. Discretization of the Lagrange top. J. Phys. Soc. Japan 69 (2000) 3193–3199. Reasons for this omission: discretization of the Euler top seemed to be an isolated curiosity; discretization of the Lagrange top seemed to be completely incomprehensible, if not even wrong. Renewed interest stimulated by a talk by T. Ratiu at the Oberwolfach Workshop “Geometric Integration”, March 2006, who claimed that HK-type discretizations for the Clebsch system and for the Kovalevsky top are also integrable. Yuri B. Suris Hirota-Kimura Discretizations
Hirota-Kimura’s discrete time Euler top ˙ � x 1 − x 1 = ǫα 1 ( � x 2 x 3 + x 2 � x 1 = α 1 x 2 x 3 , x 3 ) , ˙ � x 2 − x 2 = ǫα 2 ( � x 3 x 1 + x 3 � x 2 = α 2 x 3 x 1 , x 1 ) , � ˙ � x 3 − x 3 = ǫα 3 ( � x 1 x 2 + x 1 � x 3 = α 3 x 1 x 2 , x 2 ) . Features: ◮ Equations are linear w.r.t. � x = ( � x 1 , � x 2 , � x 3 ) T : 1 − ǫα 1 x 3 − ǫα 1 x 2 , A ( x , ǫ ) � x = x , A ( x , ǫ ) = − ǫα 2 x 3 1 − ǫα 2 x 1 − ǫα 3 x 2 − ǫα 3 x 1 1 x = f ( x , ǫ ) = A − 1 ( x , ǫ ) x . result in an explicit (rational) map: � ◮ The map is reversible (therefore birational): f − 1 ( x , ǫ ) = f ( x , − ǫ ) . Yuri B. Suris Hirota-Kimura Discretizations
◮ Explicit formulas rather messy: x 1 = x 1 + 2 ǫα 1 x 2 x 3 + ǫ 2 x 1 ( − α 2 α 3 x 2 1 + α 3 α 1 x 2 2 + α 1 α 2 x 2 3 ) � , ∆( x , ǫ ) x 2 = x 2 + 2 ǫα 2 x 3 x 1 + ǫ 2 x 2 ( α 2 α 3 x 2 1 − α 3 α 1 x 2 2 + α 1 α 2 x 2 3 ) � , ∆( x , ǫ ) x 3 = x 3 + 2 ǫα 3 x 1 x 2 + ǫ 2 x 3 ( α 2 α 3 x 2 1 + α 3 α 1 x 2 2 − α 1 α 2 x 2 3 ) � , ∆( x , ǫ ) where ∆( x , ǫ ) = det A ( x , ǫ ) = 1 − ǫ 2 ( α 2 α 3 x 2 1 + α 3 α 1 x 2 2 + α 1 α 2 x 2 3 ) − 2 ǫ 3 α 1 α 2 α 3 x 1 x 2 x 3 . (Try to see reversibility directly from these formulas!) Yuri B. Suris Hirota-Kimura Discretizations
◮ Two independent integrals: I 1 ( x , ǫ ) = 1 − ǫ 2 α 2 α 3 x 2 I 2 ( x , ǫ ) = 1 − ǫ 2 α 3 α 1 x 2 1 2 , . 1 − ǫ 2 α 3 α 1 x 2 1 − ǫ 2 α 1 α 2 x 2 2 3 ◮ Invariant volume measure and bi-Hamiltonian structure found in: M. Petrera, Yu. Suris. On the Hamiltonian structure of the Hirota-Kimura discretization of the Euler top. Math. Nachr., 2008 (to appear), arXiv: 0707.4382[math-ph] . Yuri B. Suris Hirota-Kimura Discretizations
Geometry and Integrability ◮ H. Jonas. Deutung einer birationalen Raumtransformation im Bereiche der sphärischen Trigonometrie . Math. Nachr., 6 (1951) 303–314. y = cos � Let x = cos a , y = cos b , z = cos c and � x = cos � a , � b , � z = cos � c be the cosines of the sides of two spherical triangles α = β + � with complementary angles: α + � β = γ + � γ = π . Then: x + � x + y � z + � yz = 0 , y + � y + z � x + � zx = 0 , z + � z + x � y + � xy = 0 . “Integration” of this involution in terms of elliptic functions. Yuri B. Suris Hirota-Kimura Discretizations
Hirota-Kimura or Kahan? ◮ W. Kahan. Unconventional numerical methods for trajectory calculations (Unpublished lecture notes, 1993). ˙ x = Q ( x ) + Bx ( � x − x ) /ǫ = Q ( x , � x ) + B ( x + � x ) , � where B ∈ R n × n , Q : R n → R n is a quadratic function, and Q ( x , � x ) = Q ( x + � x ) − Q ( x ) − Q ( � x ) is the corresponding symmetric bilinear function. x = f ( x , ǫ ) = A − 1 ( x , ǫ ) x , Note: equations for � x always linear, � the map is always reversible and birational, f − 1 ( x , ǫ ) = f ( x , − ǫ ) . Yuri B. Suris Hirota-Kimura Discretizations
Illustration: Lotka-Volterra system Kahan’s integrator for the Lotka-Volterra system: � � ˙ � x − x = ǫ ( � x + x ) − ǫ ( � xy + x � x = x ( 1 − y ) , y ) , � ˙ y = y ( x − 1 ) , � y − y = ǫ ( � xy + x � y ) − ǫ ( � y + y ) . Explicitly: x = x ( 1 + ǫ ) 2 − ǫ ( 1 + ǫ ) x − ǫ ( 1 − ǫ ) y � 1 − ǫ 2 − ǫ ( 1 − ǫ ) x + ǫ ( 1 + ǫ ) y , y = y ( 1 − ǫ ) 2 + ǫ ( 1 + ǫ ) x + ǫ ( 1 − ǫ ) y � 1 − ǫ 2 − ǫ ( 1 − ǫ ) x + ǫ ( 1 + ǫ ) y . Yuri B. Suris Hirota-Kimura Discretizations
1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Left: three orbits of Kahan’s discretization with ǫ = 0 . 1, right: one orbit of the explicit Euler with ǫ = 0 . 01. ◮ J.M. Sanz-Serna. An unconventional symplectic integrator of W.Kahan. Applied Numer. Math. 16 (1994) 245–250. A sort of an explanation of a non-spiralling behavior: Kahan’s integrator for the Lotka-Volterra system in Poisson. Yuri B. Suris Hirota-Kimura Discretizations
Hirota-Kimura’s discrete time Lagrange top ω 1 = ( 1 − α ) ω 2 ω 3 + z 0 γ 2 , ˙ ω 2 = − ( 1 − α ) ω 3 ω 1 − z 0 γ 1 , ˙ ω 3 = 0 , ˙ � γ 1 = ω 3 γ 2 − ω 2 γ 3 , ˙ γ 2 = ω 1 γ 3 − ω 3 γ 1 , ˙ γ 3 = ω 2 γ 1 − ω 1 γ 2 , ˙ ω 1 − ω 1 = ǫ ( 1 − α )( � � ω 2 ω 3 + ω 2 � ω 3 ) + ǫ z 0 ( � γ 2 + γ 2 ) , ω 2 − ω 2 = − ǫ ( 1 − α )( � ω 3 ω 1 + ω 3 � ω 1 ) − ǫ z 0 ( � γ 1 + γ 1 ) , � � ω 3 − ω 3 = 0 , � γ 1 − γ 1 = ǫ ( � ω 3 γ 2 + ω 3 � γ 2 ) − ǫ ( � ω 2 γ 3 + ω 2 � γ 3 ) , � � γ 2 − γ 2 = ǫ ( � ω 1 γ 3 + ω 1 � γ 3 ) − ǫ ( � ω 3 γ 1 + ω 3 � γ 1 ) , � γ 3 − γ 3 = ǫ ( � ω 2 γ 1 + ω 2 � γ 1 ) − ǫ ( � ω 1 γ 2 + ω 1 � γ 2 ) , which gives a birational map ( � ω, � γ ) = f ( ω, γ, ǫ ) . Yuri B. Suris Hirota-Kimura Discretizations
Hirota-Kimura’s “method” for finding integrals Consider the expression A = ω 2 1 + ω 2 2 − B γ 3 − C γ 2 3 , and determine A , B , C by requiring that they are conserved quantities. For this aim, solve the system of three equations for these three unknowns: γ 2 ω 2 ω 2 A + B � γ 3 + C � 3 = � 1 + � 2 , A + B γ 3 + C γ 2 3 = ω 2 1 + ω 2 2 , 2 2 2 A + B γ 3 + C γ 3 = ω 1 + ω 2 � � � � ) = f − 1 ( ω, γ, ǫ ) . Then check with ( � ω, � γ ) = f ( ω, γ, ǫ ) and ( ω , γ � � that A , B , C = A , B , C ( ω, γ, ǫ ) are conserved quantities, indeed. Proceed similarly to determine the conserved quantities D , . . . , M from D = ω 1 γ 1 + ω 2 γ 2 − E γ 3 − F γ 2 K = γ 2 1 + γ 2 2 − L γ 3 − M γ 2 3 , 3 . Does this make any sense for you??? Yuri B. Suris Hirota-Kimura Discretizations
Nevertheless, Hirota-Kimura’s “method” turns out to be not only valid in this case but also remarkably deep and general (as everything coming from R. Hirota). How should it be interpreted? Solve (symbolically) the system ( A + B γ 3 + C γ 2 3 ) ◦ f i ( ω, γ, ǫ ) = ( ω 2 1 + ω 2 2 ) ◦ f i ( ω, γ, ǫ ) with i = − 1 , 0 , 1. Verify that A = A ◦ f , B = B ◦ f , C = C ◦ f . Alternatively, one can solve the above system with i = 0 , 1 , 2, and then check that the solutions coincide. But then this system should be satisfied for all i ∈ Z . This is a very special feature of both the map f and the set of functions ( 1 , γ 3 , γ 2 3 , ω 2 1 + ω 2 2 ) . Also the sets of functions ( 1 , γ 3 , γ 2 ( 1 , γ 3 , γ 2 3 , γ 2 1 + γ 2 3 , ω 1 γ 1 + ω 2 γ 2 ) , 2 ) have this property. It is formalized in the following definition. Yuri B. Suris Hirota-Kimura Discretizations
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