Complex Structures in Algebra, Geometry, Topology, Analysis and Dynamical Systems Zalman Balanov (University of Texas at Dallas) 1 October 2, 2014 1 joint work with Y. Krasnov (Bar Ilan University) Zalman Balanov (University of Texas at Dallas) 2
1. OUTLINE • Five problems: (i) Existence of bounded solutions to quadratic ODEs; (ii)“Fundamental Theorem of Algebra” in non-associative algebras; (iii) “Intermediate Value Theorem” in R 2 ; (iv) Existence of maps with positive Jacobian; (v) Surjectivity of polynomial maps. • Quadratic ODEs of natural phenomena: (i) Euler equations (solid mechanics); (ii) Kasner equations (gen. relativity theory); (iii) Volterra equation (population dynamics); (iv) Aris equations (second order chemical reactions); (v) Ginzburg-Landau nonlinearity (superconductivity); (vi) Geodesic equation. • Complex structures in algebras as a common root of the above 5 problems • Applications Zalman Balanov (University of Texas at Dallas) 3
2. FIVE PROBLEMS 2.1. Bounded solutions to quadratic systems. “Undergraduate case”. Assume A : R n → R n is a linear operator and consider the system dx dt = Ax . (1) Proposition 1 System (1) has a periodic solution iff the following non-hyperbolicity condition is satisfied: Condition (A): A has a purely imaginary eigenvalue. Zalman Balanov (University of Texas at Dallas) 4
Assume now that we are given a quadratic system dx dt = Q ( x ) (2) with Q : R n → R n a homogeneous (polynomial) map of degree 2 (i.e Q ( λ x ) = λ 2 Q ( x ) for all λ ∈ R and x ∈ R n or, that is the same, the coordinate functions of Q are quadratic forms in n variables). QUESTION A. What is an analogue of Condition(A) (non-hyperbolicity) for the quadratic system (2) in the context relevant to Proposition 1 (existence of bounded/periodic solutions)? Zalman Balanov (University of Texas at Dallas) 5
2.2. Fundamental theorem of algebra in non-associative algebras Undergraduate fact: any complex polynomial f ( z ) = a 0 + a 1 z + a 2 z 2 + ... + a n z n ( n > 0) has at least one root z o ∈ C , i.e. f ( z o ) = 0. bf Remark. From the algebraic viewpoint, the set C of complex numbers has the following properties: (i) it is a real 2-dimensional vector space; (ii) elements of C can be multiplied in such a way that a ( α b + β c ) = α ab + β ac ∀ a , b , c ∈ C ; α, β ∈ R (3) and ab = ba ∀ a , b ∈ C . (4) Zalman Balanov (University of Texas at Dallas) 6
Definition 2 Any n -dimensional real vector space equipped with the commutative bi-linear multiplication (see (3) and (4)) is called a (commutative) algebra. Remarks. (i) The above definition does NOT require from an algebra to be associative. (ii) By obvious reasons, given a commutative real two-dimensional algebra A , one cannot expect that any polynomial equation in A has a (non-zero) root. QUESTION B: Let A be a commutative real two-dimensional algebra. To which extent should be A close to C to ensure that a “reasonable” polynomial equation in A has a (non-zero) root? Zalman Balanov (University of Texas at Dallas) 7
2.3. “Intermediate Value Theorem in R 2 Undergraduate fact (Intermediate Value Theorem): Assume: (i) f : [ a , b ] → R is a continuous function; (ii) f ( a ) · f ( b ) < 0. Then, the equation f ( x ) = 0 has at least one solution. Question: Given a continuous map Φ : B → R 2 , where B stands for a closed disc in R 2 , what is an analogue of condition (ii) providing that the equation Φ( u ) = 0 has at least one solution? Zalman Balanov (University of Texas at Dallas) 8
Remarks. (i) The above map Φ assigns to each u ∈ B a vector Φ( u ). (ii) Denote by Γ the boundary of B and assume Φ( u ) � = 0 for all u ∈ Γ. Choose a point M ∈ Γ and force it to travel along Γ and to return back. Since: (a) Γ is a closed curve, and (b) Φ is a continuous vector field, the vector Φ( M ) will make an integer number of rotations (called topological index and denoted by γ (Φ , Γ)). Zalman Balanov (University of Texas at Dallas) 9
Zalman Balanov (University of Texas at Dallas) 10
Proposition 3 Assume: (i) Φ : B → R 2 is a continuous map with no zeros on Γ ; (ii) γ (Φ , Γ) � = 0 . Then, the equation Φ( u ) = 0 has at least one solution inside B. Question C . Which requirements on Φ do provide condition (ii) from Proposition 3? Zalman Balanov (University of Texas at Dallas) 11
2.4. Existence of maps with positive Jacobian determinant Undergraduate fact: Let f : C → C defined by f ( x + iy ) = u ( x , y ) + iv ( x , y ) be a complex analytic map (i.e. the Cauchy-Riemann conditions ∂ u ∂ x = ∂ v ∂ u ∂ y = − ∂ v ∂ y , ∂ x are satisfied). Then, the Jacobian determinant Jf ( x , y ) is non-negative for all x + iy ∈ C . Definition 4 Let f : R 2 → R 2 be a (real) smooth map. We call f positively quasi-conformal (resp. negatively quasi-conformal) if Jf ( x , y ) > 0 (resp. Jf ( x , y ) < 0) for all ( x , y ) ∈ R 2 . Question D. Do there exist easy to verify conditions on f providing its positive/negative quasi-conformness? Zalman Balanov (University of Texas at Dallas) 12
2.5. Surjective quadratic maps Obvious observation: 1 -dimensional case. Let f : R → R be a quadratic map, i.e. f ( x ) = ax 2 , a ∈ R . Then, f is not surjective. Question E. Let Φ : R n → R n be a quadratic map, i.e. its coordinate functions are quadratic forms in n variables. Under which conditions is Φ surjective? Zalman Balanov (University of Texas at Dallas) 13
3.6. Summing up: We arrive at the following Main Question: What is the connection between Question A (Quadratic differential systems), Question B (algebra), Question C (topology), Question D (geometric analysis), and Question E (algebraic(?) geometry or “geometric” algebra)? Main goal of my talk: To answer the Main Question. By-product: to illustrate the obtained results with applications to quadratic systems of practical meaning. Zalman Balanov (University of Texas at Dallas) 14
3. EXAMPLES OF QUADRATIC ODEs of REAL LIFE PHENOMENA 3.1. Euler equations (see [Arnold]) ω 1 = (( I 3 − I 2 ) / I 1 ) ω 2 ω 3 ˙ ω 2 = (( I 1 − I 3 ) / I 2 ) ω 1 ω 3 ˙ ω 3 = (( I 2 − I 1 ) / I 3 ) ω 1 ω 2 ˙ describes the motion of a rotating rigid body with no external forces (here the (non-zero) principal moments of inertia I j satisfy I 1 � = I 2 � = I 3 � = I 1 and I j stands for the j-th component of the angular velocity along the principal axes). Zalman Balanov (University of Texas at Dallas) 15
3.2. Kasner equations (see [Kasner,KinyonWalcher]) x = yz − x 2 ˙ y = xz − y 2 ˙ z = xy − z 2 ˙ describe the so-called Kasner’s metrics being the exact solution to the Einstein’s general relativity theory equations in vacuum under special assumptions. Zalman Balanov (University of Texas at Dallas) 16
3.3. Volterra equations (see [HofbauerSigmund]) x 1 = x 1 L 1 ( x 1 , ..., x n ) ˙ x 2 = x 2 L 2 ( x 1 , ..., x n ) ˙ ... x n = x n L n ( x 1 , ..., x n ) ˙ where L i ( x 1 , ..., x n ) is a linear non-degenerate form (also known as predator-pray equations), describe dynamics of biological systems in which n species interact. Zalman Balanov (University of Texas at Dallas) 17
3.4. Aris equations (see [Aris]) � x = a 1 x 2 + 2 a 2 xy + a 3 y 2 ˙ y = b 1 x 2 + 2 b 2 xy + b 3 y 2 ˙ describe a dynamics of the so-called second order chemical reactions (i.e. the reactions with a rate proportional to the concentration of the square of a single reactant or the product of the concentrations of two reactants). Zalman Balanov (University of Texas at Dallas) 18
3.5. More “academic” examples 3.5.1 Given λ, µ, c ∈ R with c � = 0, define a system (see [KinyonSagle]) “typical among quadratic three-dimensional ones admitting (non-zero) periodic solutions compatible with derivations of the corresponding fields. x 0 = λ x 2 0 + ( x 2 1 + x 2 2 ) x 1 = − 2 cx 0 x 2 x 2 = 2 cx 0 x 1 3.5.2 Let H be the 4-dimensional algebra of quaternions. Given q = q 0 + q 1 i + q 2 j + q 3 k ∈ H , define a conjugate to q by q := q 0 − q 1 i − q 2 j − q 3 k . The following ODEs were considered in [MawhinCampos] q = � q � α q β q γ , ˙ q ∈ H , 1 ≤ α + β + γ. If α = 2, β = 1 and γ = 0, then one obtains a type of nonlinearities arising in Ginzburg-Landau equation which comes from the theory of superconductivity (cf. [B’ethuelBrezisH’elein]). Zalman Balanov (University of Texas at Dallas) 19
3.6. Geodesic equations. Let Φ be a regular surface in R 3 parametrized by r ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v )) . Put E = E ( u , v ) := x 2 u + y 2 u + z 2 u , F = F ( u , v ) := x u x v + y u y v + z u z v , G = G ( u , v ) := x 2 v + y 2 v + z 2 v and take the so-called First Fundamental Form dr 2 := Edu 2 + 2 Fdudv + Gdv 2 . Properties of Φ depending only on dr 2 constitute the so-called intrinsic geometry of Φ. In particular, an important problem of intrinsic geometry is to study the behavior of geodesic curves on Φ, i.e. solutions to the differential system: 2 E u ′ 2 − E v u ′′ = − E u E u ′ v ′ + G u � 2 E v ′ 2 (5) 2 E u ′ 2 − G u v ′′ = E v G u ′ v ′ + G v 2 G v ′ 2 Zalman Balanov (University of Texas at Dallas) 20
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