INVARIANTS OF

KINETIC DIFFERENTIAL EQUATIONS^{∗}

Andrea HALMSCHLAGER,

László SZENTHE and

János TÓTH

Budapest University of Technology and Economics,

Department of Mathematical Analysis July 11, 2003

Contents

1 Introduction 2 2 Algebraic invariants 3 3 Application of algebraic invariants 6 4 Acknowledgement 7

∗This paper is in final form and no version of it will be submitted for publication elsewhere.

1 Introduction

∑_{m=1}^{M} μ(m, r)X_{m} ⟶^{k_{r}} ∑_{m=1}^{M} ν(m, r)X_{m}  (r = 1, 2, …, R) (1)

• k_{r} ∈ ℝ^{+} are the reaction rate coefficients,

• μ(m, r) ∈ ℕ_{0} and ν(m, r) ∈ ℕ_{0} are the stoichiometric coefficients.

Mass action type deterministic model (a polynomial differential equation):

ẏ_{m} = ∑_{r=1}^{R}(ν(m, r) − μ(m, r))k_{r} ∏_{p=1}^{M} y_{p}^{μ(p,r)} (m = 1, 2, …, M),

• dependent variables: concentrations of species (y_{m} := [X_{m}])

• independent variable is: time.

Negative cross-effect Lorenz equation:

ẋ = −σ(x − y)
ẏ = rx − y −xz
ż = xy − bz  (σ, b, r ∈ ℝ^{+})

Rössler equation:

ẋ = x − xy −y
ẏ = x^{2} − ay
ż = bx − cz + d  (a, b, c, d ∈ ℝ^{+})

No transformation of the form MA (where M is an orthogonal and A is a positive definite diagonal transformation) transforms the Lorenz equation into a kinetic one, and there exists no universal transformation to a similar Rössler model transforming it to a kinetic equation at all the values of the parameters.

2 Algebraic invariants

ẏ^{i} = ∑_{ω∈Ω} a_{j_{1}...j_{ω}}^{i}y^{j_{1}} … y^{j_{ω}} (i, j_{1}, …, j_{ω} = 1, 2, …, n; a_{j_{1}...j_{ω}}^{i} ∈ ℝ)

(2)

Ω is finite; degree:=max(Ω)

Definition 1 (Negative cross-effect)

∃ a_{j}^{i} < 0 such that j_{i} = 0.

GL(n, ℝ) := {q ∈ ℝ^{n×n}; det(q) ≠ 0} : general linear group Q ⊂ GL(n, ℝ) is a subgroup If y = (y^{1}, y^{2}, …, y^{n}) is the vector of unknown variables of (2)

then for

ŷ := qy

we have

˙ŷ^{r} = ∑_{ω∈Ω} b_{r_{1}...r_{ω}}^{r}ŷ^{r_{1}} … ŷ^{r_{ω}} (r, r_{1}, …, r_{ω} = 1, …, n; b_{r_{1}...r_{ω}}^{r} ∈ ℝ).

b = B(a, q)

Definition 2 (Polynomial invariant) I : A → ℝ is a polynomial invariant of (2) under Q, if ∃λ : Q → ℝ s.t.

I(B(a, q)) = λ(q)I(a) (q ∈ Q, a ∈ A).

Definition 3 (Absolute/relative invariant) λ(q) = 1. (Otherwise: relative.)

Definition 4 (Orthogonal/special orthogonal group)

O(n, ℝ) = {q ∈ ℝ^{n×n}, qq^{T} = I}
SO(n, ℝ) = {q ∈ ℝ^{n×n}, qq^{T} = I, det(q) = 1}

SO ⊂ O ⊂ GL

Definition 5 (Reducible polynomial invariant)

Definition 6 (Generating system)

B := {I_{θ}; I_{θ} is a polynomial invariant of (2) under Q, θ ∈ Θ}

Example 1 A generating system of invariants of the system

ẏ^{i} = ∑ a_{j}^{i}y^{j} (Ω = {1}; i, j = 1, 2; a_{j}^{i} ∈ ℝ).

under Q = GL is I_{1}(a) = a_{1}^{1} + a_{2}^{2} = tr(a) I_{2}(a) = tr(a^{2})

I_{3}(a) = tr(a^{3}) (I_{3} is reducible as tr(a^{3}) = tr(a^{2})tr(a) − \frac{1}{2}tr(a)(tr^{2}(a) − tr(a^{2})))

I_{4}(a) = ... (reducible) ⋮ Consequently {I_{1}, I_{2}} is a minimal generating system. Definition 7 (Minimal generating system) Theorem 1 A base of polynomial invariants of equation (2) under GL(n, ℝ) (under O(n, ℝ)) can be obtained by alternation followed by full contraction (by generalized alternation followed by generalized full contraction) of the indices of the products of the coefficient tensors.

A minimal generating system of polynomial invariants of not more than the second degree of the equation

ẏ^{j} = a_{α}^{j}y^{α} + a_{αβ}^{j}y^{α}y^{β} (j, α, β = 1, 2, …, n)

is:

I_{1} = a_{α}^{α}  GL  1
I_{2} = a_{q}^{p}ε^{p}q  SO  2!
I_{3} = a_{α,p}^{α}ε^{p}  GL  1!
I_{4} = a_{β}^{α}a_{α}^{β}  GL  2
I_{5} = a_{β}^{α}a_{β}^{α}  O  3
I_{6} = a_{q}^{p}a_{s}^{r}ε^{pqrs}  SO  4!
I_{7} = a_{q}^{p}a_{αr}^{α}ε^{pqr}  SO  3!
I_{8} = a_{q}^{α}a_{αr}^{p}ε^{pqr}  SO  3!
I_{9} = a_{r}^{p}a_{αα}^{q}ε^{pqr}  SO  3!
I_{10} = a_{αβ}^{α}a_{γγ}^{β}  O  3
I_{11} = a_{αγ}^{α}a_{βγ}^{β}  O  2
I_{12} = a_{ββ}^{α}a_{γγ}^{α}  O  2
I_{13} = a_{βγ}^{α}a_{βγ}^{α}  O  2
I_{14} = a_{βγ}^{α}a_{αγ}^{β}  O  2
I_{15} = a_{αp}^{α}a_{ββ}^{q}ε^{pq}  SO  2!
I_{16} = a_{αq}^{p}a_{αs}^{r}ε^{pqrs}  SO  4!

ε^{p_{1}...p_{n}} = \begin{cases}1 & if (p_{1}, …, p_{n}) is an even permutation of (1, …, n) \\ −1 & if (p_{1}, …, p_{n}) is an odd permutation of (1, …, n) \\ 0 & if ∃ i, j ∈ (1, …, n) such that i ≠ j but p_{i} = p_{j} \end{cases}

ε is not defined if p_{1}, …, p_{n} are not from the first n positive integers.

3 Application of algebraic invariants

Some remarks:

1. Let us start with the following model (equation (2)):

ẏ^{j} = ∑_{ω∈Ω} a_{j_{1}j_{2}...j_{ω}}^{j}y^{j_{1}}y^{j_{2}}...y^{j_{ω}} (j, j_{1}, …, j_{ω} = 1, 2, …, n) (3)

and its (orthogonal or nonsingular) transform

˙ŷ^{j} = ∑_{ω∈Ω} b_{r_{1}r_{2}...r_{ω}}^{r}ŷ^{r_{1}}ŷ^{r_{2}}...ŷ^{r_{ω}} (r, r_{1}, …, r_{ω} = 1, 2, …, n). (4)

2. Invariants are looked for which take on values of different sign when calculated from the coefficients of (3) and from those of (4).

3. We examine mainly second degree, 3-dimensional equations and investigate the polynomial invariants of not more than second degree.

4. The invariants I_{2}, I_{3}, I_{6}, I_{15} and I_{16} are not related to 3-dimensional equations, so they cannot be used.

5. I_{1} and I_{11} are polynomials of coefficients irrelevant from the point of view of negative cross-effect (see Definition 1), they cannot be used either.

6. The invariants I_{5}, I_{12}, I_{13} are sums of squares, therefore they are nonnegative for all possible coefficients.

7. The sign of the invariants I_{7}, I_{8}, I_{9}, I_{10}, I_{14} is not unambiguously defined that is why they generally cannot be used either.

8. The sign of the invariant I_{4} is always nonnegative when calculated from the coefficients of a kinetic differential equation as

I_{4} = a_{β}^{α}a_{α}^{β} = (a_{1}^{1})^{2} + (a_{2}^{2})^{2} + (a_{3}^{3})^{2} + 2(a_{2}^{1}a_{1}^{2} + a_{3}^{1}a_{1}^{3} + a_{3}^{2}a_{2}^{3}),

thus I_{4} might be appropriate.

Example 2

ẋ = −y − z
ẏ = x + ay  (5)
ż = b − cz + xz

Here I_{4} = a^{2} + c^{2} − 2 so thus a and c such that a^{2} + c^{2} < 2 than I_{4} < 0. Accordingly (5) cannot be transformed into a kinetic differential equation by an orthogonal or nonsingular transformation.

4 Acknowledgement

The authors thank the French Ministry for Territorial Planning and Environment BCRD AB2001, and for the National Scientific Research Foundation (No. T037491) for partial financial support.

References

[1] Lorenz, P.: Deterministic nonperiodic flow, J. Atm. Sci. 20 (1963), 130–142.

[2] Schuman, B.; Tóth J.: No limit cycle in two species second order kinetics, Bull. sci. math. 127 (2003) 222–230.

[3] Sibirsky, K. S.: Introduction to the algebraic theory of invariants of differential equations, Stiinţa, Kishinev, 1982.

[4] Szenthe, L.: Algebraic invariants of chaotic and kinetic differential eqations, (Thesis, L. Eötvös University).

[5] Tóth, J., Hárs, V.: Orthogonal transforms of the Lorenz- and Rössler-equations, Physica D 19 (1986), 135–144.