[Paper Review] New special functions solving nonlinear autonomous dynamical systems

L. Brenig|ArXiv.org|Oct 14, 2009

Quantum chaos and dynamical systems3 citations

TL;DR

This paper introduces a new class of special functions defined by Taylor series whose coefficients are generated by a combinatorial tensor generalizing the factorial, enabling a universal closed-form solution for quasi-polynomial dynamical systems. The solution reduces any such system to a canonical Lotka-Volterra form, revealing deep connections between nonlinear dynamics, special functions, and combinatorics.

ABSTRACT

A general solution is found for a large class of time continuous autonomous nonlinear dynamical systems, the so-called quasi-polynomial systems. This solution is expressed in terms of a new type of special functions defined via their Taylor series. The coefficients of these Taylor series are generated by a tensor that generalizes the factorial function and has a combinatorial meaning. The existence of these functions raises the question of the relation between them and the chaotic behaviour of the solutions that may appear for the quasi-polynomial dynamical systems.

Motivation & Objective

• To develop a universal solution framework for a broad class of nonlinear autonomous dynamical systems, specifically quasi-polynomial (QP) systems.
• To address the lack of closed-form general solutions for nonlinear ODEs, which typically suffer from factorial explosion in Taylor coefficients.
• To establish a link between the combinatorial structure of solution coefficients and the chaotic behavior observed in QP systems.
• To unify the analysis of diverse nonlinear systems in physics, chemistry, and social sciences by reducing them to a canonical Lotka-Volterra form.
• To introduce a new class of special functions defined by Taylor series with coefficients governed by a tensor generalization of the factorial function.

Proposed method

• Represents quasi-polynomial dynamical systems in a standard quasi-polynomial (QP) form using constant matrices A and B, with equations of the form $ \dot{x}_i = x_i \sum_{j=1}^N A_{ij} \prod_{k=1}^n x_k^{B_{jk}} $.
• Applies quasi-monomial transformations $ x_i = \prod_{k=1}^n \tilde{x}_k^{C_{ik}} $, which preserve the QP structure and lead to a canonical Lotka-Volterra system via matrix transformations $ \tilde{A} = C^{-1}A $, $ \tilde{B} = BC $.
• Derives the general solution as a Taylor series $ x_i(t) = \sum_{k=0}^\infty C_i(k) \frac{t^k}{k!} $, where the coefficient $ C_i(k) $ is expressed via a multilinear sum over matrix products involving the fundamental matrix $ M = BA $.
• Identifies a rank-N tensor $ \delta_{ij_1}(\delta_{ij_2} + \delta_{i_1j_2}) \cdots $ as a generalization of the factorial function, governing the combinatorial structure of the solution coefficients.
• Establishes that the matrix $ BA $ is invariant under quasi-monomial transformations, enabling classification of QP systems into equivalence classes.
• Uses recursive time differentiation of the Lotka-Volterra system to derive explicit expressions for the Taylor coefficients $ c_i(k) $, which are then transformed back to the original QP variables.

Experimental results

Research questions

• RQ1 Can a universal closed-form solution be derived for a broad class of nonlinear autonomous ODEs, particularly quasi-polynomial systems?
• RQ2 What is the combinatorial structure underlying the Taylor series coefficients of solutions to quasi-polynomial systems?
• RQ3 How does the tensor generalization of the factorial function relate to the dynamics and potential chaotic behavior of QP systems?
• RQ4 To what extent can the solution of any QP system be reduced to the canonical Lotka-Volterra form, and what invariants are preserved under such transformations?
• RQ5 What is the relationship between the special functions defined by these Taylor series and known classes like hypergeometric functions, particularly in terms of asymptotic and integral representations?

Key findings

• A universal general solution exists for all quasi-polynomial dynamical systems, expressed as a Taylor series with explicitly computable coefficients.
• The solution coefficients $ C_i(k) $ for the original QP system are derived from the Lotka-Volterra form via transformation rules involving the matrices $ A $, $ B $, and the initial conditions.
• The tensor $ \delta_{ij_1}(\delta_{ij_2} + \delta_{i_1j_2}) \cdots $ acts as a combinatorial generalization of the factorial, counting and structuring the terms in the coefficient expansion.
• The matrix $ BA $ is an invariant under quasi-monomial transformations, enabling classification of QP systems into equivalence classes and facilitating reduction to the canonical Lotka-Volterra form.
• The solution method reduces the complexity of solving arbitrary QP systems to solving a canonical Lotka-Volterra system, preserving the dynamics via diffeomorphism.
• The new special functions defined by the Taylor series exhibit structural analogies to hypergeometric functions, suggesting potential for asymptotic and integral representations, and offering a bridge to chaotic dynamics.