Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hopf algebras in dynamical systems theory   The theory of exact and of approximate solutions for non-autonomous linear
differential equations forms a wide field with strong ties to physics and
applied problems. This paper is meant as a stepping stone for an exploration of
this long-established theme, through the tinted glasses of a (Hopf and
Rota-Baxter) algebraic point of view. By reviewing, reformulating and
strengthening known results, we give evidence for the claim that the use of
Hopf algebra allows for a refined analysis of differential equations. We
revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern
approach involving Lie idempotents. Approximate solutions to differential
equations involve, on the one hand, series of iterated integrals solving the
corresponding integral equations; on the other hand, exponential solutions.
Equating those solutions yields identities among products of iterated Riemann
integrals. Now, the Riemann integral satisfies the integration-by-parts rule
with the Leibniz rule for derivations as its partner; and skewderivations
generalize derivations. Thus we seek an algebraic theory of integration, with
the Rota-Baxter relation replacing the classical rule. The methods to deal with
noncommutativity are especially highlighted. We find new identities, allowing
for an extensive embedding of Dyson-Chen series of time- or path-ordered
products (of generalized integration operators); of the corresponding Magnus
expansion; and of their relations, into the unified algebraic setting of
Rota-Baxter maps and their inverse skewderivations. This picture clarifies the
approximate solutions to generalized integral equations corresponding to
non-autonomous linear (skew)differential equations.

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The categories of this paper are: nlin
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The categories of this paper are: dg-ga
1
The categories of this paper are: cs
11
The categories of this paper are: hep-lat
20
The categories of this paper are: acc-phys
29
The categories of this paper are: funct-an
28
The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
27
The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: bayes-an
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The categories of this paper are: physics