 
Summary: Journalof MathematicalSciences, Vol. 71, No. 3, 1994
VOLTERRA SERIES AND PERMUTATION GROUPS
A. A. Agrachev and R. V. Gamkrelidze UDC 512.542.7+517.972.1
Algebraic structures, connected with the asymptotic expansions of perturbations of smooth dynamical systems,
are investigated; first of all, the socalled shuffle multiplication for permutations and for iterated integrals.
1. INTRODUCTION. VARIATIONS OF A DYNAMICAL SYSTEM
1. We consider a system of differential equations
x=/'t (x), x~M,
on a manifold M of class C ~ with a fixed initial condition x(O) = xo and then we perturb this system by adding to the right
hand side a vector field egt(x), where e is a small parameter. We obtain the system
=?, (x) +eg, (x), x (0) =Xo. (1.1)
Let xe(t), 0 < t < 1, be the solution of the system (1.1). We pose the classical question: how to find the tangent to the curve
e , xe(1) for e = 0. The answer can be found in any manual on ordinary differential equations: one has to solve the linear
system of equations in variations
: Oft]
~= Ox I~,,) '+gt(x~ ~(0)=:0.
Then ~(1) is the desired tangent vector in the case when ~(1) # 0. However, we are concerned with those more interesting
cases when ~(1) = 0. In this case one has to consider higher terms in the Taylor expansion of the curve e , xc(1) (higher
variations of the system (1)). However, as it is known, the higher terms of the Taylor expansion do not have an invariant
meaning: they are correctly defined only in fixed local coordinates. Under nonlinear changes of coordinates, the various
