Chronological calculus

Chronological calculus is a formalism for the analysis of flows of non-autonomous dynamical systems. It was introduced by A. Agrachev and R. Gamkrelidze in the late 1970s. The scope of the formalism is to provide suitable tools to deal with non-commutative vector fields and represent their flows as infinite Volterra series. These series, at first introduced as purely formal expansions, are then shown to converge under some suitable assumptions.

Operator representation of points, vector fields and diffeomorphisms
Let $$M$$ be a finite-dimensional smooth manifold.

Chronological calculus works by replacing a non-linear finite-dimensional object, the manifold $$M$$, with a linear infinite-dimensional one, the commutative algebra $$C^\infty(M)$$. This leads to the following identifications:
 * Points $$q\in M$$ are identified with nontrivial algebra homomorphisms
 * $$\hat{q}:C^\infty(M)\to\mathbb{R}$$ defined by $$\hat{q}(a)=a(q)$$.

satisfying the Leibnitz rule $$\hat{V}(ab)=\hat{V}(a)b+a\hat{V}(b)$$.
 * Diffeomorphisms $$P:M\to M$$ are identified with $$C^\infty(M)$$-automorphisms $$\hat{P}:C^\infty(M)\to C^\infty(M)$$ defined by $$(\hat{P}a)(q)=a(P(q))$$.
 * Tangent vectors $$v\in T_q M$$ are identified with linear functionals $$\hat{v}:C^\infty(M)\to\mathbb{R}$$ satisfying the Leibnitz rule $$\hat{v}(ab)=\hat{v}(a)b(q)+a(q)\hat{v}(b)$$ at $$q$$.
 * Smooth vector fields $$V\in\text{Vec}(M)$$ are identified with linear operators $$\hat{V}:C^\infty(M)\to C^\infty(M)$$

In this formalism, the tangent vector $$V(P(q))$$ is identified with the operator $$q\circ P\circ V$$.

We consider on $$C^\infty(M)$$ the Whitney topology, defined by the family of seminorms
 * $$\displaystyle{

\|a\|_{s,K}:=\sup\left\{\left|\frac{\partial^\alpha a}{\partial q^\alpha}(q)\right|\,\mid q\in K,\,\alpha\in\N^{{\rm dim}(M)},\;|\alpha|\leq s \right\}, } $$

Regularity properties of families of operators on $$C^\infty(M)$$ can be defined in the weak sense as follows: $$t\to A_t$$ satisfies a certain regularity property if the family $$t\to A_t(a)$$ satisfies the same property, for every $$a\in C^\infty(M)$$. A weak notion of convergence of operators on $$C^\infty(M)$$ can be defined similarly.

Volterra expansion and right-chronological exponential
Consider a complete non-autonomous vector field $$(t,q)\mapsto X_t(q)$$ on $$M$$, smooth with respect to $$q$$ and measurable with respect to $$t$$. Solutions to $$\dot q(t)=X_t(q(t))$$, which in the operator formalism reads

define the flow of $$X_t$$, i.e., a family of diffeomorphisms $$t\to P^t$$, $$P^0=\mathrm{Id}$$. The flow satisfies the equation

Rewrite $$ as a Volterra integral equation $$P^t=\mathrm{Id}+\int_0^t P^{\tau_1}\circ X_{\tau_1}d\tau_1$$.

Iterating one more time the procedure, we arrive to
 * $$\begin{align}

P^t&=\mathrm{Id}+\int_0^t P^{\tau_1}\circ X_{\tau_1}d\tau_1=\mathrm{Id}+\int_0^t \left(\mathrm{Id}+\int_0^{\tau_1} P^{\tau_2}\circ X_{\tau_2}d\tau_2\right)\circ X_{\tau_1}d\tau_1\\ &=\mathrm{Id}+\int_0^t X_{\tau_1}d\tau_1+\int_0^t\int_0^{\tau_1}P^{\tau_2}\circ X_{\tau_2}\circ X_{\tau_1}d\tau_2d\tau_1. \end{align}$$

In this way we justify the notation, at least on the formal level, for the right chronological exponential

where $$\Delta_n(t)=\{(\tau_1,\dots,\tau_n)\in\mathbb R^n\,\mid\,0\leq \tau_1\leq\dots\leq \tau_n\leq t\}$$ denotes the standard $$n$$-dimensional simplex.

Unfortunately, this series never converges on $$C^\infty(M)$$; indeed, as a consequence of Borel's lemma, there always exists a smooth function $$a\in C^\infty(M)$$ on which it diverges. Nonetheless, the partial sum
 * $$\displaystyle{

S_m(t)=\mathrm{Id}+\sum_{n=1}^{m}\int\dots\int_{\Delta_n(t)}\ X_{\tau_n}\circ\dots\circ X_{\tau_1}d\tau_n\dots d\tau_1 }$$

can be used to obtain the asymptotics of the right chronological exponential: indeed it can be proved that, for every $$a\in C^\infty(M)$$, $$s\geq 0$$ and $$K\subset M$$ compact, we have

for some $$C>0$$, where $$K'=\cup_{s\in [0,t]}P^s(K)$$. Also, it can be proven that the asymptotic series $$S_m(t)$$ converges, as $$m\to+\infty$$, on any normed subspace $$L\subset C^\infty(M)$$ on which $$X_t$$ is well-defined and bounded, i.e.,
 * $$\displaystyle{

X_t(L)\subset L,\quad \|X_t\|_{L}:=\sup\left\{\|X_t(a)\|_{L}\,\mid\, a\in L,\,\|a\|_{L}\leq 1\right\}<+\infty. }$$

Finally, it is worth remarking that an analogous discussion can be developed for the left chronological exponential $$Q^t$$, satisfying the differential equation
 * $$\displaystyle{

\frac{d}{dt}Q^t=X_t\circ Q^t,\; Q^0=\mathrm{Id}. }$$

Variation of constants formula
Consider the perturbed ODE
 * $$\displaystyle{

\frac{d}{dt}P^t=P^t\circ(X_t+Y_t),\quad P^0=\mathrm{Id}. }$$

We would like to represent the corresponding flow, $$P^t=\overrightarrow{\mathrm{exp}}\int_0^t(X_\tau+Y_\tau)d\tau$$, as the composition of the original flow $$\overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau$$ with a suitable perturbation, that is, we would like to write an expression of the form
 * $$\displaystyle{P^t=\overrightarrow{\mathrm{exp}}\int_0^t(X_\tau+Y_\tau)d\tau=R^t\circ \overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau.}$$

To this end, we notice that the action of a diffeomorphism $$S$$ on $$M$$ on a smooth vector field $$W$$, expressed as a derivation on $$C^\infty(M)$$, is given by the formula
 * $$\displaystyle{

S_*W=S^{-1}\circ W\circ S=\mathrm{Ad}S^{-1}(W). }$$

In particular, if $$S^t=\overrightarrow{\mathrm{exp}}\int_0^t V_\tau d\tau$$, we have
 * $$\begin{align}

\frac{d}{dt}(\mathrm{Ad}S^t) W&=\frac{d}{dt}S^t\circ W\circ S^{-t}=S^t\circ\left( V_t\circ W-W\circ V_t\right)\circ S^{-t}\\&=\mathrm{Ad}S^t[V_t,W]=(\mathrm{Ad}S^t)\mathrm{ad}V_tW.\end{align}$$

This justifies the notation
 * $$\displaystyle{

\mathrm{Ad}S^t=\overrightarrow{\mathrm{exp}}\int_0^t\mathrm{ad}V_\tau d\tau. }$$

Now we write
 * $$\displaystyle{

\frac{d}{dt}P^t=P^t\circ(X_t+Y_t)=P^t\circ X_t+R^t\circ\overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau\circ Y_t}$$

and
 * $$\displaystyle{\frac{d}{dt}P^t=\dot{R}^t\circ\overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau+R^t\circ\overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau\circ X_t=\dot{R}^t\circ\overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau+P^t\circ X_t}$$

which implies that
 * $$\displaystyle{

\dot{R}^t=R^t\circ \left(\overrightarrow{\mathrm{exp}}\int_0^t\mathrm{ad}X_\tau d\tau\right) Y_t,\quad R^0=\mathrm{Id}. }$$

Since this ODE has a unique solution, we can write
 * $$\displaystyle{

R^t=\overrightarrow{\mathrm{exp}}\int_0^t\left(\overrightarrow{\mathrm{exp}}\int_0^\tau\mathrm{ad}X_\theta d\theta\right) Y_\tau d\tau,}$$

and arrive to the final expression, called the variation of constants formula:

Finally, by virtue of the equality $$(\mathrm{Ad}P)\overrightarrow{\mathrm{exp}}\int_0^tV_\tau d\tau=\overrightarrow{\mathrm{exp}}\int_0^t(\mathrm{Ad}P)V_\tau d\tau$$, we obtain a second version of the variation of constants formula, with the unperturbed flow $$\overrightarrow{\mathrm{exp}}\int_0^tX_\tau d\tau$$ composed on the left, that is,