Invariance of dynamical system under a transformation

Question

I have come across an interesting property of a dynamical system, being transformed by a map, but i haven't been able to figure out why this is happening (for quite some time now actually). Any help is greatly appreciated. Here goes then:

Let M be a n-D manifold and $\dot x=F(x)u_1, F\in \mathbb{R}^{n\times m}, x \in \mathbb{R}^{n}, u_1 \in \mathbb{R}^{m}$ be a control system evolving on M (F is the system matrix i.e. state transition function, and $u_1$ is the input of the system. For all practical purposes $u_1$ is an m-vector from an input space $\mathbb{R}^{m}$). Now let $x=\Psi (y)$ be a coordinate change on M and $u_2=M(y)u_1$ a transformation of the input $u_1$ of the first system. By applying these maps on the system, you get the new equations $\dot y=F(y)u_2$. As you may notice, F is the same in both systems. The problem is why is this happening i.e. for what systems and transformations does this property hold?

A little more elaboration

It is useful to investigate the maps more closely. In the general case one has

$\dot x=D\Psi \dot y$
$\dot x= F(x)u_1$

thus
$\dot y=D\Psi ^{-1} F(x)u_1$, (1)

where $D\Psi$ is the Jacobian matrix of $\Psi$. In our case it actually turns out that:

$\dot y=F(y)M(y)u_1$. (2)

You can then consider that $u_2=M(y)u_1$ and get the final system,

$\dot y=F(y)u_2$,

that is, the same system. By (1),(2) you get,

$D\Psi ^{-1} F(x)u_1=F(y)M(y)u_1 \Rightarrow (D\Psi ^{-1} F(x)-F(y)M(y))u_1=0$.

Since this holds for every $u_1$, you have the condition,

$F(\Psi (y))=D\Psi F(y)M(y)$

So, what does this condition imply? What systems F and maps $\Psi$ hold this property (of system invariance)? I should note that F is nonlinear and a case study where this actually happens is the kinematic model of a unicycle robot i.e. this. Any ideas?

Answer 1

Let me reply taking M to be the identity (indeed M is somewhat cosmetic to the discussion).

The identity $F\circ Ψ=DΨ\circ F$ is what one considers for example in the Grobman-Hartman theorem, passing from a dynamics to its linearization say at a fixed point. The possibilities for F are endless; $F$ would then be a topological conjugacy, perhaps locally, although maybe not very regular, more precisely at most Hölder continuous in general. Moreover, $F$ need not satisfy any invariance properties, which if I understand correctly is your main concern.

Answer 2

This is not a surprising fact for driftless systems to have symmetries. In our previous work we considered only nonlinear systems with drift and classified the symmetries accordingly. In case of a driftless system \dot x=F(x)u you can interpret $F$ (that's what it is) as a distribution of vector fields $f_1, \dots, f_m$ (column vectors of the matrix) and whenever such distribution is involutive ($[f_i, f_j]=\lambda_1(x)f_1+\cdots+\lambda_m(x)f_m$ then necessarily the system admits nontrivial symmetries (this is due to the Frobenius Theorem). That's the case you have here with the unicycle with $f_1=(\cos \theta, \sin \theta, 0), f_2=(0,0, 1)$. I can write in more extensive way but I hope you would get the idea from there.

Issa