I am working on stability of nonlinear switched systems and recently, I have proven that switched systems with homogeneous, cooperative, Irreducible and commuting vector fields , i.e., vector fields with Lie bracket equal to 0, are D-stable under some condition. I was trying to find an example for such systems but surprisingly, I could not find any in the papers which have dealt with them. Does anybody know a good example of commuting nonlinear vector fields? And are there any significance to such systems (from physical point of view)?
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$\begingroup$ The answers below give a general construction of commuting vector fields on $\mathbb{R}^n$ and you can make them homogeneous and nonlinear. But what do adjectives "irreducible" and "cooperative" mean? $\endgroup$– Victor ProtsakCommented Jun 12, 2010 at 4:00
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2 Answers
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Any set of nonlinear co-ordinates gives you a corresponding set of commuting vector fields. So any of the co-ordinates listed in the "See also" section of
http://en.wikipedia.org/wiki/Elliptic_coordinates
gives an example.
answered Jun 11, 2010 at 19:06
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You can take any two commuting vector fields, e.g. $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ in the plane, then conjugate by a diffeomorphism. The resulting two vector fields will commute, but need not be linear (at least if I understand what you call linearity of a vector field).
answered Jun 11, 2010 at 16:35
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$\begingroup$ More generally, $\partial/\partial u$ and $\partial/\partial v$ , where $(u,v)$ is a local coordinate system, is an example – think of polar coordinates, for example. Actually, “more generally” is an oxymoron here, since you get the same examples that Benoît is referring to. But it may be a more helpful way to think of it. $\endgroup$ Commented Jun 11, 2010 at 17:25