1. I would like to find a commonly used name for dynamical systems x' = f(x) such that f(x+y) = f(x) (dynamics of the system is invariant subject to particular coordinate change, the function describing the system is state-periodic)
2. I would like to find a name for a set of such points.
3. Additionally for non-autonomous dynamical system x' = f(x,u) I would like to find a name for a set of fixed-points y such that f(x+y,0) = f(x,0) = f(y,0) = 0.
Any help is much appreciated. Thank you very much.
If you take f(x) = sin(x), then sin(x + 2 pi) =sin(x) for all values of x. In general,
sin(x + 2 n pi) = sin(x) for all positive or negative integer values of n.
Since T = 2 pi is the smallest positive number satisfying sin(x + y) = sin(x), T = 2 pi is called the period of the sine function.
Similarly, what you are considering as f(x + y) = f(x) is to check whether the function f(x) is periodic or not. If this condition holds, the smallest positive value of y may be defined as the period of the function f(x).
Does "x" belongs to an euclidean vector space? Is "y" a constant vector?
If this is the case, then you are probably referring to "translation invariant vector fields". You can form the quotient manifold of the vector space under the action of this translation, and then x=x+y in this quotient (as equivalence classes). In practice though, depending on the dimension of the vector space and the vector y, the actual construction may be quite tricky.
If x belongs to a manifold different from an euclidean space, then the situation might change, depending on the manifold and on what y is.
Thank you all for your answers.
I wanted to write that my system is translation invariant in particular space and I wanted to call the fixed points as zero input fixed points (fixed points of the zero input dynamics) in the translation invariant space.
However I first wanted to check if common name existed.
Just to give motivation for the question. I am doing a position control of the crane system, where the mathematical model of the plant is given as x' = f(x,u), where x is a vector which belongs in Rn and u is one dimensional input to the system.
For purpose of the explanation let's assume that n=2 and that the first element of the vector represents the position while second element is velocity. The plant has infinitely many fixed points given as x = [y 0]T So the controller u=K(x) can bring the plant to any position y. Furthermore the plant can stay in any such fixed point with zero control input u=0. For such the controller I constructed an zero centered invariant set. And I want to say that the center of such set can be always translated to the new zero input fixed point inside translation invariant space.
The non-linear function is periodic in y, and the system is translation invariant in space defined by all possible [y,0].
Try to look for pendulum-like systems or systems with a cylindrical phase space.
- Badges
- Science topic
- Similar topics
- Mathematics
- Applied Mathematics
- Differential Equations