Prove that a closed trajectory is invariant under the identity mapping of two sy

Question

Prove that a closed trajectory is invariant under the identity mapping of two systems of differential equations that have the same qualitative structure

Explanation / Answer

If the autonomous di?erential equation has a closed orbit and t ? f(t) is a solution with its initial value on this orbit, then it is clear that there is some T > 0 such that f(T) = f(0). In fact, as we will show in the next section, even more is true: The solution is T-periodic; that is, f(t + T) = f(t) for all t ? R. For this reason, closed orbits of autonomous systems are also called periodic orbits. Another important special type of orbit is called a rest point. To de?ne this concept, note that if f(x0) = 0 for some x0 ? R n , then the constant function f : R ? R n de?ned by f(t) = x0 is a solution of the di?erential equation . Geometrically, the corresponding orbit consists of exactly one point.

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