Consider a general 2D system:
x' = f(x,y)
y’ = g(x,y)
Do you know any such system which has a limit cycle while there is no equilibrium in it? For example the famous Van Der Pol is not an answer; Because, although it has a limit cycle, it has an equilibrium in the origin (0,0)
x' = y
y’ = ay(1-x^2)-x
There is not such system. As an application of Poincaré - Bendixsom theorem, if z(t) is a closed orbit of a planar system, then there exist singular points inside the region delimited by z(t).
Not possible. A limit cycle (even a cycle chain) must surround at least one equilibrium.
it is impossible. According to index theory,(http://www.scholarpedia.org/article/Periodic_orbit), one limit cycle must surround one or more saddle or focus.
The definition of the limit cycle requires that a limit cycle must surround at least an equilibrium.
However this is impossible in R^2,but it would be interesting to find a vector field on R^3 with an attracting limit cycle and there is no a singularity in R^3. Can one introduce an example of this situation? Can one introduce an algebraic(polynomial) example with this property?
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