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I know that discrete time dynamical systems such as $x_{n+1} = rx_n(1-x_n)$ exhibit 2-cycles for some parameter values of r. I'm curious if there exist continuous time dynamical systems that exhibit 2-cycles. In this case, we would have 2 fixed points: $Q_1$ and $Q_2$ so that there's a heteroclinic orbit from $Q_1$ to $Q_2$ and another one from $Q_2$ to $Q_1$. Does such a system exist (even in multiple dimensions)? I'm trying to come up with such a system, but I've been unable to -- I know a number of models where there exists 3-cycles and above (such as an n-dimensional logistic model), but none with a 2-cycle.

EDIT: I realize that there were simple examples now of a 2-cycle. I would like to add a second question: Does there exist a system with both a 2-cycle in the x-y plane and a 3-cycle in the positive x-y-z region?

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Consider $$ \eqalign{\dot{x} &= y \cr \dot{y} &= x (x^2 - 1)\cr}$$ which has heteroclinic orbits joining the fixed points $(-1,0)$ and $(1,0)$.

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  • $\begingroup$ Ah yes, that is true. That is an extension of the simple harmonic oscillator, yes? I am now curious as to this: can one add a dimension $\dot{z}$ so that there is a 2-cycle in the x-y dimension and a 3-cycle in the x-y-z dimension? $\endgroup$
    – Brenton
    Commented Jun 2, 2015 at 18:55

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