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Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the delay logistic equation $\frac{dN}{dt}=rN(1-\frac{N(t-\tau)}{K})$ and one-dimensional discrete dynamical systems such as the logistic map $x_{n+1}=rx_n(1-x_n)$, whereas this does not occur for first-order autonomous ordinary differential equations like the logistic equation $\frac{dN}{dt}=rN(1-\frac{N}{K})$)? Does the time-delay $\tau$ play a similar role to the discrete time-step in allowing solutions to break free of the restrictions of monotonic flow on a line? If so, can this be expressed precisely, or it merely a loose analogy?

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