From Strogatz's Nonlinear Dynamics and Chaos (2nd ed), Section 2.6:
If a fixed point is regarded as an equilibrium solution, the approach
to equilibrium is always monotonic — overshoot and damped oscillations
can never occur in a first-order system. For the same reason, undamped
oscillations are impossible. Hence there are no periodic solutions to
$\dot{x} = f ( x )$.
I was tinkering around with a logistic map tool that I found here and observed this plot for the values: $x_0 = 0.2, r = 2.7$.
Is this an example of an overshoot / transient oscillations? Am I misunderstanding something? Why do we observe these transient oscillations at the beginning?
Best Answer
Discrete dynamical systems and continuous dynamical systems behave wildly differently. In particular, there are vastly more possible behaviors of discrete dynamical systems in low dimensions than there are for continuous dynamical systems. Prof. Strogatz is referring to the continuous case. The logistic map can produce oscillations of every period, and has periods of chaos. By contrast, chaos is not a phenomenon that's possible for continuous systems until you get to 3 dimensions.