Pomeau and Manneville described three routes to intermittency where a nearly periodic system shows irregularly spaced bursts of chaos.
Since the time spent near the periodic orbit depends sensitively on how closely the system entered its vicinity (in turn determined by what happened during the chaotic period) the length of each phase is unpredictable.
Another kind, on-off intermittency, occurs when a previously transversally stable chaotic attractor with dimension less than the embedding space begins to lose stability.
In highly turbulent flows, intermittency is seen in the irregular dissipation of kinetic energy [5] and the anomalous scaling of velocity increments.
Intermittent behavior has also been experimentally demonstrated in circuit oscillators and chemical reactions.
Intermittency in logistic map with
. The trajectory alternates between almost period-3 trajectories and chaotic trajectories. At
a stable period-3 trajectory emerges.
The intermittency in logistic map can be understood by looking at the cobweb diagram for logistic map (iterated three times). In the cobweb diagram, there are almost-tangencies where the trajectory can be trapped for a long time.
Intermittent jumping between two potential wells in the driven
Duffing oscillator
. This is an example of crisis-induced intermittency.
Lorenz attractor
showing intermittency. The system spends long periods close to the bright periodic orbit, occasionally moving away for phases of chaotic dynamics that cover the rest of the attractor. This is an example of Pomeau–Manneville dynamics.
