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An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar

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Figure

Figure 1. Phase portrait for the system defined by (10). The stable and unstableis shown as a dashed line
Figure 2. Top row: Gainleft) or 1the sudden jump in G := ∥x(t)∥22/d2 verses θ for T = 2 and d = 10−4 (top left)or 0.9 (top right)
Figure 3. As T (labelled on each curve) increases, the signature of the transitionat dc becomes increasingly clear.Note only for T→ ∞ is the final plateau∥x(T)∥22 = ∥X1 − X0∥22 ≈ 198.5 as x(t) ‘overshoots’: see Figure 1.
Figure 4. Checkpointing: during the (backward-in-time) calculation ofvelocity ν(x, t) the u(x, t) is recalculated in short sections from each checkpoint (shown as dotson the forward-in-time calculation.
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