our totally informal, free-for-all tricontinental “morning” chat:
The Battle of Taylor-Couette II
Back in 2007 Jonathan Halcrow, John Gibson and collaborators were proud to be able to show that in a minimal cell state space truncated to 61,506 dimensions, turbulent plane Couette flow at Re=400 fits neatly into the Starship Enterprise (click here).
Video of the Zoom meetup
Zoom IE + Predrag recap, including hyperlinks to source materials
Jonathan's used Claude AI to reproduce and extend his PhD thesis on plane Couette flow. Now he discovered 131 relative periodic orbits using a JAX-based solver that achieved approximately 30x speed improvement over existing C++ implementations. Jonathan verified his results against previous literature and developed a symbolic dynamics approach using a 4-letter alphabet based on different flow phases (streaky, roll-like, wave-like, and high-dissipation states). Predrag discussed the limitations of forward-in-time versus space-time approaches for symbolic dynamics in spatiotemporal systems. The conversation evolved into a broader discussion about computational methods, with Predrag explaining the advantages of spatiotemporal optimization approaches using space-time solutions rather than traditional time-stepping methods.
Next steps
- Try the Gauss-Newton method with adjoint of the Jacobian (as suggested by Matt Golden) to improve convergence of the solver.
- Share the JAX solver code with John for further analysis and understanding the speed of the new time-stepping algorithm.
- Consider running the solver internally at Google to leverage more computational resources for finding longer periodic orbits.
- Investigate further the relationship between equilibria in the smaller box and periodic orbits in the wider box.
- Refine the symbolic dynamics and the set of symbols used for partitioning the state space.
- Explore the possibility of pushing the solver to larger spatial domains and consider a space-time approach for finding solutions.
Summary
Plane Couette Flow Research Update
Jonathan presented his work on revisiting his 18-year-old thesis on plane Couette flow, which studied equilibrium solutions and periodic orbits in minimal turbulent cells. He used Claude to implement a new solver in Python and JAX, which was much faster than his, Gibson and Schreiber previous Fortran and c++ implementations. While his current work is preliminary and 100% verified, he is still catching up on recent literature and welcomes feedback on potential duplicates.
JAX Implementation Results Presentation
Jonathan presented results from his implementation using JAX, which showed significant speed improvements over ChannelFlow, particularly when using GPU and batching. He successfully reproduced solutions from previous papers and his thesis, achieving agreement down to three decimal points on dissipation and leading eigenvalues. Jonathan also explored periodic orbits in a wider box geometries, finding that equilibria from the smaller box fold before reaching the wider box, with relative periodic orbits becoming dominant instead.
Periodic Orbits Discovery Presentation
Jonathan presented his work on finding periodic orbits, discussing the discovery of 131 orbits total, with most having time periods below T=120. He explained his methodology of finding orbits through long trajectories and recurrence searching, with some recent attempts at symbolic dynamics. Predrag clarified that the orbits are primarily relative periodic rather than periodic, and Jonathan confirmed that RPO21 matches an orbit previously found by Kawahara and Kita.
Turbulence Analysis and Orbits
Jonathan presented his analysis of turbulence in the plane Coutte, focusing on periodic orbits and their role in determining escape rates from turbulent states. He identified that most orbits are dominated by a single RPO21 orbit and found that the neighborhood of PO2 serves as a key gate controlling whether turbulence persists or the flow relaminarizes. Jonathan developed a symbolic dynamics approach using letters to represent different flow states, though noted it requires remembering multiple symbols for accurate predictions.
JAX 2DMHD Solver Approaches
Matt Golden shared insights about using JAX to implement Gauss-Newton instead of Newton GMRes for 2DMHD solvers, noting it offers a larger basin of convergence and is computationally efficient. Dmitriy mentioned Stokes preconditioning as another approach for speedup, though clarified it only applies to traveling waves, not periodic orbits. The discussion then shifted to Jonathan's work on partitioning state space, where he explained using maximum visited points across orbits to define letter classifications, with Chris confirming this was done using L2 distance and asking about the use of a library of solutions, to which Jonathan responded that a nearest visitations approach worked better than using periodic orbits directly.
Turbulence Prediction Methods Discussion
Chris and Jonathan discussed the effectiveness of using proximity to solutions for predicting dynamics in turbulence. Jonathan acknowledged that while the chosen points in state space showed some correlation with known turbulence patterns, a more careful analysis using periodic orbits might yield better results.
Predrag explained that forward-in-time, one-dimensional sequence symbolic dynamics approaches are not suitable for problems with spatial translational invariance, and recommended using instead spatiotemporal symbol arrays, as demonstrated in previous work on the Kuramoto-Sivashinsky equation by Matt Gudorf and others.
Spatiotemporal Solutions and Periodic Domains
Predrag and Chris discussed spatiotemporal solutions and periodic domains in three dimensions, with Predrag explaining how solutions can be treated as building blocks or "Lego blocks" that can be combined and optimized. Predrag outlined a method involving spatiotemporal periodic domains, Bravais cells, and primitive cells to systematically find solutions.
The conversation ended with Predrag confessing that ever since his student Freddy Christiansen outperformed him in their computations, his life has been a struggle to catch up in computational work.
Code Performance Improvement Discussion
The group discussed the performance improvements in Jonathan's code implementation, with Jonathan explaining that a factor of 4 speed-up came from changing the time-stepping algorithm while maintaining the same accuracy. Jonathan noted that the time step size difference was a key factor, though he couldn't fully explain the methodological differences between the implementations. The discussion also touched on the importance of examining solutions in state space and using appropriate coordinates to understand the dynamics of the system, though Jonathan mentioned that equilibrium solutions in the current geometry were less prominent and not as dynamically important as in smaller geometries.
Unstable Directions in Symmetric Systems
Jonathan and Predrag discussed the challenges of representing unstable directions in systems with translational symmetry. Predrag explained techniques like Poincare sectioning and slicing to simplify the analysis by removing translational invariance, suggesting these methods could help find a better representation than Jonathan's initial attempt using the unstable manifold of RPO21. They also discussed how to handle multiple trajectories in spatiotemporal cases, including the concept of tori shadowing each other in space-time.
Space-Time Optimization in Dynamical Systems
Predrag and Jonathan discussed the challenges and benefits of using space-time optimization methods in dynamical systems, particularly for the 1D Kuramoto-Sivashinsky system. Predrag explained that while forward-in-time solutions face exponential instabilities, space-time approaches using optimization algorithms can successfully find stable solutions in larger domains. Jonathan expressed interest in exploring larger spatial cells and potentially using tiling of solutions found in smaller boxes to extend their work to larger spaces.
Spatio-Temporal Problem Solving Approach
Predrag discussed the approach to solving spatio-temporal problems by considering solutions in state space rather than plumber's image boxes, and proposed using a Lagrangian approach instead of Newton's method. He explained how to identify periodic solutions and build a library of space-time tiles that can be used to extend solutions beyond a given window. Chris asked about extending solutions in temporal direction, and Predrag clarified that the approach considers solutions in multiple directions, not just forward in time.
Space-Time vs Dynamical Systems Discussion
Predrag and Chris discussed the concept of thinking in space-time versus traditional dynamical systems approaches, with Predrag explaining how periodic orbits can be viewed as solutions in space-time and how observables are evaluated on these solutions.
The Salon concluded with a discussion of Messi's retirement and Argentine World Cup show, with Predrag expressing what in Argentina is a minority opinion about the team's performance.
Links to earlier Salons and other matters of importance
Alex Blumenthal Math colloquium, Sept 15, 2026
Jonathan Halcrow & Fable, Invariant solutions salon, Sept 11, 2026
