Dear All,
I am working on understanding the transition from laminar to turbulent behaviour in Navier-Stokes equations by the approach of dynamical systems theory.
I have derived a three-dimensional dynamical system model on the level surface of some energy function from the NS equations (with minimal dissipation hypothesis). Based on this model, I obtained a devil's staircases behaviour similar to those from the circle map. However, quite noticeable, which is in contrast to the conventional devil's staircase observed from the sine circle map, is the overlaps between the quasiperiodic curve and the phase locking plateaus, see attached figures. However, such an overlap is only apparent, indicating a successive hop between periodic and quasiperiodic behaviours. Moreover, the decrease in the values of the phase locking plateaus does not follow the Farey sequence, instead of following (...4/5,4/6,4/7,4/8.4/9.4/10...). While increasing the nonlinear coupling of the system, the sequence becomes (...8/10,8/11,8/12,8/13,8/14,8/15,8/16...). I would like to know if similar behaviours have been observed in experiments, or it is merely mathematical creatures? Even in the sense of mathematics, these seem against the dynamical system theory, yet following some certain rules. Could anyone point me out, please?
In the plot, Re stands for Reynolds number, and Ro stands for the rotation number, which is defined in a limit sense for an angle variable phi, where phi is the angle between velocity and vorticity vectors (subjected to a coordinate shift).
Thank you.
Very kind wishes,
Wang Zhe
Maybe you are observing twice as many phase locking intervals because of your degree of accuracy.
There is a geometrical picture that naturally produces all of the rational numbers in REDUCED FORM which you may find useful.
step one: put unit diameter tangent circles resting on the horizontal axis touching at integers values.
step two: put big as possible smaller tangent to previous circles inside the curved triangular regions described by these circles and the axis.By symmetry these touch at rational numbers: odd integers divided by two.their diameters are one quarter or the square of the denominator of the fraction at the resting point namely one fourth.
step three : repeat step two putting in largest circles in each of the created curved triangular regions. now the resting points are obtain by "farey addition" of adjacent fractions, namely add numerators to get the numerator, add denominators to get the denominator. this process always gives reduced fractions. now the denominators equal three and the diameters are one ninth.
continue recursively with step four , step five ,...... etc
by examining any fixed circle one sees a 1/x sequence of circles going down towards the tangent resting point.
this fits with your graphs.
this beautiful figure is part of the ancient apolllonian packing whose hausdorff dimension 1.3.... is mentioned numerically in mandelbrot,s book on fractals and which is discussed rigorously in the sense of proof, in my paper
" Hausdorff measures old and new and limit sets of geometrically finite kleinian groups"
which also has nice pictures.
i would need more information about the dynamics you use to venture a dynamical systems comment.
dennis sullivan
Dear Prof Sullivan,
Thank you for the wonderful reply! I have slightly modified the system before repeating the simulation with increased precision and Re range, see attached.
For k=0.05, the steps decrease in value following the Apollonian packing sequence, whereas, for k=0.15, the emergence of the tertiary steps {4/5, 4/7, 4/9, 4/11..} disturbs the sequence. In order to get a proper conjecture on the sequence, I have simulated on different k values. However, each simulation takes around 39 hours to run (on my machine), and I will update when finished.
The details of the dynamical system were sent to you in an email. Besides the devil's staircases, numerical integrations reveal an interesting slow dynamics. It would be greatly helpful if you could comment on it.
Thank you.
Very kind wishes,
Wang Zhe
Dear All,
Notice that the trace of the Jacobian of the system is nonlinear
Re-1(v2+5f1z4+f2r4)+(k/2)sin(2phi)r2z2
thus the system can be either dissipative (exhibiting chaos) or conservative (exhibiting quasiperiodicity), sensitive dependent on the position in the phase space.
For a suitable range of initial conditions, trajectories converge to the (presumably) chaotic attractor or torus knot, while other initial conditions lead to a coexisting invariant torus, resulting in an apparent overlapping between the quasiperiodicity curve and the phase locking plateaus.
The attached figure reveals a (presumably) strange attractor (blue: forward integration; green: backward integration), coexisting with an invariant torus (red).
Wang Zhe
Moreover, in the same system, I have observed the period quadrupling in the Poincare section. The attached figure combines points before the bifurcation (black), and those after the bifurcation (red). With increased nonlinearity k, period sextupling, octupling... then become observable.
With increased nonlinearity k, period sextupling, octupling... then become observable, but I have yet to observe period doubling. Strange and sigh...
I was wondered if there is any theoretical explanation behind. Kindly point me out, please.
Kind wishes,
Wang Zhe
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