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Dan Lucas, Keele University, Stabilisation of exact coherent structures by time-delay feedback

October 19 | 3:00 pm - 4:00 pm EDT

Time-delayed feedback control, attributed to Pyragas (1992 Phys. Lett. 170), is a method known to stabilise periodic orbits in low dimensional chaotic dynamical systems. A system of the form dx/dt=f(x) has an additional term G(x(t)-x(t-T)) introduced where G is some ‘gain matrix’ and T a time delay. This form of the delay term is such that it will vanish for any orbit of period T, therefore making it also an orbit of the uncontrolled system. This non-invasive feature makes the method attractive for stabilising exact coherent structures in fluid turbulence. Here we begin by validating the method for the basic flow in Kolmogorov flow; a two-dimensional incompressible Navier-Stokes flow with a sinusoidal body force. The linear predictions for stabilisation are well captured by direct numerical simulation. By applying an adaptive method to adjust the streamwise translation of the delay, a travelling wave solution is able to be stabilised up to relatively high Reynolds number. We also find that by using the symmetries of the underlying nonlinear solution, several exact coherent structures may be stabilised with this method. This means such solutions can be obtained simply through time stepping a slightly modified set of equations, therefore circumventing the usual convergence algorithms. We finish by discussing perspectives on using this approach for stabilising periodic orbits and show some preliminary results stabilising travelling waves in pipe flow.

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October 19
3:00 pm - 4:00 pm EDT
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