Fig. 1 Phase portrait of the unforced Duffing oscillator. The red dots denote the three fixed points admitted by the system. The blue (resp. orange) thick line depicts the stable (resp. unstable) manifold of the saddle point located at the origin. Gray lines highlight a few trajectories exhibited ... More

Fig. 2 Strange attractor for the Rössler system with parameters a  =   0.1, b  =   0.1 and c  =   14. Colored lines depict a τ 1 (red) and τ 2 (blue) unstable periodic orbit. More

Fig. 3 Illustration of the optimal perturbation analysis for the plane Couette flow at Re   =   300 for streamwise wavenumber α  = 0: ( a ) schematic of the flow, ( b ) optimal gain curve for different spanwise wavenumbers β , ( c ) optimal perturbation in-plane velocity ( v , w ), and ( d ) op... More

Fig. 4 Eigenvalue patterns associated with the standard bifurcations encountered for fixed points ( a ) and ( b ),and for limit cycles ( c )–( e ). In each case, the shaded region indicates the stable part of the spectrum (i.e., the lower complex half-plane for fixed-point stability, and the unit ... More

Fig. 5 Time to solution (in CPU minutes) versus different pairs of Krylov basis size m and integration time τ for fixed point computation: ( a ) base flow of 2D flow past a circular cylinder at Re   =   80, where the initial condition is assumed to be the base flow at Re   =   40; ( b ) base f... More

Fig. 6 Residual deflation as a function of the total computation time (both computed in the same hardware) for time-delayed feedback (open gray circles) and Newton GMRES (filled black diamonds) for the harmonically forced jet presented in Sec. 4.2 . To obtain a continuous signal over time for the... More

Fig. 7 Time to solution (in CPU minutes) versus different pairs of Krylov base size m and integration time τ for the computation of eigenvalues: ( a ) 2D flow past a circular cylinder at Re   =   80 ( T = 1 / 0.125 ) and ( b ) 2D open-cavity at Re   =   4700 ( T = 1 / 1.676 ... More

Fig. 8 The flow in a two-dimensional annular thermosyphon: ( a ) temperature field (ϑ) of the bilateral symmetric fixed point and ( b ) real part of the temperature field ( ℜ ( ϑ ) ) of the unstable steady mode at Ra ≃ 499 More

Fig. 9 Eigenvalues for the destabilization of the fixed point for the thermal convection on an annular thermosyphon More

Fig. 10 The flow in a two-dimensional annular thermosyphon: ( a ) temperature field (ϑ) of an unstable steady convection cell and ( b ) real part of the temperature field ( ℜ ( ϑ ) ) of the unsteady unstable mode at Ra   =   16,100 More

Fig. 11 Eigenvalues of the flow of the steady convection cell. The dashed line represents the frequency f  =   7.67 from a DNS at Ra   =   16,100. More

Fig. 12 The harmonically forced jet: ( a ) vorticity component of the stabilized unstable limit cycle at supercritical Re   =   2000 and ( b ) spatial distribution of the subharmonic Floquet mode. Inflow forcing with 5% amplitude and S t D = 0.6 . More

Fig. 13 Evolution of Floquet multipliers of the harmonic forced axisymmetric jet. The single leading unstable eigenmode is associated with the vortex pairing mechanism and a period-doubling bifurcation. The almost superposed black star represents the reference value from Ref. [ 101 ]. More

Fig. 14 Radial velocity signal at x , r = ( 5 , 0.5 ) at subcritical Re   =   1370 (black) and supercritical Re   =   2000 (red): ( a ) signal evolution and ( b ) discrete Fourier transform (DFT) spectrum; ( c ) Phase portrait of the system in coordinates ( v , v ˙ , v ... More

Fig. 15 Streamwise component of the sensitivity to base flow modifications ∇ U λ of the leading eigenvalue λ at Reynolds number Re   =   50. Spatial distribution of ( a ) the growth rate sensitivity ∇ U σ and ( b ) the frequency sensitivity ∇ U ω . More

Fig. 16 Normalized modulus of the sensitivity to a steady force ∇ F λ of the leading eigenvalue λ at Reynolds number Re =   50. Spatial distribution of ( a ) the growth rate sensitivity ∇ F σ and ( b ) the frequency sensitivity ∇ F ω . More

Fig. 17 Modulus of the dominant Floquet multiplier as a function of the Reynolds number for the flow past a circular cylinder More

Fig. 18 The flow past a circular cylinder: semitransparent vorticity magnitude contours ( ω = 0.35 ) of the limit cycle at supercritical Re   =   190 superimposed with streamwise vorticity contours ( ω x = ± 0.18 ) of the real part of the unstable steady mode More

Fig. 19 Evolution of Floquet multipliers for the flow past two side-by-side cylinders. A pair of modes associated with the flip-flop instability leaves the unit cycle increasing its moduli (both growth rate and frequency) as a function of Re in a Neimark–Sacker bifurcation. The black star represen... More