Nonlinear Dynamical Regimes and Control of Turbulence through the Complex Ginzburg-Landau Equation

Abstract

The dynamical behavior of pulse and traveling hole in a one-dimensional system depending on the boundary conditions, obeying the complex Ginzburg-Landau (CGL) equation, is studied numerically using parameters near a subcritical bifurcation. In a spatially extended system, the criterion of Benjamin-Feir-Newell (BFN) instability near the weakly inverted bifurcation is established, and many types of regimes such as laminar regime, spatiotemporal regime, defect turbulence regimes, and so on are observed. In finite system by using the homogeneous boundary conditions, two types of regimes are detected mainly the convective and the absolute instability. The convectively unstable regime appears below the threshold of the parameter control, and beyond, the absolute regime is observed. Controlling such regimes remains a great challenge; many methods such as the nonlinear diffusion parameter control are used. The unstable traveling hole in the onedimensional cubic-quintic CGL equation may be effectively stabilized in the chaotic regime. In order to stabilize defect turbulence regimes, we use the global time-delay auto-synchronization control; we also use another method of control which consists in modifying the nonlinear diffusion term. Finally, we control the unstable regimes by adding the nonlinear gradient term to the system. We then notice that the chaotic system becomes stable under strong nonlinearity.

Investigation of auto-oscilational regimes of the system by dynamic nonlinearities

“…As a first approximation 𝑧 𝑝.𝑝 (𝑡) chooses a harmonic signal with a constant component (16). To find its parameters 𝐴 𝑧0 , 𝐴 𝑧1 and 𝜔 1 we use the first and second equations of system (17), the first equation of system (18) where 𝐴 𝑧𝑘𝑙 and 𝐴 𝑧𝑘(𝑙−1)harmonic amplitude values 𝑧(𝑡) respectively on l -th and (l -1)th iteration.…”

“…Using the first Lyapunov method, it was found that the oscillations found are stable, i.e., self-oscillations. Let's find the coefficients 𝑐 00 and 𝑐 11 of the NDP link, described by (4), providing obtaining in the system of Figure 1 self-oscillations in the form (16) with given parameters 𝐴 𝑧0 =3, and 𝐴 𝑧1 =11. The frequency is determined during the synthesis.…”

“…For coefficient synthesis 𝑐 00 and 𝑐 11 according to the given parameters of self-oscillations, (6) of the balance of amplitudes and phases are used. For definiteness, let us consider the case of the presence of self-oscillations in the system in the form (16) provided that the components 𝑦 𝑠𝑘 , 𝑘 > 1, less 𝑦 𝑠0 , and 𝑦 𝑠1 . Then the system of equations for determining the coefficients 𝑐 00 and 𝑐 11 and unknown frequency ω1 according to the given parameters 𝐴 𝑧0 and 𝐴 𝑧1 is combined as (25):…”

Non-Hermitian spatiotemporal potentials for turbulence control in parabolic and fractional dispersion

“…Different attempts to control turbulence have been proposed, such as delayed global feedback, [3][4][5] local injections, 6,7 localized inhomogeneities, ?, 5 and a new approach based on a dynamical non-Hermitian potentials. 8,9 The non-Hermitian potential is intended to asymmetrically affect the excitation cascade responsible of turbulence, see Fig 1 . The present article presents a generalization of the non-Hermitian turbulence control, exploring the robustness of the turbulence control mechanism to systems with fractional dispersion.…”

Studying the Recurrent Sequence Generated by Power Function using QUATTRO-20

“…Trigonometric functions (sine and cosine) naturally represent an oscillatory behavior (vibrations, waves, etc) with periodic components. Phenomenon of recurrence allows describing the evolution of complex systems in several dynamical regimes where the characteristic regime parameters repeat or oscillate over time [1,2]. This is particularly relevant in audio processing, where signals often exhibit the repeating patterns.…”