Winding Number Instability in the Phase-Turbulence Regime of the Complex Ginzburg-Landau Equation

Abstract

We give a statistical characterization of states with nonzero winding number in the Phase Turbulence (PT) regime of the one-dimensional Complex Ginzburg-Landau equation. We find that states with winding number larger than a critical one are unstable, in the sense that they decay to states with smaller winding number. The transition from Phase to Defect Turbulence is interpreted as an ergodicity breaking transition which occurs when the range of stable winding numbers vanishes. Asymptotically stable states which are not spatio-temporally chaotic are described within the PT regime of nonzero winding number. PACS: 05.45.+b,82.40.Bj,05.70.Ln Spatio-temporal complex dynamics [1,2] is one of the present focus of research in nonlinear phenomena. Much effort has been devoted to the characterization of different dynamical phases and transitions between them for model equations such as the Complex Ginzburg-Landau Equation (CGLE) [1,[3][4][5][6][7][8][9][10][11]. One of the main questions driving these studies is whether concepts brought from statistical mechanics can be useful for describing complex nonequilibrium systems [3,12]. In this paper we give a characterization of the spatio-temporal configurations that occur in the Phase Turbulence (PT) regime of the CGLE (described below), for a finite system, in terms of a global wavenumber. This quantity plays the role of an order parameter classifying different phases. We show that in the PT regime there is an instability such that a conservation law for the global wavenumber occurs only for wavenumbers within a finite range that depends on the point in parameter space. Our study is statistical in the sense that averages over ensembles of initial conditions are used. Our results allow a characterization of the transition from PT to Defect or Amplitude Turbulence (DT) (another known dynamical regime of the CGLE) as the line in parameter space in which the range of conserved global wavenumbers shrinks to zero.The CGLE is an amplitude equation for a complex field A(x, t) describing universal features of the dynamics of extended systems near a Hopf bifurcation [1,7]. Figure 1). FIG. 1.Regions of the parameter [c1, c2]-space for the CGLE displaying different kinds of regular and chaotic behavior. Lines L1, L3 were determined in [3][4][5].

Nonlinear analysis of the Eckhaus instability: modulated amplitude waves and phase chaos with nonzero average phase gradient

“…For systems with periodic boundary conditions the average phase gradient of the whole system can only be changed, if a space-time defect occurs : |A(x, t)| drops to zero and ϕ x locally diverges at a defect. Persistent phase chaos with conserved ν ≤ ν M = 0 has been observed in numerical simulations of the CGLE (1) [20,21]. The maximum conserved average phase gradient ν M decreases as function of the coefficients c 1 , c 3 [20,21] and vanishes at the apparent transition from phase to defect chaos.…”

“…Here a(z) and φ(z) represent coherent structures [29]. Coherent structures have been studied extensively [21,[27][28][29][30] and play an important role in various regimes of the CGLE [4,[19][20][21][27][28][29][30][31].…”

“…modulated amplitude waves) coexist with the unstable plane waves. Numerical simulations [4,[19][20][21][22] provided examples of such stable modulated amplitude waves (MAWs). Stable MAWs have also been observed in experiments on surface-tension-driven hydrothermal waves [6] as well as on the Taylor-Dean system [11] and on internal waves excited by the Marangoni effect [12].…”

“…In particular, these states can be either chaotic or regular depending on the initial conditions and on the parameters c 1 , c 3 and ν. In this paper we will focus on MAWs with ν = 0 and on the dynamical regime associated to them, that is referred to as "wound-up" phase chaos [20]. It will be shown that MAWs and wound-up phase chaos exist between the dashed and the full curve in Fig.…”

“…Additional investigations of the existence domains of MAWs revealed qualitatively similar results for fixed c 1 = 0.4, 1.2, 2.1 and 5 and variable c 3 as well as for fixed c 3 = 0.83 and variable c 1 . Two of these choices were studied by numerical simulations in [20]. A similarity transformation maps coherent structures onto each other along curves (c 1 + c 3 )/(1 − c 1 c 3 ) = const in coefficient space [14].…”