Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

Abstract

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We introduce and describe periodic coherent structures of the CGLE, called Modulated Amplitude Waves (MAWs). MAWs of various period P occur naturally in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures occur which evolve toward defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.PACS numbers: 47.52.+j, 03.40.Kf, 05.45.+b, 47.54.+r Spatially extended systems can exhibit, when driven away from equilibrium, irregular behavior in space and time: this phenomenon is commonly referred to as spatio-temporal chaos [1]. The one-dimensional complex Ginzburg-Landau equation (CGLE):describes pattern formation near a Hopf bifurcation and has become a popular model to study spatiotemporal chaos [1][2][3][4][5][6][7][8][9][10][11][12][13]. As a function of c 1 and c 3 , the CGLE exhibits two qualitatively different spatiotemporal chaotic states known as phase chaos (when A is bounded away from zero) and defect chaos (when the phase of A displays singularities where A = 0). The transition from phase to defect chaos can either be hysteretic or continuous; in the former case, it is referred to as L 3 , in the latter as L 1 (Fig. 1). Despite intensive studies [5][6][7][8][9][10][11][12][13], the phenomenology of the CGLE and in particular its "phase"-diagram [5,7] are far from being understood. Moreover, it is under dispute whether the L 1 transition is sharp, and whether a pure phase-chaotic (i.e. defect-free) state can exist in the thermodynamic limit [9]. It is the purpose of this paper to elucidate these issues by presenting the mechanism which creates defects in transient phase chaotic states. Our analysis consists of four parts: (i) We describe a family of Modulated Amplitude Waves (MAWs), i.e., pulse-like coherent structures with a characteristic spatial period P . (ii) A bifurcation analysis of these MAWs reveals that their range of existence is limited by a saddle-node (SN) bifurcation. For all c 1 , c 3 within a certain range, we define P SN as the period of the MAW for which this bifurcation occurs. (iii) We show that for P > P SN , i.e., beyond the SN bifurcation, near-MAW structures display a nonlinear evolution to defects. It is found that, in phase chaos, near-MAWs with various P 's are created and annihilated perpetually. L1, L2 and L3 transitions (after [7]). Between the L2 and L3 curves, there is the hysteretic regime where either phase or defect chaos can occur; in the latter case, defects persist up to the L2 transition. Notice how the L1 and L3 transitions to defect chaos lie above our lower (P → ∞) bounds. Also shown are the SN locations for P = 20, 50.The transition to def...

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

“…Recently, this problem has been extensively studied by numerical analysis based on an equivalent ODE system [35]. Both ν = 0 and ν = 0 cases have been treated by Brusch and his collaborators in Refs.…”

“…Powerful theoretical and numerical analysis of such patterns have been reported recently, introducing the concept of modulated amplitude waves (MAWs) [35][36][37], that will be presented in section 4. The authors use a single CGL equation which has only three degrees of freedom, i.e., the real coefficients c 1 and c 2 and the mean wavenumber q [50].…”

“…This situation is due both to its relevance to many experimental systems, even far from threshold, and to the variety of spatio-temporal chaos regimes is exhibits. Theoretical studies and numerical simulations have indeed revealed regimes of phase turbulence [30,31], amplitude turbulence [30] related to localized defects solutions such as the Bekki-Nozaki or homoclonic holes [32][33][34] and, more recently, modulated amplitude waves [35][36][37].…”

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

“…For ν above the diamonds regular states were observed after a transient phase chaotic dynamics but below the diamonds most initial conditions led to persistent spatio-temporal chaos [21]. L 1 denotes the lower bound for the occurrence of defect chaos in the thermodynamic limit as calculated in [27,28].…”

“…MAWs and wound-up phase chaos with ν > 0 can be observed between the dashed and the full curve. Defect chaos can occur only above the full curve [27,28] which denotes the saddle-node bifurcation of MAWs with ν = 0 and P → ∞. The vertical dot-dashed line indicates the cut of the parameter space at c 1 = 3.5 studied in this paper.…”