Characterization of the Transition from Defect to Phase Turbulence

Egolf, David A.; Greenside, Henry

doi:10.1103/physrevlett.74.1751

Cited by 73 publications

(111 citation statements)

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“…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.…”

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confidence: 99%

“…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).…”

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confidence: 99%

“…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.…”

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Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

et al. 2000

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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...

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Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

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“…We limit our analysis to fixed c 1 = 3.5 since most previous numerical work has been done at this value [21,26]. The results will be presented as projections of the P direction onto the (c 3 , ν) plane as well as in cuts through the parameter space spanned by c 3 , ν and P .…”

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“…In addition, the CGLE exhibits two qualitatively different spatiotemporal chaotic states known as phase chaos (when the modulus of the field |A| is bounded away from zero) and defect chaos (when the phase of A displays singularities where |A| = 0) [23][24][25][26]. The subclass of MAWs with zero average phase gradient is important for understanding the transition from phase to defect chaos (see full curve in Fig.…”

Section: Introductionmentioning

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

Brusch

Torcini

Bär

2003

Physica D: Nonlinear Phenomena

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We analyze the Eckhaus instability of plane waves in the one-dimensional complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are quasiperiodic solutions of the CGLE that emerge near the Eckhaus instability of plane waves and cease to exist due to saddle-node bifurcations (SN). These MAWs can be characterized by their average phase gradient ν and by the spatial period P of the periodic amplitude modulation. A numerical bifurcation analysis reveals the existence and stability properties of MAWs with arbitrary ν and P . MAWs are found to be stable for large enough ν and intermediate values of P . For different parameter values they are unstable to splitting and attractive interaction between subsequent extrema of the amplitude. Defects form from perturbed plane waves for parameter values above the SN of the corresponding MAWs. The break-down of phase chaos with average phase gradient ν = 0 ("wound-up phase chaos") is thus related to these SNs. A lower bound for the break-down of wound-up phase chaos is given by the necessary presence of SNs and an upper bound by the absence of the splitting instability of MAWs.

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Dissipative structures and weak turbulence

Manneville

Chaos — The Interplay Between Stochastic and Deterministic Behaviour

391

494

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Characterization of the Transition from Defect to Phase Turbulence

Cited by 73 publications

References 15 publications

Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

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

Dissipative structures and weak turbulence

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