Modulational Stability of Travelling Waves in 2D Anisotropic Systems

Dangelmayr, Gerhard; Oprea, Iuliana

doi:10.1007/s00332-007-9009-3

Cited by 11 publications

(8 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An important question is which of the observed behaviour is found in a spatiotemporal setting, when the normal form is extended to the system of globally coupled Ginzburg Landau equations introduced in [32] and investigated to some extent in [22,33,36]. A preliminary study for fixed values of the parameters in region A of Figure 2(c) [22] showed that the Ginzburg Landau dynamics depends on the initial conditions.…”

Section: Discussionmentioning

confidence: 97%

“…When the spatial periodicity constraint is removed, the representation (2), with the z j considered time and space dependent, leads to a globally coupled system of PDEs of the Ginzburg Landau type, whose spatially uniform solutions are governed by (1) [32]. Analytical and numerical studies of these spatiotemporal equations are presented in [22,33,36]. In this article we do not consider the Ginzburg Landau system, but point out that the dynamics of (1) is also present in this spatiotemporal system, although the stability of the attractors of (1) in the spatiotemporal setting requires further investigation.…”

Section: The Normal Formmentioning

confidence: 99%

“…When the spatial periodicity is removed, the normal form has to be extended to a system of globally coupled partial differential equations of the Ginzburg Landau type [32,33], whose solutions represent slowly varying (in time and space) envelopes of the two pairs of counterpropagating travelling waves created at the instability. In this spatiotemporal setting, the solutions of the normal form are spatially uniform solutions of the Ginzburg Landau system.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Complex dynamics near a Hopf bifurcation with symmetry: A parameter study

Dangelmayr

2010

Dynamical Systems

View full text Add to dashboard Cite

The eight-dimensional normal form for a Hopf bifurcation with O(2) Â O(2)-symmetry is investigated for a range of parameters in which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is bounded. By using symmetry-adapted variables, the dimension of the phase space is reduced to four. In the reduced phase space, periodic solutions are revealed as fixed points and quasiperiodic solutions as periodic orbits. Two parameter regimes are identified: A region in which trajectories in the reduced phase space are attracted by fixed points and periodic orbits and bistability occurs, and a region with complex dynamics. The characteristics of the complex dynamics are chaos, intermittency and periodic windows terminating in period doubling cascades, and both periodic and chaotic attractors occur in symmetric and asymmetric forms. The phase space geometry is suggestive of a kind of in-out intermittency involving two invariant subspaces. By using an appropriate cross-section, a reduced description of the dynamics is given in terms of a three-dimensional Poincare´map. Plots of the iterates of this map indicate that they reside in a one-dimensional manifold. In the initial range of the region with complex dynamics, the bifurcation diagrams (Poincare´map iterates versus a parameter) show the typical characteristics of unimodal map families. The symmetry of attractors is diagnosed using normalized asymmetry measures calculated from the iterates of the Poincare´map.

show abstract

Section: Discussionmentioning

confidence: 97%

Section: The Normal Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Complex dynamics near a Hopf bifurcation with symmetry: A parameter study

Dangelmayr

2010

Dynamical Systems

View full text Add to dashboard Cite

show abstract

“…Numerical simulations for other sets of parameters indicate that the GCCGL equations ͑1͒ are a promising way to investigate also other types of complex spatiotemporal structures such as grain boundaries, worms, and other localized patterns. 19,36 Overall, ͑1͒ appears to be an appropriate phenomenological description of complex high-dimensional spatiotemporal phenomena that are characteristic of anisotropic systems.…”

Section: Discussionmentioning

confidence: 98%

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection

Oprea

Triandaf

Dangelmayr

et al. 2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.

show abstract

“…The dynamical equations for these envelopes follow from a weakly nonlinear, multiple scale analysis. We summarize here the basic procedure of this analysis and refer to [15,16,43] for details.…”

Section: The System Of Globally Coupled Complex Ginzburg Landau Equatmentioning

confidence: 99%

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

Oprea¹,

Dangelmayr²

2019

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

In this paper we investigate the transition from periodic solutions to spatiotemporal chaos in a system of four globally coupled Ginzburg Landau equations describing the dynamics of instabilities in the electroconvection of nematic liquid crystals, in the weakly nonlinear regime. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with O(2) × O(2) symmetry. Both the amplitude system and the normal form are studied theoretically and numerically for values of the parameters including experimentally measured values of the nematic liquid crystal Merck I52. Coexistence of low dimensional and extensive spatiotemporal chaotic patterns, as well as a temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, and a kind of spatiotemporal intermittency that is characteristic for anisotropic systems are identified and characterized. A low-dimensional model for the intermittent dynamics is obtained by perturbing the eight-dimensional normal form by imperfection terms that break a continuous translation symmetry.

show abstract

Modulational Stability of Travelling Waves in 2D Anisotropic Systems

Cited by 11 publications

References 39 publications

Complex dynamics near a Hopf bifurcation with symmetry: A parameter study

Complex dynamics near a Hopf bifurcation with symmetry: A parameter study

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

Contact Info

Product

Resources

About