Iuliana Oprea

A BIFURCATION STUDY OF WAVE PATTERNS FOR ELECTROCONVECTION IN NEMATIC LIQUID CRYSTALS

We present the results of a bifurcation analysis of electroconvection in a planar layer of nematic liquid crystals, based on the recently introduced weak electrolyte model, which is an extension of the standard model to an electrodiffusion model with two active ion species. We show numerically that in certain regions of the space of material parameters a primary instability involving four oblique traveling rolls can occur. Near threshold the model equations are reduced to a system of normal form ODEs that admits six distinct basic wave patterns, and allows to classify the stability of these waves in terms of five nonlinear coefficients. For parameters matched to 152 and MBBA I, the stable wave patterns are traveling rolls and alternating waves. Approaches towards a refined stability analysis based on an extension of the ODE normal form to a system of globally coupled Ginzburg Landau equations are briefly discussed.

Dynamics and Bifurcation of Patterns in Dissipative Systems

Instabilities, Bifurcation, and the Role of Symmetry: Symmetry and Pattern Formation in the Visual Cortex (M Golubitsky et al.) Validity of the Ginzburg-Landau Approximation in Pattern Forming Systems with Time Periodic Forcing (N Breindl et al.) Pattern Formation on a Sphere (P C Matthews) Convergence Properties of Fourier Mode Representations of Quasipatterns (A M Rucklidge) Localized Patterns, Waves, and Weak Turbulence: Phase Diffusion and Weak Turbulence (J Lega) Pattern Formation and Parametric Resonance (D Armbruster & T-C Jo) Rogue Waves and the Benjamin-Feir Instability (C M Schober) Modelling and Characterization of Spatio-Temporal Complexity: A Finite-Dimensional Mechanism Responsible for Bursts in Fluid Mechanics (E Knobloch) Biological Lattice Gas Models (M S Alber et al.) Characterizations of Far from Equilibrium Structures Using Their Contours (G Nathan and G H Gunaratne) Internal Dynamics of Intermittency (R Sturman & P Ashwin) and other articles.

Structurally Stable Heteroclinic Cycles and the Dynamo Dynamics

Heteroclinic cycles, i.e. trajectories that connect a finite number of saddle points of a dynamical system until they eventually come back to the same saddle point, are structurally unstable. They occur as bifurcation phenomena. However it has been shown that additional structure in the dynamical systems may lead to structurally stable behavior of these cycles. This is typically the case for Hamiltonian systems where it has been well known for a long time. In addition, symmetry in the equations will also force heteroclinic cycles to be structurally stable. This fundamentally is accomplished by the fact that symmetric systems will have invariant subspaces. Hence a connection between two saddles will become structurally stable if the restriction of one of the saddles to an invariant subspace leads to a sink in that subspace and hence the restriction of the flow to the invariant subspace may generate a saddle — sink connection. For a bibliography on the subject see [8] in the present volume and for a comprehensive introduction, see [6]. The prototypical example has been studied by Busse et al [4] and analysed as a robust heteroclinic cycle by Guckenheimer and Holmes [7]. Figure 1 illustrates the case: We consider a 3-d system where all coordinate planes and all coordinate axes are invariant subspaces. This can for instance be obtained for a system that has the symmetry group generated by reflections through the planes of coordinates and by cyclic permutation of these axes of coordinates. Now, assume that there exists a saddle on a coordinate axis with a 2-d stable manifold and a 1-d unstable manifold. By the permutation symmetry we will have saddles on each of the axes and if there is a heteroclinic orbit connecting 2 saddles in one invariant subspace there will be a whole cycle of these orbits connecting a saddle back to itself (see figure 1). In some parameter regimes the cycle is attracting. A time series for any of the three variables will show the variable to level off at a particular value until it transits to another value in a very short time where it will stay again for a long time etc. Upon addition of noise a“stochastic limit cycle” is created, whereby the transition times between saddles is exponentially distributed and unlike an attracting heteroclinic cycle without noise, there exists a finite mean period [11]. This dynamical behavior of relatively long quiescent behavior randomly followed by a quick transition to another long quiescent behavior makes this an attractive model for the behavior of magnetic reversals. The following sections will discuss our program to flesh out this model with more and more concrete physical details.

Storing cycles in Hopfield-type networks with pseudoinverse learning rule: Admissibility and network topology

Cyclic patterns of neuronal activity are ubiquitous in animal nervous systems, and partially responsible for generating and controlling rhythmic movements such as locomotion, respiration, swallowing and so on. Clarifying the role of the network connectivities for generating cyclic patterns is fundamental for understanding the generation of rhythmic movements. In this paper, the storage of binary cycles in Hopfield-type and other neural networks is investigated. We call a cycle defined by a binary matrix Σ admissible if a connectivity matrix satisfying the cycles transition conditions exists, and if so construct it using the pseudoinverse learning rule. Our main focus is on the structural features of admissible cycles and the topology of the corresponding networks. We show that Σ is admissible if and only if its discrete Fourier transform contains exactly r=rank(Σ) nonzero columns. Based on the decomposition of the rows of Σ into disjoint subsets corresponding to loops, where a loop is defined by the set of all cyclic permutations of a row, cycles are classified as simple cycles, and separable or inseparable composite cycles. Simple cycles contain rows from one loop only, and the network topology is a feedforward chain with feedback to one neuron if the loop-vectors in Σ are cyclic permutations of each other. For special cases this topology simplifies to a ring with only one feedback. Composite cycles contain rows from at least two disjoint loops, and the neurons corresponding to the loop-vectors in Σ from the same loop are identified with a cluster. Networks constructed from separable composite cycles decompose into completely isolated clusters. For inseparable composite cycles at least two clusters are connected, and the cluster-connectivity is related to the intersections of the spaces spanned by the loop-vectors of the clusters. Simulations showing successfully retrieved cycles in continuous-time Hopfield-type networks and in networks of spiking neurons exhibiting up-down states are presented.

Modulational Stability of Travelling Waves in 2D Anisotropic Systems

Abstract The modulational stability of travelling waves in 2D anisotropic systems is investigated. We consider normal travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for two envelopes of left- and right-travelling waves, and oblique travelling waves, which are described by solutions of a globally coupled Ginzburg–Landau system for four envelopes associated with two counterpropagating pairs of travelling waves in two oblique directions. The Eckhaus stability boundary for these waves in the plane of wave numbers is computed from the linearized Ginzburg–Landau systems. We identify longitudinal long and finite wavelength instabilities as well as transverse long wavelength instabilities. The results of the stability calculations are confirmed through numerical simulations. In these simulations we observe a rich variety of behaviors, including defect chaos, elongated localized structures superimposed to travelling waves, and moving grain boundaries separating travelling waves in different oblique directions. The stability classification is applied to a reaction–diffusion system and to the weak electrolyte model for electroconvection in nematic liquid crystals.

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

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.

Parametrically forced pattern formation

Pattern formation in a nonlinear damped Mathieu-type partial differential equation defined on one space variable is analyzed. A bifurcation analysis of an averaged equation is performed and compared to full numerical simulations. Parametric resonance leads to periodically varying patterns whose spatial structure is determined by amplitude and detuning of the periodic forcing. At onset, patterns appear subcritically and attractor crowding is observed for large detuning. The evolution of patterns under the increase of the forcing amplitude is studied. It is found that spatially homogeneous and temporally periodic solutions occur for all detuning at a certain amplitude of the forcing. Although the system is dissipative, spatial solitons are found representing domain walls creating a phase jump of the solutions. Qualitative comparisons with experiments in vertically vibrating granular media are made. (c) 2001 American Institute of Physics.

Steady State–Hopf Mode Interactions at the Onset of Electroconvection in the Nematic Liquid Crystal Phase V

We report on a new mode interaction found in electroconvection experiments on the nematic liquid crystal mixture Phase V in planar geometry. The mode interaction (codimension two) point occurs at a critical value of the frequency of the driving AC voltage. For frequencies below this value the primary pattern-forming instability at the onset voltage is an oblique stationary instability involving oblique rolls, and above this value it is an oscillatory instability giving rise to normal traveling rolls (oriented perpendicular to and traveling in the director direction). The transition has been confirmed by measuring the roll angle and the dominant frequency of the time series, as both quantities exhibit a discontinuous jump across zero when the AC frequency is varied near threshold. The globally coupled system of Ginzburg–Landau equations that qualitatively describe this mode interaction is constructed, and the resulting normal form, in which slow spatial variations of the mode amplitudes are ignored, is analyzed. This analysis shows that the Ginzburg–Landau system provides the adequate theoretical description for the experimentally observed phenomenon. The experimentally observed patterns at and higher above the onset allow us to narrow down the range of the parameters in the normal form.

INTERMITTENCY AND CHAOS NEAR HOPF BIFURCATION WITH BROKEN 𝕆(2) × 𝕆(2) SYMMETRY

The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2) symmetry is perturbed by imperfection terms that break a continuous translation symmetry. The parameters of the fully symmetric normal form are fixed to values for which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is attracted by a chaotic attractor resulting from a period doubling cascade of periodic orbits. By using symmetry-adapted variables, the dimension of the phase space of the normal form is reduced to four and the dimension of the perturbed normal form is reduced to five. In the reduced phase space, periodic solutions are revealed as fixed points, and quasiperiodic solutions as periodic orbits. For the perturbed normal form, parameter regimes with different types of chaotic dynamics are identified when the imperfection parameter is varied. The characteristics of this complex dynamics are symmetry breaking and increasing, various period doubling cascades, intermittency and crises, and switching between symmetry-conjugated chaotic saddles. In particular, the perturbed system serves as a low dimensional model for the complicated switching dynamics found in simulations of the globally coupled system of Ginzburg–Landau equations extending the 𝕆(2) × 𝕆(2)-symmetric normal form to account for spatial modulations. In addition, this system can be considered as a low dimensional model for the dynamics of perturbed waves in anisotropic systems with imperfect geometries due to the presence of sidewalls.