Zig-zag chaos - a new spatiotemporal pattern in nonlinear dynamics
|
|
- Roland McBride
- 5 years ago
- Views:
Transcription
1 Zig-zag chaos - a new spatiotemporal pattern in nonlinear dynamics Iuliana Oprea 1, 2 1 Faculty of Mathematics, University of Bucharest, Romania 2 Department of Mathematics, Colorado State University, US Abstract In this paper we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, bifurcating 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 travelling wave envelopes. The Ginzburg Landau system is derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique travelling rolls. The chaotic nature of the pattern and the resemblance with the observed experimental spatiotemporal chaos in the nematic electroconvection of the nematic I52 are confirmed through a combination of techniques that include time series analysis of the amplitudes of the dominant modes, statistical descriptions and normal form theory, showing good agreement between theory and experiments. 1 Introduction Spatiotemporal chaos (STC), a deterministic pattern with unpredictable, irregular spatial and temporal variation, occurs in a wide variety of physical systems when many spatial degrees of freedom contribute to the dynamics (see [13] and references therein for a recent review). While there is nowadays a concise description of chaos in systems with few degrees of freedom, the problem of finding general approaches for the characterization of complex dynamics in high dimensional chaotic systems, as well as the existence of universal organizing principles for STC are still unsolved [1, 12, 13]. Due to the short characteristic times and large aspect ratios [2] nematic electroconvection is a well suited experimental system to find and study periodic and more complex spatiotemporal patterns like worms, defects and spatiotemporal chaos. It has been predicted by experimentalists [4] that, since STC evolves continuously from the uniform state, it is possible to use coupled complex Ginzburg Landau equations, one for each mode, with coefficients derived from the equations of motion for the nematic electroconvection, to elucidate and characterize its dynamics. A theoretical treatment is now available, via the weak electrolyte model(wem) [9, 5]. We found numerically[11], in the analysis of a system of four globally coupled complex Ginzburg Landau equations (GCCGLE) derived from WEM in [5], a complex spatiotemporal pattern very similar to the experimentally observed STC described in [3, 14], with complex spatiotemporal behavior preserving portions of structures similar to zig and zag families of travelling waves, connected by chaotic excursions in space and time. We will refer to it as zig-zag-chaos. In this paper we present a quantitative and qualitative analysis of the computed pattern. The paper is structured as follows. In Sec. 2 we present the derivation of the GCGLE; in Sec. 3 we describe the principal characteristics of the zig-zag-chaos. Sec. 4 is dedicated to conclusions and open questions. 1
2 2 The system of four globally coupled complex Ginzburg Landau equations The recently developed mathematical model for the nematic electroconvection, the weak electrolyte model (WEM) [9], consists of partial differential equations for the velocity field, the director, the electric potential and charge, derived from the Navier-Stokes equations for an anisotropic electrically conducting fluid, the conservation of charge, Poisson s law, and a partial differential equation for the conductivity. The WEM equations are extremely complicated, therefore a weakly nonlinear analysis near the onset is particularly useful. In our previous work [5, 10], we have analyzed stable basic wave patterns predicted by the WEM near a Hopf bifurcation of the basic state. In our Ginzburg-Landau analysis of the WEM equations, the complex amplitudes of these waves are considered as slowly varying wave envelopes A i (x i, t), i = 1, 2, 3, 4, that modulate the travelling waves in the four oblique directions. These envelopes depend on characteristic wave variables x 1 = (x +, y + ), x 2 = (x, y + ), x 3 = (x, y ), x 4 = (x +, y ), where x ± = x ± v x t, y ± = y ± v y t with (v x, v y ) the critical group velocities. The rescaled form of the dynamical equations for the envelopes results in a system of four globally coupled complex Ginzburg-Landau equations (GCCGLE) [6] for the envelopes A i. These equations are formulated in terms of slow variables ξ ± = εx ±, η ± = εy ±, τ = ε 2 t, where the small parameter ε 2 measures the distance of the bifurcation parameter from the onset value. The equation for A 1 has the form A 1t (x +, y +, t) = {a 0 + D( x+, y+ ) + a 1 A 1 (x +, y +, t) 2 + a 2 < A 2 (s, y +, t) 2 > + a 3 < A 3 (x + + s, y + + s, t) 2 + a 4 < A 4 (x +, s, t) 2 >}A 1 (x +, y +, t) + (1) a 5 < A 2 (x + + s, y +, t)a 3 (x + + s, y + + s, t)a 4 (x +, y + + s, t) >, where D is a second order differential operator, D( x+, y+ ) = D 1 x D 2 x+ y+ + D 3 y 2 +, and the brackets denote averages over s. The equations for A 2, A 3, A 4 follow from Eq. 1 by reflection operations, see [5, 10] for details. The linear and nonlinear coefficients occurring in (1) are computed numerically from the underlying WEM equations for various choices of the parameters. To any solution of the GCCGLE we associate a spatiotemporal pattern by setting u(x, t) = Re{A 1 (x +, y +, t)e i(ω ct+p c x+q c y) + A 2 (x, y +, t)e i(ω ct p c x+q c y) + A 3 (x, y, t)e i(ωct pcx qcy) + A 4 (x +, y, t)e i(ωct+pcx qcy) }. (2) If spatial variations are ignored, the globally coupled system for the A j reduces to the normal form for a Hopf bifurcation with O(2) O(2) symmetry, that has six basic solutions corresponding to six basic wave patterns: travelling (TW) and standing waves, two types of travelling rectangles, standing rectangles, and alternating waves, which alternate periodically between differently oriented standing waves. Each of these waves can occur with different orientations, for example, a TW can propagate in any of the four directions (±p c, ±q c ), and their stability in the ODE setting is classified in terms of the nonlinear coefficients a j (see [6]). The basic periodic solutions of the above normal form induce wave solutions of the original system (1) with critical wave numbers (p c, q c ). The globally coupled system of complex Ginzburg Landau equations (1) allows to extend these solutions to families of wave solutions with nearby critical wave numbers (p c + εp, q c + εq). 3 Characteristics of the zig-zag chaos The first experiments on zig-zag spatiotemporal chaos at the onset, in the weakly nonlinear regime, for the nematic liquid crystal I52, have been reported by Dennin, Cannel and Ahlers in [3]. Similar 2
3 patterns have been observed in the nematic electroconvection for the nematic I52 in [14]. The zigzag spatiotemporal chaos observed in [3] consists of a superposition of four degenerate modes, namely right - and left- relatively fast travelling waves with wave vectors at angles Θ and π Θ with respect to the director (the average molecular alignment). Several statistics tools have been used in [3, 4] in the analysis of the demodulated video images to reveal the chaotic aspects of the observed pattern. The authors have suggested that, since STC evolves continuously from the uniform state, in the weakly nonlinear regime, an appropriate system of four globally coupled Ginzburg Landau equations which describe the dynamics of the envelopes of the four modes could be derived from the equations of motion, allowing quantitative and qualitative comparison between theory and experiment. They are available now in the form of the four GCCGLE described in the previous section. In the numerical simulation of the four GCCGLE (2) we found numerically, for various sets of parameters, different types of STC and other localized structures evolving at the onset. Among them we found a complex spatiotemporal pattern very similar to the experimentally observed STC described in [3, 14], with complex spatiotemporal behavior preserving portions of structures similar to left- and right- families of travelling waves, connected by chaotic excursions in space and time, which we called the zig-zag chaos. Details of the computation and values of the nonlinear coefficients a i, 1 i 5 can be found in [11]. One of the most striking characteristics of this pattern, which can be clearly seen in a movie, is an alternation between phases of irregular, chaotic behavior and phases of regular behavior. This alternation is illustrated in the three pattern snapshots of Figure 1. The pattern shown in Figure 1a was recorded at t = 2700, which is during a chaotic phase. Here we can see an irregular distribution of zig and zag waves, as well as transitions between them, and regions of low and high intensity. The patterns shown in Figures 1b,c, recorded at t = 3120 and t = 3127, occur during a regular phase. During these phases the dynamics is approximately that of an alternating wave that extends throughout the spatial domain, which is clearly apparent in the snapshots. The time series of the three most energetic mode amplitudes a j (m, n, t) of the spectral decomposition of unknowns in (2) are shown in Figure 2. The sharp downward peaks of the dominant central mode, combined with upward peaks of one or both of the other modes, signify the chaotic phases. To confirm the chaotic dynamics we have computed the Lyapunov exponents of these time series using a three dimensional embedding technique[11]. All time series turned out to have one positive and two negative Lyapunov exponents. Although the average energy content of the non central mode amplitudes is small, they still have a significant effect on the shapes of the pattern during the irregular phases. The envelope patterns A j (x, y, t) in physical space show an interesting and aesthetic spatiotemporal variation on the slow time (T/ε 2 ) and spatial (λ x /ε,λ y /ε) scales. Most of the times two envelopes exhibit regular structures, whereas one or sometimes two of the other envelopes appear spatially irregular. Both the regular and irregular structures are changing permanently, and there are regular irregular transitions from one envelope to another. A snapshot of the A j corresponding to the pattern of Figure 1a is shown in Figure 3. Here we see that A 1, A 2 appear spatially irregular, whereas A 3 and A 4 exhibit a D 2 D 4 and an O(2) D 4 translation invariance, respectively. The alternation between regular and irregular phases strongly suggests that the dynamics of our pattern can be described as chaotic excursions from a torus. In general, a system with three continuous symmetries such as our 2D GCCGLE can have structurally stable invariant tori of dimension up to five, however, the dominance of the (0, 0) modes indicates that the torus is the AW torus that occurs already in the O(2) O(2) normal form. The nature of the excursions from the torus is not yet clear. One possibility is that the parameters are close to a homoclinic bifurcation and we see a higher dimensional version of Shilnikov type chaos. Another possibility is that there exists a structurally stable homoclinic orbit in higher dimensions that is not present in 3
4 the O(2) O(2) normal form. To clearly decide about the nature of the pattern dynamics requires further investigation, including parameter variations and numerical studies as well as the analysis of the modulational stability of the family of AW solutions. In the case of a homoclinic bifurcation we can expect that variations of the parameters may cause the disappearance of the intermittent regular phases and lead to a fully spatiotemporally chaotic dynamics. We note that, when comparing movies, the pattern dynamics of u(x, t) during the irregular phases appears quite similar to the near onset electroconvection pattern recorded recently at Kent State University for the nematic liquid crystal I52 [14], after applying the demodulation techniques described in [3] to the images. To be able to clearly diagnose spatiotemporal chaos and to distinguish a chaotic spatiotemporal pattern from other types of complex spatiotemporal behavior, we have also use the method of Karhunen-Loève (KL) decomposition[8] combined with time-series analysis of the resulting mode amplitudes. The spectrum of eigenvalues of this decomposition, as well as the amplitudes of these spatial modes, give insight in the character of the solution and can be used to diagnose chaotic behavior. Results of the characterization of the zig-zag chaos based on the KL decomposition, a long used technique in the signal analysis and processing, will be presented elsewhere[11]. Since the KL mode decomposition distinguishes between temporal behavior, reflected in the KL modes amplitudes, and spatial behavior reflected in the KL modes themselves, this method seems to fit very well the type of spatiotemporal pattern we consider here. 4 Discussion and conclusions The overall characterization of the zig-zag chaos we develop in this paper offer deeper insight into the spatiotemporal structure of the pattern than a simple statistical analysis as performed in the experimental case. Our example shows a richness of spatial structures essential to the dynamics, since 10 KL modes are needed to fully describe the solution[11]. The dynamics of the zig-zag chaos is complex and the correlation analysis shows that even if we are not in the presence of an extensive STC, however the temporal and spatial complexity are not independent of each other, as it might happen in systems where the system size is comparable with the typical correlation length. The availability of the amplitude equations (1) allows us to find new approaches in the characterization of the zig-zag chaos. A preliminary bifurcation analysis shows that two cases are possible: either the zig-zag chaos represents a structurally stable homoclinic orbit, with families of alternating waves as saddle, or the system is close to a homoclinic bifurcation generating chaotic dynamics, like a higher dimensional Shilnikov saddle-node bifurcation dynamics. This conjecture deserves further studies: an Eckhaus stability analysis for the structurally stable homoclinic orbit and/or the addition of higher order terms in (1) or its ODE associated normal form, combined with numerical simulations, might give a definite answer. Preliminary numerical computations for other sets of parameters indicate that the GCGL equations (1) seem to be a promising way to investigate more complex spatiotemporal structures as grain boundaries, worms and other localized patterns [10]. Overall, (1) seems to be an appropriate phenomenological description of highly dimensional complex spatiotemporal phenomena. An exciting task for future research is to compare the quantitative, experimentally determined statistical measures of these states with the calculations based on the equations of motion and the tools developed in this paper. Such work, relevant to real experimental data, is in process[7] 4
5 (a) (b) (c) Figure 1: Three pattern snapshots. a: Typical pattern observed in a chaotic phase. b,c: Two patterns in a time range in which the dynamics is approximately an alternating wave. In this and the other 2D patterns shown, maxima and minima are mapped to red and blue, respectively. 5
6 a 1 (m,n) a 2 (m,n) (a) t/t t/t (b) a 3 (m,n) t/t (c) a 4 (m,n) t/t (d) Figure 2: Time series a j (m, n, t) for the three modes (m, n) with the highest energy content. Blue: dominant mode (0, 0), red and green: second and third mode ( 1, 0) and (0, 1) for a 1, (0, 1) and (0, 1) for a 2, (0, 4) and (0, 4) for a 3, and ( 1, 0) and (1, 0) for a 4. As time unit we use the basic period T = 2π/ω c corresponding to 32 time steps. 6
7 Figure 3: Envelope snapshot A j (x, y), j = 1..4, associated with the pattern shown in Figure 1a. 7
8 References [1] M. Cross and P. C. Hohenberg. Spatiotemporal chaos. Science, 263:1569, [2] L. Kramer and W. Pesch. Electrohydrodynamic instabilities in nematic liquid crystals. In A. Buka and L. Kramer, editors, Pattern Formation in Liquid Crystals, page 221. Springer, Berlin, [3] M. Dennin, G. Ahlers, and D.S. Cannell. Spatiotemporal chaos in electroconvection. Science, 272:388, 1996; [4] Dennin, M., Cannell, D.S. and G. Ahlers, Patterns of electroconvection in a nematic liquid crystal, Phys. Rev. Let. 57, 1(1998), p [5] G. Dangelmayr and I. Oprea. A bifurcation study of wave patterns for electroconvection in nematic liquid crystals. Mol. Cryst. Liq. Cryst, 413:2441, [6] G. Dangelmayr and M. Wegelin. Hopf bifurcations in anisotropic systems. In M. Golubitsky, D. Luss, and S. Strogatz, editors, Pattern Formation in Continuous and Coupled Systems, pages Springer, IMA Vol. in Math. and Appl. 115, [7] I. Oprea, G. Dangelmayr, J. Gleesson, in preparation, 2006 [8] D. Armbruster, R. Heiland, and E. Kostelich. Kltool: A tool to analyze spatiotemporal complexity. Chaos, 4:421, [9] M. Treiber and L. Kramer. Mol. Cryst.Liq.Cryst., 261:311, [10] G. Dangelmayr and I. Oprea. Modulational Stability of Travelling Waves in 2D Anisotropic Systems. Submitted, [11] I. Oprea, I. Triandaf, G. Dangelmayr, I. Schwartz Qualitative and quantitative characterization of zig-zagspatiotemporal chaos in a system of amplitude equations for the nematic electroconvection Submitted Chaos, [12] Greenside, H., Spatiotemporal chaos in large systems, the scaling of complexity with size, arxiv:chao-dyn/ v1 [13] H.G.Schuster, W. Just, Deterministic chaos, an Introduction, Wiley, [14] J. Gleeson, G. Accaria, in preparation Acknowledgment This research has been supported by the National Science Foundation and the Romanian Ministry of Education and Research through CERES programm (Contract N0. C4-187/2004) 8
A Theory of Spatiotemporal Chaos: What s it mean, and how close are we?
A Theory of Spatiotemporal Chaos: What s it mean, and how close are we? Michael Dennin UC Irvine Department of Physics and Astronomy Funded by: NSF DMR9975497 Sloan Foundation Research Corporation Outline
More informationPatterns of electroconvection in a nematic liquid crystal
PHYSICAL REVIEW E VOLUME 57, NUMBER 1 JANUARY 1998 Patterns of electroconvection in a nematic liquid crystal Michael Dennin, 1 David S. Cannell, 2 and Guenter Ahlers 2 1 Department of Physics and Astronomy,
More informationSpatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal
Spatiotemporal chaos in electroconvection of a homeotropically aligned nematic liquid crystal Sheng-Qi Zhou and Guenter Ahlers Department of Physics and iqcd, University of California, Santa Barbara, California
More informationA Study in Pattern Formation: Electroconvection in Nematic Liquid Crystals
UNIVERSITY OF CALIFORNIA Santa Barbara A Study in Pattern Formation: Electroconvection in Nematic Liquid Crystals A Dissertation submitted in partial satisfaction of the requirements for the degree of
More informationSpatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts
Spatio-Temporal Chaos in Pattern-Forming Systems: Defects and Bursts with Santiago Madruga, MPIPKS Dresden Werner Pesch, U. Bayreuth Yuan-Nan Young, New Jersey Inst. Techn. DPG Frühjahrstagung 31.3.2006
More informationarxiv:chao-dyn/ v1 5 Aug 1998
Extensive Scaling and Nonuniformity of the Karhunen-Loève Decomposition for the Spiral-Defect Chaos State arxiv:chao-dyn/9808006v1 5 Aug 1998 Scott M. Zoldi, Jun Liu, Kapil M. S. Bajaj, Henry S. Greenside,
More informationSquare patterns and their dynamics in electroconvection
Square patterns and their dynamics in electroconvection Elżbieta Kochowska *, Nándor Éber, Wojciech Otowski #, Ágnes Buka Research Institute for Solid State Physics and Optics, Hungarian Academy of Sciences,
More informationSpatiotemporal Dynamics
KITP, October 2003: Rayleigh-Bénard Convection 1 Spatiotemporal Dynamics Mark Paul, Keng-Hwee Chiam, Janet Scheel and Michael Cross (Caltech) Henry Greenside and Anand Jayaraman (Duke) Paul Fischer (ANL)
More informationBIFURCATION TO TRAVELING WAVES IN THE CUBIC-QUINTIC COMPLEX GINZBURG LANDAU EQUATION
BIFURCATION TO TRAVELING WAVES IN THE CUBIC-QUINTIC COMPLEX GINZBURG LANDAU EQUATION JUNGHO PARK AND PHILIP STRZELECKI Abstract. We consider the 1-dimensional complex Ginzburg Landau equation(cgle) which
More informationExample of a Blue Sky Catastrophe
PUB:[SXG.TEMP]TRANS2913EL.PS 16-OCT-2001 11:08:53.21 SXG Page: 99 (1) Amer. Math. Soc. Transl. (2) Vol. 200, 2000 Example of a Blue Sky Catastrophe Nikolaĭ Gavrilov and Andrey Shilnikov To the memory of
More informationSelf-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model
Letter Forma, 15, 281 289, 2000 Self-Replication, Self-Destruction, and Spatio-Temporal Chaos in the Gray-Scott Model Yasumasa NISHIURA 1 * and Daishin UEYAMA 2 1 Laboratory of Nonlinear Studies and Computations,
More informationarxiv:physics/ v1 14 Nov 2006
Controlling the stability transfer between oppositely traveling waves and standing waves by inversion symmetry breaking perturbations A. Pinter, M. Lücke, and Ch. Hoffmann Institut für Theoretische Physik,
More informationThe Complex Ginzburg-Landau equation for beginners
The Complex Ginzburg-Landau equation for beginners W. van Saarloos Instituut Lorentz, University of Leiden P. O. Box 9506, 2300 RA Leiden The Netherlands This article appeared in Spatio-temporal Patterns
More informationAnomalous scaling on a spatiotemporally chaotic attractor
Anomalous scaling on a spatiotemporally chaotic attractor Ralf W. Wittenberg* and Ka-Fai Poon Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6 Received 28 June
More informationPhase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos
Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 682 688 c International Academic Publishers Vol. 35, No. 6, June 15, 2001 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional
More informationIm(v2) Im(v3) Re(v2)
. (a)..1. Im(v1) Im(v2) Im(v3)....... Re(v1) Re(v2).1.1.1 Re(v3) (b) y x Figure 24: (a) Temporal evolution of v 1, v 2 and v 3 for Fluorinert/silicone oil, Case (b) of Table, and 2 =,3:2. (b) Spatial evolution
More informationPattern Formation and Chaos
Developments in Experimental Pattern Formation - Isaac Newton Institute, 2005 1 Pattern Formation and Chaos Insights from Large Scale Numerical Simulations of Rayleigh-Bénard Convection Collaborators:
More informationB5.6 Nonlinear Systems
B5.6 Nonlinear Systems 5. Global Bifurcations, Homoclinic chaos, Melnikov s method Alain Goriely 2018 Mathematical Institute, University of Oxford Table of contents 1. Motivation 1.1 The problem 1.2 A
More informationOnset of electroconvection of homeotropically aligned nematic liquid crystals
Onset of electroconvection of homeotropically aligned nematic liquid crystals Sheng-Qi Zhou, 1 Nándor Éber, 2 Ágnes Buka, 2 Werner Pesch, 3 and Guenter Ahlers 1 1 Department of Physics and iqcd, University
More informationExistence of Secondary Bifurcations or Isolas for PDEs
Existence of Secondary Bifurcations or Isolas for PDEs Marcio Gameiro Jean-Philippe Lessard Abstract In this paper, we introduce a method to conclude about the existence of secondary bifurcations or isolas
More informationRayleigh-Bénard convection in a homeotropically aligned nematic liquid crystal
PHYSICAL REVIEW E VOLUME 58, NUMBER 5 NOVEMBER 1998 Rayleigh-Bénard convection in a homeotropically aligned nematic liquid crystal Leif Thomas, 1 Werner Pesch, 2 and Guenter Ahlers 1 1 Department of Physics
More informationThe influence of noise on two- and three-frequency quasi-periodicity in a simple model system
arxiv:1712.06011v1 [nlin.cd] 16 Dec 2017 The influence of noise on two- and three-frequency quasi-periodicity in a simple model system A.P. Kuznetsov, S.P. Kuznetsov and Yu.V. Sedova December 19, 2017
More informationCONVECTIVE PATTERNS IN LIQUID CRYSTALS DRIVEN BY ELECTRIC FIELD An overview of the onset behaviour
CONVECTIVE PATTERNS IN LIQUID CRYSTALS DRIVEN BY ELECTRIC FIELD An overview of the onset behaviour Agnes Buka Research Institute for Solid State Physics and Optics of the Hungarian Academy of Sciences
More informationConvection Patterns. Physics 221A, Spring 2017 Lectures: P. H. Diamond Notes: Jiacong Li
Convection Patterns Physics 1A, Spring 017 Lectures: P. H. Diamond Notes: Jiacong Li 1 Introduction In previous lectures, we have studied the basics of dynamics, which include dimensions of (strange) attractors,
More informationProblem Set Number 2, j/2.036j MIT (Fall 2014)
Problem Set Number 2, 18.385j/2.036j MIT (Fall 2014) Rodolfo R. Rosales (MIT, Math. Dept.,Cambridge, MA 02139) Due Mon., September 29, 2014. 1 Inverse function problem #01. Statement: Inverse function
More informationarxiv:patt-sol/ v1 14 Jan 1997
Travelling waves in electroconvection of the nematic Phase 5: A test of the weak electrolyte model Martin Treiber, Nándor Éber, Ágnes Buka, and Lorenz Kramer Universität Bayreuth, Theoretische Physik II,
More informationPhase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265 269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
More informationSpatiotemporal Chaos in Rayleigh-Bénard Convection
Spatiotemporal Chaos in Rayleigh-Bénard Convection Michael Cross California Institute of Technology Beijing Normal University June 2006 Janet Scheel, Keng-Hwee Chiam, Mark Paul Henry Greenside, Anand Jayaraman
More informationExperiments with Rayleigh-Bénard convection
Experiments with Rayleigh-Bénard convection Guenter Ahlers Department of Physics and IQUEST, University of California Santa Barbara CA 93106 USA e-mail: guenter@physics.ucsb.edu Summary. After a brief
More informationTime-periodic forcing of Turing patterns in the Brusselator model
Time-periodic forcing of Turing patterns in the Brusselator model B. Peña and C. Pérez García Instituto de Física. Universidad de Navarra, Irunlarrea, 1. 31008-Pamplona, Spain Abstract Experiments on temporal
More informationSymmetry Properties of Confined Convective States
Symmetry Properties of Confined Convective States John Guckenheimer Cornell University 1 Introduction This paper is a commentary on the experimental observation observations of Bensimon et al. [1] of convection
More informationSPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE. Itishree Priyadarshini. Prof. Biplab Ganguli
SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to
More informationIsotropic and anisotropic electroconvection
Physics Reports 448 (2007) 115 132 www.elsevier.com/locate/physrep Isotropic and anisotropic electroconvection Ágnes Buka a,, Nándor Éber a, Werner Pesch b, Lorenz Kramer b a Research Institute for Solid
More informationIntroduction LECTURE 1
LECTURE 1 Introduction The source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly great generality is in
More informationStrongly asymmetric soliton explosions
PHYSICAL REVIEW E 70, 036613 (2004) Strongly asymmetric soliton explosions Nail Akhmediev Optical Sciences Group, Research School of Physical Sciences and Engineering, The Australian National University,
More informationWinterbottom, David Mark (2006) Pattern formation with a conservation law. PhD thesis, University of Nottingham.
Winterbottom, David Mark (26) Pattern formation with a conservation law. PhD thesis, University of Nottingham. Access from the University of Nottingham repository: http://eprints.nottingham.ac.uk/118/1/thesis_-_dm_winterbottom.pdf
More informationStatistics of defect trajectories in spatio-temporal chaos in inclined layer convection and the complex Ginzburg Landau equation
CHAOS VOLUME 14, NUMBER 3 SEPTEMBER 2004 Statistics of defect trajectories in spatio-temporal chaos in inclined layer convection and the complex Ginzburg Landau equation Cristián Huepe and Hermann Riecke
More informationInterfacial waves in steady and oscillatory, two-layer Couette flows
Interfacial waves in steady and oscillatory, two-layer Couette flows M. J. McCready Department of Chemical Engineering University of Notre Dame Notre Dame, IN 46556 Page 1 Acknowledgments Students: M.
More informationMathematical Problems in Liquid Crystals
Report on Research in Groups Mathematical Problems in Liquid Crystals August 15 - September 15, 2011 and June 1 - July 31, 2012 Organizers: Radu Ignat, Luc Nguyen, Valeriy Slastikov, Arghir Zarnescu Topics
More informationChapter 3. Gumowski-Mira Map. 3.1 Introduction
Chapter 3 Gumowski-Mira Map 3.1 Introduction Non linear recurrence relations model many real world systems and help in analysing their possible asymptotic behaviour as the parameters are varied [17]. Here
More informationFinal Report 011 Characterizing the Dynamics of Spat io-temporal Data
Final Report 11 Characterizing the Dynamics of Spat io-temporal Data PrincipaJ Investigator: Eric J. Kostelich Co-Principal Investigator: H. Dieter Armbruster DOE Award number DE-FG3-94ER25213 Department
More informationGeorgia Institute of Technology. Nonlinear Dynamics & Chaos Physics 4267/6268. Faraday Waves. One Dimensional Study
Georgia Institute of Technology Nonlinear Dynamics & Chaos Physics 4267/6268 Faraday Waves One Dimensional Study Juan Orphee, Paul Cardenas, Michael Lane, Dec 8, 2011 Presentation Outline 1) Introduction
More informationPattern formation in Nikolaevskiy s equation
Stephen Cox School of Mathematical Sciences, University of Nottingham Differential Equations and Applications Seminar 2007 with Paul Matthews, Nottingham Outline What is Nikolaevskiy s equation? Outline
More informationScaling laws for rotating Rayleigh-Bénard convection
PHYSICAL REVIEW E 72, 056315 2005 Scaling laws for rotating Rayleigh-Bénard convection J. D. Scheel* and M. C. Cross Department of Physics, California Institute of Technology 114-36, Pasadena, California
More informationarxiv:chao-dyn/ v1 12 Feb 1996
Spiral Waves in Chaotic Systems Andrei Goryachev and Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON M5S 1A1, Canada arxiv:chao-dyn/96014v1 12
More informationTorus Doubling Cascade in Problems with Symmetries
Proceedings of Institute of Mathematics of NAS of Ukraine 4, Vol., Part 3, 11 17 Torus Doubling Cascade in Problems with Symmetries Faridon AMDJADI School of Computing and Mathematical Sciences, Glasgow
More informationChaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 6 JUNE 2000 Chaotic and turbulent behavior of unstable one-dimensional nonlinear dispersive waves David Cai a) and David W. McLaughlin Courant Institute
More informationUniversity of California, Irvine. Dissertation. Doctor of Philosophy. in Physics. by Carina Kamaga
University of California, Irvine Domain Coarsening in Electroconvection Dynamics of Topological Defects in the Striped System Dissertation submitted in partial satisfaction of the requirements for the
More informationSymbolic dynamics and chaos in plane Couette flow
Dynamics of PDE, Vol.14, No.1, 79-85, 2017 Symbolic dynamics and chaos in plane Couette flow Y. Charles Li Communicated by Y. Charles Li, received December 25, 2016. Abstract. According to a recent theory
More informationRoute to chaos for a two-dimensional externally driven flow
PHYSICAL REVIEW E VOLUME 58, NUMBER 2 AUGUST 1998 Route to chaos for a two-dimensional externally driven flow R. Braun, 1 F. Feudel, 1,2 and P. Guzdar 2 1 Institut für Physik, Universität Potsdam, PF 601553,
More informationPattern Formation and Spatiotemporal Chaos
Pattern Formation and Spatiotemporal Chaos - Chennai, 2004 1 Pattern Formation and Spatiotemporal Chaos Insights from Large Scale Numerical Simulations of Rayleigh-Bénard convection Collaborators: Mark
More informationcomplicated [6]. In the presence of an anisotropy this degeneracy is lifted and Eq.(1) is appropriate. Examples from hydrodynamics are convection inst
Ordered defect chains in the 2D anisotropic complex Ginzburg-Landau equation Roland Faller and Lorenz Kramer Physikalisches Institut, Universitat Bayreuth, D-95440 Bayreuth, Germany November 21, 1997 Abstract.
More informationDynamics of a mass-spring-pendulum system with vastly different frequencies
Dynamics of a mass-spring-pendulum system with vastly different frequencies Hiba Sheheitli, hs497@cornell.edu Richard H. Rand, rhr2@cornell.edu Cornell University, Ithaca, NY, USA Abstract. We investigate
More informationThe world of the complex Ginzburg-Landau equation
The world of the complex Ginzburg-Landau equation Igor S. Aranson Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439 Lorenz Kramer Physikalisches Institut, University of Bayreuth,
More informationLecture 7: The Swift-Hohenberg equation in one spatial dimension
Lecture 7: The Swift-Hohenberg equation in one spatial dimension Edgar Knobloch: notes by Vamsi Krishna Chalamalla and Alban Sauret with substantial editing by Edgar Knobloch January 10, 2013 1 Introduction
More informationFlow patterns and nonlinear behavior of traveling waves in a convective binary fluid
PHYSICAL REVIE% A VOLUME 34, NUMBER 1 JULY 1986 Flow patterns and nonlinear behavior of traveling waves in a convective binary fluid Elisha Moses and Victor Steinberg Department of Nuclear Physics, Weizmann
More information7 Two-dimensional bifurcations
7 Two-dimensional bifurcations As in one-dimensional systems: fixed points may be created, destroyed, or change stability as parameters are varied (change of topological equivalence ). In addition closed
More informationPattern formation in the damped Nikolaevskiy equation
PHYSICAL REVIEW E 76, 56 7 Pattern formation in the damped Nikolaevskiy equation S. M. Cox and P. C. Matthews School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7
More informationMAS212 Assignment #2: The damped driven pendulum
MAS Assignment #: The damped driven pendulum Sam Dolan (January 8 Introduction In this assignment we study the motion of a rigid pendulum of length l and mass m, shown in Fig., using both analytical and
More information13.1 Ion Acoustic Soliton and Shock Wave
13 Nonlinear Waves In linear theory, the wave amplitude is assumed to be sufficiently small to ignore contributions of terms of second order and higher (ie, nonlinear terms) in wave amplitude In such a
More informationLOCALIZED PATTERNS OF THE CUBIC-QUINTIC SWIFT-HOHENBERG EQUATIONS WITH TWO SYMMETRY-BREAKING TERMS. Zhenxue Wei
Ann. of Appl. Math. 34:1(2018), 94-110 LOCALIZED PATTERNS OF THE CUBIC-QUINTIC SWIFT-HOHENBERG EQUATIONS WITH TWO SYMMETRY-BREAKING TERMS Yancong Xu, Tianzhu Lan (Dept. of Math., Hangzhou Normal University,
More informationPhysics 212: Statistical mechanics II Lecture XI
Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is
More informationLecture 2 Supplementary Notes: Derivation of the Phase Equation
Lecture 2 Supplementary Notes: Derivation of the Phase Equation Michael Cross, 25 Derivation from Amplitude Equation Near threshold the phase reduces to the phase of the complex amplitude, and the phase
More informationExperiments on Thermally Driven Convection
Experiments on Thermally Driven Convection Guenter Ahlers ABSTRACT Experimental results for thermally driven convection in a thin horizontal layer of a nematic liquid crystal heated from below or above
More informationTORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR
TORSION PENDULUM: THE MECHANICAL NONLINEAR OSCILLATOR Samo Lasič, Gorazd Planinšič,, Faculty of Mathematics and Physics University of Ljubljana, Slovenija Giacomo Torzo, Department of Physics, University
More informationA New Dynamic Phenomenon in Nonlinear Circuits: State-Space Analysis of Chaotic Beats
A New Dynamic Phenomenon in Nonlinear Circuits: State-Space Analysis of Chaotic Beats DONATO CAFAGNA, GIUSEPPE GRASSI Diparnto Ingegneria Innovazione Università di Lecce via Monteroni, 73 Lecce ITALY giuseppe.grassi}@unile.it
More informationTransition from the macrospin to chaotic behaviour by a spin-torque driven magnetization precession of a square nanoelement
Transition from the macrospin to chaotic behaviour by a spin-torque driven magnetization precession of a square nanoelement D. Berkov, N. Gorn Innovent e.v., Prüssingstr. 27B, D-07745, Jena, Germany (Dated:
More informationPulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach
PHYSICAL REVIEW E, VOLUME 63, 056602 Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach N. Akhmediev, 1 J. M.
More informationBIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs
BIFURCATION PHENOMENA Lecture 4: Bifurcations in n-dimensional ODEs Yuri A. Kuznetsov August, 2010 Contents 1. Solutions and orbits: equilibria cycles connecting orbits compact invariant manifolds strange
More informationMathematical Foundations of Neuroscience - Lecture 7. Bifurcations II.
Mathematical Foundations of Neuroscience - Lecture 7. Bifurcations II. Filip Piękniewski Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland Winter 2009/2010 Filip
More informationHow fast elements can affect slow dynamics
Physica D 180 (2003) 1 16 How fast elements can affect slow dynamics Koichi Fujimoto, Kunihiko Kaneko Department of Pure and Applied Sciences, Graduate school of Arts and Sciences, University of Tokyo,
More informationTopological Bifurcations of Knotted Tori in Quasiperiodically Driven Oscillators
Topological Bifurcations of Knotted Tori in Quasiperiodically Driven Oscillators Brian Spears with Andrew Szeri and Michael Hutchings University of California at Berkeley Caltech CDS Seminar October 24,
More information19 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, 2-7 SEPTEMBER 2007 NONLINEAR DYNAMICS IN PARAMETRIC SOUND GENERATION
9 th INTERNATIONAL CONGRESS ON ACOUSTICS MADRID, -7 SEPTEMBER 7 NONLINEAR DYNAMICS IN PARAMETRIC SOUND GENERATION PACS: 43.5.Ts, 43.5.+y V.J. Sánchez Morcillo, V. Espinosa, I. Pérez-Arjona and J. Redondo
More informationChapter 4. Transition towards chaos. 4.1 One-dimensional maps
Chapter 4 Transition towards chaos In this chapter we will study how successive bifurcations can lead to chaos when a parameter is tuned. It is not an extensive review : there exists a lot of different
More informationTakens embedding theorem for infinite-dimensional dynamical systems
Takens embedding theorem for infinite-dimensional dynamical systems James C. Robinson Mathematics Institute, University of Warwick, Coventry, CV4 7AL, U.K. E-mail: jcr@maths.warwick.ac.uk Abstract. Takens
More informationINTRODUCTION. Nonlinear Analysis: Modelling and Control, Vilnius, IMI, 1999, No 4, pp
Nonlinear Analysis: Modelling and Control, Vilnius, IMI, 1999, No 4, pp.113-118 WHY PATTERNS APPEAR SPONTANEOUSLY IN DISSIPATIVE SYSTEMS? K. Staliûnas Physikalisch-Technische Bundesanstalt, D-38023 Braunschweig,
More informationarxiv:chao-dyn/ v1 5 Mar 1996
Turbulence in Globally Coupled Maps M. G. Cosenza and A. Parravano Centro de Astrofísica Teórica, Facultad de Ciencias, Universidad de Los Andes, A. Postal 26 La Hechicera, Mérida 5251, Venezuela (To appear,
More informationDelay-induced chaos with multifractal attractor in a traffic flow model
EUROPHYSICS LETTERS 15 January 2001 Europhys. Lett., 57 (2), pp. 151 157 (2002) Delay-induced chaos with multifractal attractor in a traffic flow model L. A. Safonov 1,2, E. Tomer 1,V.V.Strygin 2, Y. Ashkenazy
More informationMonte Carlo Simulation of Ferroelectric Domain Structure: Electrostatic and Elastic Strain Energy Contributions
Monte Carlo Simulation of Ferroelectric Domain Structure: Electrostatic and Elastic Strain Energy Contributions B.G. Potter, Jr., B.A. Tuttle, and V. Tikare Sandia National Laboratories Albuquerque, NM
More informationOne dimensional Maps
Chapter 4 One dimensional Maps The ordinary differential equation studied in chapters 1-3 provide a close link to actual physical systems it is easy to believe these equations provide at least an approximate
More informationLooking Through the Vortex Glass
Looking Through the Vortex Glass Lorenz and the Complex Ginzburg-Landau Equation Igor Aronson It started in 1990 Project started in Lorenz Kramer s VW van on the way back from German Alps after unsuccessful
More informationBifurcations of Traveling Wave Solutions for a Generalized Camassa-Holm Equation
Computational and Applied Mathematics Journal 2017; 3(6): 52-59 http://www.aascit.org/journal/camj ISSN: 2381-1218 (Print); ISSN: 2381-1226 (Online) Bifurcations of Traveling Wave Solutions for a Generalized
More informationCHALMERS, GÖTEBORGS UNIVERSITET. EXAM for DYNAMICAL SYSTEMS. COURSE CODES: TIF 155, FIM770GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET EXAM for DYNAMICAL SYSTEMS COURSE CODES: TIF 155, FIM770GU, PhD Time: Place: Teachers: Allowed material: Not allowed: August 22, 2018, at 08 30 12 30 Johanneberg Jan Meibohm,
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationarxiv: v1 [cond-mat.stat-mech] 6 Mar 2008
CD2dBS-v2 Convergence dynamics of 2-dimensional isotropic and anisotropic Bak-Sneppen models Burhan Bakar and Ugur Tirnakli Department of Physics, Faculty of Science, Ege University, 35100 Izmir, Turkey
More informationFebruary 24, :13 Contribution to NDES2005 seehafer revised HIERARCHICAL MODELLING OF A FORCED ROBERTS DYNAMO
HIERARCHICAL MODELLING OF A FORCED ROBERTS DYNAMO REIK DONNER, FRED FEUDEL, NORBERT SEEHAFER Nonlinear Dynamics Group, University of Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany reik@agnld.uni-potsdam.de
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationStochastic differential equations in neuroscience
Stochastic differential equations in neuroscience Nils Berglund MAPMO, Orléans (CNRS, UMR 6628) http://www.univ-orleans.fr/mapmo/membres/berglund/ Barbara Gentz, Universität Bielefeld Damien Landon, MAPMO-Orléans
More informationComplex Patterns in a Simple System
Complex Patterns in a Simple System arxiv:patt-sol/9304003v1 17 Apr 1993 John E. Pearson Center for Nonlinear Studies Los Alamos National Laboratory February 4, 2008 Abstract Numerical simulations of a
More informationChapitre 4. Transition to chaos. 4.1 One-dimensional maps
Chapitre 4 Transition to chaos In this chapter we will study how successive bifurcations can lead to chaos when a parameter is tuned. It is not an extensive review : there exists a lot of different manners
More informationWHAT IS A CHAOTIC ATTRACTOR?
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties
More informationI seminari del giovedì. Transizione di fase in cristalli liquidi II: Ricostruzione d ordine in celle nematiche frustrate
Università di Pavia Dipartimento di Matematica F. Casorati 1/23 http://www-dimat.unipv.it I seminari del giovedì Transizione di fase in cristalli liquidi II: Ricostruzione d ordine in celle nematiche frustrate
More informationElectroconvection with and without the Carr-Helfrich effect in a series of nematics
electronic-liquid Crystal Communications January 23, 2004 Electroconvection with and without the Carr-Helfrich effect in a series of nematics Elżbieta Kochowska 1, Szilárd Németh 1, Gerhard Pelzl 2 and
More informationINTRODUCTION TO CHAOS THEORY T.R.RAMAMOHAN C-MMACS BANGALORE
INTRODUCTION TO CHAOS THEORY BY T.R.RAMAMOHAN C-MMACS BANGALORE -560037 SOME INTERESTING QUOTATIONS * PERHAPS THE NEXT GREAT ERA OF UNDERSTANDING WILL BE DETERMINING THE QUALITATIVE CONTENT OF EQUATIONS;
More information= w. These evolve with time yielding the
1 Analytical prediction and representation of chaos. Michail Zak a Jet Propulsion Laboratory California Institute of Technology, Pasadena, CA 91109, USA Abstract. 1. Introduction The concept of randomness
More informationarxiv: v1 [nlin.ao] 2 Mar 2018
Stable amplitude chimera states in a network of locally coupled Stuart-Landau oscillators K. Premalatha, V. K. Chandrasekar, 2 M. Senthilvelan, and M. Lakshmanan ) Centre for Nonlinear Dynamics, School
More informationInteraction of an Intense Electromagnetic Pulse with a Plasma
Interaction of an Intense Electromagnetic Pulse with a Plasma S. Poornakala Thesis Supervisor Prof. P. K. Kaw Research collaborators Prof. A. Sen & Dr.Amita Das. v B Force is negligible Electrons are non-relativistic
More informationFINITE TIME BLOW-UP FOR A DYADIC MODEL OF THE EULER EQUATIONS
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 357, Number 2, Pages 695 708 S 0002-9947(04)03532-9 Article electronically published on March 12, 2004 FINITE TIME BLOW-UP FOR A DYADIC MODEL OF
More informationTHREE DIMENSIONAL SYSTEMS. Lecture 6: The Lorenz Equations
THREE DIMENSIONAL SYSTEMS Lecture 6: The Lorenz Equations 6. The Lorenz (1963) Equations The Lorenz equations were originally derived by Saltzman (1962) as a minimalist model of thermal convection in a
More informationAPPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems
APPLIED PARTIM DIFFERENTIAL EQUATIONS with Fourier Series and Boundary Value Problems Fourth Edition Richard Haberman Department of Mathematics Southern Methodist University PEARSON Prentice Hall PEARSON
More information