Controlling chaos by a modified straight-line stabilization method

By adjusting external control signal, rather than some available parameters of the system, we modify the straight-line stabilization method for stabilizing an unstable periodic orbit in a neighborhood of an unstable fixed point formulated by Ling Yang et al., and derive a more simple analytical expression of the external control signal adjustment. Our technique solves the problem that the unstable fixed point is independent of the system parameters, for which the original straight-line stabilization method is not suitable. The method is valid for controlling dissipative chaos, Hamiltonian chaos and hyperchaos, and may be most useful for the systems in which it may be difficult to find an accessible system parameter in some cases. The method is robust under the presence of weak external noise. Received 10 January 2001     

Spatial orders appearing at instabilities of synchronous chaos of spatiotemporal systems

Various spatial orders introduced by the instabilities of synchronous chaotic state of spatiotemporal systems are investigated by considering coupled map lattice and chaotic partial differential equation. In particular, the motions of on-off intermittent states at the onset of the instabilities are studied in detail. The chaotic desynchronized patterns can be described by a simple universal form, including three parts: the synchronous chaos; a spatially ordered pattern, determined by the unstable mode of the reference synchronous chaos; and on-off intermittency of the scale of this given pattern. Received 31 July 2002 / Received in final form 20 November 2002 Published online 31 December 2002     

Observations of deterministic chaos in financial time series by recurrence plots, can one control chaotic economy?

Several economical time series such as exchange rates US$/British Pound, USA Treasure Bonds rates and Warsaw Stock Index WIG have been investigated using the method of recurrence plots. The percentage of recurrence REC and the percentage of determinism DET have been calculated for the original and for shuffled data. We have found that in some cases the values of REC and DET parameters are about 20% lower for the surrogate data which indicates the presence of unstable periodical orbits in the considered data. A similar result has been obtained for the chaotic Lorenz model contaminated by noise. Our investigations suggest that real economical dynamics is a mixture of deterministic and stochastic chaos. We show how a simple chaotic economic model can be controlled by appropriate influence of time-delayed feedback. Received 13 October 2000     

Revisiting the role of correlation coefficient to distinguish chaos from noise

The correlation coefficient vs. prediction time profile has been widely used to distinguish chaos from noise. The correlation coefficient remains initially high, gradually decreasing as prediction time increases for chaos and remains low for all prediction time for noise. We here show that for some chaotic series with dominant embedded cyclical component(s), when modelled through a newly developed scheme of periodic decomposition, will yield high correlation coefficient even for long prediction time intervals, thus leading to a wrong assessment of inherent chaoticity. But if this profile of correlation coefficient vs. prediction horizon is compared with the profile obtained from the surrogate series, correct interpretations about the underlying dynamics are very much likely. Received 8 March 1999     

An integrable model in general relativity

Melnikov-method-based theoretical results are demonstrated concerning the relative effectiveness of any two weak excitations in suppressing homoclinic/heteroclinic chaos of a relevant class of dissipative, low-dimensional and non-autonomous systems for the main resonance between the chaos-inducing and chaos-suppressing excitations. General analytical expressions are derived from the analysis of generic Melnikov functions providing the boundaries of the regions as well as the enclosed area in the amplitude/initial phase plane of the chaos-suppressing excitation where homoclinic/heteroclinic chaos is inhibited. The relevance of the theoretical results on chaotic attractor elimination is confirmed by means of Lyapunov exponent calculations for a two-well Duffing oscillator. Received 21 May 2002 / Received in final form 13 September 2002 Published online 29 November 2002     

Coupling induced topological transition in a ring of chaotic oscillators

A ring of diffusively coupled R?ssler oscillators, which can develop the conventional rotating wave from high-dimensional chaos by increasing the coupling ɛ continuously is studied. The chaotic generator for the rotating wave emerges around ɛ = ɛ, where the topological transition induced by the coupling not only changes the topological structure of all the oscillators, which share a common strange attractor, but also changes them into being different from each other. Starting from this transition, infinitely long range temporal correlation and spatial order in the style of antiphase state are established gradually, which gives rise to the chaotic generator of the rotating wave. Received 15 March 2001     

Theoretical and experimental study of discrete behavior of Shilnikov chaos in a CO 2 laser

The discrete distribution of homoclinic orbits has been investigated numerically and experimentally in a CO₂ laser with feedback. The narrow chaotic ranges appear consequently when a laser parameter (bias voltage or feedback gain) changes exponentially. Up to six consecutive chaotic windows have been observed in the numerical simulation as well as in the experiments. Every subsequent increase in the number of loops in the upward spiral around the saddle focus is accompanied by the appearance of the corresponding chaotic window. The discrete character of homoclinic chaos is also demonstrated through bifurcation diagrams, eigenvalues of the fixed point, return maps, and return times of the return maps. Received 28 September 2000 and 27 October 2000     

Coarsening process in one-dimensional surface growth models

Surface growth models may give rise to instabilities with mound formation whose typical linear size L increases with time (coarsening process). In one dimensional systems coarsening is generally driven by an attractive interaction between domain walls or kinks. This picture applies to growth models for which the largest surface slope remains constant in time (corresponding to model B of dynamics): coarsening is known to be logarithmic in the absence of noise ( L(t) ∼ ln t) and to follow a power law ( L(t) ∼t ^1/3) when noise is present. If the surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard equation: here we study the late stages of coarsening through a linear stability analysis of the stationary periodic configurations and through a direct numerical integration. Analytical and numerical results agree with regard to the conclusion that steepening of mounds makes deterministic coarsening faster : if α is the exponent describing the steepening of the maximal slope M of mounds ( M ^α∼L) we find that L(t) ∼t ⁿ: n is equal to for 1≤α≤2 and it decreases from to for α≥2, according to n = α/(5α - 2). On the other side, the numerical solution of the corresponding stochastic equation clearly shows that in the presence of shot noise steepening of mounds makes coarsening slower than in model B: L(t) ∼t ^1/4, irrespectively of α. Finally, the presence of a symmetry breaking term is shown not to modify the coarsening law of model α = 1, both in the absence and in the presence of noise. Received 28 September 2001 and Received in final form 21 November 2001     

Effect of tangential interface motion on the viscous instability in fluid flow past flexible surfaces

The stability of linear shear flow of a Newtonian fluid past a flexible membrane is analysed in the limit of low Reynolds number as well as in the intermediate Reynolds number regime for two different membrane models. The objective of this paper is to demonstrate the importance of tangential motion in the membrane on the stability characteristics of the shear flow. The first model assumes the wall to be a “spring-backed” plate membrane, and the displacement of the wall is phenomenologically related in a linear manner to the change in the fluid stresses at the wall. In the second model, the membrane is assumed to be a two-dimensional compressible viscoelastic sheet of infinitesimal thickness, in which the constitutive relation for the shear stress contains an elastic part that depends on the local displacement field and a viscous component that depends on the local velocity in the membrane. The stability characteristics of the laminar flow in the limit of low are crucially dependent on the tangential motion in the membrane wall. In both cases, the flow is stable in the low Reynolds number limit in the absence of tangential motion in the membrane. However, the presence of tangential motion in the membrane destabilises the shear flow even in the absence of fluid inertia. In this case, the non-dimensional velocity (Λ_t) required for unstable fluctuations is proportional to the wavenumber k ( Λ _t∼k) in the plate membrane type of wall while it scales as k² in the viscoelastic membrane type of wall ( Λ _t∼k ²) in the limit k→ 0. The results of the low Reynolds number analysis are extended numerically to the intermediate Reynolds number regime for the case of a viscoelastic membrane. The numerical results show that for a given set of wall parameters, the flow is unstable only in a finite range of Reynolds number, and it is stable in the limit of large Reynolds number. Received 8 November 2000 and Received in final form 20 March 2001     

Bulk singularities at critical end points: a field-theory analysis

A class of continuum models with a critical end point is considered whose Hamiltonian [φ,ψ] involves two densities: a primary order-parameter field, φ, and a secondary (noncritical) one, ψ. Field-theoretic methods (renormalization group results in conjunction with functional methods) are used to give a systematic derivation of singularities occurring at critical end points. Specifically, the thermal singularity ∼ | t|^{2 - α} of the first-order line on which the disordered or ordered phase coexists with the noncritical spectator phase, and the coexistence singularity ∼ | t|^{1 - α} or ∼ | t|^β of the secondary density <ψ> are derived. It is clarified how the renormalization group (RG) scenario found in position-space RG calculations, in which the critical end point and the critical line are mapped onto two separate fixed points _CEP ^* and _λ ^*, translates into field theory. The critical RG eigenexponents of _CEP ^* and _λ ^* are shown to match. _CEP ^* is demonstrated to have a discontinuity eigenperturbation (with eigenvalue y = d), tangent to the unstable trajectory that emanates from _CEP ^* and leads to _λ ^*. The nature and origin of this eigenperturbation as well as the role redundant operators play are elucidated. The results validate that the critical behavior at the end point is the same as on the critical line. Received 18 January 2001     

Dynamic exponent in extremal models of pinning

The depinning transition of a front moving in a time-independent random potential is studied. The temporal development of the overall roughness w(L,t) of an initially flat front, , is the classical means to have access to the dynamic exponent. However, in the case of front propagation in quenched disorder via extremal dynamics, we show that the initial increase in front roughness implies an extra dependence over the system size which comes from the fact that the activity is essentially localized in a narrow region of space. We propose an analytic expression for the exponent and confirm this for different models (crack front propagation, Edwards-Wilkinson model in a quenched noise etc.). Received 27 August 1999     

Anomalous dynamical light scattering in soft glassy gels

We compute the dynamical structure factor S(q,τ) of an elastic medium where force dipoles appear at random in space and in time, due to “micro-collapses” of the structure. Various regimes are found, depending on the wave vector q and the collapse time θ. In an early time regime, the logarithm of the structure factor behaves as (qτ)^3/2, as predicted in (L. Cipelletti et al., Phys. Rev Lett. 84, 2275 (2000)) using heuristic arguments. However, in an intermediate-time regime we rather obtain a (qτ)^5/4 behaviour. Finally, the asymptotic long-time regime is found to behave as q ^3/2τ. We also give a plausible scenario for aging, in terms of a strain-dependent energy barrier for micro-collapses. The relaxation time is found to grow with the age t_w, quasi-exponentially at first, and then as t _w ^4/5 with logarithmic corrections. Received 15 April 2002     

Non-equilibrium behavior at a liquid-gas critical point

Second-order phase transitions in a non-equilibrium liquid-gas model with reversible mode couplings, i.e., model H for binary-fluid critical dynamics, are studied using dynamic field theory and the renormalization group. The system is driven out of equilibrium either by considering different values for the noise strengths in the Langevin equations describing the evolution of the dynamic variables (effectively placing these at different temperatures), or more generally by allowing for anisotropic noise strengths, i.e., by constraining the dynamics to be at different temperatures in d _|| - and d _⊥-dimensional subspaces, respectively. In the first, isotropic case, we find one infrared-stable and one unstable renormalization group fixed point. At the stable fixed point, detailed balance is dynamically restored, with the two noise strengths becoming asymptotically equal. The ensuing critical behavior is that of the standard equilibrium model H. At the novel unstable fixed point, the temperature ratio for the dynamic variables is renormalized to infinity, resulting in an effective decoupling between the two modes. We compute the critical exponents at this new fixed point to one-loop order. For model H with spatially anisotropic noise, we observe a critical softening only in the d _⊥-dimensional sector in wave vector space with lower noise temperature. The ensuing effective two-temperature model H does not have any stable fixed point in any physical dimension, at least to one-loop order. We obtain formal expressions for the novel critical exponents in a double expansion about the upper critical dimension d _c = 4 - d _|| and with respect to d _|| , i.e., about the equilibrium theory. Received 4 April 2002 Published online 13 August 2002     

Eigenstates of an operating quantum computer: hypersensitivity to static imperfections

We study the properties of eigenstates of an operating quantum computer which simulates the dynamical evolution in the regime of quantum chaos. Even if the quantum algorithm is polynomial in number of qubits n_q, it is shown that the ideal eigenstates become mixed and strongly modified by static imperfections above a certain threshold which drops exponentially with n_q. Above this threshold the quantum eigenstate entropy grows linearly with n_q but the computation remains reliable during a time scale which is polynomial in the imperfection strength and in n_q. Received 7 March 2002/ Received in final form 3 May 2002 Published online 19 July 2002     

Coulomb repulsion versus Hubbard repulsion in a disordered chain

We study the difference between on site Hubbard and long range Coulomb repulsions for two interacting particles in a disordered chain. The system size L (in units of the lattice spacing) is of the order of the one particle localization length and the energies are taken near the band center. In the two cases, the limits of weak and strong interactions are characterized by uncorrelated energy levels and are separated by a crossover regime where the states are more extended and the spectra more rigid. U denoting the interaction strength and t the kinetic energy scale, the crossovers take place for interaction energy to kinetic energy ratios U/t and U/(2tL) of order one, for Hubbard and Coulomb repulsions respectively. While Hubbard repulsion can only yield weak critical chaos with intermediate spectral statistics, Coulomb repulsion can drive the two particle system to quantum chaos with Wigner-Dyson spectral statistics. The interaction matrix elements are studied to explain this difference. Received 21 March 2000 and Received in final form 5 February 2001     

Anomalous dynamical light scattering in soft glassy gels

We compute the dynamical structure factor S(q,τ) of an elastic medium where force dipoles appear at random in space and in time, due to “micro-collapses” of the structure. Various regimes are found, depending on the wave vector q and the collapse time θ. In an early-time regime, the logarithm of the structure factor behaves as (qτ)^3/2, as predicted in L. Cipelletti, S. Manley, R.C. Ball, D.A. Weitz, Phys. Rev. Lett. 84, 2275 (2000) using heuristic arguments. However, in an intermediate-time regime we rather obtain a (qτ)^5/4 behaviour. Finally, the asymptotic long-time regime is found to behave as q ^3/2τ. We also give a plausible scenario for aging, in terms of a strain-dependent energy barrier for micro-collapses. The relaxation time is found to grow with the age t _w, quasi-exponentially at first, and then as t _w ^4/5 with logarithmic corrections. Received 23 July 2001     

Tensor polarization of 12C[2+ 1] in the 16O(13C,12C)17O reaction at 50 MeV

The ¹⁶O(¹³C,¹²C)¹⁷O reaction at 50 MeV has been investigated using the kinematical coincidence method. Polarization tensors t ₂₀ and t ₄₀ of ¹²C[2⁺ ₁] for the quantization axis taken along the direction of propagation have been measured by analyzing the energy spectrum of ¹²C[2⁺ ₁], modulated by the effect of γ ray emission. The deduced t ₄₀ values significantly deviate from zero, contrary to the prediction of the distorted-wave Born approximation theory based on one-step p shell neutron stripping without spin-dependent interactions. The phenomenological spin–orbit interaction necessary to reproduce the magnitude of measured t ₄₀ is found to be much larger than the folding model prediction. It is shown that the experimental polarization tensors as well as the cross sections can be reproduced by introducing multi-step processes involving excitations in ¹²C and ¹³C without introducing spin-dependent interactions. Received: 2 August 1999 / Revised version: 3 February 2000     

Statistical properties of eigenvalues for an operating quantum computer with static imperfections

We investigate the transition to quantum chaos, induced by static imperfections, for an operating quantum computer that simulates efficiently a dynamical quantum system, the sawtooth map. For the different dynamical regimes of the map, we discuss the quantum chaos border induced by static imperfections by analyzing the statistical properties of the quantum computer eigenvalues. For small imperfection strengths the level spacing statistics is close to the case of quasi-integrable systems while above the border it is described by the random matrix theory. We have found that the border drops exponentially with the number of qubits, both in the ergodic and quasi-integrable dynamical regimes of the map characterized by a complex phase space structure. On the contrary, the regime with integrable map dynamics remains more stable against static imperfections since in this case the border drops only algebraically with the number of qubits. Received 19 June 2002 / Received in final form 30 September 2002 Published online 17 Decembre 2002 RID="a" ID="a"e-mail: dima@irsamc.ups-tlse.fr RID="b" ID="b"UMR 5626 du CNRS     

Aging and non-linear glassy dynamics in a mean field model

The mean field approach of glassy dynamics successfully describes systems which are out-of-equilibrium in their low temperature phase. In some cases an aging behaviour is found, with no stationary regime ever reached. In the presence of dissipative forces however, the dynamics is indeed stationary, but still out-of-equilibrium, as inferred by a significant violation of the fluctuation dissipation theorem. The mean field dynamics of a particle in a random but short-range correlated environment, offers the opportunity of observing both the aging and driven stationary regimes. Using a geometrical approach previously introduced by the author, we study here the relation between these two situations, in the pure relaxational limit, i.e. the zero temperature case. In the stationary regime, the velocity (v)-force (F) characteristics is a power law v∼F ⁴, while the characteristic times scale like powers of v, in agreement with an early proposal by Horner. The cross-over between the aging, linear-response regime and the non-linear stationary regime is smooth, and we propose a parametrization of the correlation functions valid in both cases, by means of an “effective time”. We conclude that aging and non-linear response are dual manifestations of a single out-of-equilibrium state, which might be a generic situation. Received 7 May 2000 and Received in final form 22 August 2000     

Eigenvalues and eigenfunctions of a clover plate

We report a numerical study of the flexural modes of a plate using semi-classical analysis developed in the context of quantum systems. We first introduce the Clover billiard as a paradigm for a system inside which rays exhibit stable and chaotic trajectories. The resulting phase space explored by the ray trajectories is illustrated using the Poincare surface of section, and shows that it has both integrable and chaotic regions. Examples of the stable and the unstable periodic orbits in the geometry are presented. We numerically solve the biharmonic equation for the flexural vibrations of the Clover shaped plate with clamped boundary conditions. The first few hundred eigenvalues and the eigenfunctions are obtained using a boundary elements method. The Fourier transform of the eigenvalues show strong peaks which correspond to ray periodic orbits. However, the peaks corresponding to the shortest stable periodic orbits are not stronger than the peaks associated with unstable periodic orbits. We also perform statistics on the obtained eigenvalues and the eigenfunctions. The eigenvalue spacing distribution P(s) shows a strong peak and therefore deviates from both the Poisson and the Wigner distribution of random matrix theory at small spacings because of the C_4v symmetry of the Clover geometry. The density distribution of the eigenfunctions is observed to agree with the Porter-Thomas distribution of random matrix theory. Received 12 February 2001 and Received in final form 17 April 2001