Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ > 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λ^q that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.     

Persistence of the Feigenbaum Attractor in One-Parameter Families

Consider a map ψ₀ of class C ^r for large r of a manifold of dimension n greater than or equal to 2 having a Feigenbaum attractor. We prove that any such ψ₀ is a point of a local codimension-one manifold of C ^r transformations also exhibiting Feigenbaum attractors. In particular, the attractor persists when perturbing a one-parameter family transversal to that manifold at ψ₀. We also construct such a transversal family for any given ψ₀, and apply this construction to prove a conjecture by J. Palis stating that a map exhibiting a Feigenbaum attractor can be perturbed to obtain homoclinic tangencies. Received: 4 August 1998 / Accepted: 11 May 1999     

Critical Slowing Down in One-Dimensional Maps and Beyond

This is a brief review on critical slowing down near the Feigenbaum period-doubling bifurcation points and its consequences. The slowing down of numerical convergence leads to an “operational” fractal dimension D=2/3 at a finite order bifurcation point. There is a cross-over to D ₀=0.538... when the order goes to infinity, i.e., to the Feigenbaum accumulation point. The problem of whether there exists a “super-scaling” for the dimension spectrum D _q ^W that does not depend on the primitive word W underlying the period-n-tupling sequence seems to remain open     

Chaotic oscillation in diffusion flame induced by radiative heat loss

We numerically investigate the dynamic behavior of flame front instability in a diffusion flame caused by radiative heat loss from the viewpoint of nonlinear dynamics. As the Damköhler number increases at a high activation temperature, the dynamic behavior of the flame front undergoes a significant transition from a steady-state to high-dimensional deterministic chaos through the period-doubling cascade process known as the Feigenbaum transition. The existence of high-dimensional chaos in flame dynamics is clearly demonstrated using a sophisticated nonlinear time series analysis technique based on chaos theory.     

Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations

To understand the nonlinear dynamical behaviour of a one-dimensional pulsating detonation, results obtained from numerical simulations of the Euler equations with simple one-step Arrhenius kinetics are analysed using basic nonlinear dynamics and chaos theory. To illustrate the transition pattern from a simple harmonic limit-cycle to a more complex irregular oscillation, a bifurcation diagram is constructed from the computational results. Evidence suggests that the route to higher instability modes may follow closely the Feigenbaum scenario of a period-doubling cascade observed in many generic nonlinear systems. Analysis of the one-dimensional pulsating detonation shows that the Feigenbaum number, defined as the ratio of intervals between successive bifurcations, appears to be in reasonable agreement with the universal value of d = 4.669. Using the concept of the largest Lyapunov exponent, the existence of chaos in a one-dimensional unsteady detonation is demonstrated.     

Fractal boundary of domain of analyticity of the Feigenbaum function and relation to the Mandelbrot set

The universal map for the period-doubling transition to chaos is studied numerically in the complex plane. The boundary of the domain of analyticity of this function is obtained graphically and is shown to be a fractal with self-similar properties obtained by rescaling with the universal constants and. In the complex parameter plane, this domain is shown asymptotically to be similar to part of the Mandelbrot set.     

Dynamic analysis and synchronization control of an unusual chaotic system with exponential term and coexisting attractors

This paper reports a new four-dimensional chaotic system consisting of an exponential nonlinear term, two quadratic nonlinear terms and five linear terms. The system has only one equilibrium and performs stability, periodicity and chaos with the variation of the parameters. It losses its stability with the occurrence of Hopf bifurcation and goes into chaos via period-doubling bifurcation. One more interesting feature of the system is that it can generate multiple coexisting attractors for different initial conditions, such as two strange attractors with one limit cycle, one strange attractor with two limit cycles, etc. The dynamic properties of the system are presented by numerical simulation includes bifurcation diagrams, Lyapunov exponent spectrum and phase portraits. An electronic circuit is constructed to implement the chaotic attractor of the system. Based on the linear quadratic regulator (LQR) method, the synchronization control of the system is investigated.     

Fibonacci order in the period-doubling cascade to chaos

In this contribution, we describe how the Fibonacci sequence appears within the Feigenbaum scaling of the period-doubling cascade to chaos. An important consequence of this discovery is that the ratio of successive Fibonacci numbers converges to the golden mean in every period-doubling sequence and therefore the convergence to ϕ, the most irrational number, occurs in concert with the onset of deterministic chaos.     

Chaos and generalised multistability in a mesoscopic model of the electroencephalogram

We present evidence for chaos and generalised multistability in a mesoscopic model of the electroencephalogram (EEG). Two limit cycle attractors and one chaotic attractor were found to coexist in a two-dimensional plane of the ten-dimensional volume of initial conditions. The chaotic attractor was found to have a moderate value of the largest Lyapunov exponent (3.4 s⁻¹ base e) with an associated Kaplan-Yorke (Lyapunov) dimension of 2.086. There are two different limit cycles appearing in conjunction with this particular chaotic attractor: one multiperiodic low amplitude limit cycle whose largest spectral peak is within the alpha band (8-13 Hz) of the EEG; and another multiperiodic large-amplitude limit cycle which may correspond to epilepsy. The cause of the coexistence of these structures is explained with a one-parameter bifurcation analysis. Each attractor has a basin of differing complexity: the large-amplitude limit cycle has a basin relatively uncomplicated in its structure while the small-amplitude limit cycle and chaotic attractor each have much more finely structured basins of attraction, but none of the basin boundaries appear to be fractal. The basins of attraction for the chaotic and small-amplitude limit cycle dynamics apparently reside within each other. We briefly discuss the implications of these findings in the context of theoretical attempts to understand the dynamics of brain function and behaviour.     

Multiparameter Critical Situations, Universality and Scaling in Two-Dimensional Period-Doubling Maps

We review critical situations, linked with period-doubling transition to chaos, which require using at least two-dimensional maps as models representing the universality classes. Each of them corresponds to a saddle solution of the two-dimensional generalization of Feigenbaum-Cvitanovi? equation and is characterized by a set of distinct universal constants analogous to Feigenbaum’s α and δ. One type of criticality designated H was discovered by several authors in 80-th in the context of period doubling in conservative dynamics, but occurs as well in dissipative dynamics, as a phenomenon of codimension 2. Second is bicritical behavior, which takes place in systems allowing decomposition onto two dissipative period-doubling subsystems, each of which is brought by parameter tuning onto a threshold of chaos. Types of criticality designated as FQ and C occur in non-invertible two-dimensional maps. We present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems. In particular, we indicate a possibility for realization of the H type criticality without vanishing dissipation, but with its compensation in a self-oscillatory system. Next, we present a number of examples (coupled Hénon-like maps, coupled driven oscillators, coupled chaotic self-oscillators), which manifest bicritical behavior. For FQ-type we indicate possibility to arrange it in non-symmetric systems of coupled period-doubling subsystems, e.g. in Hénon-like maps and in Chua’s circuits. For C-type we present examples of its appearance in a driven Rössler oscillator at the period-doubling accumulation on the edge of syncronization tongue and in a model map with the Neimark–Sacker bifurcation     

Universality of fractal dimension at the onset of period-doubling Chaos

We present numerical evidence for the hypothesis that, at the threshold of period-doubling chaos in a dynamical system, the fractal dimension of the associated strange attractor assumes a universal value.     

On bifurcations and transition to chaos in a Josephson junction

Chaos in the dc and rf current driven Josephson junction is investigated by measurements on a phase-locked loop. The method allows a direct display of the Poincaré map and the bifurcation diagram, as the parameter space is searched. In some regions of the parameter space the Feigenbaum period-doubling route to chaos is observed upon a change of any parameter.     

Oscillations in the Determination of Renyi Dimension

To describe the degree of chaos of a strange attractor created by a nonlinear map, GRASSBERGER [1] revived Renyi Dimension D_q which is a common generalization of both metric capacity D₀ and information dimension D₁. But often, the numerical determination of D_q is complicated by an oscillating effect first investigated by BADII and PUOLITI [5] for a dimension similar to D₀. This effect is studied in the present paper. We consider the conditions for the appearance of oscillations and the dependence on q, and we give some numerical examples.     

Stretched exponential relaxation on the hypercube and the glass transition

We study random walks on the dilute hypercube using an exact enumeration Master equation technique, which is much more efficient than Monte Carlo methods for this problem. For each dilution p the form of the relaxation of the memory function q(t) can be accurately parametrized by a stretched exponential over several orders of magnitude in q(t). As the critical dilution for percolation is approached, the time constant tends to diverge and the stretching exponent drops towards 1/3. As the same pattern of relaxation is observed in a wide class of glass formers, the fractal like morphology of the giant cluster in the dilute hypercube appears to be a good representation of the coarse grained phase space in these systems. For these glass formers the glass transition may be pictured as a percolation transition in phase space. Received 16 June 2000 and Received in final form 13 October 2000     

Nonlinear optical conductivity of two-dimensional semiconductors with Rashba spin-orbit coupling in terahertz regime

We explain how specific dynamical properties give rise to the limit distribution of sums of deterministic variables at the transition to chaos via the period-doubling route. We study the sums of successive positions generated by an ensemble of initial conditions uniformly distributed in the entire phase space of a unimodal map as represented by the logistic map. We find that these sums acquire their salient, multiscale, features from the repellor preimage structure that dominates the dynamics toward the attractors along the period-doubling cascade. And we explain how these properties transmit from the sums to their distribution. Specifically, we show how the stationary distribution of sums of positions at the Feigebaum point is built up from those associated with the supercycle attractors forming a hierarchical structure with multifractal and discrete scale invariance properties.     

Effects of external noise on the circle map and the transition to chaos in Josephson junctions

We investigate the effects of external current noise on a microwave-driven Josephson junction. We show that the circle return map for the superconducting phase difference is stable with respect to the external noise and find that the effects of fluctuations on the route to chaos described with the circle map can be opposite to those for the Feigenbaum period-doubling cascade: increasing noise can here act as a control parameter triggering a periodically oscillating junction chaotic by generating an inflection point in the return (circle) map. This may prove important also for other physical systems, including charge density waves.     

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (x_n+1 = 1-μx_n^1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.     

On the metric properties of the Feigenbaum attractor

The two standard literature definitions of the function associated with the Feigenbaum attractor are not equivalent. The method due to Vulet al. and Feigenbaum is used to calculate the Haussdorff dimension of the Feigenbaum attractor, using as input the trajectory scaling functions. The two calculations yield the same Hausdorff dimensionD=0.5380451435 to within the accuracy of the computation.     

Period-doubling bifurcations of the thermocapillary convection in a floating half zone

This study experimentally explored the fine structures of the successive period-doubling bifurcations of the time-dependent thermocapillary convection in a floating half zone of 10 cSt silicone oil with the diameter d0=3.00 mm and the aspect ratio A=l/d0=0.72 in terrestrial conditions.The onset of time-dependent thermocapillary convection predominated in this experimental configuration and its subsequent evolution were experimentally detected through the local temperature measurements.The experimental results revealed a sequence of period-doubling bifurcations of the time-dependent thermocapillary convection,similar in some way to one of the routes to chaos for buoyant natural convection.The critical frequencies and the corresponding fractal frequencies were extracted through the real-time analysis of the frequency spectra by Fast-Fourier-Transfor-mation(FFT).The projections of the trajectory onto the reconstructed phase-space were also provided.Furthermore,the experimentally predicted Feigenbaum constants were quite close to the theoretical asymptotic value of 4.669 [Feigenbaum M J.Phys Lett A,1979,74:375-378].     

Nonextensivity at the edge of chaos of a new universality class of one-dimensional unimodal dissipative maps

We introduce a new universality class of one-dimensional unimodal dissipative maps. The new family, from now on referred to as the (z₁, z₂)-logarithmic map, corresponds to a generalization of the z-logistic map. The Feigenbaum-like constants of these maps are determined. It has been recently shown that the probability density of sums of iterates at the edge of chaos of the z-logistic map is numerically consistent with a q-Gaussian, the distribution which, under appropriate constraints, optimizes the nonadditive entropy S_q. We focus here on the presently generalized maps to check whether they constitute a new universality class with regard to q-Gaussian attractor distributions. We also study the generalized q-entropy production per unit time on the new unimodal dissipative maps, both for strong and weak chaotic cases. The q-sensitivity indices are obtained as well. Our results are, like those for the z-logistic maps, numerically compatible with the q-generalization of a Pesin-like identity for ensemble averages.