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.     

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.     

Circular-like maps: sensitivity to the initial conditions, multifractality and nonextensivity

Dissipative one-dimensional maps may exhibit special points (e.g., chaos threshold) at which the Lyapunov exponent vanishes. Consistently, the sensitivity to the initial conditions has a power-law time dependence, instead of the usual exponential one. The associated exponent can be identified with 1/(1-q), where q characterizes the nonextensivity of a generalized entropic form currently used to extend standard, Boltzmann-Gibbs statistical mechanics in order to cover a variety of anomalous situations. It has been recently proposed (Lyra and Tsallis, Phys. Rev. Lett. 80, 53 (1998)) for such maps the scaling law , where and are the extreme values appearing in the multifractal function. We generalize herein the usual circular map by considering inflexions of arbitrary power z, and verify that the scaling law holds for a large range of z. Since, for this family of maps, the Hausdorff dimension d_f equals unity for all z in contrast with q which does depend on z, it becomes clear that d_f plays no major role in the sensitivity to the initial conditions. Received 5 February 1999     

Non-Smooth Bifurcation and Chaos in a DC-DC Buck Converter

A direct-current-direct-current （DC-DC） buck converter with integrated load current feedback is studied with three kinds of Poincaré maps. The external corner-collision bifurcation occurs when the crossing number per period varies, and the internal corner-collision bifurcations occur along with period-doubling and period-tripling bifurcations in this model. The multi-band chaos roots in external corner-collision bifurcation and often grows into 1-band chaos. A new kind of chaotic sliding orbits, which is more complex for non-smooth systems, is also found in this model.     

Torus destruction via global bifurcations in a piecewise-smooth, continuous map with square-root nonlinearity

It has been shown recently that torus formation in piecewise-smooth maps can occur through a special type of border collision bifurcation in which a pair of complex conjugate Floquet multipliers “jump” from the inside to the outside of the unit circle. It has also been shown that a large class of impacting mechanical systems yield piecewise-smooth maps with square-root singularity. In this Letter we investigate the dynamics of a two-dimensional piecewise-smooth map with square-root type nonlinearity, and describe two new routes to chaos through the destruction of two-frequency torus. In the first scenario, we identify the transition to chaos through the destruction of a loop torus via homoclinic bifurcation. In the other scenario, a change of structure in the torus occurs via heteroclinic saddle connections. Further parameter changes lead to a homoclinic bifurcation resulting in the creation of a chaotic attractor. However, this scenario is much more complex, with the appearance of a sequence of heteroclinic and homoclinic bifurcations.     

Lynden-Bell and Tsallis distributions for the HMF model

Systems with long-range interactions can reach a Quasi Stationary State (QSS) as a result of a violent collisionless relaxation. If the system mixes well (ergodicity), the QSS can be predicted by the statistical theory of Lynden-Bell (1967) based on the Vlasov equation. When the initial condition takes only two values, the Lynden-Bell distribution is similar to the Fermi-Dirac statistics. Such distributions have recently been observed in direct numerical simulations of the HMF model (Antoniazzi et al. 2006). In this paper, we determine the caloric curve corresponding to the Lynden-Bell statistics in relation with the HMF model and analyze the dynamical and thermodynamical stability of spatially homogeneous solutions by using two general criteria previously introduced in the literature. We express the critical energy and the critical temperature as a function of a degeneracy parameter fixed by the initial condition. Below these critical values, the homogeneous Lynden-Bell distribution is not a maximum entropy state but an unstable saddle point. Known stability criteria corresponding to the Maxwellian distribution and the water-bag distribution are recovered as particular limits of our study. In addition, we find a critical point below which the homogeneous Lynden-Bell distribution is always stable. We apply these results to the situation considered in Antoniazzi et al. For a given energy, we find a critical initial magnetization above which the homogeneous Lynden-Bell distribution ceases to be a maximum entropy state. For an energy U=0.69, this transition occurs above an initial magnetization M_x=0.897. In that case, the system should reach an inhomogeneous Lynden-Bell distribution (most mixed) or an incompletely mixed state (possibly fitted by a Tsallis distribution). Thus, our theoretical study proves that the dynamics is different for small and large initial magnetizations, in agreement with numerical results of Pluchino et al. (2004). This new dynamical phase transition may reconcile the two communities by showing that they study different regimes.     

Statistical dynamics at critical bifurcations in Duffing-van der Pol oscillator

We study the characteristic features of certain statistical quantities near critical bifurcations such as onset of chaos, sudden widening and band-merging of chaotic attractor and intermittency in a periodically driven Duffing-van der Pol oscillator. At the onset of chaos the variance of local expansion rate is found to exhibit a self-similar pattern. For all chaotic attractors the variance Σ_n(q) of fluctuations of coarse-grained local expansion rates of nearby orbits has a single peak. However, multiple peaks are found just before and just after the critical bifurcations. On the other hand, Σ_n (q) associated with the coarse-grained state variable is zero far from the bifurcations. The height of the peak of Σ_n(q) is found to increase as the control parameter approached the bifurcation point. It is maximum at the bifurcation point. Power-law variation of maximal Lyapunov exponent and the mean value of the state variablex is observed near sudden widening and intermittency bifurcations while linear variation is seen near band-merging bifurcation. The standard deviation of local Lyapunov exponent λ(X,L) and the local mean valuex(L) of the coordinatex calculated after everyL time steps are found to approach zero in the limitL → ∞ asL ^-Β. Β is sensitive to the values of control parameters. Further weak and strong chaos are characterized using the probability distribution of ak-step difference quantity δx_k = x_i+k x _i.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

Hyperchaos--chaos--Hyperchaos Transition in a Class of On--Off Intermittent Systems Driven by a Family of Generalized Lorenz Systems

Blowout bifurcation in nonlinear systems occurs when a chaotic attractor lying in some symmetric subspace becomes transversely unstable. A class of five-dimensional continuous autonomous systems is considered, in which a two-dimensional subsystem is driven by a family of generalized Lorenz systems. The systems have some common dynamical characters. As the coupling parameter changes, blowout bifurcations occur in these systems and brings on change of the systems＇ dynamics. After the bifurcation the phenomenon of on-off intermittency appears. It is observed that the systems undergo a symmetric hyperchaos-chaos-hyperchaos transition via or after blowout bifurcations. An example of the systems is given, in which the drive system is the Chen system. We investigate the dynamical behaviour before and after the blowout bifurcation in the systems and make an analysis of the transition process. It is shown that in such coupled chaotic continuous systems, blowout bifurcation leads to a transition from chaos to hyperchaos for the whole systems, which provides a route to hyperchaos.     

Andronov-Hopf bifurcations in planar, piecewise-smooth, continuous flows

An equilibrium of a planar, piecewise-C¹, continuous system of differential equations that crosses a curve of discontinuity of the Jacobian of its vector field can undergo a number of discontinuous or border-crossing bifurcations. Here we prove that if the eigenvalues of the Jacobian limit to λL±iωL on one side of the discontinuity and −λR±iωR on the other, with λL,λR>0, and the quantity Λ=λL/ωL−λR/ωR is nonzero, then a periodic orbit is created or destroyed as the equilibrium crosses the discontinuity. This bifurcation is analogous to the classical Andronov-Hopf bifurcation, and is supercritical if Λ<0 and subcritical if Λ>0.     

Synchronized Anti-Phase and In-Phase Oscillations of Intracellular Calcium Ions in Two Coupled Hepatocytes System

We study the dynamic behaviour of two intracellufar calcium oscillators that are coupled through gap junctions both to Ca^2＋ and inositol（1,4,5）-trisphosphate （IP3）. It is found that synchronized anti-phase and in-phase oscillations of cytoplasmic cadcium coexist in parameters space. Especially, synchronized anti-phase oscillations only occur near the onset of a Hopf bifurcation point when the velocity of IP3 synthesis is increased. In addition, two kinds of coupling effects, i.e., the diffusions of Ca^2＋ and IP3 among cells on synchronous behaviour, are considered. We fnd that small coupling of Ca^2＋ and large coupling of IP3 facilitate the emergence of synchronized anti-phase oscillations. However, the result is contrary for the synchronized in-phase case. Our findings may provide a qualitative understanding about the mechanism of synchronous behaviour of intercellular calcium signalling.     

A generalized Kolmogorov-Sinai-like entropy under Markov shifts in symbolic dynamics

In this paper, a generalized Kolmogorov-Sinai-like entropy ( entropy) in the sense of Tsallis is proposed with a nonextensive parameter q under Markov shifts, which contains the classical Kolmogorov-Sinai (KS) entropy and the Rényi entropy as well as Bernoulli shifts as special cases. To verify the formula of this entropy, a one-dimensional iterative system is chosen as an example of Markov shifts, and its entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method. The numerical results show that this entropy is monotonically decreasing as q increases.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Weak chaos and metastability in a symplectic system of many long-range-coupled standard maps

We introduce, and numerically study, a system of N symplectically and globally coupled standard maps localized in a d=1 lattice array. The global coupling is modulated through a factor r^-α, being r the distance between maps. Thus, interactions are long-range (nonintegrable) when 0≤α≤1, and short-range (integrable) when α>1. We verify that the largest Lyapunov exponent λ_M scales as λ_M ∝ N^-κ(α), where κ(α) is positive when interactions are long-range, yielding weak chaos in the thermodynamic limit N↦∞ (hence λ_M→0). In the short-range case, κ(α) appears to vanish, and the behaviour corresponds to strong chaos. We show that, for certain values of the control parameters of the system, long-lasting metastable states can be present. Their duration t_c scales as t_c ∝N^β(α), where β(α) appears to be numerically in agreement with the following behavior: β>0 for 0 ≤α< 1, and zero for α≥1. These results are consistent with features typically found in nonextensive statistical mechanics. Moreover, they exhibit strong similarity between the present discrete-time system, and the α-XY Hamiltonian ferromagnetic model.     

Simple stochastic models showing strong anomalous diffusion

We show that strong anomalous diffusion, i.e. where is a nonlinear function of q, is a generic phenomenon within a class of generalized continuous-time random walks. For such class of systems it is possible to compute analytically where n is an integer number. The presence of strong anomalous diffusion implies that the data collapse of the probability density function cannot hold, a part (sometimes) in the limit of very small , now . Moreover the comparison with previous numerical results shows that the shape of is not universal, i.e., one can have systems with the same but different F. Received 14 April 2000     

Classical dynamics of polar diatomic molecules in external fields

In this paper, we present the study of the global classical dynamics of a rigid diatomic molecule in the presence of combined electrostatic and nonresonant polarized laser fields. In particular, we focus on the collinear field case, which is an integrable system because the z-component P_φ of the angular momentum is conserved. The study involves the complete analysis of the stability of the equilibrium points, their bifurcations and the evolution of the phase flow as a function of the field strengths and P_φ. Finally, the influence of the bifurcations on the orientation of the quantum states is studied.     

Multi-affinity and multi-fractality in systems of chaotic elements with long-wave forcing

Multi-scaling properties in quasi-continuous arrays of chaotic maps driven by long-wave random force are studied. The spatial pattern of the amplitude X(x,t) is characterized by multi-affinity, while the field defined by its coarse-grained spatial derivative exhibits multi-fractality. The strong behavioral similarity of the X- and Y-fields respectively to the velocity and energy dissipation fields in fully-developed fluid turbulence is remarkable, still our system is unique in that the scaling exponents are parameter-dependent and exhibit nontrivial q-phase transitions. A theory based on a random multiplicative process is developed to explain the multi-affinity of the X-field, and some attempts are made towards the understanding of the multi-fractality of the Y-field. Received 16 November 1998     

Dynamic behaviours of 2∞ attractor and q-phase transitions at bifurcations in logistic map

Dynamic behaviours of the 2^∞ attractor at the accumulation of period doubling in the logistic map are studied by the sum of the local expansion rates S_n(x₁) of nearby orbits. The variance 〈[S_n(x)]²〉 and algebraic exponent ß_n(x₁) = S_n(x₁)/ln(n) exhibits self-similar structures. The critical bifurcations such as intermittency, band merging and crisis-sudden widening of the chaotic attractor are studied in terms of a q-weighted average Λ(q), (− ∞ < q < ∞) of the coarse-grained local expansion rates Λ of nearby orbitals.     

Noise-induced bifurcations and chaos in the average motion of globally-coupled oscillators

A system of coupled master equations simplified from a model of noise-driven globally coupled bistable oscillators under periodic forcing is investigated. In the thermodynamic limit, the system is reduced to a set of two coupled differential equations. Rich bifurcations to subharmonics and chaotic motions are found. This behavior can be found only for certain intermediate noise intensities. Noise with intensities which are too small or too large will certainly spoil the bifurcations. In a system with large though finite size, the bifurcations to chaos induced by noise can still be detected to a certain degree. Received 6 April 1999 and Received in final form 1 November 1999     

Fractal formations in the Lattice Limit Cycle model

We examine the fractal patterns arising in the Lattice Limit Cycle model, when it is restricted on square and fractal lattices. We show that, for processes taking place on regular 2d substrates, the fractal dimensions depend on the kinetic constants and we have observed a sharp phase-transition from uniform 2d spatial distributions (d_f=2) when the kinetic parameters are near the Hopf bifurcation point, to a inside the limit cycle region. For processes taking place on substrates which contain inactive sites, we observe nucleation of homologous species around inactive regions leading to poisoning, when the active sites are distributed in a fractal manner on the substrate. This is less frequent in cases where the active sites are distributed uniformly and randomly on the lattice leading, normally, to non-trivial steady states.