Q-phase Transitions Research Articles

A Solvable Chaotic Map of q-Phase Transitions Caused by Diverging Local Expansion Rates

In chaotic dynamical systems, non-hyperbolicity or bifurcations such as an onset of type-I intermittency (a tangent bifurcation) and a crisis, which can be observed, for example, at both ends of the period-three window of the logistic map, cause characteristic large fluctuations in a finite range of the local expansion rate, whose long time average is equal to the maximum Lyapunov exponent. This appears as a non-analytic rate function of the local expansion rate. Some researchers call it a ‘‘q-phase transition’’. 1) For a one-dimensional chaotic map xnþ1 ¼ f ðxnÞ, the local expansion rate � ðxnÞ is given by � ðxn Þ¼ log jdf ðxnÞ=dxnj .I f the map has a smooth extremum at x ¼ x� , � ðx� Þ goes to negative infinity. Although a smooth extremum is a special case of non-hyperbolicity in a one-dimensional map, diverging local expansion rates due to non-hyperbolicity in general chaotic dynamical systems such as higher-dimensional maps and differential equations also cause a nonanalytic rate function of the local expansion rate. 1) The Lorenz plot with the classical parameters 2) has a cuspshaped maximum, at which the local expansion rate goes to

Fluctuating dynamics at the quasiperiodic onset of chaos, Tsallis [formula omitted]-statistics and Mori's [formula omitted]-phase thermodynamics

We analyze the fluctuating dynamics at the golden-mean transition to chaos in the critical circle map and find that trajectories within the critical attractor consist of infinite sets of power laws mixed together. We elucidate this structure assisted by known renormalization group (RG) results. Next we proceed to weigh the new findings against Tsallis’ entropic and Mori's thermodynamic theoretical schemes and observe behavior to a large extent richer than previously reported. We find that the sensitivity to initial conditions ξ t has the form of families of intertwined q-exponentials, of which we determine the q-indexes and the generalized Lyapunov coefficient spectra λ q . Further, the dynamics within the critical attractor is found to consist of not one but a collection of Mori's q-phase transitions with a hierarchical structure. The value of Mori's ‘thermodynamic field’ variable q at each transition corresponds to the same special value for the entropic index q. We discuss the relationship between the two formalisms and indicate the usefulness of the methods involved to determine the universal trajectory scaling function σ and/or the occurrence and characterization of dynamical phase transitions.

Crossover from Critical to Chaotic Attractor Dynamics in Logistic and Circle Maps

A key feature of the Tsallis statistics is its crossover to the Boltzmann-Gibbs (BG) statistics. We illustrate the mechanisms for this crossover at or near critical attractors in one-dimensional maps. First, we consider the intermittency route to chaos and explain how the crossover is due to either the feedback feature from chaotic regions into the neighborhood of the tangency, to a shift from tangency, or to perturbation by noise. We describe the role of the crossover in the related dynamics of critical clusters at thermal critical states. Secondly, we consider the onset of chaos via period doubling bifurcations and discuss two different ways to bring about the crossover. One corresponds to a shift of the map into a2 n -band chaotic attractor and the other to a perturbation of the Feigenbaum attractor with additive noise. We recall that in the latter case we obtain the properties of glassy dynamics close to vitrification in structural glass formers. Finally, we comment briefly on the equivalent properties for the quasiperiodic onset of chaos and its application to the localization transition in incommensurate systems. In all cases we indicate that the critical dynamics is based on a set of Mori’s q-phase transitions and find that the value of q at each transition corresponds to the same special value for the entropic index q.

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 Y(x, t) := |(X(x + δ, t) − X(x, t))/δ| 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.

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 ( x 1) of nearby orbits. The variance 〈[ S n ( x)] 2〉 and algebraic exponent ß n ( x 1) = S n ( x 1)/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.

Thermodynamics and multicorrelations of intermittent dynamics

Statistical properties of type-I intermittency are investigated from the standpoint of the long-time description and time correlations of intermittent time series by utilizing the So-Ose-Mori mapping. It is shown that the two coexisting characteristics, laminar state and burst, can be singled out in a clear-cut way by introducing the ensemble processing with a parameter q. For q\ensuremath{\rightarrow}-\ensuremath{\infty}, the relevant statistical quantities describe the statistics of a purely laminar state and, for q\ensuremath{\rightarrow}\ensuremath{\infty}, they describe the statistics of a purely burst behavior. In particular the dynamical statistical characteristics (temporal correlations) turn out to exhibit clear-cut change at the so-called q-phase transition point ${\mathit{q}}_{\mathit{c}}$ in the limit of the intermittency transition of the control parameter. For q${\mathit{q}}_{\mathit{c}}$, the spectral density takes a power-law form of the frequency, the exponent 2 being independent of q. On the other hand, for qg${\mathit{q}}_{\mathit{c}}$ it has a white-noise spectrum. These are characteristics of the pure laminar state and burst, respectively.

On the scaling laws of higher-order q-phase transitions

The q-phase transition points in the context of the thermodynamical formalism of dynamical systems arise via the degeneracy of eigenvalues of the corresponding transfer operator. The scaling behaviour near bifurcation points of dynamical systems is investigated by a mean-field-like expansion for the characteristic equation of this operator. Scaling relations in the vicinity of q-phase-transition points, which are brought about by a doubly (respectively triply) degenerated eigenvalue, are explicitly derived. For the characteristic function (topological pressure) this relation reads phi (q) approximately=ln nu *+ delta a phi ((q-q*)/ delta a) where the exponent a=1, 1/2, 1/3 of the bifurcation parameter delta depends on general properties of the phase-transition point. The approach explains the universal features of the scaling behaviour.

General derivation of scaling functions near q-phase transition points

Scaling relations of characteristic functions in the context of the thermodynamical formalism of dynamical systems are considered. Starting from the characteristic equation of the transfer operator a Ginzburg-Landau-like expansion in the vicinity of a bifurcation point is proposed. The phase transition based on a doubly degenerated eigenvalue is discussed in detail and explicit expressions for the scaling functions are derived.

Structures of chaos and q-phase transitions

Abstract A statistical-mechanical formalism of chaos based on the geometry of invariant sets in phase space is discussed to show that chaotic dynamical systems can be treated by a formalism analogous to that of thermodynamic systems if one takes a relevant coarse-grained quantity, but their statistical laws are quite different from those of thermodynamic systems. This is a generalization of statistical mechanics for dealing with dissipative and hamiltonian dynamical systems of a few degrees of freedom. Thus the sum of the local expansion rate of nearby orbits along a relevant orbit over a long but finite time has been introduced in order to describe and characterize (1) a drastic change of the structure of a chaotic region at a bifurcation and associated anomalous phenomena, (2) the critical KAM torus, diffusion and repeated sticking of a chaotic orbit to a critical torus. Here a q-phase transition, analogous to the ferromagnetic phase transition, plays an important role.

Statistical mechanics and crossover scaling for Pomeau-Manneville type intermittent chaos

On the basis of the statistical mechanics formalism, we study analytically intermittent chaos, concerning ourselves with the q-phase transition near the bifurcation point. The crossover exponents for the Renyi entropies, the singularity spectrum and the q-weighted rate of bursts are rigorously obtained.

Linearities of the f( ) Spectrum at Bifurcations of Chaos in Dissipative Differential Systems

It is shown that the singularity spectrum I(a) of the natural invariant measure in the driven damped pendulum exhibits three types of linear slopes at bifurcation points of chaos, which agree with theoretical estimates of the linear slopes . . Chaotic attractors of dissipative differential systems have fascinating multi­ fractal structures. I) Recently it has been shown that the structures could be char­ acterized by the generalized dimensions D(q), (-=<q<=) and the singularity spectra I(a) of the natural invariant measures. 2 ) For two-dimensional maps, it has been found that D(q) and I(a) have close relation to the q-weighted average A(q) of the coarse-grained expansion rates A of nearby orbits along the local unstable manifolds and their spectrum ¢I(A).3H) In our previous papers,6H) therefore, chaotic attractors at ~arious bifurcation points have been studied in terms of A(q) and ¢I(A), and singular local structures of chaotic attractors have been shown to bring about linearities in ¢I(A) which cause q-phase transitions of A(q). For two-dimensional maps, it has also been shown that the linearities in ¢I(A) lead to linearities in I(a) which causeq-phase transitions of D(q).IO) This indicates that the metric structures of chaotic attractors can be characterized by the linearities in l(a).5) In this paper, therefore, we shall investigate the I(a) spectrum and its linearities for dissipative differential systems. As a typical example of dissipative differential systems which exhibit chaotic behavior, we take the driven damped pendulum, whose equation of motion is

Dynamic Scaling Laws for the Intermittent Switching between Two Bands

The bifurcations of chaos exhibit various fascinating features. I ),2) At the bifurcation point the chaotic attractor suffers a drastic change, and anomalous fluctuations emerge for various quantities. Many investigations have been devoted to the study of the bifurcations, where dynamic scaling laws in certain sense are expected to hold, as shown in the previous papers.)-6) A cascade of band merging after the onset of chaos in the logistic map leads to a chaotic attractor which consists of two bands. Furthermore, the two bands collide with an unstable fixed point and continuously merge. into one band. I ),2) Then a remarkable linear slope is formed in the spectrum ¢(~) for fluctuations of a coarsegrained coordinate ~.7),2) As the bifurcation parameter is further increased, ¢(~) deviates from the linearity but this deviation obeys a dynamic scaling, as. will be shown in the present paper. Recently the characterization of chaos has successfully been made by the use of dynamic structure functions, which reflect the local structures of a strange attractor.) When bands merge, the q-weighted avergage ~(q) of ~ exhibits a q-phase transition at q=l in the limit r---'>oo, where r is the mean lifetime of the intraband motions in each band. For a finite but large r, ~(q) obeys a dynamic scaling for q around q=l with rlq-11:S1, as will be shown later. We take the logistic map,

Q-Phase Transitions and Dynamic Scaling Laws at Attractor-Merging Crises in the Driven Damped Pendulum

q-Phase Transitions and Dynamic Scaling Laws at Attractor-Merging Crises in the Driven Damped Pendulum

The q-Phase Transition and Critical Scaling Laws near the Band-Crisis Point of Chaotic Attractors

The q-Phase Transition and Critical Scaling Laws near the Band-Crisis Point of Chaotic Attractors

Q-Phase Transitions and Critical Phenomena just after Band Crisis

Just after a band crisis, a new type of q-phase transition occurs, and the fluctuation spectrum of coarse-grained local expansion rates has a remarkable linear part. As the crisis point is approached from above, the deviation from the linear part obeys a dynamic scaling law.

Critical Scaling for Continuous q-Phase Transitions in Intermittent Phenomena

Critical Scaling for Continuous q-Phase Transitions in Intermittent Phenomena

Q-Phase Transitions in Chaotic Attractors of Differential Equations at Bifurcation Points

Most nonlinear ordinary differential equations exhibit chaotic attractors which have singular local structures at their bifurcation points. By taking the driven damped pendulum and the Duffing equation, such chaotic attractors are studied in terms of the q-phase transitions of a q-weighted average A(q), (-oo<q<oo) of the coarse-grained expansion rates A of nearby orbits along the unstable manifolds. We take their Poincare maps in order to obtain the expansion rates A and their spectrum g\(A) explicitly. It is shown that q-phase transitions occur at crises in the differential equations. Just before the crises, qp-phase transitions occur due to the collisions of the attractors with unstable periodic orbits. Numerical values of the transition points qp thus obtained agree fairly well with theoretical predictions. q.-phase transitions occur just after the crises where the chaotic . attractors are suddenly spread over chaotic repellers. Thus it turns out that the q-phase transitions of A(q) are useful for characterizing the chaotic attractors of differential equations at their bifurcation points.

Singular Local Structures of Chaotic Attractors and q-Phase Transitions of Spatial Scaling Structures

Bifurcations of chaotic attractors, such as the destruction of the Hénon attractor and the saddle-node bifurcation, are shown to produce singular local structures which bring about q-phase transitions of the generalized dimensions D(q), (-∞≪ q ≪ ∞) and the q-weighted average α(q) of the singularity exponents α of the natural invariant measure.

Scaling Structures of Chaotic Attractors and q-Phase Transitions at Crises in the Henon and the Annulus Maps

Singular local structures of the chaotic attractors at the bifurcation points for three types of crises in the Henon and the annulus maps are studied theoretically and numerically in terms of the spectrum heAl of the coarse· grained local expansion rates A of nearby orbits along the local unstable manifold, and are shown to produce linear parts in heAl with slopes qa=2, qp< 1/2 and qT= 1. These linear slopes bring about three types of discontinuous phase transitions of the q-weighted average A(q), (-oo<q<oo) of A at q=qa, qp, qT, respectively, as q is varied. The q-phase transition at q = qa is due to the homoclinic tangencies, whereas that at q = qp is caused by a collision of the chaotic attractor with an unstable periodic orbit, where the slope qp and the singularity exponent a of the natural invariant measure at the periodic orbit are given in terms of the eigenvalues of certain periodic orbits. Just after the merging of two chaotic attractors into one chaotic attractor, a q-phase transition occurs at q=qT due to the intermittent hopping motions between two phase-space regions formerly occupied by the old attractors.

Singular Local Structures of Chaotic Attractors Due to Collisions with Unstable Periodic Orbits in Two-Dimensional Maps

Chaotic attractors exhibit fascinating orbital structures in which complexity and order are locally interwoven in a nested manner. Recently it has turned out that their structures can be characterized by the generalized dimensions D(q),(-co<q<co) and the singularity spectra I(a) of the natural invariant measures.1),2) It has also turned out for two-dimensional maps that D(q) and I(a) are closely related to the q-weighted average A(q) of the coarse-grained local expansion rates A of nearby orbits along the local unstable manifold and their spectrum h(A) so that the structures of chaotic attractors can also be characterized by A(q) and h(A).3H) Indeed, it has been shown that the singular local structures at the bifurcation points of the band mergings and crises bring about discontinuous transitions of A(q) as q is varied, so that their different phases represent different local structures of chaotic attractors.)-9) These q-phase transitions are caused by linear parts in h(A) created by singular local structures. Such a linear part is produced by a collision of the chaotic attractor with an unstable periodic orbit which gives a new maximum Amax of A. In twodimensional maps, such as the Henon and dissipative standard maps, s1,1ch a collision is accompanied by the accumulation of homoclinic or heteroclinic tangency points to the periodic orbie),9)-1l) In this paper we shall formulate the scaling exponent a ) of the natural invariant measure at the periodic orbit and the slope qp= h'(A) of the linear part in terms of the eigenvalues of certain periodic orbits and the order z of the tangency. This will lead to a new kind of relations between two exponents a and A on the periodic orbits. For a chaotic orbit {Xm}, (m=O, 1, 2, ... ) generated by a twoodimensional dissipative map Xm+l=F(Xm), let ,h(Xm) be the local expansion rate of nearby orbits at Xm along the local unstable manifold, i.e., ,h(Xm)= 10gIDF(Xm)ul(Xm)1 with Ul(Xm) being the ,unit vector tangent to the local unstable manifold at X m,3) and, using the temporal coarse-graIning first introduced by Fujisaka,12) define the coarse-grained local expansion rate3H)