Characterization of Local Structures of Chaotic Attractors in Terms of Coarse-Grained Local Expansion Rates

Abstract

809Chaotic attractors at the bifurcation points of band mergings (or splittings), crises and saddle· node bifurcations have singular local structures which produce coherent large fluctuations of the coarse·grained local expansion rates of nearby orbits. Such local structures are studied in terms of a weighted average A(q), (-00< q< (0) of the coarse·grained local expansion rates along the local unstable manifold with a q·dependent weight. By taking invertible two·dimensional and noninverti· ble one·dimensional maps, it is shown that, as q is varied, A(q) exhibits discontinuous phase transi· tions at discrete values of q; qa, qp, .... Three types of such phase transitions are discussed. One is that caused by the homoc1inic tangencies with qa=2.0. The second is that due to the accumulation of homoc1inic tangency points at unstable periodic points with qp"'" -0.8. The third is that due to the intermittent hopping motions between two repellers with qT=LO. Different phases of the phase transitions represent different local structures. Thus the q-weighted average A(q) turns out to provide a useful means for characterizing singular local structures of chaotic attractors. at NERL on June 14, 2015 http://ptp.oxfordjournals.org/ Downloaded from = {l/Zn(q)}<An(Xo)exp{ -n(q-1)A n (X o )}> , (2'12)O"n(q)=-dAn(q)/dq, (2'13) ={n/Zn(q)}<{An(Xo)-An(q)pexp{n(q-1)A n (X o )}> .Since An'(q)~O, An(q) is a monotone non-increasing function of q. O"n(q) is the at NERL on June 14, 2015

Role of unstable periodic orbits in phase transitions of coupled map lattices

“…Since UPOs form the skeleton of the chaotic invariant set [5], this means a system accompanying q-phase transitions has the invariant set with a global anomalous structure. It is worth remarking that q-phase transitions in chaos with few DOFs indicate local singularities of the attractor, where hyperbolicity is lost [6], whereas q-phase transitions in extended chaos treated here signify global ones, that are non-analyticities in the distribution function of UPOs, which arise without losing hyperbolicity. The non-analyticity of the UPO distribution in the 1D Bernoulli CML can be explicitly confirmed if we consider the case k ≪ 1, in which we can refer to the exact solution of the 2D Ising model [29] as is seen in Eq.…”

“…Therefore the 1D Bernoulli CML is shown to exhibit phase transitions in the 2-dimensional space-time. These transitions, brought about by varying the temperature parameters in the thermodynamic formalism, are called q-phase transitions in the context of dynamical systems with few DOFs [6]. Moreover the existence of the q-phase transitions can be analytically shown in the weak-interaction limit k → 0.…”

“…This is not a transition dealt with in statistical mechanics which involves large fluctuations of thermodynamic quantities and occurs only in the thermodynamic limit, but a transition with large dynamical fluctuations of observables which occurs in the long-time limit. It has been shown that the large fluctuations reflect the dynamics and qphase transitions indicate a singular local structure of the chaotic attractor, such as homoclinic tangencies of stable and unstable manifolds and band crises [6].…”

Scaling Structures and Statistical Mechanics of Type I Intermittent Chaos

“…Then it would be important to study a relevant bifurcation of chaos which brings about an eminent change of a chaotic attractor at the bifurcation point. The purpose of the present series of papers 8 ), 9) is, therefore, to characterize the chaotic attractors at the bifurcation points for various bifurcations by the scaling structures, and then elucidate the scaling structures from the orbital structures of the critical attractors.…”

Classification of Collisions of Chaotic Attractors with Unstable Periodic Orbits in Terms of the q-Phase Transitions

“…(4) (5) The spectrum h(A) is given by the Legendre transform h(A)=qA -<lJ=(q) with q=h'(A) for n~=.2) Inserting (2) into (3) and taki~g the most dominant integrand for n~=, we also obtain h(A)=A -¢(A).3)-6) Let us take the Henon mapl8)…”