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1.
U. Tırnaklı C. Tsallis M. L. Lyra 《The European Physical Journal B - Condensed Matter and Complex Systems》1999,11(2):309-315
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 df equals unity for all z in contrast with q which does depend on z, it becomes clear that df plays no major role in the sensitivity to the initial conditions.
Received 5 February 1999 相似文献
2.
X.-G. Chao J. Dai W.-X. Wang D.-R. He 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2006,40(3):423-430
This article reports a sudden chaotic attractor change in a system described by a conservative and
dissipative map concatenation. When the driving parameter passes a critical value, the chaotic
attractor suddenly loses stability and turns into a transient chaotic web. The iterations spend
super-long random jumps in the web, finally falling into several special escaping holes. Once in
the holes, they are attracted monotonically to several periodic points. Following Grebogi, Ott, and
Yorke, we address such a chaotic attractor change as a crisis. We numerically demonstrate
that phase space areas occupied by the web and its complementary set (a fat fractal forbidden net)
become the periodic points' “riddled-like” attraction basins. The basin areas are dominated by
weaker dissipation called “quasi-dissipation”. Small areas, serving as special escape holes, are
dominated by classical dissipation and bound by the forbidden region, but only in each periodic
point's vicinity. Thus the crisis shows an escape from a riddled-like attraction basin. This feature
influences the approximation of the scaling behavior of the crisis's averaged lifetime, which is
analytically and numerically determined as 〈τ〉∝(b-b0)γ, where b0
denotes the control parameter's critical threshold, and γ≃-1.5. 相似文献
3.
A quasi-crisis in a quasi-dissipative system 总被引:3,自引:0,他引:3
X. -M. Wang Y. -M. Wang K. Zhang W. -X. Wang H. Chen Y. -M. Jiang Y. -Q. Lu J. -S. Mao D. -R. He 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2002,19(1):119-124
A system concatenated by two area-preserving maps may be addressed as “quasi-dissipative", since such a system can display
dissipative behaviors. This is due to noninvertibility induced by discontinuity in the system function. In such a system,
the image set of the discontinuous border forms a chaotic quasi-attractor. At a critical control parameter value the quasi-attractor
suddenly vanishes. The chaotic iterations escape, via a leaking hole, to an emergent period-8 elliptic island. The hole is the intersection of the chaotic quasi-attractor and
the period-8 island. The chaotic quasi-attractor thus changes to chaotic quasi-transients. The scaling behavior that drives
the quasi-crisis has been investigated numerically.
Received 29 May 2001 and Received in final form 6 November 2001 相似文献
4.
A. Robledo 《Pramana》2005,64(6):947-956
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 λq 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. 相似文献
5.
Dequan Li 《Physics letters. A》2008,372(4):387-393
This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the z-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the z-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation. 相似文献
6.
L. G. Moyano A. P. Majtey C. Tsallis 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,52(4):493-500
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 λM scales as λM ∝
N-κ(α), where κ(α) is positive when interactions are
long-range, yielding weak chaos in the thermodynamic
limit N↦∞ (hence λM→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 tc scales as tc ∝Nβ(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. 相似文献
7.
Peter Reimann 《Journal of statistical physics》1996,82(5-6):1467-1501
We study one-dimensional single-humped maps near the boundary crisis at fully developed chaos in the presence of additive weak Gaussian white noise. By means of a new perturbation-like method the quasi-invariant density is calculated from the invariant density at the crisis in the absence of noise. In the precritical regime, where the deterministic map may show periodic windows, a necessary and sufficient condition for the validity of this method is derived. From the quasi-invariant density we determine the escape rate, which has the form of a scaling law and compares excellently with results from numerical simulations. We find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz>1. Finally, we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak Gaussian noise. Within this extended class of noises and for single-humped maps with any fixed orderz>0 of the maximum, in the scaling law for the escape rate both the critical exponents and the scaling function are universal. 相似文献
8.
V. Schwämmle F. D. Nobre C. Tsallis 《The European Physical Journal B - Condensed Matter and Complex Systems》2008,66(4):537-546
The stability of q-Gaussian distributions as particular solutions of the
linear diffusion equation and its generalized nonlinear form, , the porous-medium equation, is investigated through both numerical
and analytical approaches. An analysis of the kurtosis of the distributions strongly suggests that an initial q-Gaussian, characterized by an index qi, approaches asymptotically the
final, analytic solution of the porous-medium equation, characterized by an index q, in such a way that the relaxation rule for
the kurtosis evolves in time according to a q-exponential, with a relaxation index qrel ≡qrel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (qi ≥ 5/3) into a finite-variance
one (q < 5/3), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light
on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary
states to the ultimate thermal equilibrium
state. 相似文献
9.
G. Vilasi 《The European Physical Journal B - Condensed Matter and Complex Systems》2002,30(2):207-210
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 相似文献
10.
We present an analytic perturbative method for calculatingf(α) and the generalized dimensionD
q
of the critical invariant circle of the polynomial circle map. The scaling behaviour is found to depend onz, the exponent defining the map. The asymptotic bounds of the scaling constantsα(z) andδ(z) are verified analytically. 相似文献
11.
Systems of strongly coupled chaotic maps generically exhibit collective behavior emerging out of extensive chaos. We show how the well-known renormalization group (RG) of unimodal maps can be extended to the coupled systems, and in particular to coupled map lattices (CMLs) with local diffusive coupling. The RG relation derived for CMLs is nonperturbative, i.e., not restricted to a particular class of configurations nor to some vanishingly small region of parameter space. After defining the strong-coupling limit in which the RG applies to almost all asymptotic solutions, we first present the simple case of coupled tent maps. We then turn to the general case of unimodal maps coupled by diffusive coupling operators satisfying basic properties, extending the formal approach developed by Collet and Eckmann for single maps. We finally discuss and illustrate the general consequences of the RG: CMLs are shown to share universal properties in the space-continuous limit which emerges naturally as the group is iterated. We prove that the scaling properly ties of the local map carry to the coupled systems, with an additional scaling factor of length scales implied by the synchronous updating of these dynamical systems. This explains various scaling laws and self-similar features previously observed numerically. 相似文献
12.
D. Braun 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2000,11(1):3-12
I investigate the propagator of the Wigner function for a dissipative chaotic quantum map. I show that a small amount of dissipation
reduces the propagator of sufficiently smooth Wigner functions to its classical counterpart, the Frobenius-Perron operator,
if . Several consequences arise: the Wigner transform of the invariant density matrix is a smeared out version of the classical
strange attractor; time dependent expectation values and correlation functions of observables can be evaluated via hybrid quantum-classical formulae in which the quantum character enters only via the initial Wigner function. If a classical phase-space distribution is chosen for the latter or if the map is iterated sufficiently
many times the formulae become entirely classical, and powerful classical trace formulae apply.
Received 7 October 1999 相似文献
13.
T. Tél 《Zeitschrift für Physik B Condensed Matter》1982,49(2):157-160
An equation is proposed for describing stable and unstable manifolds for a wide class of two-dimensional invertible maps. Several branches of the stable and unstable manifolds of the dissipative mapx
n+1
=1–a|x
n
|+bz
n
,z
n+1
=x
n
are constructed explicitly. The limiting case when the strange attractor disappears is discussed. 相似文献
14.
X.-M. Wang Z.-J. Fang J.-F. Zhang 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2006,37(3):417-422
The multiple Devil's staircase, which describes phase-locking
behavior, is observed in a discontinuous nonlinear circle map.
Phase-locked steps form many towers with similar structure in
winding number(W)-parameter(k) space. Each step belongs to a
certain period-adding sequence that exists in a smooth curve. The
Collision modes that determine steps and the sequence of mode
transformations create a variety of tower structures and their
particular characteristics. Numerical results suggest a scaling
law for the width of phase-locked steps in the
period-adding (W=n/(n+i), n,i∈int) sequences, that is,
Δk(n)∝n-τ (τ>0). And the study indicates
that the multiple Devil's staircase
may be common in a class of discontinuous circle maps. 相似文献
15.
Haibo Xu Guangrui Wang Shigang Chen 《The European Physical Journal B - Condensed Matter and Complex Systems》2001,22(1):65-69
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 相似文献
16.
A.N. Pisarchik R. Meucci F.T. Arecchi 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2001,13(3):385-391
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 相似文献
17.
A. Bershadskii T. Nakano D. Fukayama T. Gotoh 《The European Physical Journal B - Condensed Matter and Complex Systems》2000,18(1):95-101
Using results of a direct numerical simulation (DNS) of 3D turbulence we show that the observed generalized scaling (i.e. scaling moments versus moments of different orders) is consistent with a lognormal-like distribution of turbulent energy dissipation fluctuations
with moderate amplitudes for all space scales available in this DNS (beginning from the molecular viscosity scale up to largest ones). Local multifractal thermodynamics has been developed to interpret the data obtained using the generalized scaling, and a new interval
of space scales with inverse cascade of generalized energy has been found between dissipative and inertial intervals of scales
for sufficiently large values of the Reynolds number.
Received 21 July 2000 相似文献
18.
Buncha Munmuangsaen 《Physics letters. A》2009,373(44):4038-4043
A new chaotic attractor is presented with only five terms in three simple differential equations having fewer terms and simpler than those of existing seven-term or six-term chaotic attractors. Basic dynamical properties of the new attractor are demonstrated in terms of equilibria, Jacobian matrices, non-generalized Lorenz systems, Lyapunov exponents, a dissipative system, a chaotic waveform in time domain, a continuous frequency spectrum, Poincaré maps, bifurcations and forming mechanisms of its compound structures. 相似文献
19.
J.P. Salas M. I narrea A.I. Pascual 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2002,20(1):45-54
In the frame work of classical mechanics, we study the nonlinear dynamics of a single ion trapped in a Penning trap perturbed
by an electrostatic sextupolar perturbation. The perturbation is caused by a deformation in the configuration of the electrodes.
By using a Hamiltonian formulation, we obtain that the system is governed by three parameters: the z-component of the canonical angular momentum P
φ - which is a constant of the motion because the perturbation we assume is axial-symmetric -, the parameter δ that determines
the ratio between the axial and the cyclotron frequencies, and the parameter a which indicates how far from the ideal design the electrodes are. We study the case P
φ = 0. By means of surfaces of section, we show that the phase space structure is made of three fundamental families of orbits:
arch, loop and box orbits. The coexistence of these kinds of orbits depends on the parameter δ. The escape is also explained on the basis of
the shape of the potential energy surface as well as of the phase space structure.
Received 6 September 2001 / Received in final form 19 March 2002 Published online 28 June 2002 相似文献
20.
M. Agop P. E. Nica P. D. Ioannou A. Antici V. P. Paun 《The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics》2008,49(2):239-248
The scale relativity model was extended for the motions on fractal curves of fractal dimension D F and third order terms in the equation of motion of a complex speed field. It results that, in a fractal fluid, the convection, dissipation and dispersion are compensating at any scale (differentiable or non-differentiable), whereas a generalized Schrödinger type equation is obtained for an irrotational movement of the fractal fluid. For D F = 2 and the dissipative approximation of the motions, the fractal model of atom is build: the real part of the complex speed field describes the electron motion on stationary orbits according to a quantification condition, while the imaginary part of the complex speed field gives the electron energy quantification. For D F = 3 and the dispersive approximation of motions, some properties of the matter are explained: at the differentiable scale the flowing regimes (non-quasi-autonomous and quasi-autonomous) of the fractal fluids are separated by the experimental “0.7 structure”, while for the non-differentiable scale the fractal potential acts as an energy accumulator and controls through coherence the transport phenomena. Moreover, the compatibility between the differentiable and non-differentiable scales implies a Cantor space-time, and consequently a fractal at any scale. Thus, some properties of the matter (the anomaly of nano-fluids thermal conductivity, the superconductivity etc.) can be explained by this model. 相似文献