49 research outputs found
Generalized dimensions of Feigenbaum's attractor from renormalization-group functional equations
A method is suggested for the computation of the generalized dimensions of
fractal attractors at the period-doubling transition to chaos. The approach is
based on an eigenvalue problem formulated in terms of functional equations,
with a coefficient expressed in terms of Feigenbaum's universal fixed-point
function. The accuracy of the results is determined only by precision of the
representation of the universal function.Comment: 6 pages, 2 table
Directed transport of two interacting particles in a washboard potential
We study the conservative and deterministic dynamics of two nonlinearly
interacting particles evolving in a one-dimensional spatially periodic
washboard potential. A weak tilt of the washboard potential is applied biasing
one direction for particle transport. However, the tilt vanishes asymptotically
in the direction of bias. Moreover, the total energy content is not enough for
both particles to be able to escape simultaneously from an initial potential
well; to achieve transport the coupled particles need to interact
cooperatively. For low coupling strength the two particles remain trapped
inside the starting potential well permanently. For increased coupling strength
there exists a regime in which one of the particles transfers the majority of
its energy to the other one, as a consequence of which the latter escapes from
the potential well and the bond between them breaks. Finally, for suitably
large couplings, coordinated energy exchange between the particles allows them
to achieve escapes -- one particle followed by the other -- from consecutive
potential wells resulting in directed collective motion. The key mechanism of
transport rectification is based on the asymptotically vanishing tilt causing a
symmetry breaking of the non-chaotic fraction of the dynamics in the mixed
phase space. That is, after a chaotic transient, only at one of the boundaries
of the chaotic layer do resonance islands appear. The settling of trajectories
in the ballistic channels associated with transporting islands provides
long-range directed transport dynamics of the escaping dimer
The effect of noise on the dynamics of a complex map at the period-tripling accumulation point
As shown recently (O.B.Isaeva et al., Phys.Rev E64, 055201), the phenomena
intrinsic to dynamics of complex analytic maps under appropriate conditions may
occur in physical systems. We study scaling regularities associated with the
effect of additive noise upon the period-tripling bifurcation cascade
generalizing the renormalization group approach of Crutchfield et al.
(Phys.Rev.Lett., 46, 933) and Shraiman et al. (Phys.Rev.Lett., 46, 935),
originally developed for the period doubling transition to chaos in the
presence of noise. The universal constant determining the rescaling rule for
the intensity of the noise in period-tripling is found to be
Numerical evidence of the expected scaling is
demonstrated.Comment: 9 pages, 4 figure
Emergence of continual directed flow in Hamiltonian systems
We propose a minimal model for the emergence of a directed flow in autonomous
Hamiltonian systems. It is shown that internal breaking of the spatio-temporal
symmetries, via localised initial conditions, that are unbiased with respect to
the transporting degree of freedom, and transient chaos conspire to form the
physical mechanism for the occurrence of a current. Most importantly, after
passage through the transient chaos, trajectories perform solely regular
transporting motion so that the resulting current is of continual ballistic
nature. This has to be distinguished from the features of transport reported
previously for driven Hamiltonian systems with mixed phase space where
transport is determined by intermittent behaviour exhibiting power-law decay
statistics of the duration of regular ballistic periods
Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4
We gain tight rigorous bounds on the renormalisation fixed point for period
doubling in families of unimodal maps with degree critical point. We use a
contraction mapping argument to bound essential eigenfunctions and eigenvalues
for the linearisation of the operator and for the operator controlling the
scaling of added noise. Multi-precision arithmetic with rigorous directed
rounding is used to bound operations in a space of analytic functions yielding
tight bounds on power series coefficients and universal constants to over
significant figures.Comment: 15 pages, 8 figure
From collective periodic running states to completely chaotic synchronised states in coupled particle dynamics
We consider the damped and driven dynamics of two interacting particles
evolving in a symmetric and spatially periodic potential. The latter is exerted
to a time-periodic modulation of its inclination. Our interest is twofold:
Firstly we deal with the issue of chaotic motion in the higher-dimensional
phase space. To this end a homoclinic Melnikov analysis is utilised assuring
the presence of transverse homoclinic orbits and homoclinic bifurcations for
weak coupling allowing also for the emergence of hyperchaos. In contrast, we
also prove that the time evolution of the two coupled particles attains a
completely synchronised (chaotic) state for strong enough coupling between
them. The resulting `freezing of dimensionality' rules out the occurrence of
hyperchaos. Secondly we address coherent collective particle transport provided
by regular periodic motion. A subharmonic Melnikov analysis is utilised to
investigate persistence of periodic orbits. For directed particle transport
mediated by rotating periodic motion we present exact results regarding the
collective character of the running solutions entailing the emergence of a
current. We show that coordinated energy exchange between the particles takes
place in such a manner that they are enabled to overcome - one particle
followed by the other - consecutive barriers of the periodic potential
resulting in collective directed motion
Self-similar correlations in a barrier billiard
Abstract We give a renormalization analysis of the self-similarity of autocorrelation functions in symmetric barrier billiards for golden mean trajectories. For the special case of a half-barrier we present a rigorous calculation of the asymptotic height of the main peaks in the autocorrelation function. Fundamental to this work is a detailed analysis of a functional recurrence equation which has previously been used in the analysis of fluctuations in the Harper equation and of correlations in strange non-chaotic attractors and in quantum two-level systems
Directed current in the Holstein system
We propose a mechanism to rectify charge transport in the semiclassical
Holstein model. It is shown that localised initial conditions, associated with
a polaron solution, in conjunction with a nonreversion symmetric static
electron on-site potential constitute minimal prerequisites for the emergence
of a directed current in the underlying periodic lattice system. In particular,
we demonstrate that for unbiased spatially localised initial conditions,
violation of parity prevents the existence of pairs of counter-propagating
trajectories, thus allowing for a directed current despite the
time-reversibility of the equations of motion. Occurrence of long-range
coherent charge transport is demonstrated