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 $\gamma=12.2066409...$ 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 $4$ 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 $320$ 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