J. Kamphorst Leal da Silva

RELAXATION AND TRANSIENTS IN A TIME-DEPENDENT LOGISTIC MAP

We study the one-dimensional logistic map with control parameter perturbed by a small periodic function. In the pure constant case, scaling arguments are used to obtain the exponents related to the relaxation of the trajectories at the exchange of stability, period-doubling and tangent bifurcations. In particular, we evaluate the exponent z which describes the divergence of the relaxation time τ near a bifurcation by the relation τ ~ | R - Rc |-z. Here, R is the control parameter and Rc is its value at the bifurcation. In the time-dependent case new attractors may appear leading to a different bifurcation diagram. Beside these new attractors, complex attractors also arise and are responsible for transients in many trajectories. We obtain, numerically, the exponents that characterize these transients and the relaxation of the trajectories.

Dynamical properties of a particle in a classical time-dependent potential well

We study numerically the dynamical behavior of a classical particle inside a box potential that contains a square well which depth varies in time. Two cases of time dependence are investigated: periodic and stochastic. The periodic case is similar to the one-dimensional Fermi accelerator model, in the sense that KAM curves like islands surrounded by an ergodic sea are observed for low energy and invariant spanning curves appear for high energies. The ergodic sea, limited by the first spanning curve, is characterized by a positive Lyapunov exponent. This exponent and the position of the lower spanning curve depend sensitively on the control parameter values. In the stochastic case, the particle can reach unbounded kinetic energies. We obtain the average kinetic energy as function of time and of the iteration number. We also show for both cases that the distributions of the time spent by the particle inside the well and the number of successive reflections have a power law tail.

Transients in a time-dependent logistic map

We study the one-dimensional logistic map with parametric perturbation. Using a small periodic function as the perturbation, new attractors may appear. Beside these new attractors, complex attractors exist and are responsible for transients in many trajectories. We observe that each one of these transients is characterized by a power law decay. We find the exponent related to this decay.

On the equivalence of finite size scaling renormalization group and phenomenological renormalization

It is shown that the recently proposed finite size scaling renormalization group, when using systems infinite in one dimension and finite in the others, is equivalent to the Nightingale (correlation length) phenomenological renormalization. The equivalence, however, is concerned only with the critical coupling and thermal critical exponent; the finite size scaling renormalization group approach provides other exponents in a more precise and elegant fashion through the flux diagram of the recursion relations in the space spanned by the parameters of the Hamiltonian. In addition, a new set of magnetic exponents, which are so accurate as the thermal ones, can now be obtained in an easier way.

Some exact results from the mean field renormalization group

Abstract The mean field renormalization group is applied to the study of the directed self-avoiding random walk in d dimensions and to the spin-½ ferromagnetic Ising model on the Bethe lattice. In both cases the known exact results are obtained.

Mean-field renormalization group approach to two-dimensional site directed percolation

Two-dimensional site directed percolation is studied by the mean-field renormalization group approach to bulk and surface critical properties. Extrapolation techniques for the exponents and the percolation threshold are described. A very good estimate of the percolation threshold (pc=0.7055+or-0.0001) and the first evaluation of the surface exponent yhs=0.653+or-1 are obtained.

Mean-field and non-mean-field behaviors in scale-free networks with random Boolean dynamics

We study two types of simplified Boolean dynamics over scale-free networks, both with synchronous update. Assigning only Boolean functions AND and XOR to the nodes with probability

The kinetics of fragmentation processes with mass loss

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Renormalization group study of the two-dimensional site-diluted Ising model

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On the critical dynamics of one-dimensional Potts models

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