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Register a new userAsymptotic expansions for the escape rate of stochastically perturbed unimodal maps
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The escape rate of a stochastic dynamical system can be found as an expansion in powers of the noise strength. In previous work the coefficients of such an expansion for a one-dimensional map were fitted to a general form containing a few parameters. These parameters were found to be related to the fractal structure of the repeller of the system. The parameter alpha, the noise dimension, remains to be interpreted. This report presents new data for alpha showing that the relation to the dimensions is more complicated than predicted in earlier work and oscillates as a function of the map parameter, in contrast to other dimension-like quantities.
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The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed to have Gaussian distribution. In this paper, we address the problem of escape induced by bounded noise. We show that the dynamics of escape from an attractors basin is equivalent to that of a closed system with an appropriately chosen hole. Using this equivalence, we show that there is a minimum noise amplitude above which escape takes place, and we derive analytical expressions for the scaling of the escape rate with noise amplitude near the escape transition. We verify our analytical predictions through numerical simulations of a two-dimensional map with noise.
The breeding method is a computationally cheap procedure to generate initial conditions for ensemble forecasting which project onto relevant synoptic growing modes. Ensembles of bred vectors, however, often lack diversity and align with the leading Lyapunov vector, which severely impacts their statistical reliability. In previous work we developed stochastically perturbed bred vectors (SPBVs) and random draw bred vectors (RDBVs) in the context of multi-scale systems. Here we explore when this method can be extended to systems without scale separation, and examine the performance of the stochastically modified bred vectors in the single scale Lorenz 96 model. In particular, we show that the performance of SPBVs crucially depends on the degree of localisation of the bred vectors. It is found that, contrary to the case of multi-scale systems, localisation is detrimental for applications of SPBVs in systems without scale-separation when initialised from assimilated data. In the case of weakly localised bred vectors, however, ensembles of SPBVs constitute a reliable ensemble with improved ensemble forecasting skills compared to classical bred vectors, while still preserving the low computational cost of the breeding method. RDBVs are shown to have superior forecast skill and form a reliable ensemble in weakly localised situations, but in situations when they are strongly localised they do not constitute a reliable ensemble and are over-dispersive.
We consider several generalizations of the classical $gamma$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for
We present analytical expressions for the time-dependent and stationary probability distributions corresponding to a stochastically perturbed one-dimensional flow with critical points, in two physically relevant situations: delayed evolution, in which the flow alternates with a quiescent state in which the variate remains frozen at its current value for random intervals of time; and interrupted evolution, in which the variate is also re-set in the quiescent state to a random value drawn from a fixed distribution. In the former case, the effect of the delay upon the first passage time statistics is analyzed. In the latter case, the conditions under which an extended stationary distribution can exist as a consequence of the competition between an attractor in the flow and the random re-setting are examined. We elucidate the role of the normalization condition in eliminating the singularities arising from the unstable critical points of the flow, and present a number of representative examples. A simple formula is obtained for the stationary distribution and interpreted physically. A similar interpretation is also given for the known formula for the stationary distribution in a full-fledged dichotomous flow.
We numerically investigate the sensitivity to initial conditions of asymmetric unimodal maps $x_{t+1} = 1-a|x_t|^{z_i}$ ($i=1,2$ correspond to $x_t>0$ and $x_t<0$ respectively, $z_i >1$, $0<aleq 2$, $t=0,1,2,...$) at the edge of chaos. We employ three distinct algorithms to characterize the power-law sensitivity to initial conditions at the edge of chaos, namely: direct measure of the divergence of initially nearby trajectories, the computation of the rate of increase of generalized nonextensive entropies $S_q$ and multifractal analysis. The first two methods provide consistent estimates for the exponent governing the power-law sensitivity. In addition to this, we verify that the multifractal analysis does not provide precise estimates of the singularity spectrum $f(alpha)$, specially near its extremal points. Such feature prevents to perform a fine check of the accuracy of the scaling relation between $f(alpha)$ and the entropic index $q$, thus restricting the applicability of the multifractal analysis for studing the sensitivity to initial conditions in this class of asymmetric maps.
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