On the concept of complexity in random dynamical systems

We introduce a measure of complexity in terms of the average number of bits per time unit necessary to specify the sequence generated by the system. In random dynamical system, this indicator coincides with the rate K of divergence of nearby trajectories evolving under two different noise realizations. The meaning of K is discussed in the context of the information theory, and it is shown that it can be determined from real experimental data. In presence of strong dynamical intermittency, the value of K is very different from the standard Lyapunov exponent computed considering two nearby trajectories evolving under the same randomness. However, the former is much more relevant than the latter from a physical point of view as illustrated by some numerical computations for noisy maps and sandpile models.Comment: 35 pages, LaTe

Stochastic Resonance in Deterministic Chaotic Systems

We propose a mechanism which produces periodic variations of the degree of predictability in dynamical systems. It is shown that even in the absence of noise when the control parameter changes periodically in time, below and above the threshold for the onset of chaos, stochastic resonance effects appears. As a result one has an alternation of chaotic and regular, i.e. predictable, evolutions in an almost periodic way, so that the Lyapunov exponent is positive but some time correlations do not decay.Comment: 9 Pages + 3 Figures, RevTeX 3.0, sub. J. Phys.

Lack of self-average in weakly disordered one dimensional systems

We introduce a one dimensional disordered Ising model which at zero temperature is characterized by a non-trivial, non-self-averaging, overlap probability distribution when the impurity concentration vanishes in the thermodynamic limit. The form of the distribution can be calculated analytically for any realization of disorder. For non-zero impurity concentration the distribution becomes a self-averaging delta function centered on a value which can be estimated by the product of appropriate transfer matrices.Comment: 17 pages + 5 figures, TeX dialect: Plain TeX + IOP macros (included

Statistical Mechanics of Shell Models for 2D-Turbulence

We study shell models that conserve the analogues of energy and enstrophy, hence designed to mimic fluid turbulence in 2D. The main result is that the observed state is well described as a formal statistical equilibrium, closely analogous to the approach to two-dimensional ideal hydrodynamics of Onsager, Hopf and Lee. In the presence of forcing and dissipation we observe a forward flux of enstrophy and a backward flux of energy. These fluxes can be understood as mean diffusive drifts from a source to two sinks in a system which is close to local equilibrium with Lagrange multipliers (``shell temperatures'') changing slowly with scale. The dimensional predictions on the power spectra from a supposed forward cascade of enstrophy, and from one branch of the formal statistical equilibrium, coincide in these shell models at difference to the corresponding predictions for the Navier-Stokes and Euler equations in 2D. This coincidence have previously led to the mistaken conclusion that shell models exhibit a forward cascade of enstrophy.Comment: 25 pages + 9 figures, TeX dialect: RevTeX 3.

Predictability in Systems with Many Characteristic Times: The Case of Turbulence

In chaotic dynamical systems, an infinitesimal perturbation is exponentially amplified at a time-rate given by the inverse of the maximum Lyapunov exponent $\lambda$. In fully developed turbulence, $\lambda$ grows as a power of the Reynolds number. This result could seem in contrast with phenomenological arguments suggesting that, as a consequence of `physical' perturbations, the predictability time is roughly given by the characteristic life-time of the large scale structures, and hence independent of the Reynolds number. We show that such a situation is present in generic systems with many degrees of freedom, since the growth of a non-infinitesimal perturbation is determined by cumulative effects of many different characteristic times and is unrelated to the maximum Lyapunov exponent. Our results are illustrated in a chain of coupled maps and in a shell model for the energy cascade in turbulence.Comment: 24 pages, 10 Postscript figures (included), RevTeX 3.0, files packed with uufile

Characterization of chaos in random maps

We discuss the characterization of chaotic behaviours in random maps both in terms of the Lyapunov exponent and of the spectral properties of the Perron-Frobenius operator. In particular, we study a logistic map where the control parameter is extracted at random at each time step by considering finite dimensional approximation of the Perron-Frobenius operatorComment: Plane TeX file, 15 pages, and 5 figures available under request to [email protected]

Characterization of a periodically driven chaotic dynamical system

We discuss how to characterize the behavior of a chaotic dynamical system depending on a parameter that varies periodically in time. In particular, we study the predictability time, the correlations and the mean responses, by defining a local--in--time version of these quantities. In systems where the time scale related to the time periodic variation of the parameter is much larger than the ``internal'' time scale, one has that the local quantities strongly depend on the phase of the cycle. In this case, the standard global quantities can give misleading information.Comment: 15 pages, Revtex 2.0, 8 figures, included. All files packed with uufile

Superfast front propagation in reactive systems with anomalous diffusion

We study a reaction diffusion system where we consider a non-gaussian process instead of a standard diffusion. If the process increments follow a probability distribution with tails approaching to zero faster than a power law, the usual qualitative behaviours of the standard reaction diffusion system, i.e., exponential tails for the reacting field and a constant front speed, are recovered. On the contrary if the process has power law tails, also the reacting field shows power law tail and the front speed increases exponentially with time. The comparison with other reaction-transport systems which exhibit anomalous diffusion shows that, not only the presence of anomalous diffusion, but also the detailed mechanism, is relevant for the front propagation.Comment: 4 pages and 4 figure

Exact solution of a 2d random Ising model

The model considered is a d=2 layered random Ising system on a square lattice with nearest neighbours interaction. It is assumed that all the vertical couplings are equal and take the positive value J while the horizontal couplings are quenched random variables which are equal in the same row but can take the two possible values J and J-K in different rows. The exact solution is obtained in the limit case of infinite K for any distribution of the horizontal couplings. The model which corresponds to this limit can be seen as an ordinary Ising system where the spins of some rows, chosen at random, are frozen in an antiferromagnetic order. No phase transition is found if the horizontal couplings are independent random variables while for correlated disorder one finds a low temperature phase with some glassy properties.Comment: 10 pages, Plain TeX, 3 ps figures, submitted to Europhys. Let

Micro market based optimisation framework for decentralised management of distributed flexibility assets

Continuously changing electricity demand and intermittent renewable energy sources pose challenges to the operation of power systems. An alternative to reinforcing the grid infrastructure is to deploy and manage distributed energy storage systems. In this work, a micro-energy market is proposed for smart domestic energy trading in the low-voltage distribution systems in the context of high penetration of photovoltaic systems and battery energy storage systems. In addition, a micro-balancing market is proposed to address the congestions due to unforeseen energy imbalance. Centralised and decentralised management strategies are simulated in real time, based on generation and demand forecasts. In addition, electric vehicles are also simulated as potential storage solutions to improve grid operation. A techno-economic evaluation informs key stakeholders, in particular grid operators on strategies for a sustainable implementation of the proposed strategies. The results show that the micro-energy market reduces the energy cost for all grid users by 4.1–20.2%, depending on their configuration. In addition, voltage deviation, peak electricity demand and reverse power flow have been reduced by 12.8%, 7.7% and 85.6% respectively, with the proposed management strategies. The micro-balancing market has been demonstrated to keep the voltage profile and thermal characteristic within the set limit in case of contingency