Juan C. Vallejo

WADA BASINS AND UNPREDICTABILITY IN HAMILTONIAN AND DISSIPATIVE SYSTEMS

Prediction is one of the fundamental goals of science. When prediction is lost, it can be thought that one of the foundations of science is shattered. The notion of a chaotic system and the sensitive dependence on initial conditions implies a certain lack of prediction on the time evolution along an orbit. However we do not speak here about this temporal prediction, but about an extreme dependence on the initial conditions that fractal structures in phase space impose, and that obstructs the prediction of the final state of the system.

Characterization of the local instability in the Hénon–Heiles Hamiltonian

Several prototypical distributions of finite-time Lyapunov exponents have been computed for the two-dimensional Henon– Heiles Hamiltonian system. Different shapes are obtained for each dynamical state. Even when an evolution is observed in the morphology of the distributions for the smallest integration intervals, they can still serve for characterizing the dynamical state of the system.  2003 Elsevier Science B.V. All rights reserved.

Predictability of orbits in coupled systems through finite-time Lyapunov exponents

The predictability of an orbit is a key issue when a physical model has strong sensitivity to the initial conditions and it is solved numerically. How close the computed chaotic orbits are to the real orbits can be characterized by the shadowing properties of the system. The finite-time Lyapunov exponents distributions allow us to derive the shadowing timescales of a given system. In this paper we show how to obtain information about the predictability of the orbits even when using arbitrary initial orientation for the initial deviation vectors. As a model to test our results, we use a system of two coupled Rossler oscillators. We analyze the dependence of the shadowing time on the coupling strength and internal nature of the oscillators. The main focus rests on the dependence of these results on the length of the finite-time intervals and the computation of the most appropriate interval for a better forecast. We emphasize the importance of extracting information from all of the relevant exponents to obtain an insight into the sources of the nonhyperbolicity of the system.

Controlling chaos in a fluid flow past a movable cylinder

Abstract The model of a two-dimensional fluid flow past a cylinder is a relatively simple problem with a strong impact in many applied fields, such as aerodynamics or chemical sciences, although most of the involved physical mechanisms are not yet well known. This paper analyzes the fluid flow past a cylinder in a laminar regime with Reynolds number, Re , around 200, where two vortices appear behind the cylinder, by using an appropriate time-dependent stream function and applying non-linear dynamics techniques. The goal of the paper is to analyze under which circumstances the chaoticity in the wake of the cylinder might be modified, or even suppressed. And this has been achieved with the help of some indicators of the complexity of the trajectories for the cases of a rotating cylinder and an oscillating cylinder.

Forecasting and Chaos

Forecasting is the process of making predictions of the future based on past and present data. One of the key questions of the scientific method is the possibility of making predictions using a model and to confront them with new observations as a test of its goodness. It seems rather natural to think that with an adequate increase in numerical computational facilities, the errors could be neglected and that from a set of initial conditions which are known with enough precision, one could predict the future state of a dynamical system. However, every model has inherent inaccuracies leading its results to deviate from the true solution. Furthermore, the computational issues constitute another source of errors. When a system shows a strong sensitivity to the initial conditions, then the system is chaotic. And when the nonlinearity is present, one prediction can be destroyed by an initial error, even by a very small one. Fortunately, the presence of chaos does not always imply a low predictability. An orbit can be chaotic and still be predictable, in the sense that the chaotic orbit is followed, or shadowed, by a real orbit, thus making its predictions physically valid.

Dynamical Regimes and Time Scales

The key factor to build the finite-time distributions is finding the most adequate interval length, to be large enough to ensure a satisfactory reduction of the local fluctuations, but small enough to reveal slow trends. This length is different for every orbit. There are different time scales to take into account in every model, including transient behaviours that could be of interest. By a proper selection of the total integration time, we can characterise the dynamics using small finite-time interval lengths. But increasing those lengths, we see how the distributions stop tracing the flow at local scales and begin to describe the flow at global scales, that is, the global regime.

Role of dark matter haloes on the predictability of computed orbits

Aims. The predictability of a system indicates how often a computed orbit is close to a real orbit of the system, independent of its stability or chaotic nature. We explore the effect of dark halo shapes on the predictability of computed orbits in a Milky Way mean field model. We also present the sources for the low predictability found in some orbits. Methods. We derived a predictability index from the distributions of the finite-time Lyapunov exponents. We computed those distributions and analysed the evolution of their shapes when the finite-time interval sizes are varied. The predictability index can be computed using the interval lengths corresponding to the timescales when the flow dynamics leaves the local regime and enters the global regime. Results. These analyses reveal that not all chaotic orbits have the same predictability and that the predictability of some orbits is more affected than others by the orientation and shape of the dark halo. We show that the lowest predictability may be linked to strong unstable dimension variability.