The Forecast of Predictability and Instability in Physical Modelling

the last century, the numerical approach to solving physical problems has gained in relevance with the increase of computational facilities. Methods derived from chaos theory and nonlinear dynamics techniques are quite useful in solving real problems where chaos is present and a strong dependence on initial conditions is a key issue. Predictability refers to the assessment of the likely errors in a forecast, either qualitatively or quantitatively. It refers directly to the stability/instability of the true orbit, but also to the coincidence of the calculated orbit itself, or reliability. Instability is a very well-known factor when considering a given solution to the model. In a broad sense, an orbit is unstable if it is strongly dependent on initial conditions. More precisely, an orbit is chaotic if the dynamics is bounded, has at least one positive Lyapunov exponent and the omega set is not periodic and does not consist solely of equilibrium points and connecting arcs. The larger the instability, the larger is the likehood that the real orbit will diverge from the calculated one. Typically, the reliability time goes with the inverse of the Lyapunov exponent, or Lyapunov time. As all numerical calculations have inherent inaccuracies, beyond certain timescales even the best method will diverge from the true orbit. The usual steps for analysing a given system involve calculating the instability for a given set of initial conditions. It happens that the standard definition of the Lyapunov exponent has a very long convergence time (if any). Due to the slow convergence of the asymptotic value, many others numerical indexes are used, but the basic idea remains the same. However, we can also refer to the numerical reliability of the system understood as the confidence of the computed orbit and the true orbit during an interval of time, independently of its instability. Essentially, computers move from one true orbit to another nearby orbit because of rounding errors and internal floating number representations. However, the computed orbit (called pseudo-orbit) may still lead to correct predictions because of the existence of a nearby exact solution. Otherwise, the calculated orbit may be very distant. The true orbit is called a shadow, and the existence of shadow orbits is a very strong property, with the shadowing time being a valid limit for the predictability of the system. A basic requirement for shadowing is that the system be hyperbolic. In case of nonhy-perbolicity, the point may …