An increase in fractal dimension is shown to occur in filtered chaotic signals. The dependence of the Lyapunov dimension on the filter parameters is used to predict the behavior of the information dimension which is directly evaluated for two experimental systems: the NMR laser and Rayleigh-Benard convection. Good quantitative agreement with the theoretical predictions is found. The understanding of the role of filtering not only clarifies aspects relevant in the calculation of fractal dimensions, but also yields an indirect but precise way to evaluate Lyapunov exponents.PACS numbers: 05.45.+b, 47.20.Tg, Experimental chaotic signals are more and more frequently characterized by the evaluation of the fractal dimension of the underlying strange attractor, reconstructed in a suitable embedding space. In order to increase the signal-to-noise ratio, the experimental data are often filtered, with use of either analog or digital techniques. Moreover, the finite instrumental bandwidth of the measuring apparatus may produce similar effects. Here we show that filtering processes introduce additional Lyapunov exponents and may cause an increase of the fractal dimension D(q). The results of direct evaluations of the information dimension for two sets of experimental data are also presented. The increase of D(q) with decreasing filter-bandwidth 77 confirms quantitatively the theoretical predictions. A precise estimate of the Lyapunov exponents is also obtained. Consider a physical system modeled by evolution equations of the form u(f)=F(u), where xx(t) is the state vector in phase space. The nonlinear differentiate function F(u) determines the time behavior of u(/). In the case of an experimental system, a single component x(t) of uG) is usually measured, and its values are recorded as a scalar time series {x(t)\. We concentrate, for simplicity, on an ideal linear low-pass filter, whose action can be described by our adding to the original differential model the further equation(1) where z(t) is the output of the filter and 77 the cutoff frequency. Since z(t) is coupled to the system through x(r), a faithful description of the dynamics can be obtained by the reconstruction of the attractor in an Edimensional "embedding" space through points Zi = {z(ti),z(ti+te), . . . ,z(ti + (E-\)At)}, At being a suitably chosen sampling time. If we indicate with X\ > X2 > • • • > A,£ the Lyapunov exponents of the original system, the Lyapunov dimension Z)L of the unfiltered attractor is given by Z>L "./ + £&*/1 X/ + i I, where j is the largest index for which the sum over k is nonnegative. x If the filter is present, a new Lyapunov exponent Xj = -77 should be taken into account, while the others remain unaffected, as a result of the form of Eq.(1). As a consequence, the dimension D^ remains unchanged as long as 77 > | A. y -+11 • To discuss what happens when the latter inequality is no longer satisfied, we assume, for simplicity, that only three Lyapunov exponents (one of which is equal to 0) determine the information dimension of the unfil...
