SINGULAR CONTINUOUS SPECTRA IN DISSIPATIVE DYNAMICS

We demonstrate the occurrence of regimes with singular continuous (fra ctal) Fourier spectra in autonomous dissipative dynamical systems. The particular example is an ordinary-differential-equation system at the accumulation points of bifurcation sequences associated with the crea tion of complicated homoclinic orbits. Two different mechanisms respon sible for the appearance of such spectra are proposed. In the first ca se, when the geometry of the attractor is symbolically represented by the Thue-Morse sequence, both the continuous-time process and its disc rete Poincare map have singular power spectra. The other mechanism is due to the logarithmic divergence of the first return times near the s addle point; here the Poincare map possesses the discrete spectrum, wh ile the continuous-time process displays the singular one. A method is presented for computing the multifractal characteristics of the singu lar continuous spectra with the help of the usual Fourier analysis tec hnique.