Singular continuous spectra in dissipative dynamics

We demonstrate the occurrence of regimes with singular continuous (fractal) 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 creation of complicated homoclinic orbits. Two different mechanisms responsible for the appearance of such spectra are proposed. In the first case, when the geometry of the attractor is symbolically represented by the Thue-Morse sequence, both the continuous-time process and its discrete Poincar\'e map have singular power spectra. The other mechanism is due to the logarithmic divergence of the first return times near the saddle point; here the Poincar\'e map possesses the discrete spectrum, while the continuous-time process displays the singular one. A method is presented for computing the multifractal characteristics of the singular continuous spectra with the help of the usual Fourier analysis technique.