Afraimovich–Pesin dimension for Poincaré recurrences in one- and two-dimensional deterministic and noisy chaotic maps

The relation between Lyapunov exponents, the Kolmogorov–Sinai entropy (KS-entropy) and the Afraimovich–Pesin dimension (AP-dimension) has been numerically analyzed in one- and two-dimensional chaotic maps. In our simulations we show that without noise the AP-dimension corresponds to the KS-entropy. In the presence of noise, the AP-dimension corresponds to the relative metric entropy. Since in a deterministic case the relative metric entropy corresponds to the KS-entropy the obtained results enable us to conclude that for considered chaotic maps of different dimension the AP-dimension corresponds to the relative metric entropy in both deterministic and stochastic cases.

► Afraimovich–Pesin dimension does not correspond to the Lyapunov exponent in one- and two-dimensional chaotic maps with noise. ► Afraimovich–Pesin dimension corresponds to the relative metric entropy in one- and two-dimensional chaotic maps with noise. ► Afraimovich–Pesin dimension corresponds to the Kolmogorov–Sinai entropy in one- and two-dimensional chaotic maps without noise.