Studying hyperbolicity in chaotic systems

Hyperbolic chaotic attractor in amplitude dynamics of coupled self-oscillators with periodic parameter modulation

“…To verify the hyperbolicity of the attractor, we apply the numerical approach (suggested e.g., in Refs. [34][35][36]). In this procedure, the angles between the directions of growth of small perturbations are calculated forward and backward in time at points of a reference trajectory.…”

Statistical properties of dynamical chaos

“…Stable and unstable manifolds of the attractor trajectory intersect transversally. This property is preserved in the presence of small external noise [59]. However, the Lorenz attractor demonstrates a bifurcational transition into the nonhyperbolicity mode [61].…”

“…In this case the continuous limit transition D → 0 does not exist for the probability density of noisy nonhyperbolic systems [14]. Moreover, the probability characteristics of nonhyperbolic chaos are very sensitive to even the slightest changes of the system parameters [13,42,59,73].…”

“…For three-dimensional systems the procedure becomes more complicated because in this case manifolds are twodimensional. In [59], we proposed a method for diagnosing the hyperbolicity for three-dimensional differential systems. It was experimentally established that such systems as the Rössler system, Chua's circuit [60], and the Anishchenko-Astakhov oscillator are typically nonhyperbolic ones; that is, they are structurally unstable [11].…”

“…Our numeric results have also shown that the presence of small additive noise sources does not significantly affect the structure of angle probability distributions for either nearly hyperbolic and nonhyperbolic attractors. The character of a chaotic set remains the same under influence of noise [59].…”