Narrow Results

In this paper, the dynamics of maps representing classes of controlled sampled systems with backlash are examined. First, a bilinear one-dimensional map is considered, and the analysis shows that, depending on the value of the control parameter, all orbits originating in an attractive set are either periodic or dense on the attractor. Moreover, the dense orbits have sensitive dependence on initial data, but behave rather regularly, i.e. they have quasiperiodic subsequences and the Lyapunov exponent of every orbit is zero. The inclusion of a second parameter, the processing delay, in the model leads to a piecewise linear two-dimensional map. The dynamics of this map are studied using numerical simulations which indicate similar behavior as in the one-dimensional case.

We study numerically statistical distributions of sums of orbit coordinates, viewed as independent random variables in the spirit of the Central Limit Theorem, in weakly chaotic regimes associated with the excitation of the first (k = 1) and last (k = N) linear normal modes of the Fermi–Pasta–Ulam-α system under fixed boundary conditions. We show that at low energies (E = 0.19), when k = 1 linear mode is excited, chaotic diffusion occurs characterized by distributions that are well approximated for long times (t > 10⁹) by a q-Gaussian Quasi-Stationary State (QSS) with q ≈ 1.4. On the other hand, when k = N mode is excited at the same energy, diffusive phenomena are absent and the motion is quasi-periodic. In fact, as the energy increases to E = 0.3, the distributions in the former case pass through shorterq-Gaussian states and tend rapidly to a Gaussian (i.e. q → 1) where equipartition sets in, while in the latter we need to reach up to E = 4 to see a sudden transition to Gaussian statistics, without any passage through an intermediate QSS. This may be explained by different energy localization properties and recurrence phenomena in the two cases, supporting the view that when the energy is placed in the first mode weak chaos and "sticky" dynamics lead to a more gradual process of energy sharing, while strong chaos and equipartition appear abruptly when only the last mode is initially excited.

We present strong numerical evidence for the existence of “islands” (small, localized, disjoint regions of support) for a two-dimensional map which was studied before for various ranges of parameters. Once this evidence is accepted, it leads to a combinatorial computer-assisted proof for the existence of an ACIM supported on all 175 of these islands.