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Dynamic stability of beams under axial forces – Lyapunov exponents for general fluctuating loads
Published in W. B. Krätzig, O. T. Bruhns, H. L. Jessberger, K. Meskouris, H.-J. Niemann, G. Schmid, F. Stangenberg, A. N. Kounadis, G. I. Schuëller, Structural Dynamics, 1991
Modern concepts of stability investigations apply the projection on a unit hypersphere. In the two-dimensional case, this projection is performed by introducing polar coordinates by which the displacement and the velocity of the parametric oscillator are replaced by phase and amplitude processes. The phase equation is decoupled from the amplitude process. It is a nonlinear and inhomogeneous equation leading to a stationary rotation in the phase plane, if the damping ratio of the oscillator is less than one. Evaluated in a limited angle range, the phase process possesses an invariant measure which is periodic and nonsingular. According to the multiplicative ergodic theorem of Oseledec (1968), this stationary distribution determines the top Lyapunov exponent of the system and therewith the exponential growth behaviour of the amplitude process.
Overview of dynamical systems and chaos
Published in Marcio Eisencraft, Romis Attux, Ricardo Suyama, Chaotic Signals in Digital Communications, 2018
Numerically, chaos was found by John von Neumann (1903–1957) and Stanislaw M. Ulam (1909–1984) by simulating in a computer the logistic map with p = 4, which was used, in 1947, as a random number generator. The title of their short paper is “On Combination of Stochastic and Deterministic Processes,” because apparently random sequences were obtained from a deterministic equation [44]. In time-continuous systems, numerical chaotic solutions were reported in 1963 by Edward N. Lorenz (1917–2008) in the manuscript “Deterministic Non-periodic Flow,” about a hydrodynamic model for meteorology, which was solved in a computer [22]. In this context, the butterfly effect is a poetic metaphor of sensitivity to initial conditions: the flight of a butterfly certainly causes tiny alterations in the atmosphere; however, if this system produces chaotic solutions, then such minor changes can result in completely different weather scenarios. Observe that, in simulations, rounding and truncation errors always occur; therefore, a numerically calculated chaotic solution will diverge from the true solution with the same initial conditions. Under certain circumstances, the shadowing lemma assures that there is a true solution with slightly different initial coordinates that stays near (“shadows”) the numerically computed solution (e.g., [16]). Thus, the characterization of a chaotic system from data obtained via simulations can be valid. This characterization is usually based on its spatio-temporal “statistical” properties (as in truly stochastic processes). Some tools employed are analysis of power spectrum density, entropy derived from Information Theory, and invariant measure provided by Ergodic Theory (e.g., [11]). Other tools are Lyapunov exponent and Hausdorff dimension, which will be defined in the next sections.
Discrete spectrum for group actions
Published in Dynamical Systems, 2023
The study of complex functions can be traced back to the work of Morse and Hedlund [3], who studied the complex functions of symbolic systems and proved that the boundedness of the complex functions is equivalent to the ultimate period of the system. By the well-known Halmos-von Neumann Representation Theorem [2], an ergodic system with discrete spectrum is the minimal rotation over a compact abelian metric group. For -actions, Huang et al. [4] introduced the measure complexity of an invariant measure by using the mean metric. They showed that if an invariant measure has discrete spectrum, then the measure complexity of this invariant measure is bounded. Later in [5] Huang, Li, Thouvenot, Xu and Ye proved that the converse statement remains true. Remark that the characterization of discrete spectrum using measure complexity was also proved by Vershik, Zatitskiy and Petrov in [6] via studying the class of so-called admissible metrics in a Lebesgue space. For amenable group actions, Yu et al. [7] introduced the measure complexity of an invariant measure by using the mean metric along Følner sequences. They proved that an invariant measure has discrete spectrum if and only if it has bounded measure complexity.
Weighted Hardy’s inequalities and Kolmogorov-type operators
Published in Applicable Analysis, 2019
A. Canale, F. Gregorio, A. Rhandi, C. Tacelli
For Ornstein–Uhlenbeck type operators , , , perturbed by multipolar inverse square potentials a weighted multipolar Hardy inequality and related existence and nonexistence results were stated in [7]. In such a case, the invariant measure for these operators is .