On the forward dynamical behavior of nonautonomous systems

Abstract

This paper is concerned with the forward dynamical behavior of nonautonomous systems. Under some general conditions, it is shown that in an arbitrary small neighborhood of a pullback attractor of a nonautonomous system, there exists a family of sets {Aε(p)} p∈P of phase space X, which is forward invariant such that {Aε(p)} p∈P uniformly forward attracts each bounded subset of X. Furthermore, we can also prove that {Aε(p)} p∈P forward attracts each bounded set at an exponential rate.

Model reduction for constrained mechanical systems via spectral submanifolds

“…For an efficient extraction of the FRS, fast computation of periodic orbits is necessary. Periodic orbits of low-dimensional nonlinear systems can be obtained via various numerical methods such as numerical integration, shooting methods [12,13], collocation schemes [14] and harmonic balance techniques [15]. However, the computational costs of these methods are prohibitive for high-dimensional systems such as finite element (FE) models [16,17].…”

“…In practice, some samples of Ω near the external resonance can be taken and SSM solutions zptq in ( 22) and ( 20) can be compared to decide whether TI SSM solutions are sufficient or TV SSM solutions are required. Substituting relations ( 20), (13), and ( 14) or ( 16) into (7), the functional A L 2 pzptqq can be simplified as…”

“…More details about the expansion and consolidation stages can be found in Chapter 13 of [14] (see Figs. 13.2-13.3 of [14] for schematic plots).…”

“…As an alternative, we use a shooting method combined with parameter continuation to calculate the FRCs of the full system. Specifically, we use a coco-based shooting toolbox [13] for validation, where the Newmark scheme is used for the forward simulation. We use 1,000 time steps per excitation period for numerical integration [17].…”

“…We now validate the FRS obtain via our SSM-based ROM against the FRCs of the full system for the forc- ing amplitude samples ϵ P t0.02, 0.06, 0.1u. Similarly to the previous example, we use the shooting method combined with parameter continuation [13] to calculate the FRCs of the full system. As seen in Fig.…”