Limit-quasiperiodic

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

If a finite-dimensional system has an almost-periodic solution, that is not quasi-periodic, then the coefficients An are linear compositions of a finite number of basis frequencies. .., ujrn with rational factors. Such solutions are called limit-quasiperiodic. For this case Pontryagin [112] had proven that the dimension m of the minimal set must satisfy the following inequality... 

In particular, for a system of third order we have m = 1, i.e. its limit-quasiperiodic solutions have the form... 

The structure of the minimal set of a limit-quasiperiodic trajectory is a fractal. In other words, it is characterized locally as a direct product of an m-dimensional disk and a zero-dimensional Cantor set K. Obviously, in the limit-periodic case, it has the form of a direct product of an interval and K. 

As for attractive minimal sets, it follows from Pugh s theorem that they are structurally unstable. Although the minimal sets composed of recurrent and limit-quasiperiodic orbits are by far not key players in the nonlinear dynamics, quasiperiodic motions have always been of major interest because they model many oscillating phenomena having a discrete spectrum. 

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

We apply Eq. (4.111) to the case of coherent modulation of quasiperiodic form (see Eq. (4.56)). Without a limitation of the generality, we can assume that = 1. We then find, using Eq. (4.111), that the rates Re g)(f) tend to the long-time limits... 

One can define fhe long-time limit of the quasiperiodic modulation, when... 

It is readily seen that the set of equations (76) consists of three equations of motion in the real variables ReIm c, w. If, (x) = constant, chaos in the system does not appear since the set (76) becomes a two-dimensional autonomous system. The maximal Lyapunov exponents for the systems (75) and (72)-(74) plotted versus the pulse duration T are presented in Fig. 36. We note that within the classical system (75) by fluently varying the length of the pulse T, we turn order into chaos and chaos into order. For 0 < T < 0.84 and 1.08 < 7) < 7.5, the maximal Lyapunov exponents Li are negative or equal to zero and, consequently, lead to limit cycles and quasiperiodic orbits. In the points where L] = 0, the system switches its periodicity. The situation changes dramatically if,... 

In the present volume we discuss only two limits of coherent and incoherent transitions, and have not considered the intermediate region in which the coherence is destroyed. In the classical dynamics of polyatomics, the periodicity of motion is destroyed due to increasing coupling with the bath modes, which leads to quasiperiodic motion and... 

The solutions of these equations (the trajectories) will for long times (i.e., after transient effects associated with switching on the external parameters have decayed) approach so-called limit sets, which may be classified into fixed points (stationary states), limit cycles (periodic oscillations), mixedmode oscillations, quasiperiodic oscillations, and chaotic behavior. Transitions between these states may occur upon variation of the external parameters pk and are called bifurcations. Experimental evidence for these effects with the system CO + 02/Pt(110) will be briefly presented without going further into details of the underlying general theory (see 16, 17). 

Fig. 16 are only rarely strictly periodic, because usually rather small fluctuations in the external parameters are sufficient to trigger abrupt changes. However, in principle, mixed-mode oscillations belong to the category of multiple-periodic limit cycles. If the behavior is governed by two incommensurate frequencies, i.e., the ratio of two periodicities is an irrational number. This situation is denoted by quasiperiodicity and has been realized experimentally with periodically forced oscillations, as will be described next. 

Thus some of the fluid elements move on aperiodic chaotic trajectories and others on quasiperiodic orbits. The quasiperiodic orbits are invariant surfaces in the phase space that form the boundaries of the chaotic layers and limit the motion of the chaotic trajectories. There is a similar structure around each elliptic periodic orbit resulting from broken resonant tori that are also surrounded by invariant tori forming isolated islands inside the chaotic region. 

Intrinsic non-RRKM behavior occurs when an initial microcanonical ensemble decays nonexponentially or exponentially with a rate constant different from that of RRKM theory. The former occurs when there is a bottleneck (or bottlenecks) in the classical phase space so that transitions between different regions of phase space are less probable than that for crossing the transition state [fig. 8.9(e)]. Thus, a micro-canonical ensemble is not maintained during the unimolecular decomposition. A limiting case for intrinsic non-RRKM behavior occurs when the reactant molecule s phase space is metrically decomposable into two parts, for example, one part consisting of chaotic trajectories which can decompose and the other of quasiperiodic trajectories which are trapped in the reactant phase space (Hase et al., 1983). If the chaotic motion gives rise to a uniform distribution in the chaotic part of phase space, the unimolecular decay will be exponential with a rate constant k given by... 

A number of studies have successfully applied the theory of nonlinear dynamics to studies of atomic and molecular motions some representative references follow. In perhaps the first such study, in 1955 DeVogelare and Boudart considered aspects of nonlinear dynamics in models of bimolecular exchange reactions. Poliak, Pechukas, and Child explored periodic orbits and their associated asymptotic limit sets in exchange reactions. - De Leon, Berne, and Rosenberg considered quasiperiodicity and chaos in unimolecular... 

In Figure 10 the chaotic region is extremely small. However, in Figures 11 and 12 we show a second system s phase space map as a function of en-ergy.35,119 xhjs system exhibits a mode-mode resonance at low energies, with a hyperbolic fixed point located near the center of the Poincare map. Note in Figure 11 that as the energy increases, the measure of quasiperiodic phase space decreases and approaches a limit in which most of the tori are destroyed, with... 

The torus attractor has associated with it two distinct kinds of behavior quasiperiodic and periodic. These two kinds of behavior are, in turn, associated with two distinct relationships between the two natural periods (or frequencies) in the system. The first frequency corresponds to the now unstable limit cycle... 

A different example is the H3 system. Here the ground state of H3 is an equilateral triangle. Recent experiments of Carrington and Kennedy indicate that has a high density of quasibound states embedded in the continuum above the H ground electronic state dissociation limit. These states or at least some of them should be representable as stable periodic and quasiperiodic orbits. 

We have presented the first clear evidence of quasiperiodicity in a chemical dynamical system. The shape of the torus as well ais its evolution is consistent with the nature of some generic instabilities of this chemical system. Some preliminary experiments in our laboratory indicate that similair tori co ild be obtained after a Hopf bifurcation of a limit cycle in perfect agreement with the theoretical predictions. 

The homogenized failure surface obtained has been coupled with finite element limit analysis. Both upper- and lower-bound approaches have been developed, with the aim to provide a complete set of numerical data for the design and/or the stmctural assessment of complex structures. The finite element lower-bound analysis is based on an equilibrated triangular element, while the upper bound is based on a triangular element with discontinuities of the velocity field in the interfaces. Recent developments include the extension of the model to blast analysis (Milan et al. 2009), to quasiperiodic masoruy (Milan et al. 2010), and to FRP strengthening (Milan and Lourengo 2013). 

---