Irrational frequency ratio

Sufficiently irrational tori here means those tori whose frequency ratio uj /u>2 is such that... 

The corresponding trajectories can be best visualized as motion restricted to a two-dimensional torus, as shown in Fig. 1. If the frequency ratio, or the winding number ( i/( 2, is a rational number, the two DOFs are in resonance and an individual trajectory will close on itself on the torus. By contrast, if coi/a)2 is an irrational number, then as time evolves a single trajectory will eventually cover the torus. The motion in the latter case is called conditionally periodic. 

The solid curve between regions I and II is the boundary bottleneck between these two regions. Davis (1985) has suggested that the trajectory which defines this boundary has a frequency ratio with the worst irrational number. Such numbers are well known (Berry, 1978) and the golden mean. 

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... 

A quasi-periodic solution to a system of ODEs is characterized by at least two frequencies that are incommensurate (their ratio is an irrational number) (Bohr, 1947 Besicovitch, 1954). Several such frequencies may be present on high-order tori, but for the two-dimensional forced systems we examine, we may have no more than two distinct frequencies (a two-torus, T2). A quasi-periodic solution is typically bora when a pair of complex conjugate FMs of a periodic trajectory leave the unit circle at some angle , where /2ir is irrational. Such a solution is also expected when we periodically perturb an autonomously oscillating system with a frequency incommensurate to its natural frequency. 

The previous examples illustrate the special case of harmonic spectra. Aharmonic spectra result when the ratio of the carrier to modulation is irrational e.g.,coc/com = /2. There is no fundamental frequency for the spectra, with aharmonic character arising from reflected side frequencies that do not fall at positive frequency locations (see Figure 9.28f). 

Pulsed flow from both inlets with irrational ratio of frequencies... 

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. 

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