The definition of instantaneous frequency using the analytic signal has a phys- ical meaning only for signals that have a single frequency component varying in time (monochromatic). Although it seems intuitively plausible to consider functions with more than one frequency component varying in time, the defini- tion of instantaneous frequency is an ill posed problem in this case, as it has been pointed out often (see for instance [53]). In the case of two components
f(t) = a1expi φ1(t) +a2expi φ2(t), several physical restrictions lead to the defi-
nition of the instantaneous frequency as the average of the derivatives of the two phases, but only in the case that a1 = a2. Oliveira and Barroso [38] introduce
a heterodyne definition of frequency to obtain the average of the two frequencies agreeing with the definition of instantaneous frequency in the case a1 6=a2, and
also give conditions for the case ofn components.
In practice when two frequency time-dependent components are present, we would like to determine both components, and not only the average. In Hamil- tonian systems, the coupling between the different dynamical variables produces that several frequency components are present in each degree of freedom of the
trajectory, and we want to obtain the exact value of at least one of them. For us, the case of multicomponent signals can be addressed with a useful definition of instantaneous frequency, rather than a rigorous one.
Consider the signalf(t) = (2−.0005(t−45)2) exp(2π i(.5t+.6 sin(t/4))). This
is a monochromatic function with variable phase and amplitude. We see that the variation of the amplitude is not as important as the oscillations due to the complex exponential; therefore, the definition of the instantaneous frequency as the derivative of the phase coincides with our intuition of what the frequency should be. This turns out to be correct since the function f, as a function of complex time t, is entire and therefore it satisfies the definition of analytic signal (Definition 1). In Figure 2.2 a) we plotted the real part of f, and the density plot of the modulus of its wavelet transform is represented in b); note that the maximum of this surface for each time value b corresponds to the instantaneous frequency at that given time, as it can be seen in c).
Consider now the function with two frequency components represented in Fig- ure 2.3. The function to analyze is f(t) = (1−.0001(t−45)2) exp(2π i(.5t+
.6 sin(t/4))) +.6 exp(2π i(1.2t+.6 cos(t/4))). Note in the density plot of the mod- ulus of the wavelet transform (Figure 2.3 b) that there are two ridges corresponding to two frequency components; the one with highest intensity corresponds to the component with higher amplitude (around frequency .5), and the band with less intensity corresponds to the component with lower amplitude (around frequency 1.2). This is, the modulus of the wavelet transform has two local maxima for each point in time. In this case, we consider as “the dominant frequency” the one with higher amplitude, corresponding to the absolute maximum. Hence, we define the instantaneous frequency as the one producing the absolute maximum of the modulus of the wavelet transform for each time.
This procedure seems to be appropriate for the computation of the instanta- neous frequency of numerical solutions of Hamiltonian systems. When dealing with solutions of a general Hamiltonian system (2.3) we want to assign an instantaneous frequency to a complex time-series of the formzk(t) =xk(t)+i yk(t) (k= 1, . . . , n).
a) b) c) 0 10 20 30 40 50 60 70 80 90 100 −2 −1 0 1 2 f(t)=(2−.0005(t−45)2)exp(2π i (t/2 + .6*sin(t/4))) Re(f(t)) 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 Modulus of the CWT t Frequency 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 t ω (t)
Figure 2.2: a) Analytic signal with variable phase and amplitude, and b) modulus of its wavelet transform. In c), we can see that the maximum of the modulus of the wavelet transform coincides with the instantaneous frequency of the function defined as the derivative of the instantaneous phase.
Due to the coupling terms, these time-series generally have several frequency com- ponents. Our aim is to identify n basic frequencies that describe quasiperiodic motions, and to determine how the frequencies vary in time for the case of chaotic trajectories. Therefore, in our heuristic solution for the multicomponent case, we consider only the absolute maximum of the modulus of the wavelet transform of
zk to obtain just one time-varying frequency, and disregard other local maxima.
Although it seems that we lose information by doing this, usually the location of other local maxima corresponds to the absolute maximum of another variable of the trajectory,zl. Therefore, we will end up with as many time-varying frequencies
a) b) c) 0 10 20 30 40 50 60 70 80 90 100 −2 −1 0 1 2
f(t)=(1−.0001*(t−45)2)exp(2π i(t/2+.6*sin(t/4)))+0.6exp(2π i(1.2t+.6cos(t/4)))
Re(f(t)) 10 20 30 40 50 60 70 80 90 100 0.5 1 1.5 Modulus of the CWT t Frequency 0 10 20 30 40 50 60 70 80 90 100 0 0.5 1 1.5 t ω (t)
Figure 2.3: a) Two frequency components with variable phase and amplitude, and b) modulus of its wavelet transform. In c), we show how the instantaneous frequency is determined by the absolute maximum (in frequency) for each point in time.
as degrees-of-freedom of the system.