# Development of Parameters towards Voice Bifurcations
## 1 Introduction  
Normal healthy voice exhibits a nearly periodic behavior, which can be readily observed in its acoustic or glottal source signals via spectral visualization techniques such as spectrogram, power spectrum, or cepstrogram. Voice pathology, however, often introduces interference to the vibration of vocal folds, forcing them to oscillate without collision or out of sync or sometimes even chaotically [1, 2]. Such disruptions directly translate to a loss of voice quality, often described as hoarse, rough, breathy, or strained. The pathological interference of vocal fold oscillation leads to an anomaly in voice signals. To classify voice signals, the Workshop on Acoustic Voice Analysis [3] in 1994 recommended the use of the following voice signal types:  
* Type I--nearly periodic;
* Type II--contain intermittency, strong subharmonics, or modulations; and
* Type III--chaotic or random. These types are used throughout the paper. Classification of these types were suggested to be done via visual inspection of the spectrogram of the voice signal under study. To date, no commercial or publicly accessible software is available to classify the voice types automatically, although there is a long-standing active research effort [4, 5, 6, 7, 8] applying concepts from dynamical systems theory. This paper approaches this classification problem based on the Type-II voice characteristics. Type II encompasses a broad spectrum of voicing patterns. It could be summarized as follows: A voice signal consists of clean tonal components but not all componentsbelong to the harmonic series with an obvious speaking fundamental frequency. These extraneous (or non-harmonic) tonal components, which could appear intermittently during phonation, disturb the audibly pleasant harmonic composition of a healthy voice and contribute to the perceived roughness in the voice [9]. The non-harmonic components include period-\(n\) subharmonics and biphonia--the coexistence of two harmonic series with unrelated fundamental frequencies. The pathological Type-II voice may also exhibit intermittency or mid-phonon changes [10; 11; 12], switching between multiple modes of oscillation, with or without non-harmonic components. The variation in oscillation modes can be explained by the dynamic systems theory [10; 13; 14; 15]. Dynamical systems may have bifurcation points, where a small change in system parameters results in a large change in system behavior. For the voice system, the system parameters include the subglottal pressure and the vocal fold mass and stiffness. Bifurcation diagrams are often used to visualize these bifurcation points. With an asymmetric vocal fold model of superior nerve paralysis, a two-parameter bifurcation diagram in the asymmetry factor and subglottal pressure (Figure 7 in [13]) suggests a large number of possible oscillation modes (or attractors of a dynamical system) that represent all three voice types. There is one oscillation mode each for Type I and III, and the rest are different Type-II modes with different phase-locking modes (i.e., different subharmonic periods and biphonia frequency ratios) [15]. Asymmetric vocal folds vibration mode can switch suddenly from Type I to Type II or III, or vice versa as the subglottal pressure fluctuates during phonation. The model also suggests that severe asymmetry may lead to multiple Type-II modes during phonation. These observations from the theoretical model are in agreement with the observation of pathological voice, which can exhibit mid-phonon voice change [10; 11; 12]. It is possible for the bifurcations to occur in a rapid succession, and one such case is included in this paper (Case 3). Some of its oscillation modes in its acoustic signal are captured in Figure 1.  
Figure 1: Snapshots of acoustic signal with mid-phonon bifurcation. At \(t_{0}=0.2\), the voice mostly presents Type-I behavior. It is, however, preceded by a Type-II mode with substantial subharmonics at \(t_{0}=0.1\) and is then followed by Type-II voice with strong biphonia at \(t_{0}=0.3\).