A chaotic systems analysis of rhythms in feeling states.

Formal research as well as casual observation argues that fluctuations in mood are both complex and irregular (e.g., Almagor & Ehrlich, 1990). Moreover, moods clearly vary in a fashion typical of systems that are at least partially chaotic, that is, for a given individual they are not predictable in detail but can be characterized in terms of their global or overall pattern. The discovery that a system is chaotic is important for at least two reasons. First, it means that apparently random fluctuations which typify the system's behavior are not caused by chance alone, nor experimental error, but the chaotic regime itself (e.g., Rapp, 1993). Second, it implies that even extremely complex system behavior may be the product of an interaction of surprisingly few variables (e.g., Crutchfield, Doyne, Packard, & Shaw, 1986). With these thoughts in mind, we designed the present study to investigate the appropriateness and utility of approaching mood fluctuation as chaotic.

Chaos systems analysis has recently become an important line of development in both the biological and behavioral sciences (e.g., Abraham, Abraham, & Shaw, 1990; Basar, 1990; Crutchfield et al., 1986; Rapp, Bashore, Martinerie, Albano, Zimmerman, Mees, 1989; Harth, 1983; Levine & Fitzgerald 1992; Loye & Eisler, 1987). The present investigation is one of the first to bring its methodology to the study of moods. This approach is consistent with the recent trend toward idiographic strategies in the study of affect, as traditional statistical procedures have failed to deal adequately with notable individual differences (Diener & Iran-Nejad, 1986). For this reason time series analyses have, for example, proven useful (Cooper & McConville, 1990; Larsen, 1987; Wessman & Ricks, 1966).

Positive and negative affect tend to be global characteristics of mood (Almagor & Ehrlich, 1990; Russell, 1978). Thus, investigators have theorized that mood variability can be understood in a two-dimensional model consisting of a positive-negative affect polarity plus a second dimension, affect intensity (Mehrabian & Russell, 1974; Russell, 1979); through in a sophisticated factor analytic approach, Watson and Tellegan (1985) have also shown that intensity can be subsumed into orthogonal positive and negative affect dimensions. Such multidimensional models are reminiscent of Wundt's 1897 three-dimensional theory of feeling, in which the poles of the principal axes were pleasure-pain, excitement-relaxation, and strain/anticipation-relaxation. Pilot studies in our laboratory led to the selection of the first two of these three dimensions for the present investigation, both because of their conceptual simplicity and because of the ease with which experienced observers rated them from direct introspection.

Method

Subjects

These were four upper-division male full-time undergraduate students and one male faculty member, 20, 24, 42, 48, and 50 years of age. None held outside jobs or experienced other major schedule disruptions during the study.

Materials and Procedure

Participants indicated their mood state on two Likert scales each waking 30 min for a 3-week period. The scales were straight lines 4 cm in length. The first scale contained the pleasure-pain (pleasant-unpleasant) axis, ranging from extreme pleasure to extreme pain, and the second contained the excitement-relaxation axis, ranging from extreme excitement to extreme relaxation. During the study each participant carried a booklet that contained each day's scales on a separate page. A reference adjective list provided a broad range of descriptors describing graduated degrees of pleasure and pain, and excitement and relaxation. These scales were developed during previous investigations in which participants found they could respond to them quickly and with confidence (Winkler & Combs, 1993; Winkler, Combs, & Daley, 1992).

Analyses and Discussion

Attractor Reconstructions

One way to display the temporal pattern of a chaotic or semichaotic system is to graph its progress through a two-dimensional or higher state space. Such plots disclose the form of the underlying pattern, or attractor, into which the system tends to settle. For empirical data this process of attractor reconstruction may take the form of a sequential plotting of pairs of values of the dependent variable, separated by a constant delay or lag. This was done with the present variables for each participant. Using Schaffer's Dynamical Software for the PC (Schaffer, Truty, & Fulmer, 1988) each participant's data was first smoothed by interpolating four points between each successive pair of Likert values and performing a three-point running average on the result. The subsequent data sets were lagged three units against themselves (the equivalent to .3 hour) and plotted as reconstructed attractors. The results are strikingly similar to preliminary attractor reconstructions reported by Hannah (1991) for a comparable investigation of mood fluctuations.

These reconstructions have a similar appearance to chaotic attractors, suggesting the presence of chaos in the mood fluctuations. To assure ourselves that this appearance was not caused by chance alone, we performed several reconstructions of the data sets after randomizing the order of the scores. (For a discussion of the assets and liabilities of this and related procedures, see, e.g., Rapp's 1993 article). This Monte Carlo procedure yielded attractor-like reconstructions of a distinctly different appearance, especially apparent in the actual screen images, which were color coded for velocity.

Individual differences in attractor patterns emerged both in terms of shapes and sizes. For instance, on the pleasure-pain scale the plots of Participants 1 through 4 are skewed modestly toward small values, indicating a tendency of mood to oscillate in the unpleasant range, while the plot of Participant 5 is roughly symmetrical. On both the pleasure-pain end the excitement-relaxation scales, Plots 1 and 5 display numerous large smooth swings (maximum values ranging as high as 37 and minimum values dropping as low as 4), indicating long swings in mood--evidently circadian swings for Participant 5--whereas Plots 3 and 4 on both scales tend toward smaller oscillations near the center, with only occasional large departures. For Participant 3 the total range was small, between roughly 10 and 30. Following this up we noted a tendency for the older participants, 1, 2, and 3, to produce somewhat smaller attractors, representing less extreme mood fluctuations, than did the younger ones, 4 and 5. Whether this represents a true developmental difference, as common sense might suggest, or an age-related bias in the rating process, or even just random differences between individuals could not be determined from the present small sample of participants.

The type of attractor represented by Plots 1, 2 and 5 on both scales is composed of many small oscillations nested in larger mood swings, whereas Plots 3 and 4, especially on the pleasure-pain scale, are dominated by smaller oscillations spanning less than the entire range.

Fractal Dimension 

Chaotic systems are highly complex and exhibit fractal properties. Applications of fractal geometry have been used to characterize psychological phenomena such as learning curves (Bendler & Shlesinger, 1991), as well as biological phenomena such as pulmonary blood flow (Glenny & Robertson, 1991), the surface properties of cells (Kenough, Hyam, Pink, & Quinn, 1991), and characteristics of the nervous system (King, 1991). Insomuch as the above mood attractors are produced by psychological and biological factors as well as purely situational ones, it would not be surprising to find a fractal component to them. A common way to probe for such a component is to compute fractal dimension estimates. Such estimates would be expected to be smaller than the attractor, or embedding, dimensions (in this case we have 2 dimensional attractors), whereas stochastically random distributions (as opposed to chaotic ones) yield estimates near the value of the dimension. This is because stochastically random sequences are more complex in the sense that they are not constrained, as is deterministic chaos.

A type of fractal dimension, the information dimension, was computed for these data using Sarraille and DiFalco's (1992) FD3 program for the PC.

Fractal dimension estimates were surprisingly similar for all participants, and to those previously reported by Hannah (Casti, 1992). As anticipated, the values are notably smaller than the embedding dimension, or 2. Moreover, when the same scores are randomized to produce first approximations of stochastic distributions, the resultant values are found to be consistently smaller. The fact that they are only slightly smaller suggests the presence of a partially chaotic process mingled with more constrained processes such as circadian, ultradian, or infradian rhythms (Peters, 1991). The reality of such rhythms is evidenced below.

Table 1

Fractal Information Dimension Estimates

               Information Dimension      Information Dimension
                   (Nonrandom)                 (Random)
                 P/P(a)   E/R(b)              P/P      E/R

Subject 1      1.63498   1.54763           1.76553   1.69670
Subject 2      1.75333   1.57248           1.79213   1.65831
Subject 3      1.63393   1.58397           1.72664   1.65923
Subject 4      1.60223   1.59151           1.64783   1.70301
Subject 5      1.59218   1.59497           1.75097   1.72901

a P/P = pleasure/pain scale values.
b E/R = excitement/relaxation scale values.

Frequency Spectra

As would be expected if mood vacillations approximate chaotic attractors (Abraham & Shaw, 1992), they display the presence of cyclic components (Almagor & Ehrlich, 1990; Hall, Sing, & Romanoski, 1991; Larsen & Diener, 1987; Larsen & Kasimatis, 1990; Rossi, 1986; Rossi & Rossi, 1977; Watson & Tellegen, 1985). Spectral analyses were used to disclose the dominant frequency components in each subject's data set. These were computed using the Dynamical Software. The resulting power spectra are shown in Figure 3.

Each participant's record produced unique patterns. Those of Participant 5, for instance, display a single prominent peak at a periodicity of about 16 hours. This represents a circadian rhythm, because this individual, like most, was awake about 16 hours a day. The circadian rhythm is seen in the long smooth lines of his attractors, noted previously. Other participants also displayed more or less prominent 16-hour-peak periodicities. For the pleasure-pain scale these are evident to some degree in the spectra of all participants. For the excitation-relaxation scale this periodicity is strongly apparent in the spectra of Participant 2 as well as 5.

Participant 4 presents a dominant peak at about 9 hours on the pleasure-pain dimension and a lesser prominence at about this interval on the excitation-relaxation scale. This would seem to indicate the presence of two daily maxima in the participant's records, separated by about 9 hours. Time series plots confirmed this to be the case. It was in no way apparent, however, that this rhythm was driven by his daily schedule. It may well be an enduring characteristic of the individual.

Several participants displayed multiple peaks. Such was the case with Participants 2, 3, and 4. On the pleasure-pain scale, Participant 3 exhibits rhythms at 4 and 8 hours, as well as the 16-hour circadian cycle. Participant 2 exhibits a very similar pattern for the excitation-relaxation spectrum. In contrast, for Participant 1 neither spectra displays clearly dominant periodicities. This is consistent with his own informal report that he maintains almost no circadian periodicity in his personal life. This seems to be an enduring personal characteristic and not the result of work or school schedules during the study.

Individual Differences

The present study reveals a complex mood process at least partially undergirded by chaotic dynamics. Further, it provides additional evidence that temporal patterns in affect are markedly individual in character (Larsen, 1987; Larsen & Kasimatis, 1990). Such individual patterns can be displayed topographically as attractor reconstructions and can be subjected to frequency analyses as well as measures of chaoticity such as the fractal dimension. Future research is suggested in the direction of gaining a clearer understanding of the extent to which fluctuations in affect are fundamentally chaotic, and also in the direction of seeking fundamental types of rhythmic patterns as seen in attractor reconstructions. This would require larger numbers of participants and, if possible, longer sequences of data.

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