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Journal of Evaluation in Clinical Practice ISSN 1365-2753

Dynamics of violence
David Katerndahl MD MA,1 Sandra Burge PhD,1 Robert Ferrer MD MPH,1 Johanna Becho BA2 and
Robert Wood DrPH3
1
Professor, 2Research Associate, 3Biostatistician, Department of Family and Community Medicine, University of Texas Health Science Center at
San Antonio, San Antonio, TX, USA

Keywords Abstract
battered women, domestic violence,
intimate partner violence, non-linear Rationale, aims and objectives Three behavioural models suggest different dynamic
dynamics, systems theory patterns of intimate partner violence (IPV). However, few studies permit assessment of IPV
dynamics. The purpose of this study was to estimate the degree of non-linearity in daily
Correspondence violence between partners over a 3-month period, identify their specific dynamic patterns
Dr David Katerndahl and determine whether measures of violence severity and dynamics are interrelated.
Family and Community Medicine Methods From six primary care clinics, we enrolled 200 adult women who experienced
University of Texas Health Science Center violence in the previous month and asked them to complete daily telephone assessments
at San Antonio of household environment, marital relationship and violence using Interactive Verbal
7703 Floyd Curl Drive Response. To assess non-linearity of violence, algorithmic complexity was measured
San Antonio, TX 78229 by LZ complexity and lack of regularity was measured by approximate entropy. Lyapunov
USA exponents and correlation dimension saturation were used to approximate dynamic
E-mail: katerndahl@uthscsa.edu patterns.
Results Of the 9618 daily reports, women reported experiencing abuse on 39% of days,
Accepted for publication: 31 March 2014
while perpetrating violence themselves on 23% of days. Most (59%) displayed random
dynamics, 30% showed chaotic and 12% showed periodic dynamics. All three measures of
doi:10.1111/jep.12151
non-linearity consistently demonstrated non-linear patterns of violence. Using multivariate
analysis of variance, neither episode severity for men or women showed significant differ-
ences across dynamic types, but chaotic dynamics had the lowest frequencies of violence
in men and women while random dynamics had the highest frequencies. Approximate
entropy was positively correlated with violence frequency and burden in men and women,
but Lyapunov exponent was inversely related to violence. LZ complexity correlated posi-
tively with wife-perpetrated violence only.
Conclusions IPV is rarely a predictable, periodic phenomenon; no behavioural model
describes the violence dynamics for all violent relationships. Yet, the measures of non-
linearity and specific dynamic patterns correlate with different violent features of these
relationships.

of abuse, Ristock’s study [4] of abusive lesbian relationships sug-

Introduction gests that some display cyclic patterns while others exhibit fluc-
From 12 to 29% of primary care female patients experience tuating power dynamics. Such power dynamics may relate to the
current intimate partner violence (IPV) [1]. Women in abusive varying control strategies used by perpetrators [5]. Unfortunately,
relationships report that violence has caused physical injury with the exception of the study by Umberson et al. [6] on violent
(42%), need for medical treatment (11%), hospitalization (9%) and men, none of these studies involved daily assessments of violence.
counselling (27%), and resulted in time taken from work (18%) The dynamics of IPV from day to day can be characterized in
[2]. Most of the work on the dynamics of husband-to-wife abuse is two ways. Dynamic pattern (periodic, chaotic, random) can be
based on surveys or qualitative interviews [3,4] conducted at one identified, or dynamics can be represented as the degree of non-
point or a few points in time; little has been made to assess linearity (disproportionality between levels of input and output)
prospectively the day-to-day dynamics of abuse. Looking for pat- over time. Although a variety of theories address the prevalence
terns in dynamics has yielded contradictory findings. Although the and consequences of IPV, only three behavioural models suggest
work of Wolf-Smith and LaRossa [5] may support a cyclic pattern daily dynamic patterns, each suggesting a different pattern.

Journal of Evaluation in Clinical Practice 20 (2014) 695–702 © 2014 John Wiley & Sons, Ltd. 695
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Dynamics of violence D. Katerndahl et al.

specific path they follow. Chaotic systems are unpredictable and

The cycle of violence
do not respond predictably to interventions. Giles-Sims’ empha-
In this model, battered women are not constantly abused, nor is sis on the importance of feedback, on interdependent causal
their abuse inflicted at totally random times. Instead, battering factors and on stabilization of violence once established is con-
appears to recur in cycles. The battering cycle has three distinct sistent with chaotic dynamics [9].
phases, which vary in length and pattern across couples. In the
tension building phase, minor battering events may occur, but the
The Duluth model, also known as the power
woman alters her behaviour to keep the peace. Many couples
and control wheel
remain in this phase for long periods of time, but eventually,
tension builds, leading to an explosion. This acute battering inci- This model posits that violence is used to control people’s behav-
dent is characterized by high severity and brutality, batterers’ iour. In contrast to the cycle of violence, the Duluth model recog-
lack of control and brevity (usually a few hours). Following the nizes that abuse is a constant force in battered women’s lives. The
explosion is the calm, loving respite, where the batterer knows power and control wheel depicts eight key non-physical abusive
he has gone too far and tries to compensate for his violent behav- behaviours exhibited by men who batter: coercion and threats;
iour with loving kindness. This behaviour is usually successful at intimidation; emotional abuse; isolation; denying, blaming or
pulling the victim back into the relationship, where she remains minimizing the violence; using the children; evoking male privi-
vulnerable to future victimization. Eventually, tension builds, and lege; and economic control. Here, violence is part of an ongoing
the couple moves into phase 1 again [7]. Copel’s [8] qualitative pattern of controlling behaviours, not simply isolated incidents or
study of disabled women in violent relationships saw predictable, cyclical explosions of pent-up anger. Batterer’s use of physical
cyclic patterns but, instead of a loving respite, found only a assaults may be infrequent and ‘random’, but assaults reinforce
period of separation. Considering the cycle of violence, the three the power of other controlling tactics. These tactics undermine the
phases would yield a cyclic or periodic pattern. Although the partner’s ability to act autonomously [12]. Under this model, the
frequency of the periodicity would vary among couples due to constant force of controlling behaviours produce constant stress
variable phase length, within each couple the periodicity would within the relationship, occasionally erupting in violence. A type
remain fairly constant. Periodic dynamics, in which the system of random dynamics (pink noise or criticality) is common in
cycles its behaviour, results when actions and outcomes are complex systems. Criticality results from constant stress on a
tightly coupled, and when current behaviour is dependent on system composed of interdependent components with varying pre-
previous behaviour. Periodic systems have strong attractors dilections to respond, yielding a random pattern of responses of
influencing behaviours, and are stable and insensitive to varying intensity. Systems characterized by criticality have no
small changes in their state (insensitive to initial conditions). attractors limiting or influencing their behaviour, and may or may
Periodic systems are predictable and respond predictably to not be sensitive to initial conditions [13]. These systems are unpre-
interventions. dictable in behaviour and in response to intervention. Both chaotic
and random dynamics are said to be ‘non-linear’ because the
output on such systems is not proportional to the input and, hence,
Family systems theory
unpredictable.
Family systems theory focuses on wife battering as an ongoing Although considerable research on domestic violence has taught
interaction pattern resistant to change. The first act of violence is us a great deal about risk factors and consequences for abusive
generally not severe, and the hitter is usually contrite, so the relationships, we know little about the detailed day-to-day patterns
event does not drive the victim away. The hitter has ‘stretched’ of abuse. Three generally accepted behavioural models about the
the unspoken boundary against using violence, and the relation- dynamics of violence exist, but there is little longitudinal data to
ship held. This can lead to more violence [9]. Initially, perpetra- support one over another or to explain why different dynamics
tors are distressed and contrite about their own behaviour. may be seen in different relationships [4]. Prior research on the
However, with repetition, they become desensitized, and the dynamics of violent relationships suggests that such relationships
shock and self-reproof extinguish over time. The physical may have both predictable and unpredictable components: predict-
aggression will eventually squash resistance, whereas the deni- able in their lack of emotional reaction to relationship dynamics
gration will batter the victim’s self-concept and self-efficacy [9]. and stress [6], but unpredictable in the shifting power dynamics
Aggressive acts increase the likelihood of a person being aggres- within the relationship [4]. Prior research on dynamics of IPV in
sive again [10]. External influences may also influence aggres- primary care female patients found non-linearity, although the
sion. Capaldi and Kim’s [11] dynamic model emphasizes the dynamic pattern types (periodic, chaotic, random) occurred with
couple’s interaction within several contextual factors. Under similar frequencies [14].
family systems theory, the violence is dependent on the feedback Holtzworth-Munroe and Meehan [15] have emphasized the
loops between victim, batterer and context. Such feedback, vari- need for research on the immediate, situational and dyadic pro-
able in strength and direction, should lead to chaotic patterns in cesses in violent relationships, whereas Hardesty and Chung [16]
violence. In chaotic dynamics, the overall pattern of behaviour specifically recommended longitudinal research including daily
recurs but the specific path is unpredictable; this results when assessment of violence and risk factors. Thus, the purpose of this
actions and outcomes are separated in time, and when feedback study was to estimate the degree of non-linearity in daily husband-
within the system varies in strength and direction. Chaotic to-wife violent events over a 3-month period, identify the specific
systems also have attractors influencing their behaviour but they dynamic patterns and determine whether measures of violence
are sensitive to small changes (initial conditions) in terms of the severity and dynamics are interrelated.

696 © 2014 John Wiley & Sons, Ltd.

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D. Katerndahl et al. Dynamics of violence

Using the time series of the daily assessments of level of

Methods husband-to-wife violence, we calculated LZ complexity and
Lyapunov exponents using the Chaos Data Analyzer software
Sample
(Physics Academic Software, Raleigh, NC, USA) for each
As previously described [17], women with a recent history of subject’s time series. We calculated approximate entropy using the
husband-to-wife abuse were recruited from six primary care ApEn module in OCTAVE (Free Software Foundation, Boston,
clinics in San Antonio. Adult (≥18 years old), non-pregnant MA, USA). Entropy has been reported for a variety of psychologi-
women were asked to complete the 6-item brief Conflict Tactic cal measures [26,27]. Stable estimates require as few as 50 data
Scale [18] in the examination room while they waited to see their points for approximate entropy [28–30] and 30 data points for LZ
physician. complexity [31]; Lyapunov exponents are resistant to the effects of
missing data if non-linearly imputed data (≤15%) were used in
data sets with underlying periodic or chaotic dynamics [21]. Cor-
Procedure relational analysis suggests that LZ complexity is related to ApEn
Two hundred subjects were asked to complete a daily relationship (r = 0.43, P < 0.001) in this study, but neither correlated with
assessment for 12 weeks using Interactive Verbal Response via Lyapunov exponent.
telephone from a safe environment. Subjects could report assess-
ments at any time of the day or night, but preferably at the same Assessment of dynamic pattern
time each day. During the daily assessment, subjects answered
questions concerning the previous day’s experience. The daily Initial assignment of dynamic patterns (periodic, chaotic, random)
relationship assessment measured violence severity using a modi- was made based on whether the Lyapunov exponent was positive
fied Conflict Tactic Scale for men and women. or negative (presence or absence of sensitivity to initial condi-
Six summary measures of violence severity served as outcome tions), and whether the correlation dimension saturated with
measures. First, the frequencies of husband- and wife-perpetrated increasing embedding dimensions (presence or absence of a low-
violence were used. Second, the mean levels of episode severity dimensional attractor). A periodic pattern would have a negative
for husbands and wives were calculated across all days in which Lyapunov exponent (not sensitive to initial conditions) and the
any violence occurred. Finally, the overall levels of violence presence of an attractor. A chaotic pattern would have a positive
burden for husbands and wives were calculated as the mean daily Lyapunov exponent (sensitive to initial conditions), and the pres-
violence across all days reported. ence of an attractor. A random pattern would have a positive
Lyapunov exponent, but no attractor.
However, the above results are only suggestive of the particular
Analysis dynamic pattern. For example, the Lyapunov exponent can very
rarely be negative in a chaotic [32] or random [32,33] system
To quantitatively assess dynamics, time series data must be com- under certain circumstances. These techniques were designed for
plete. Daily reports were made on a mean of 63.2 [±15.9 standard analysing long time series (n > 1000) and thus must be interpreted
deviation (SD)] days with 50% of subjects reporting on 80% of with caution when smaller data sets are used.
days or more; missing data were imputed using the nstep pro- To confirm dynamic patterns, additional analyses were per-
cedure in the TISEAN software package [19] to maintain any formed. Time series analysis was conducted to look for an ARIMA
non-linear characteristics. The nstep approach to imputation has (autoregression, integration, moving average) model that would fit
been shown to least distort non-linear characteristics of time series the data. None of the time series data were non-stationary (with
when compared with traditional methods [20]. Unlike other autocorrelation of first lag ≥ 0.9). Although (0,0,0) models (those
approaches to handling missing data, nstep successfully corrects not needing adjustment to account for the time series) suggest
for 25–60% missing data if such data were missing at random and randomness, all other ARIMA models suggest periodic dynamics
15–40% if missing data follow a power distribution in chaotic time [33].
series, and 40% and 25%, respectively, in periodic time series [21]. To confirm the initial assignment of chaotic dynamics, surrogate
When initial data points in the time series were insufficient for testing was performed. In surrogate testing, time series data are
nstep (generally n ≤ 4), the mode of the time series was inserted randomized using phase randomization, which maintains the mag-
until the time series was long enough to use nstep. nitude at each frequency but randomizes the phase, producing a
time series with similar linear dynamics but dissimilar non-linear
dynamics [34]. For each analysis, the randomization was repeated
Assessment of non-linearity
20 times and the resulting correlation dimension was compared
Three types of complexity measurements are available and we with the correlation dimension of the original data. Because the
used one example of each type [22]. These measure non-linearity, linear dynamics are maintained, we would expect that several of
with higher coefficients suggesting either chaotic or random the surrogate data sets would have correlation dimensions less than
dynamics. (1) Algorithmic complexity (a measure of the amount of or equal to that of the original data, if that data come from a
information needed to describe the data) was measured by LZ periodic system. However, if the original data came from a non-
complexity [23]. (2) Regularity (or the lack of it) was measured by linear (chaotic) system, then few of the surrogate data sets would
approximate entropy (ApEn) [24]. Finally, (3) sensitivity to initial have correlation dimensions less than or equal to that of the origi-
conditions (speed with which two adjacent points diverge over nal data. The proportion of surrogate data sets with correlation
time) was measured with the largest Lyapunov’s exponent [25]. dimensions less than or equal to that of the original data gives a

© 2014 John Wiley & Sons, Ltd. 697

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Dynamics of violence D. Katerndahl et al.

Table 1 Dynamic patterns predictions and observations (based on initial assignment)

Predictions Observations

Periodic Chaotic Random Statistic

Measure Periodic Chaotic Random (P) (C) (R) (P-value) Post hoc

Non-linearity measures (mean)

LZ complexity Low Intermediate High 0.977 0.934 1.088 F = 7.55**** R > C,P
Approximate entropy Low Intermediate High 0.611 0.583 0.763 F = 17.35**** R > C,P
χ2 = 24.72**** –
ARIMA models (% compatible)
1,0,0 High – Low 25% – 9%
0,0,0 Low – High 0% – 32%
Surrogate testing (% with P < 0.05) – High – – 70% – – –
Vector autoregression analysis
Variance accounted (mean R2 [2]) High Intermediate Low 0.818 0.714 0.649 F = 5.92*** P>C>R
Significant predictors (proportion of possible) High Intermediate Low 0.407 0.360 0.282 F = 3.69* P,C > R
Maximal significant lag (proportion of possible) High Intermediate Low 0.926 0.978 0.917 F = 1.33 –
Number of circularly causal prior day predictors* Low Intermediate High 0/2 1/2 2/2 – –

*Pooled vector autoregressives.

#P ≤ 0.10; *P ≤ 0.05; **P ≤ 0.01; ***P ≤ 0.005; ****P ≤ 0.001.
ARIMA, autoregression, integration, moving average.

measure of statistical significance. To support a ‘chaotic’ assign- dynamics had the least. Table 1 summarizes the predictions of this
ment, surrogate testing had to yield a P ≤ 0.05. Final assignments validation process.
consisted of those initial assignments that were ‘validated’ by
ARIMA modelling and surrogate testing.
Association between dynamics and
Further validation was performed using comparisons of LZ
violence severity
complexity and ApEn across patterns as well as results predicted
by vector autoregressive (VAR) analysis. If initial pattern assign- The relationships between measures of non-linearity and violence
ments were correct, we would expect that LZ complexity and severity were assessed using Spearman correlations. The relation-
ApEn would increase when moving from periodic to random ships between dynamic pattern and violence severity were
dynamics. VAR models use multiple concurrent predictors’ time assessed using multivariate analysis of variance (MANOVA) with
series to develop models explaining the dependent variable’s time REGWF post hoc testing.
series. VARs provide useful descriptions of temporal covariability
among variables, good estimates for forecasting and sensitivity to
identification of external ‘shocks’ to the time series. VARs using
Results
STATA software (Stata Corp., College Station, TX, USA) were run Subjects were predominantly Hispanic (76%) and of low income
on each subject using the dependent variable (level of violence) (53% with <$20 000). Their mean age was 38.2 years (±11.7 SD)
and all of the predictor variables. Sequential VARs were run with and 131 (68%) had at least a high school education. Although 85
increasing numbers of lags from 1 to 7 days using only predictor (43%) were in common law marriages, the mean duration of rela-
variables that showed any day-to-day variation. We used the VAR tionship was 9.6 (±8.9 SD) years with a mean duration of abuse
model with the most numbers of possible lags as long as the was 5.5 (±6.5 SD) years.
likelihood ratios (assessing whether additional variance was
accounted for by increasing the number of lags) remained signifi-
Dynamics of violence
cant (P ≤ 0.10) to determine which model was best. These VARs
yielded β-coefficients (and their significance) for each lag for each All three measures of non-linearity suggested that, as a group,
predictor of level of violence [35]. In addition, compared with these women experienced dynamics of abuse that were generally
random dynamics, VAR analysis in periodic dynamics would be non-linear. LZ complexity, measuring algorithmic complexity, had
expected to account for more variance in IPV, have the highest a mean of 1.035 (±.225 SD). As Fig. 1 shows, the LZ complexity
proportion of significant predictors and have the maximal signifi- statistics for most women indicated random patterns, that is, meas-
cant lagged coefficient proportionally (based on the maximum lag uring above the levels for ‘benchmark’ comparison time series
length in days). Finally, using pooled VAR results (pooled using with periodic or chaotic dynamics. Although some women had
methods of Greenland [36]), we would expect that random dynam- negative exponents, suggesting periodic dynamics, most had
ics would be associated with the most circularly causal predictors Lyapunov exponents similar to those seen in chaotic time series.
(using 1-day lagged coefficients in which predictors were signifi- Finally, the mean approximate entropy across all women was
cantly associated with subsequent IPV while IPV was significantly 0.686 (±0.187 SD), suggesting non-linearity (chaotic or random
associated with the subsequent predictor) of IPV while periodic patterns). All approximate entropies were less than those of the

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D. Katerndahl et al. Dynamics of violence

Distribution of Lyapunov Exponent

Measures of non-linearity and their
100
relationships to abuse
80 Table 2 presents results of the MANOVA analysing the relation-
Frequency

60 ship between dynamic pattern and violence severity. Neither

episode severity for men or women showed significant differences
40 across patterns. However, chaotic dynamics were associated with
20 the lowest frequencies of violence for men and women while
random dynamics had the highest violence frequencies. Although
0 overall burden showed a similar pattern for male-perpetrated vio-
–2.500 –2.000 –1.500 –1.000 –0.500 0.000 0.500
Lyapunov lence, differences for overall burden in female-perpetrated vio-
Distribution of LZ Complexity lence only approached significance.
40 Table 3 presents the correlations between measures of non-
linearity and violence severity. Different patterns of non-linearity–
30 violence correlations were found depending on the measure of
non-linearity used. Although both approximate entropy and
Frequency

20 Lyapunov exponents correlated with violence frequency and

burden in both men and women, these measures were inversely
10 related to the violence; approximate entropy was positively corre-
lated with violence while Lyapunov exponent was inversely
0 related to violence. LZ complexity was only associated with
0.000 0.500 1.000 1.500 2.000 wife-perpetrated violence, but correlated positively with all three
LZ complexity
measures.
Distribution of Approximate Entropy
25
Discussion
20
Periodicity in the dynamics of abuse was rare, although the level of
Frequency

15 non-linearity observed depended on the measure used. Overall,

random dynamics were found more often than chaos and perio-
10
dicity combined. Dynamic pattern was primarily associated with
5 the frequencies of husband- and wife-perpetrated violence rather
than their episodic severities. The three measures of non-linearity
0
0.000 0.500 1.000 1.500 showed different patterns of association with the violence vari-
ApEn ables. Although both approximate entropy (suggesting regularity
versus irregularity) and Lyapunov’s exponents (instability) [37]
Figure 1 Distributions of non-linearity measures with ‘benchmark’ com- correlated with husband and wife frequency and burden, they did
parisons. P, periodic; C, chaotic; Rp, random (pink noise); Rw, random
(white noise).
so in opposite ways; LZ complexity was associated with wife-
perpetrated violence only.
Previous work has suggested that IPV is indeed a complex
phenomenon [38–40]. Although physical violence may decrease
‘benchmark’ comparison random time series. Of the 135 women in over time, psychological abuse may not [41,42]. In addition, over
whom dynamic patterns could be determined, more than half a 15-month period, relationships without abuse and those locked in
(n = 79) displayed random dynamics with 40 (29.6%) showing extreme violence tended to remain unchanged, whereas those
chaotic and 16 (11.9%) showing periodic dynamics. Table 1 pre- involving verbal abuse only or violence but less than severe control
sents predictions made for each dynamic pattern and the observed typically changed over time [43]. In fact, the heterogeneity of the
findings. Both LZ complexity and approximate entropy increased violence experience further complicates our understanding of the
across periodic-chaotic-random patterns. Not only do the ARIMA phenomenon [44,45].
models support the validity of periodic and random dynamics, but This study has several potential implications. First, whether
surrogate testing supported the classification of chaotic dynamics. considering the dynamic pattern or degree of non-linearity, the
Finally, VAR analyses generally supported the assignment of majority of women reported abuse occurring in non-linear patterns.
dynamic patterns. The proportional number of significant lagged Random dynamics was clearly the most common pattern, but the
predictors, and maximal significant lag were highest for periodic other patterns did occur. These findings suggest that, while the
dynamics and lowest for random dynamics, while circular causal- Duluth model may be the best overall description for the evolution
ity was most evident in random versus periodic dynamics. Overall, of violence in any particular household, family systems theory and
19 (83%) of 23 predictions of initial pattern assignment were the cycle of violence may also be at work; even Walker [7] sug-
supported. Limiting cases to those patterns additionally supported gested that several different cyclic patterns were possible and may
by ARIMA modelling and surrogate testing resulted in 13 (87%) of change based on stage of life and situation. In fact, the differing
15 predictions (excluding pooled VARs) in pattern assignment associations of measures of non-linearity and measures of
supported. violence [violence frequency was associated with high irregularity

© 2014 John Wiley & Sons, Ltd. 699

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Dynamics of violence D. Katerndahl et al.

Table 2 Dynamic patterns and violence

Dynamic pattern
severity (mean)
Periodic Chaotic Random
Violence (n = 16) (n = 40) (n = 79) F (P)

Husband-perpetrated violence
Frequency 0.35a 0.28a 0.48b 10.51 (0.000)
Episode severity 2.87 2.35 2.55 1.42 (0.245)
Burden 1.10a,b 0.72a 1.31b 4.45 (0.014)
Wife-perpetrated violence
Frequency 0.22a,b 0.15a 0.29b 8.01 (0.001)
Episode severity 1.97 1.78 1.86 0.17 (0.840)
Burden 0.60 0.32 0.59 2.38 (0.097)

Wilks’ lambda (F, P) = 371.41 (0.000). Superscript letters define the subgroups from the posthoc
analysis.
F, P, F-statistic, P-value.

Table 3 Non-linearity and violence severity (rs) in chaotic relationships toward the ‘edge of chaos’ could promote
spontaneous change [49]. Finally, in relationships exhibiting
Violence (n = 140) LZ complexity Lyapunov ApEn
random dynamics, multifaceted approaches or positive role models
Husband-perpetrated violence could yield results (if intervention is even possible).
Frequency 0.14 −0.28**** 0.47****
Episode severity 0.08 −0.14 0.02
Burden 0.13 −0.28**** 0.40**** Limitations
Wife-perpetrated violence
First, the sample size is small for time series analysis, especially
Frequency 0.25*** −0.27**** 0.37****
Episode severity 0.15# −0.03 0.00
for determining dynamic pattern. However, previous studies
Burden 0.25*** −0.21* 0.28****
suggest that stable measures of approximate entropy [28] and LZ
complexity [31] can be obtained with as few as 50 and 30 data
#P ≤ .10; *P ≤ .05; **P ≤ .01; ***P ≤ .005; ****P ≤ .001. points, respectively. Dynamic patterns have been assigned using
data sets with as few as 100 data points [32]; studies of corporate
innovations have used 50, 74 and 102 data points [33]. Second, it
(approximate entropy) but insensitivity to conditions (Lyapunov is unclear whether women accurately reported the level of violence
exponent)] reinforces the possibility that IPV is a complex phenom- perpetrated by their partners or themselves. However, Regan et al.
enon, not explainable by a single behavioural model. One possible [50] found that violence reports by husbands and wives were
explanation is that none of the three theories is correct; either a new highly correlated. Finally, the predominance of Hispanics within
theory is needed or IPV represents a ‘dynamic disorder’, a disorder the sample may limit the generalizability of the findings. Caetano
defined by its own dynamics rather than any ‘classical’ cause [46]. et al. [51] found that, over a 5-year period, Hispanics were 2.5
Another possibility is that the dynamic pattern may evolve over times as likely as Anglos to initiate IPV, while reporting violence
time, depending on stressors, resources and support. Thus, IPV may recurrence rate four times higher.
begin as a sudden, unexpected explosion as family systems theory
would predict, but over time such violence may become tolerated by
the couple. Eventually, the husband may realize that the violence is Conclusion
not the only strategy that works for him, and power and control
wheel strategies emerge. Finally, after years of abuse and loss of In conclusion, IPV is rarely a predictable, periodic phenomenon;
control, these strategies are no longer necessary to achieve control no one behavioural model seems to describe the violence dynam-
of the relationship and the predictable cycle of violence pattern ics for all violent relationships. Yet, the measures of non-linearity
takes over. A final possible explanation is that IPV is a conglomerate and specific dynamic patterns correlate with different violent fea-
of different couples in different environments yielding different tures of these relationships. These observations may have impor-
dynamics; no one behavioural model will ever explain every violent tant implications for our understanding of the phenomenon and for
relationship. intervention.
The observations of non-linearity and varying dynamic patterns
have potential treatment implications. First, non-specific
Acknowledgements
approaches such as control/ anti-control interventions [47], the
presence of a third individual [7] or mindfulness approaches [48] This study was funded through a grant from the National Science
could alter the violence dynamics. Second, different approaches Foundation (#0826812). Automated data collection was provided
could be applied depending on the dynamic pattern of the violence. by the University of Colorado, Department of Family Medicine
Thus, although periodic dynamics should respond in predictable Information Services group. We wish to thank Stephanie Mitchell,
ways to interventions that address stressful triggers, approaches that Kelli Giacomini, Robert Mesec and Wilson Pace for their invalu-
attempt to reinforce positive attractors or nudge negative attractors able assistance.

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