Introduction

Attention-deficit/hyperactivity disorder (ADHD) is among the most common psychiatric disorders in children and adolescents (Polanczyk et al., 2015). Both subclinical symptoms and clinical ADHD, as well as their developmental courses, are associated with a myriad of concurrent and long-term impairments in social, academic, and adaptive functioning (Pingault et al., 2011; Willcutt et al., 2012). Twin studies have demonstrated substantial genetic as well as unique environmental influences on ADHD (Pingault et al., 2015; Zheng et al., 2020). These findings highlight the potential and necessity for identifying modifiable environmental risk or protective factors to prevent, manage, and treat ADHD, with parenting practices being salient candidates. Reviews and meta-analyses have shown that children’s ADHD is negatively related to positive and supportive parenting (e.g., parental warmth, involvement), and positively related to negative parenting behaviors (e.g., harsh control, inconsistent discipline; Deault, 2010; Ellis & Nigg, 2009; Pinquart, 2017). Accordingly, cumulative evidence has shown the effectiveness of parenting interventions for ADHD (Coates et al., 2015).

Adolescents are at heightened risk for psychopathology under the confluence of biological, social, and cognitive changes during this developmental period (Lerner & Galambos, 1998). Notably, positive parent–adolescent relationships (e.g., parental warmth) decline during the transition into adolescence and across middle adolescence (Smetana & Rote, 2019). ADHD clinical diagnoses and subclinical symptoms, as well as their aetiologies and developmental courses, also demonstrate persistence and changes from childhood to adolescence (Pingault et al., 2011, 2015; Willcutt et al., 2012). Therefore, it is valuable to unravel the reciprocal links between parenting practices and ADHD symptoms in adolescence.

Daily Parenting, Parent–Adolescent Relationships, and Adolescent Adjustment

A surge of studies has investigated parenting practices, parent–adolescent relationships, and related adolescent adjustment through the collection of intensive longitudinal data (ILD) using daily diary designs or ecological momentary assessments (EMAs). ILD offer a novel and unique way to elucidate day-to-day fluctuations in parenting behaviors and parent–adolescent relationships, as well as their concurrent and subsequent associations with adolescent development (Keijsers et al., 2022). Cumulative evidence has documented day-to-day fluctuations in positive parenting (e.g., parental warmth; Xu & Zheng, 2023), negative parenting (e.g., psychological control; Aunola et al., 2013; Xu & Zheng, 2022), and family conflict (Fosco & Lydon-Staley, 2020). Short-term parenting and parent–adolescent dynamics can inform long-term adolescent adjustment. For instance, the magnitude of fluctuations in daily parenting behaviors and parent–adolescent relationships–termed lability–is positively related to later adolescent internalizing and externalizing problems (Fosco et al., 2019). Notably, the associations between parenting and adolescent adjustment at the conventional macro timescale (e.g., annually) do not necessarily generalize to the micro daily level, indicating the complex characteristics of family dynamics on different timescales (Boele et al., 2020; Keijsers et al., 2022).

Studies have increasingly explored the links between daily parenting and adolescent adjustment, although primarily on socioemotional adjustment other than ADHD. For instance, in 151 parent–adolescent dyads assessed daily over three weeks, daily adolescent internalizing problems were positively linked with same-day parent–adolescent conflict; the negative link between daily family cohesion and same-day adolescent depression only manifested in families with relatively low general levels of family cohesion across days (Fosco & Lydon-Staley, 2020). More importantly, recent studies have revealed substantial between-family differences in within-family short-term dynamics (Bülow et al., 2022), which calls for more efforts into exploring the antecedents of these between-family differences.

Different parenting behaviors could interactively influence the development of externalizing problems. For instance, children of parents who consistently showed high and stable levels of harsh discipline and low levels of warmth from kindergarten to grade 2 showed the highest levels of externalizing problems in grade 4 and 5 (Zheng et al., 2017). The interaction of different parenting behaviors in the daily context has been addressed only rarely. In one exception, adolescents reported feeling more loved on days when parents reported more warmth and less parent–adolescent conflict, and the link between daily parent–adolescent conflict and adolescents’ feeling of being loved was suppressed on days with high parental warmth (Coffey et al., 2022). Similarly, Silva et al. (2020) documented a positive link between daily parent–adolescent conflict and same-day adolescent distress, which was suppressed by high person-mean levels of parental warmth. Therefore, it is beneficial to explore how the relations between one daily parenting behavior and daily adolescent adjustment are moderated by another parenting behavior assessed either at a micro level in daily contexts or at the conventional macro level.

A small but growing body of research has used various designs to collect ILD of child ADHD symptoms and related parenting behaviors and parent–child relationships. Daily ADHD symptoms fluctuated over 50 days in a clinical sample of children and adolescents with ADHD (van der Schans et al., 2020). Earlier work using EMA over 7 days revealed that ADHD symptoms varied substantially when engaging in different activities depending on the specific contexts and situation (Whalen et al., 2006), and that momentary ADHD symptoms were positively linked with momentary maternal distress (Whalen et al., 2011). Lability in daily parental warmth over one week was positively associated with child ADHD symptoms (Li & Lansford, 2018), suggesting parenting lability as an indicator of parenting inconsistency. Notably, a recent 2-week daily diary study in parents of children with ADHD showed that daily parental self-efficacy explained the negative link between daily inattentive symptoms and parental involvement quality (Ogg et al., 2022). Daily hyperactive/impulsive symptoms suppressed the positive link between parental time/energy and involvement quality (Ogg et al., 2022). More research is needed to further elucidate the daily links between child and adolescent ADHD symptoms and various parenting practices.

Within-Person Daily Processes as Dynamical Systems

Dynamic systems theory conceptualizes human development as interrelated complex developmental systems that self-regulate and self-organize in the emergence of novel adaptation (Lewis, 2000). A system refers to the constellation of a set of variables and the dynamics that describe the interrelationships and mechanisms of change of these variables over time. ILD offer a novel opportunity for understanding daily family processes as dynamical systems and to elucidate their temporal dynamics that (co)regulate these processes. The dynamical systems approach explicitly models changes in the system as variables of interest by leveraging differential equations to estimate derivatives (Boker, 2012; Chow et al., 2017). One type of dynamical systems that could be informative for illuminating within-person fluctuations in daily parenting practices and adolescent adjustment is the damping oscillating process (Boker, 2007; Chow et al., 2005; Zheng & Hu, 2021). Two temporal dynamics describe two distinct features of such a dynamical system: the oscillating parameter η that indicates fluctuation over time, and the damping parameter ζ that indicates change in the amplitude of fluctuation over time (Fig. 1). For an adolescent with elevated ADHD symptoms, a negative oscillating parameter suggests that the system will detect a higher than average level of symptoms and self-regulate a return to maintain an equilibrium level. As the symptoms move further away from the equilibrium level, they tend to return more strongly, exhibiting the system’s sensitivity to the level of symptoms relative to the equilibrium level. In contrast, the damping parameter indicates the system’s sensitivity to change in symptoms and describes how gradually or abruptly elevated symptoms will return to the equilibrium level. A positive and negative damping parameter indicate acceleration and deceleration respectively, demonstrating either a speed up or slow down of the return to the equilibrium level over time.

Fig. 1
figure 1

A Damping Oscillating Process with Damping and Slow Oscillation (top) or without Damping but Fast Oscillation (bottom)

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There are substantial between-person differences in these temporal system dynamics. Some dynamical systems may adopt a more gradual and fine-tuning regulating mechanism whereby elevated symptoms return to the equilibrium level more smoothly, whereas some other systems may follow a quick regulating mechanism whereby elevated symptoms return much faster (Zheng & Hu, 2021). More rapid fluctuations in psychiatric symptoms as reflected in the oscillating parameter indicate higher risk for severe violent behaviors (Odgers et al., 2009). Critical slowing down in affect fluctuations suggests upcoming system phase transitions, such as between the onset and termination of depression (van de Leemput et al., 2014). Moreover, macro level contexts (e.g., family socioeconomic status) could modify the environmental sources of individual differences in these daily affective regulating processes (Zheng & Hu, 2021). Therefore, a better understanding of these between-person and between-family differences in system dynamics as well as their antecedents could facilitate the identification of maladaptive regulating mechanisms as well as macro level risk and protective factors that could prevent or ameliorate maladaptation and promote adaptive dynamic processes.

The literature has widely supported the bidirectional associations between parenting practices and child adjustment (Paschall & Mastergeorge, 2016; Shelleby & Ogg, 2020), although their reciprocal associations in daily contexts have rarely been investigated. Aunola et al. (2013) found in 1st graders that parental psychological control was positively linked with child negative emotions the next day, whereas child negative emotions were negatively linked to parental psychological control the next day in a 1-week daily diary study. Xu and Zheng (2022) revealed bidirectional cross-day associations between parental psychological control and adolescent emotional problems in a month-long daily diary study. In parent–adolescent dyads followed daily over two weeks across two waves one year apart, dyads who generally got along better with each other demonstrated stronger co-regulation of their daily happiness (Mercado et al., 2019), suggesting interrelated parent–adolescent daily affective processes modulated by family-level parent–adolescent relationships.

Coupled dynamical systems using coupled differential equations could further elaborate how two dynamic processes are interwoven to maintain a delicate balance and equilibrium level in daily life (Boker & Laurenceau, 2007; Chow et al., 2017). For instance, the parental warmth system may be sensitive to the level of (e.g., elevated) or change (e.g., increasing) in the adolescent ADHD system and consequently regulates its own level accordingly, demonstrating an adolescent-driven effect in which parents respond to adolescents’ behavioral changes. The adolescent ADHD system may also be sensitive to the level of (e.g., lower) or change (e.g., decreasing) in the parental warmth system and makes regulating changes accordingly, consistent with a parent-driven effect. Several studies have adopted coupled damping oscillating processes to investigate the co-regulation of affective processes (Steele & Ferrer, 2011) and couple interactions (Feinberg et al., 2017) in couples. This dynamical systems approach nonetheless has yet to be applied to investigate daily parenting practices and adolescent adjustment.

The Current Study

Both parenting practices and adolescent adjustment demonstrate meaningful and substantial day-to-day within-person fluctuations. Developmental changes at multiple levels during adolescence, including parent–adolescent relationship quality as well as ADHD symptoms, render adolescence a unique developmental window for elucidating their reciprocal links. ILD enable us to understand daily parenting behaviors and adolescent adjustment as dynamical systems and to uncover their daily links on a fine-grained micro timescale. Leveraging 30-day daily diary data and dynamical systems modeling in a sample of community-recruited adolescents and their parents, this study aimed to investigate perceived daily parental warmth and adolescent ADHD symptoms as coupled dynamical systems. Parental warmth has been viewed as an indicator of responsive and nurturant parenting, and broadly refers to a high-quality affectional bond in the parent–child dyad; it includes parental affection, support, and care that parents express to their children (Baumrind et al., 2010; Morris et al., 2013; Rohner et al., 2012). Emerging evidence has demonstrated substantial and meaningful within-person daily fluctuations of parental warmth (Xu & Zheng, 2023) and its link to child ADHD symptoms (Li & Lansford, 2018). Based on the literature, we expected that the two systems are coupled with each other, although there would be between-family differences in these system dynamics. Specifically, parents would be sensitive to the level of and change in ADHD symptoms so they would increase their level of warmth to down regulate elevated symptoms; adolescents would be sensitive to the level of and change in parental warmth such that the level of their ADHD symptoms would decrease when experiencing more warmth.

Second, based on the conventional and daily diary literature on the interactive role of parenting behaviors, we explored whether baseline parental non-harsh discipline could partly explain between-family differences in these system dynamics. We used parental non-harsh discipline assessed at baseline rather than harsh or inconsistent discipline at the daily level because the frequency of daily harsh or inconsistent discipline would be rather low in a community sample of adolescents. We conceptualized this measure as indicating a relatively stable between-family difference that indicates a general familial context (see Silva et al., 2020) of the parental use of discipline to control adolescents’ behaviors. Using parent-reported non-harsh discipline together with adolescent-reported daily parental warmth, these two parenting behaviors (control/discipline and warmth/responsiveness) define unique patterns or styles of parenting behaviors (Morris et al., 2013; Smetana & Rote, 2019). We expected that in families with average or higher baseline discipline, parents and adolescents would be more sensitive to each other than those in families with relatively low baseline discipline. Nonetheless, due to the scant literature, we considered this hypothesis as exploratory.

Method

Participants and Procedure

Participants were recruited from the community through multiple channels, including flyers and newsletters circulated in a western province in Canada. Recruitment information was also posted on social media (e.g., Instagram, Facebook). Trained research assistants provided study information to participants who reached out and expressed interest. Interested participants received online consent and assent forms; participants who provided consent and assent received an online baseline survey, which took approximately 45 min to complete. Five days after submitting the baseline survey, participants received an online daily survey each day at 5 pm for the following 30 consecutive days. The daily survey took approximately 15 min to complete, and the participants were instructed to fill out the daily survey before their bedtime. A $45 e-gift card was provided to the participants as compensation after completing the month-long daily survey. This study was approved by the Research Ethics Board at University of Alberta.

Total participants were 99 Canadian parent–adolescent dyads recruited from 2019 to 2020. The ages of the participating parents ranged between 30 to 60 years old (M = 43.7, SD = 6.2; 72% female, 2% missing), and adolescents’ ages ranged between 12 and 17 years old (M = 14.5, SD = 1.8; 55% female). Parents self-identified as White (58%), Asian (28%), Latino or Hispanic (4%), Black or African (2%), multiracial (2%), and other (2%; 4% missing). Adolescents self-identified as White (54%), Asian (23%), multiracial (9%), Latino or Hispanic (4%), Black or African (2%), and other (7%; 1% missing). Among the parents, 78% had a bachelor’s degree or above, 2% middle school or below, 18% high school or equivalent, and 2% missing. Parents also reported their annual income: 18% had a personal annual income below $35,000, 2% between $35,000 to $45,000, 12% between $45,000 to $55,000, 18% between $55,000 to $65,000, and 38% above $65,000 (4% missing).

Because short daily observations could yield unreliable assessment of within-person variation, participants with fewer than 10 valid daily reports were excluded from the analysis (13 adolescents, 45 total excluded observations, range 1–9 days). The final analytic sample consisted of 86 adolescents who reported on average 23.3 daily surveys (total 2,005 observations, range 10–30, 87% ≥ 20; Mage = 14.5 years, 55% female, 56% White, 22% Asian, 9% multiracial, 6% other, 5% Latino or Hispanic, 1% Black or African, 1% missing). Participants who were included vs. excluded did not show significant differences in sex, χ2(1) = 0.003, p = 0.960, or ethnicity, χ2(1) = 1.47, p = 0.230 (coded as White vs. others). There was no difference in their age, t(96) = 0.46, p = 0.320 or baseline parental non-harsh discipline, t(95) = -1.48, p = 0.070. There were no differences in the participating parent’s sex, χ2(1) = 2.40, p = 0.300, parental income, t(93) = -0.76, p = 0.450, average level of parental warmth t(97) = 0.46, p = 0.650, or adolescent ADHD symptoms, t(97) = -1.17, p = 0.240.

Measures

Baseline Parent-Reported Parental Non-Harsh Discipline

In the baseline survey, parents reported the general level of parental use of non-harsh disciplinary strategies when adolescents did something wrong, disobeyed parents, or made parents angry in the past year. Four items from the Straus et al. (1998) Parent–Child Conflict Tactics Scales were used (“Explained why something was wrong.” “Send to their room.” “Gave them something else to do instead of what they were doing wrong.” “Take away privileges, or ground them.”), and each item was rated on a 4-point Likert scale (0 = never, 1 = 1–2 times, 2 = several times, 3 = very often). Items were averaged to create a composite score, with higher scores indicating higher levels of parental non-harsh discipline (ω = 0.84).

Adolescent Perceived Daily Parental Warmth

In daily surveys, adolescents reported perceived parental warmth using six items modified from previous research (Artemis & Touloumakos, 2016; Quach et al., 2015; e.g., “My parents made me feel wanted and needed today.” “My parents were really interested in what I did today.” “My parents talked to me in a warm and loving way.”). Each item was rated on a 4-point Likert scale (1 = almost never true, 2 = rarely true, 3 = sometimes true, 4 = almost always true), and an average score was created, with higher scores indicating higher levels of perceived daily parental warmth. Multilevel confirmatory factor analysis (MLCFA) showed a satisfactory model fit (CFI = 0.99, TLI = 0.98, RMSEA = 0.02, SRMRw = 0.01, SRMRb = 0.02) with one latent factor of parental warmth at both the within- and between-level, and an averaged standardized factor loading of 0.64 and 0.93 at the within- and between-level, respectively. Following Geldhof et al. (2014), the within-level ordinal ωw was 0.90, and the between-level ordinal ωb was 0.98.

Daily Adolescent ADHD Subclinical Symptoms

Adolescents reported their daily ADHD symptoms using five items adapted from the Strengths and Difficulties Questionnaire (e.g., “I am restless. I cannot stay still for long.” “I am constantly fidgeting or squirming.” “I am easily distracted. I find it difficult to concentrate.” “I think before I do things.” “I finish the work I am doing. My attention is good.” (Goodman et al., 1998) on a 3-point Likert scale (0 = not true, 1 = somewhat true, 2 = certainly true), with the last two items reverse coded. MLCFA showed a satisfactory model fit (CFI = 0.97, TLI = 0.94, RMSEA = 0.03, SRMRw = 0.12, SRMRb = 0.03) with one latent factor of ADHD symptoms at both the within- and between-level, and an averaged standardized factor loading of 0.53 and 0.89 at the within- and between-level, respectively. We calculated an average score over all items, with higher scores indicating higher levels of ADHD symptoms. The ordinal ωw was 0.71 and ωb was 0.95.

Analytic Strategies

Latent differential equation structural models with 5-dimension time-delay embedding (Boker, 2007, 2012; Chow et al., 2005) were used to estimate the derivatives and to fit a damping oscillating model to perceived daily parental warmth and ADHD symptoms. As shown in Fig. 2, for five consecutive observations of equal spacing, three latent variables, x (deviation from the equilibrium level, also called displacement or relative level), \(\dot{x}\) (first derivative, or velocity, which represents instantaneous linear slope or change), and \(\ddot{x}\) (second derivative, or acceleration, which indicates the change in change), were estimated by specifying a fixed factor loading matrix L (Eq. (1)). Note that each column in L is the indefinite integral to its left column. The regression coefficient of the acceleration (\(\ddot{x}\)) on the velocity (\(\dot{x}\)) represents the damping parameter ζ, and the regression coefficient of the acceleration (\(\ddot{x}\)) on the displacement (x) represents the oscillating parameter η (Eq. (2)). A significant negative η suggests a dynamical system with oscillation. The amplitude of the oscillating system decreases over time if ζ < 0 (i.e., damping) and increases if ζ > 0 (i.e., amplifying). In the current model specification, the prediction residual e (assumed to have zero mean, identically and independently distributed) can include both time-independent unique variance such as random or stochastic environmental shock or disturbance, and time-dependent common variance that propagates over time but not explained in the model.

Fig. 2
figure 2

A Second-order Latent Differential Equation Model Fitting Univariate Damping Oscillating Model Using 5-dimension Time-delay Embedding with Five Consecutive Observations

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$$L=\left[\begin{array}{crc}1& -2& {\left(-2\right)}^{2}/2\\ 1& -1& {\left(-1\right)}^{2}/2\\ 1& 0& 0\\ 1& 1& {1}^{2}/2\\ 1& 2& {2}^{2}/2\end{array}\right]$$
(1)
$${\ddot{x}}_{t}=\zeta {\dot{x}}_{t}+\eta {x}_{t}+{e}_{t}$$
(2)

As shown in Fig. 3 and Eqs. (3) and (4), for bivariate coupled damping oscillating models, the model specification within each outcome (x and y) is the same as in the univariate model. Covariances between the displacement and velocity of each variable with that of the other variable were estimated as indicated by the four dotted double-arrow paths (\({\sigma }_{xy}\), \({\sigma }_{x\dot{y}}\), \({\sigma }_{\dot{x}y}\), and \({\sigma }_{\dot{x}\dot{y}}\)). Besides its own oscillating and damping parameters (\({\eta }_{1}\) and \({\zeta }_{1}\)), the acceleration of perceived parental warmth (x) was also predicted by the displacement and velocity of ADHD symptoms (y), as indicated by the dotted single-arrow paths (\({\gamma }_{1}\) and \({\kappa }_{1}\)). The same applies to ADHD symptoms (\({\eta }_{2}\) and \({\zeta }_{2}{;\gamma }_{2}\) and \({\kappa }_{2}\)) (Boker & Laurenceau, 2007; Hu et al., 2014). Finally, the covariance of the two residuals was also estimated (\({\sigma }_{\ddot{x}\ddot{y}}\)).

Fig. 3
figure 3

A Second-order Latent Differential Equation Model Fitting Coupled Damping Oscillating Model

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$${\ddot{X}}_{t}={\eta }_{1}{X}_{t}+{\zeta }_{1}{\dot{X}}_{t}+{\gamma }_{1}{Y}_{t}+{\kappa }_{1}{\dot{Y}}_{t}+{e}_{1t}$$
(3)
$${\ddot{Y}}_{t}={\eta }_{2}{Y}_{t}+{\zeta }_{2}{\dot{Y}}_{t}+{\gamma }_{2}{X}_{t}+{\kappa }_{2}{\dot{Y}}_{t}+{e}_{2t}$$
(4)

To examine between-person differences in these system dynamics between the coupled dynamical systems, we extended these models into a multilevel model where random components were estimated for each of the self-regulating (four in total) and co-regulating (four in total) dynamics and incorporated between-level predictors (i.e., baseline parental discipline) for these random components (Hu et al., 2014). Because 95% Credible Intervals (CIs) could not be used to infer significance of variance components (variance cannot be negative) in Bayesian estimation, we used the ratio (≥ 0.25) of the SD (σ) vs. fixed effects (μ) of the estimated random components following the recommendations of Bolger et al. (2019). Following convention in time series analysis, individual time series data were first within-person centered around its equilibrium level (i.e., individual average scores). All models were fit in Mplus 8 (Muthén & Muthén, 1998–2017) using Bayesian estimation with the default uninformative priors. Like the Full Information Maximum Likelihood estimator, Bayesian estimation also uses full information from all available observations. Two MCMC chains and 100,000 maximum iterations for each chain were used. The default value of 1.05 for the Potential Scale Reduction convergence criterion was used.

Results

Descriptive Statistics

Perceived parental warmth and ADHD symptoms (Table 1) were significantly and negatively correlated at both the within- (r = -0.13, p < 0.01) and between-levels (r = -0.65, p < 0.001). At the between-level, parental non-harsh discipline was significantly correlated with both perceived parental warmth (r = -0.22, p < 0.05) and ADHD symptoms (r = 0.39, p < 0.01). Based on the intra-class correlations (ICCs), despite relatively high average levels of perceived parental warmth (M = 3.22) and low average levels of ADHD symptoms (M = 0.54), both constructs demonstrated within-person fluctuations across the assessed period, with 23% and 21% variance attributed to within-person variation, respectively.

Table 1 Descriptive Statistics for Investigated Variables
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Coupled Latent Differential Equation Structural Modeling

The negative and significant estimates of the oscillating parameter (Table 2) for perceived parental warmth and ADHD symptoms (ηs = -0.38 and -0.42, ps < 0.001) suggest that both systems demonstrated within-person fluctuations over time, consistent with the ICCs’ suggestions. Parental warmth demonstrated no significant changes in the magnitude of its fluctuation (\({\zeta }_{1}\) = -0.08, ns), whereas elevated ADHD symptoms tended to stabilize to an equilibrium level over time (\({\zeta }_{2}\) = -0.15, p = 0.007). The change in ADHD symptoms was negatively linked to the acceleration in parental warmth (\({\kappa }_{1}\) = -0.33, p = 0.001), indicating an adolescent-driven effect whereby adolescents perceived that their parents reacted to change in ADHD symptoms. Figure 4 illustrates the pattern of this adolescent-driven effect with four scenarios based on model estimated trajectories. Facing elevated and increasing levels of ADHD symptoms (Fig. 4a), parental warmth which was originally at its equilibrium level fluctuates accordingly as ADHD symptoms gradually return to its equilibrium level. In contrast, when facing lower and decreasing levels of parental warmth (Fig. 4b), ADHD symptoms are barely disturbed by this change to the coupled system and generally remain at the same low level (i.e., the absence of a parent-driven effect). When both systems demonstrate changes simultaneously (Fig. 4c), they co-regulate and ADHD symptoms will return to their equilibrium level. Figure 4a–c illustrate scenarios of stabilized coupled systems where the two systems have already synchronized (these two systems already existed before the first day’s assessment), as the peaks of their fluctuations are always opposite to each other. Figure 4d demonstrates the synchronization of one initially unstabilized coupled system. When both parental warmth and ADHD symptoms are lower than their average levels, ADHD symptoms will self-regulate back to equilibrium, and drive the co-regulation of parental warmth such that their fluctuations will be synchronized gradually, as evidenced by the opposite peaks around day 15.

Table 2 Parameter Estimates of Coupled Latent Differential Equation Models
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Fig. 4
figure 4

Model Estimated Coupled Dynamical Systems of Perceived Parental Warmth (parent) and Adolescent ADHD Symptoms (adolescent) with Initial Changes in Only Adolescent ADHD Symptoms (a), Only Perceived Parental Warmth (b), or Both (c, d). Note. a-c depict coupled dynamical systems that had already synchronized (their peaks are always opposite to each other) before the first daily assessment, while d depicts coupled dynamical systems that had not yet synchronized but synchronized along the assessment period

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Regarding the covariances, lower parental warmth was correlated with elevated ADHD symptoms (r = -0.27, p < 0.001), and an increase in parental warmth was accompanied by a decrease in ADHD symptoms (r = -0.39, p < 0.001). An adolescent with elevated symptoms tended to show a decreasing or smaller increasing slope in symptoms (r = -0.10, p = 0.020). Adolescents whose symptoms were increasing also perceived their parents to have elevated warmth (r = 0.25, p < 0.001), and increasing parental warmth was accompanied by relatively lower levels of ADHD symptoms (r = -0.21, p < 0.001). No other pairwise correlations among the derivatives or their residuals were significant.

Moderated Multi-Level Coupled Dynamical Systems

All these system dynamics demonstrated significant between-person differences (i.e., random effects; Table 2) using the ≥ 0.25 ratio criterion, though the two oscillating parameters showed acceptable boundary ratios (0.23). Baseline parental non-harsh discipline significantly moderated the oscillating parameter (B = 0.09, 95% CI [0.02, 0.16], one-tail p = 0.006) and the damping parameter (B = -0.14, 95% CI [-0.26, -0.02], one-tail p = 0.014) of ADHD symptoms, but not other system dynamics. The two coefficients remained significant and in the same direction after including adolescent age, sex (0 = female), and race/ethnicity (0 = White) as covariates for the oscillating parameter (B = 0.09, 95% CI [0.01, 0.17], one-tail p = 0.011) and the damping parameter (B = -0.27, 95% CI [-0.45, -0.10], one-tail p < 0.001). Figure 5 plots model estimated (co)regulating trajectories at ± 1 SD of baseline parental non-harsh discipline. Among families where parents rarely use non-harsh discipline (top), both ADHD symptoms and parental warmth tended to fluctuate without converging to an equilibrium level compared to the group average pattern illustrated in Fig. 4c. In contrast, in families with more frequent baseline parental non-harsh discipline (bottom), ADHD symptoms tended to return to their equilibrium level faster with smaller fluctuating magnitudes when compared to the group average pattern (Fig. 4c). Parental warmth also demonstrated small fluctuating magnitudes. In other words, adolescents in families with more frequent use of baseline non-harsh discipline tended to show more stable and less fluctuating ADHD symptoms and perceived parental warmth.

Fig. 5
figure 5

Moderated Coupled Dynamical Systems of Perceived Parental Warmth (parent) and Adolescent ADHD Symptoms (adolescent) at One Standard Deviation Below (top) or Above (bottom) the Group Average of Family-level Parental Non-Harsh Discipline

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Discussion

ILD and the dynamical systems approach offer a novel opportunity for elucidating within-person fluctuations in daily parenting practices and adolescent adjustment as co-regulating dynamical systems on a micro timescale. Using month-long daily diary data and multilevel coupled differential equation modeling, this study explored the temporal system dynamics that characterize the self- and co-regulating mechanisms underlying daily dynamic processes involving perceived parental warmth and adolescent ADHD symptoms in a community sample. The findings reveal that both daily parental warmth and ADHD symptoms show meaningful day-to-day within-person fluctuations. Dynamical systems modeling characterizes them as two oscillating processes, with ADHD symptoms showing a damping in its fluctuating magnitude but parental warmth showing a stable one. The two systems are coupled by a link from an increase in ADHD symptoms to the deceleration of parental warmth. Family-level parental non-harsh discipline moderates the dynamics of the ADHD system, such that in families with more baseline non-harsh discipline, both systems tend to be more stable and fluctuate less.

Daily Parental Warmth and Adolescent ADHD Symptoms as Coupled Dynamical Systems

Consistent with previous studies (Coffey et al., 2022; Silva et al., 2020), perceived parental warmth demonstrates substantial within-person daily fluctuations, with about a quarter of its variance attributed to within-person variation. This finding is congruent with the broader parenting literature utilizing ILD and suggests that parenting practices show meaningful fluctuations on micro timescales (Boele et al., 2020; Keijsers et al., 2022). Moreover, the magnitude of its fluctuation generally remains stable over time, indicating the ebb and flow of parental warmth in daily lives which are possibly influenced by internal (e.g., parental mood, child behaviors) and external (e.g., work-life spillover) experiences (Xu & Zheng, 2023). About 20% of the variance in ADHD symptoms could be attributed to within-person variation, consistent with van der Schans et al. (2020). Notably, the ADHD system demonstrates an interesting and adaptive self-regulating mechanism in which elevated symptoms will gradually return to their equilibrium level as indicated by the negative damping pattern.

Our expectation regarding the coupled dynamical systems was partly supported. As opposed to the reciprocal associations (Paschall & Mastergeorge, 2016; Shelleby & Ogg, 2020) or parent-driven effects (Coates et al., 2015; Deault, 2010; Ellis & Nigg, 2009) implied by the literature, but more aligned with a recent daily diary study (Ogg et al., 2022), we found only an adolescent-driven effect, such that parental warmth tends to decelerate when ADHD symptoms increase. However, in contrast to the general finding of a maladaptive mechanism whereby child ADHD symptoms could decrease parental warmth and involvement (Ogg et al., 2022; Shelleby & Ogg, 2020), dynamical systems modeling revealed a more nuanced and adaptive mechanism involving temporal dynamics. Specifically, as opposed to the two more rudimentary system dynamics sensitive to the levels (i.e., \(\eta\) and \(\gamma\)), the significantly coupled path (\(\kappa\)) indicates a sensitivity to change in the system, representing a more fine-tuning system regulation dynamic (Zheng & Hu, 2021). Hence, parents as perceived by adolescents are sensitive to changes in adolescents’ ADHD symptoms in daily contexts, and accordingly will self-regulate and change their affection toward adolescents more gradually rather than fluctuate too quickly or abruptly. Although this change of parental warmth does not directly influence ADHD symptoms, it likely will influence other adolescent adjustment given its positive role in a variety of child and adolescent outcomes (Pinquart, 2017).

Several tentative explanations could explain the absence of a parent-driven effect. Methodologically, participants were from a community sample and generally demonstrated low levels of ADHD symptoms. Moreover, adolescents reported on their ADHD symptoms as well as their perceived parental warmth. Elevated ADHD symptoms could potentially distort their reports of parental warmth and subsequently obscure parent-driven effects. Theoretically, as adolescents begin to seek more autonomy and form identities (Lerner & Galambos, 1998), parent–adolescent relationships in the whole family system also begin to shift and reorganize, yielding a stronger voice for adolescents (Smetana & Rote, 2019). This normative developmental change could lead to more salient and direct adolescent-driven effects in proximal daily family socialization processes, while parent-driven effects may become more indirect and distal, acting in the macro familial context as evidenced by parental discipline.

The Role of Parental Non-Harsh Discipline in System Regulation

Consistent with recent studies (Bülow et al., 2022), families show different regulating dynamics in their daily lives regarding both parental warmth and ADHD symptoms. Despite a positive relation between parental non-harsh discipline and ADHD symptoms at the between-family level, parental non-harsh discipline moderates the self-regulation of ADHD symptoms in an adaptive way, congruent with claims that the associations between parenting and adolescent adjustment are not necessarily comparable at different timescales (Boele et al., 2020; Keijsers et al., 2022). Families with relatively high and low baseline parental non-harsh discipline demonstrate two distinct patterns in the regulation of the two systems. Specifically, among families with high baseline non-harsh discipline, adolescents tend to show more stable and less fluctuating ADHD symptoms. In contrast, in families with low baseline non-harsh discipline, ADHD symptoms and parental warmth tend to fluctuate without converging to an equilibrium. Given that faster and larger symptom fluctuations typically forewarn adverse outcomes (Chow et al., 2005; Odgers et al., 2009; van de Leemput et al., 2014), the findings uncover potential maladaptive daily family dynamics and processes in parent–adolescent interactions among families with low baseline non-harsh discipline.

Consistent with the broader literature on parenting and ADHD (Coates et al., 2015; Deault, 2010; Ellis & Nigg, 2009), parental non-harsh discipline plausibly exerts potent effects by controlling and managing the manifestation of ADHD symptoms, as evidenced by stable symptom fluctuations among families with more baseline non-harsh discipline. Parents who rarely use discipline may be neglectful and indulgent of their children. The findings also join the body of recent studies (Fosco & Lydon-Staley, 2020; Mercado et al., 2019; Silva et al., 2020) highlighting that family- or person-level parent–adolescent relationships or parenting behaviors can modulate daily parent–adolescent dynamics as well as the interactive roles of different parenting behaviors in influencing adolescent adjustment (Coffey et al., 2022; Zheng et al., 2017).

Theoretical and Practical Implications

This study makes several theoretical and practical contributions to the literature. First, adolescence is a sensitive developmental period under the confluence of changes in multiple aspects conferring opportunities as well as risks for adaptive development (Lerner & Galambos, 1998). Changes in parent–adolescent relationships (Smetana & Rote, 2019) and ADHD symptoms during adolescence (Pingault et al., 2011; Willcutt et al., 2012) call for further understanding of their dynamic links at multiple timescales. The findings show that both parental warmth and ADHD symptoms demonstrate substantial day-to-day within-person fluctuations. Moreover, there are substantial between-family differences in system dynamics. Parenting and adolescent development occur in daily contexts. Hence, future research should further investigate daily parenting practices, parent–adolescent relationships, and adolescent adjustment. Second, corroborating the dynamic system theory of human development (Lewis, 2000), daily parental warmth and ADHD symptoms could be understood as coupled dynamical systems that are further modulated by family-level non-harsh discipline. The dynamical systems approach offers a novel opportunity for uncovering how changes are interrelated by explicitly incorporating change and change in change in the modeling process. Accordingly, we revealed fine-grained and nuanced patterns regarding adolescent-driven effects between the two dynamic processes along various levels of baseline parental discipline.

Third, the current findings contribute to the parenting literature on reactive positive parenting in responding to ADHD symptoms, but at a daily level, as well as the moderating role of macro level parenting behaviors in the broader familial context. The findings highlight the interactive roles of parental warmth and parental non-harsh discipline tapping two defining dimensions–affection/responsiveness and control/discipline–of parenting styles (Morris et al., 2013; Smetana & Rote, 2019). Current findings suggest that the potential influences of parental warmth on ADHD symptoms may be less direct, but instead may operate through its influences on other family dynamics and adolescent adjustment (e.g., adolescent and family level coping mechanisms) that subsequently influence ADHD symptoms. From a practical point of view, the findings provide preliminary daily evidence supporting core components of some existing parenting interventions for ADHD (Coates et al., 2015), which target how parents react to their children in daily life as well as family-level behavioral control.

Strengths, Limitations, and Future Directions

The current study has several notable strengths. Past research generally only collected one to two weeks of daily reports. The month-long assessment of daily parental warmth and ADHD symptoms enables us to disentangle between-person differences from within-person fluctuations, and to more accurately capture within-person fluctuations with larger statistical power and greater generalizability to daily contexts compared to previous studies with shorter assessed time periods. The use of adolescent-reported parental warmth and parent-reported non-harsh discipline could partly circumvent the issue of shared method variance by the same informant as some daily diary studies have revealed (Xu & Zheng, 2022). The use of a relatively racially/ethnically diverse sample as opposed to a homogenous White sample also enhances the generalizability of our findings. To our knowledge, the current study represents the first endeavor to investigate daily parenting practices and adolescent adjustment as coupled dynamical systems in accordance with dynamic systems theory (Lewis, 2000).

Bearing strengths in mind, a few limitations should be considered when interpreting the results. First, the current sample is relatively small and the families generally had higher than average provincial socioeconomic status and parental education. Larger and more demographically representative samples are needed in future research. Second, we only examined subclinical ADHD symptoms in a community sample. Regulating dynamics likely will differ in clinical samples or at-risk families. For instance, adolescents with clinical ADHD may demonstrate non-damping or even amplifying fluctuating magnitudes. Future research is needed to explore self- and co-regulation in coupled systems in diverse populations. Relatedly, some of the items assessing ADHD symptoms (e.g., restlessness, distractibility) could also be related to internalizing problems (e.g., depression, anxiety). Future research could look into comorbidity of ADHD symptoms with other disorder symptoms at the daily level.

Third, we only measured perceived daily parental warmth, while parental non-harsh discipline was assessed at the family level, because daily discipline would be infrequent and rare in community populations of adolescents. Nonetheless, future studies could assess daily parent–adolescent closeness and conflict (Coffey et al., 2022; Fosco & Lydon-Staley, 2020) and other indicators of adolescent maladjustment (e.g., internalizing problems). Fourth, our daily measures of parental warmth and ADHD symptoms were both adolescent-reported. This could potentially incur the issue of common method variance that could inflate the identified daily links. Notably, adolescents with elevated daily ADHD symptoms could perceive their parents as being less warm on the same day. They may also be less adept at accurately reporting their own ADHD symptoms than other informants in their environment (e.g., parents). Future research employing multiple informants are needed to circumvent these issues. Fifth, we did not investigate the long-term changes and subsequent outcomes of these short-term dynamics. Measurement burst designs are needed to explore future outcomes connected to short-term dynamics. With the ongoing COVID pandemic, it is possible that these short-term coupling dynamics may have changed among some families, and that they may have led to various maladaptive developmental outcomes in the long run.

Conclusions

Daily parental warmth and adolescent ADHD symptoms both demonstrate meaningful within-person fluctuations and could be understood as coupled dynamical systems with oscillating processes. The ADHD system demonstrates an adaptive self-regulation mechanism whereby elevated symptoms will return to their normal level over time. The parental warmth system is sensitive to change in ADHD symptoms such that parents will fine-tune their warmth toward adolescents with gradual changes when adolescents demonstrate heightened symptoms. Among families with more parental non-harsh discipline, both systems tend to be more stable and show less fluctuation. ILD and dynamical systems approach offer a new lens for uncovering short-term family dynamics and adolescent maladjustment at a refined micro daily level. Future research using multiple timescales should explore antecedents and consequences of between-family differences in these short-term family dynamics involving various parenting practices and adolescent developmental outcomes.