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Dynamical Systems Principles Underlying Resistance, Resilience, and 1
Growth 2
3
Yannick Hill, 1 Department of Human Movement Sciences, Faculty of Behavioral and 4
Movement Sciences, Vrije Universiteit Amsterdam, Amsterdam Movement Sciences, 5
the Netherlands; 2 Institute of Brain and Behaviour Amsterdam, Amsterdam, the 6
Netherlands; 3 Lyda Hill Institute for Human Resilience, Colorado Springs, CO, USA 7
Ruud J. R. Den Hartigh, Department of Psychology, University of Groningen, 8
Netherlands 9
10
RUNNING HEAD: Dynamic Principles of Resistance, Resilience, and Growth 11
12
Correspondence address: Yannick Hill, Department of Human Movement Sciences, Vrije 13
Universiteit Amsterdam, Van der Boechorstraat 7, 1081 BT Amsterdam, Netherlands, Email: 14
y.hill@vu.nl 15
Dynamics of Resistance, Resilience, and Growth 1
Abstract: Resilience has traditionally been conceptualized as resisting, bouncing back from, and 16
growing from a stressor. However, recent literature has pointed out that these are different 17
processes with bouncing back coming closest to the literal meaning of the term resilience. To 18
detect whether an individual demonstrates one of these three stressor-responses, different 19
analysis strategies have been suggested. However, deeper theoretical explanations for how 20
patterns of resistance, resilience, and growth come about, have been lacking. To address this 21
gap, this paper proposes a coherent framework based on a dynamical systems approach. We first 22
discuss how adapting to stressors emerges from complex interactions between multiple levels of 23
organization within the system. These interactions unfold on different time scales: what appears 24
as resistance on slower or macro scales may actually consist of bouncing back at micro scales 25
that change much faster. Next, we discuss how the different trajectories that distinguish 26
resistance, resilience, and growth can be understood through attractor dynamics. We address 27
the fixed-point attractors, which are commonly used in the resilience literature to detect early 28
warning signals of bifurcations following resilience losses. Moreover, we describe the 29
implications of limit cycles and strange attractors which capture multiple pathways to adapt to 30
stressors that can lead to growth patterns. We conclude that resisting, bouncing back from, or 31
growing from a stressor represent distinct phenomena that can be distinguished both empirically 32
and theoretically from a dynamical systems perspective. These distinctions may drive future 33
development of theoretical models, empirical measurements, and theory-driven interventions. 34
35
Key Words: Attractor Dynamics; Complexity; Interaction-dominance; Resilience; Stress; Time 36
Series 37
Dynamics of Resistance, Resilience, and Growth 2
INTRODUCTION 38
Over the past few decades, the concept of resilience has received considerable attention 39
within the scientific literature, especially psychology (Hosseini, Barker, & Ramirez-Marquez, 40
2016). This increased interest has led to a change in how we conceptualize and measure 41
resilience. Specifically, in psychology, there is an emerging trend that has moved the definition 42
of resilience from a personality trait to a process that unfolds through person-environment 43
interactions over time. However, there is no consensus within this movement on what 44
characterizes this process (Bonanno, 2004; Den Hartigh & Hill, 2022; Kegelaers, 2023; Kiefer & 45
Pincus, 2023; Lozano Nasi, Jans, & Steg, 2023, Taleb, 2012). For example, Masten and Powell 46
(2003) stated that resilience could be comprised of three adaptational processes in response to 47
stressors: resisting the impact of a stressor, bouncing back following a perturbation caused by a 48
stressor, and growing from a stressor. Despite the popularity of such multifaceted 49
conceptualizations, it has been argued that these types of stressor responses likely represent 50
distinct processes of which bouncing back comes closest to the literal resilience definition 51
(Carver, 1998; Den Hartigh & Hill, 2022; Hill, Den Hartigh, Meijer, De Jonge, & Van Yperen, 52
2018b; Kiefer & Pincus, 2023; Kiefer, Silva, Harrison, & Araújo, 2018; Layne, Ruzek, & 53
Dixon., 2021; Lozano Nasi et al., 2023). Therefore, Den Hartigh and Hill (2022) proposed the 54
use of time series data to make clear distinctions between resistance, resilience, and growth. 55
Based on insights from physics, the authors argue that these three processes can even be 56
distinguished mathematically by mapping the trajectory following the exposure to a stressor. 57
Importantly, while the authors primarily delivered a conceptual analysis with resulting 58
methodological and practical implications, they did not discuss the theoretical underpinnings of 59
resistance, resilience, and growth. The current article aims to fill this gap. 60
Dynamics of Resistance, Resilience, and Growth 3
Following the suggestion by Den Hartigh and Hill (2022), we proceed from a dynamical 61
systems approach because “this approach may […] offer a coherent framework to understand the 62
phenomena of resistance, resilience, and growth” (p. 6). In doing so, we also adopt the idea that 63
these concepts represent three different ways in which an individual can respond to stressors. In 64
the context of dynamical systems, stressors are generally defined as stimuli that require some 65
type of adaptation by an organism (Hill, Kiefer et al., 2020; Kiefer et al., 2018; Sato et al., 2006). 66
Thus, a stressor induces a perturbation to which the system needs to adapt (Taleb, 2012). To 67
exemplify the distinction between resistance, resilience, and growth following perturbations, we 68
divided this article into three sections. First, we provide a brief introduction to dynamical 69
systems features that are of relevance to the framework we outline. Second, we explain in more 70
detail how adapting to stressors emerges in dynamical systems from changing interactions 71
between multiple variables at different temporal and spatial scales (Hill & Den Hartigh, 2023; 72
Pincus & Metten, 2010). Third, we discuss attractor dynamics that can be used to describe 73
resilience and how it changes over time when exposed to repeated perturbations (e.g., Hill et al., 74
2018a, Scheffer et al., 2009, 2012, 2018; Van de Leemput et al., 2014) as well as how resistance 75
and growth may occur (e.g., Kiefer et al., 2018). 76
INTERACTION-DOMINANT DYNAMICS 77
Classic psychological models can be classified as component driven. In terms of adapting 78
to perturbations, a component driven model would maintain that a specific perturbation causes a 79
sequence of changes in the same psychophysiological variables to result in either resistance, 80
resilience, or growth across contexts and time. Moreover, such “textbook causality” assumes that 81
the causal mechanism and the associated interactions between variables (i.e., mediation or 82
moderation effects) would also stay stable over time. This means that the response to any 83
Dynamics of Resistance, Resilience, and Growth 4
perturbation can be explained by specific variables that exert linear influences with stable 84
interactions that do not change (for an in-depth discussion, see Den Hartigh, Cox, & Van Geert, 85
2017). In this case, the same changes of resilience factors like dispositional optimism and 86
motivation (Sarkar & Fletcher, 2014) would exert similar adaptational processes to specific 87
perturbations from the environment. In contrast, a hallmark of dynamical systems is that they are 88
interaction-dominant (Den Hartigh et al., 2017; Diniz et al., 2011; Gernigon, Den Hartigh, 89
Vallacher, & Van Geert, 2023; Van Orden, Holden & Turvey, 2003). This entails that how a 90
person ultimately adapts to a stressor cannot be reduced to a specific set of fixed variables and 91
causal mechanisms (Bonanno, 2021; Hasselman, 2023; Hill et al., 2018a). Instead, the response 92
to a stressor emerges from the ongoing interactions among several variables within a system on 93
different time scales (Pincus & Metten, 2010). The influence of any specific variable cannot be 94
assessed in isolation as its influence is dependent on how it is embedded within the system. 95
Moreover, variables may display nonlinear effects, particularly when a system is approaching 96
critical stages that are marked by rather sudden changes or phase transitions. For example, slight 97
changes in emotions may have little impact on how a person responds to a perturbation, but in a 98
critical state, even slight emotional changes may cause a person to fail to positively adapt and 99
develop psychopathological symptoms (Helmich et al., 2021; Kuranova et al. 2020; Van de 100
Leemput et al., 2014). 101
A typical illustration of the interaction-dominance within humans is the constant 102
feedback loops among different physiological levels of organization. The human body is 103
structured into a few large systems, such as the cardiovascular system, which are further 104
subdivided into increasingly smaller units from specific organs to individual cells. For the human 105
body to exert behavior, it utilizes a feedback loop of information exchange including top-down 106
Dynamics of Resistance, Resilience, and Growth 5
process from the larger structures of organizations (i.e., macro scales) to the individual cells (i.e., 107
micro scales) and a bottom-up process from the individual cells to the larger structures (e.g., 108
Balagué, Hristovski, Almarcha, Garcia-Retortillo, & Ivanov, 2020, 2022). This means that 109
behavior by definition emerges from the ongoing interactions among these different levels of 110
organization within the system. Similar feedback loops are also prominent in psychology for 111
processes at different spatial and time scales (e.g., Den Hartigh, Van geert, Van Yperen, Cox, & 112
Gernigon, 2016; Granic & Patterson, 2006; Lichtwarck-Aschoff, Van Geert, Bosma, & Kunnen, 113
2008; Van der Steen, Steenbeek, Den Hartigh, & Van Geert, 2019). That is, fast-changing 114
processes that typically occur on a micro scale (e.g., an emotion) feed into the slower change 115
process on the macro scale (e.g., psychological well-being). 116
RESILIENCE ON DIFFERENT SPATIAL AND TEMPORAL SCALES 117
To guide the reader through this section, we start with a key statement that is subsequently 118
elaborated on: 119
To observe and distinguish resistance, resilience, and growth following a 120
stressor, the adaptational process needs to be examined on the adequate spatial 121
and temporal scale. 122
In the context of resilience, the hallmark of interacting spatial and temporal scales within 123
dynamical systems can be seen in the resilience changes that occur following the exposure to 124
several minor stressors (Den Hartigh et al., 2022; Hill et al., 2018a; Scheffer et al., 2018; Van de 125
Leemput et al., 2014). That is, the overall ability of a person to demonstrate resilience may 126
decline due to everyday stressors that elicit adaptational processes on a micro scale. However, at 127
the same time, the macro scale constrains the change processes on the micro scale. This implies 128
that perturbations that occur at a micro scale may not be detectable or occur with a considerable 129
Dynamics of Resistance, Resilience, and Growth 6
time-delayed on macro scale processes. If a specific component within a system changes due to 130
an external perturbation, it may spread the perturbations effect over time throughout the system 131
due to its constant interactions with other variables (Hill & Den Hartigh, 2023). For example, if 132
an athlete suffers a strong decline in their motivation after a significant loss, they may train less 133
frequently and change their diet, which can lead to a loss in confidence, which further affects 134
additional variables in the system. Furthermore, the interactions between these variables may 135
also change following the perturbation. The athlete may, for instance, consume many calories 136
independent of the training whereas before the calory intake was synched with the training 137
schedule. Over time, the changes within the variables and their interaction patterns cause the 138
performance level to decline. Thus, the initial perturbation of a single variable may cause a 139
change within the system at a micro scale over time, rather than causing an immediate change at 140
the macro scale (e.g., the performance level). Changes in resilience therefore emerge from 141
changes in the system’s structure rather than being completely determined by isolated variables 142
(Hasselman, 2023; Hill & Den Hartigh, 2023; Kiefer & Pincus, 2023, Pincus & Metten, 2010; 143
Scheffer et al., 2012). 144
Moreover, stressors that are seemingly resisted on micro scales due to measurement 145
inaccuracies (e.g., too low frequency) can have cascading effects that become visible on the 146
macro scale. For example, if emotions are only measured once per day, dynamic changes in 147
response to specific events or informative fluctuations throughout may not be captured, which 148
makes the emotions appear more stable than they actually are (cf. Reitsema et al., 2023). 149
Therefore, it is essential for the perturbation to be aligned with the selection of both a variable’s 150
spatial (e.g., emotion on a micro scale or well-being on a macro scale) and the corresponding 151
Dynamics of Resistance, Resilience, and Growth 7
temporal scale (i.e., measurement frequency) in order to distinguish resistance, resilience, or 152
growth in a time series (e.g., Den Hartigh & Hill, 2022). 153
Although resistance, resilience, and growth come about through complex dynamic 154
processes, this does not mean that all the variables on the different scales, and their interactions, 155
need to be understood in detail. On the contrary, Taken’s (1981) delay embedding theorem 156
suggests that one collective variable suffices to reconstruct the state of a system. Such a 157
collective variable needs to be well-embedded within the system and thus reflective of the 158
underlying subsystems dynamics. Put simply, a collective variable captures the complexity of the 159
system at reduced dimensionality. Collective variables can be assessed to detect early warning 160
signals of resilience losses, for instance, as it may indicate a critical change in the underlying 161
dynamics of the system (e.g., Hill, Den Hartigh, Cox, De Jonge, & Van Yperen, 2020). 162
However, while the echoes of the perturbations that likely occur on a micro scale may be 163
captured in changes of the collective variable, this does not directly capture the perturbation 164
caused by a stressor and the subsequent return process. According to Den Hartigh and Hill 165
(2022), assessing the perturbation and the return process are essential to obtain a direct measure 166
of resilience. Thus, following this perspective, changes in temporal dynamics of collective 167
variables provide a proxy rather than a direct measure of resilience. 168
A direct measure of resilience would require an assessment of the perturbation caused by 169
the stressor and the time the variable requires to return to the previous level of functioning 170
(Baretta et al., 2023; Den Hartigh et al., 2022; Hill, Van Yperen, & Den Hartigh, 2021). These 171
values can be combined to calculate the area under the curve (AUC) from the expected trajectory 172
in the absence of a perturbation, which yields a score of resilience – the smaller the AUC, the 173
more resilient is the response (Den Hartigh & Hill, 2022). This AUC may only become 174
Dynamics of Resistance, Resilience, and Growth 8
detectable on micro scales which likely change relatively fast, while on a macro scale, no 175
perturbation can be detected, which would indicate resistance to the stressor. In this case, growth 176
would be reflected in an initial resilience process that exceeds the previous and stabilizes in a 177
higher level of functioning. Alternatively, according to Carver (1998), accelerated return 178
processes in response to the same stressor causing a decrease in the AUC may also signal 179
growth. Note, however, that growth processes can occur in a discontinuous manner without ever 180
revisiting the previous state (Kiefer et al., 2018). Therefore, AUC calculations are not necessary 181
or universally possible to determine growth. Altogether, resistance, resilience, and growth can 182
thus simultaneously exist as a response to the same stressor depending on which level of 183
organization is measured. 184
Even though not explicitly building on the framework of Takens (1981), the fact that we 185
can reconstruct the dynamics of a system due to its subsystem interdependencies, has already 186
found practical applications to research on resilience within ecology (e.g., Scheffer et al., 2012) 187
and clinical psychology (e.g., Van de Leemput et al., 2014). By conceptualizing resilience as a 188
multiscale process, it may not be necessary to map out the exact process on a micro scale. 189
Instead, indicators for changing resilience can be derived from changes in the temporal dynamics 190
on macro scale variables (i.e., collective variables). For example, early warning signals of 191
resilience losses can be detected in changing fluctuation patterns (e.g., lag-1 autocorrelation) of 192
one particular collective variable. Increases in autocorrelations are assumed to occur due to 193
increased sensitivity to perturbations and reduced recovery speed following perturbations. In 194
other words, with repeated perturbation, the autocorrelation would increase, signaling resilience 195
losses. However, at the level of collective variables, the AUC changes that occur on micro scale 196
processes are not detectable and are therefore derived from indirect measures, such as 197
Dynamics of Resistance, Resilience, and Growth 9
autocorrelations. To illustrate, daily stressors may cause perturbations in emotional states that 198
can be assessed with AUC calculations (e.g., Kuranova et al., 2020). However, these 199
perturbations may not occur in a collective variable associated with a macro scale state, such as 200
mental well-being. Nevertheless, for collective variables that capture the same dynamics of 201
resilience losses, proxies like autocorrelations may instead be used to track early warning signals 202
of resilience losses. 203
Another potential issue with the use of collective variables is the time-delay at which 204
small perturbations become detectable. As previously discussed, the collective variable provides 205
a proxy of resilience losses through early warning signals, rather than a direct measure of 206
changes in resilience. While such early warning signals can be useful to implement well-timed 207
interventions, it also has to be considered that due to the time delay with which these signals 208
become apparent, irreversible changes on micro scales structures or processes may have already 209
taken place (cf. Hill & Den Hartigh, 2023). That is, key structural processes providing support 210
functions on micro scales may have already collapsed before they become detectable on macro 211
scales. For example, in a simple predator-prey model, the collapse of the prey’s population 212
coincides with a sharp increase of the predator population. However, the predator’s population 213
collapse cannot be prohibited anymore when they fail to adapt to different food resources 214
because the warning signal of their population decline occurs after the irreversible onset of the 215
collapse of the prey’s population. Translated to the domain of psychology, such time-delayed 216
effects can be found in cardiovascular diseases that are caused by psychological stress (e.g., 217
Dimsdale, 2008). Some cardiovascular damage may emerge as a consequence of chronological 218
stress. However, the physiological damage may have already occurred before an individual’s 219
psychological experience of the chronic stress emerges. 220
Dynamics of Resistance, Resilience, and Growth 10
Considering the specific time scales at which adaptational processes unfold is also critical 221
for distinguishing resistance and resilience from growth (Den Hartigh et al., 2022). A dynamical 222
system can permanently alter its underlying structure to improve its functionality (i.e., 223
demonstrate growth, Hill & Den Hartigh, 2023; Hill, Kiefer, Oudejans, Baetzner, & Den Hartigh, 224
2024; Kiefer et al., 2018; Kiefer & Pincus, 2023). However, to fully capture this growth 225
trajectory, the system needs to be monitored for a sufficient period of time. If the monitoring is 226
discontinued too early, we may only observe a return to the previous level of functioning (i.e., 227
resilience, Den Hartigh & Hill, 2022) or the changes on the system’s micro scale structures and 228
processes have not yet become detectable on the macro scale making the system appear to have 229
resisted the change (Kiefer et al., 2018). Thus, a perturbation needs to be examined not only at 230
the right spatial scale, but also at the right temporal scale to accurately distinguish resistance, 231
resilience, and growth. 232
To summarize, dynamical systems consist of different elements at various spatial and 233
temporal scales that constantly interact with each other. To distinguish resistance, resilience, and 234
growth in response to a perturbation, the adaptational process needs to be closely aligned with 235
the right spatial scale as well as the right time scale. In dynamical systems, perturbations that 236
mainly affect structures on a micro scale may seemingly be resisted or become observable at a 237
macro scale with a considerable time delay. Similarly, variables within a system may remain 238
stable, but changes in their interaction structure may cause the system to become destabilized 239
and lose resilience (Hill & Den Hartigh, 2023; Kiefer & Pincus, 2023; Pincus & Metten, 2010). 240
Thus, what appears to be resistance may be classified as resilience (or even growth) on different 241
spatial scales or on different time scales. 242
ATTRACTOR DYNAMICS 243
Dynamics of Resistance, Resilience, and Growth 11
An attractor describes the iterative state-by-state change process a system undergoes to 244
move towards a stable state (Milnor, 1985). This means that attractors provide conceptual and 245
mathematical models that capture the change process that a system undergoes following a 246
perturbation caused by a stressor. Note that stable end-states of attractors do not imply that they 247
are rigid and do not undergo any changes. Instead, attractors can also reflect structured perpetual 248
changes like cyclic movements (e.g., walking), which show global stability characterized by 249
ongoing changes (i.e., local instabilities) mimicking the different spatial and temporal scales 250
discussed in the previous section. Furthermore, attractors are not fixed over time and a system 251
state can move into and out of different attractors (i.e., non-stationarity). For example, gait 252
patterns are often disrupted following major injuries like anterior cruciate ligament (ACL) 253
ruptures (e.g., Knoll et al., 2004). It can be reasoned that this change represents a change of the 254
organization of the movement system in terms of a different attractor state and that rehabilitation 255
training focuses on changing these attractor dynamics (cf. Kiefer & Myer, 2015). In this case, the 256
return to the pre-injury attractor dynamics could be classified as a resilience trajectory, which 257
does not take place within the dynamics of a singular attractor. In this section, we first discuss 258
fixed-point attractors in relation to resilience, due to their current popularity and application in 259
the psychological literature. Then, we discuss limit cycles and strange attractors due to their 260
potential for understanding psychological resilience, resistance, and growth. 261
Fixed-Point Attractors & Limit Cycles 262
Again, we start with a key statement that is subsequently elaborated on: 263
Fixed point attractors and limit cycles typically demonstrate resilience following 264
a perturbation but can allow for global resistance and growth considering 265
changes in the basin of attraction. 266
Dynamics of Resistance, Resilience, and Growth 12
A fixed-point attractor is characterized by a single equilibrium state in which a system 267
stabilizes. This means that when a system is perturbed, it will always return to the same state. A 268
common example used to describe such attractors is a pendulum, which will always stabilize to 269
the same position no matter how it is perturbed. Given the common definition of resilience as 270
returning to the previous (stable) state following a perturbation (e.g., Hill et al., 2018b), such 271
fixed-point attractors are the most commonly used attractor in psychology for describing this 272
process (Scheffer et al., 2018). 273
Contemporary approaches to understanding resilience in psychology often use the 274
metaphor of two co-existing fixed-point attractors (e.g., Hill et al., 2018a; Scheffer et al., 2018; 275
Van de Leemput et al., 2014). One of these fixed points represents a desirable state like mental 276
well-being, while the other fixed point represents an undesirable state like psychopathological 277
symptoms (see Figure 1). Together, these two attractors form a landscape of hills and valleys, 278
where the valleys represent the desirable and undesirable state. The metaphor further suggests 279
that the current state is like a ball that moves under the laws of gravity across this landscape. 280
Thus, a person can only stabilize in the two fixed-point attractors and the states in-between are 281
classified as so-called repellers – a state where the system cannot stabilize in and will be pulled 282
away from. According to this conceptualization, resilience is demonstrated when an external 283
perturbation pushes the ball, but it rolls back and re-stabilizes in the previous state. The deeper 284
the valley (i.e., basin of attraction), the faster the ball rolls back and the system stabilizes. 285
Furthermore, next to the current position of the system within the landscape, the basin of 286
attraction also determines how strong a perturbation must be in order to push the ball over the 287
edge and force a phase transition, resulting in the stabilization within the other fixed-point 288
attractor. Note that resilience is typically coupled to positive outcomes in the psychological 289
Dynamics of Resistance, Resilience, and Growth 13
literature (e.g., Luthar & Cicchetti, 2000) and therefore, the desirable state is often taken as the 290
reference point. 291
292
293
Figure 1. Simulated time series of phase transition accompanied by changes in the underlying 294
fixed-point attractor landscape. The fixed-point attractors are depicted by the hills and valleys on 295
surrounding the time series graph. The ball represents the current state of the system. The 296
decrease in the basin of attraction of the healthy state destabilize the system leading to a sudden 297
transition to a pathological state. Note that a modeling approach using the cusp catastrophe adds 298
a third dimension to the transition given by the interaction of the two control parameters. 299
300
The two co-existing fixed-point attractors are commonly used to describe how exposure 301
to repeated stressors can cause resilience losses, which result in smaller perturbations causing 302
Dynamics of Resistance, Resilience, and Growth 14
similar phase transitions as large perturbations. Specifically, repeated stressors may weaken the 303
basin of attraction of the desirable state which causes the person to become more sensitive to 304
perturbations of future stressors and require more time to return to the equilibrium state – a 305
process called critical slowing down (Den Hartigh & Hill, 2022; Den Hartigh et al., 2022; Hill et 306
al., 2018a; Helmich et al., 2021; Kuranova et al., 2020; Scheffer et al., 2009, 2012, 2018; Van de 307
Leemput et al., 2014) or relaxation time (Hill et al., 2021; Kelso, 2010). The shallower the basin 308
of attraction in a fixed-point attractor becomes, the more time it takes for the ball to roll back and 309
stabilize in the previous state. As discussed previously, these early warning signals of resilience 310
losses have primarily been linked with changes in the temporal signature (e.g., autocorrelation) 311
of variables representing macro scale structures within a system but have rarely been linked to 312
direct measures of increasing recovery times (e.g., Hill et al., 2021; Kuranova et al., 2020). 313
While fixed-point attractors have primarily been applied to the concept of resilience in 314
psychology, they may also account for resistance and growth. First, if an individual quickly 315
returns to the previous (attractor) state after a stressor, one could argue that a shift to another 316
state is resisted. Indeed, if the measurement frequency is low, any resilience process (i.e., the 317
return to the previous state after the stressor) would not be detected. In fixed-point attractors, a 318
perturbation forces an adaptational process (i.e., resilience) while the system remains in its 319
overall attractor state, showing global stability (e.g., Scheffer et al., 2009). Thus, fixed-point 320
attractors can account for the measurement issue of resistance as they allow for local instability 321
on a micro scale following a perturbation, while demonstrating global stability. However, even 322
on macro scales, processes like critical slowing down that signify changes in resilience can be 323
identified through high-frequency assessments (Hill, Den Hartigh et al., 2020; Van de Leemput 324
et al., 2014). In other words, the system remains in its equilibrium state only in the absence of a 325
Dynamics of Resistance, Resilience, and Growth 15
perturbation or when the system’s state is assessed a macro level. 326
Although not considered as such in previous literature, fixed-point attractors can result in 327
changes of the system reflecting growth. The commonly used attractor landscape with two fixed 328
points does includes a “healthy” and “unhealthy” attractor state, but it could arguably be 329
extended to include an improved healthy state. Similar to transitions to an undesirable state, the 330
transition to the more desirable state could be either induced by a single strong perturbation or by 331
changing the basin of attraction of the current state. When the basin of attraction increases, the 332
system would re-stabilize in its original state more quickly. According to Carver, the latter 333
phenomenon may also be considered as a kind of growth (or thriving) reflecting improved 334
resilience to future perturbations. Similar conceptualizations have also been put forward in 335
biology under the term conditioning hormesis (Calabrese 2016a, 2016b; Calabrese et al., 2007; 336
Hill, Kiefer et al., 2024). For example, following the exposure to a toxoid vaccine, the immune 337
system will be able to respond more quickly and effectively when exposed to the same 338
potentially infectious agent in the future. Although increasing the basin of attraction to enhance 339
resilience (and according to some definitions thus stimulate growth) seems intuitive, this 340
approach may become problematic as the strong basin of attraction may trap the system in this 341
particular state thereby hindering growth (for an elaborate discussion, see Hill, Morison, 342
Westphal, Gerwann, & Ricca, 2024). 343
However, despite the popularity of the fixed-point attractor conceptualization in 344
resilience research, recent criticism has pointed out that this approach likely fails to fully capture 345
the complexity of adaptational processes, specifically distinguishing resilience and growth (Hill, 346
Morison et al., 2024). A more comprehensive model in this direction, for instance, is the cusp 347
catastrophe model (e.g., Guastello, 2009; Guatello & McGuigan, 2024; Pincus & Metten, 2010; 348
Dynamics of Resistance, Resilience, and Growth 16
Van der Maas & Molenaar, 1992; Ruhland & Van Geert, 1998). The cusp catastrophe model 349
describes the dynamic changes of a system when two (or more) control parameters are gradually 350
varied. Indeed, Pincus and Metten (2010) already outlined how the cusp catastrophe model can 351
be applied to resilience. Specifically, this model is particularly useful to explain sudden 352
transitions as often seen as a result of critical slowing down as well as similar discontinuous 353
growth trajectories. Moreover, the cusp catastrophe model allows for the existence of several 354
(changing) attractor states as well as gradual changes of these states. Interestingly, the two fixed-355
point attractors may only emerge when the bifurcation parameter increases, which forces the 356
system into a rigid (and fragile) state (Pincus & Metten, 2010). Overall, the cusp model provides 357
strong theoretical basis for modeling sudden transitions and has already been applied to contexts 358
such as mass-disaster (Benight et al., 2020) and team performance (Guastello & McGuigan, 359
2024) with some initial empirical validation. It therefore provides a logical extension to further 360
applications in resilience research. 361
Another promising avenue relates to so-called limit cycles. Limit cycles represent 362
behaviors that follow a periodic repetition, such as oscillators. This means that these attractors 363
are characterized by states that repeat themselves in a steady rhythm. Here, it should be noted 364
that limit cycles tend to be too simplistic to represent the oscillatory behaviors at issue. For 365
example, cardiac activity seems to emerge from dynamics of deterministic chaos rather than 366
being perfect oscillators (Babloyantz & Destexhe, 1988). Thus, we use the limit cycle for a 367
conceptual discussion, rather than suggesting that human behavior that follows periodic patterns 368
always follow limit cycles. To illustrate, organ functioning such as the heartbeat or breathing 369
consist of two phases (e.g., contraction and relaxation of the cardiac muscle tissue) that 370
perpetually repeat themselves. Such limit cycles can also be scaled to behavior, such as walking, 371
Dynamics of Resistance, Resilience, and Growth 17
as it consists of oscillating movement repetitions (Hobbelen & Wisse, 2007). Similarly, limit 372
cycles can extend to longer periods of time including weekly rhythms. For example, a person 373
may always display similar emotions, cognitions, and behaviors at similar points in time during 374
the day throughout the work week. 375
In case of small perturbations that do not disrupt the stability of the periodic cycle, the 376
system returns to the attractor and therefore demonstrates resilience. In contrast, resistance can 377
again only be considered if the global stability of the attractor is considered. For example, if a 378
person who walks down the street receives a little push and does not fall but restabilizes to keep 379
on walking, they demonstrate local resilience (e.g., briefly stumbling, but returning to the 380
previous gate pattern), but global stability or resistance (i.e., continue walking). Finally, growth 381
might be considered in light of Carver’s (1998) conceptualization of increasing recovery speed 382
due to an increase in the basin of attraction. For example, athletes may specifically train to 383
increase the basin of attraction to be able to both reduce the impact of a perturbation (e.g., a 384
tackle from an opponent) and quickly return to the previous pattern. Thus, resistance, resilience, 385
and growth also seem to be different concepts when considering limit cycles. 386
Strange Attractors 387
The guiding statement of this section is: 388
Strange attractors allow for resilience and growth in response to perturbations, 389
while resistance may only occur on the level of the attractor itself. 390
Strange attractors represent another set of attractors that are characterized by global 391
stability and local instabilities (Ruelle & Takens, 1971). However, unlike fixed-point attractors 392
or limit cycles, the basin of attraction in strange attractors displays recurring self-similarity (i.e., 393
fractal structures). This means that the system never revisits the exact same state over time. In 394
Dynamics of Resistance, Resilience, and Growth 18
other words, two points that appear to be highly similar will evolve into vastly different 395
trajectories over time (Grebogi, Ott, & Yorke, 1987). However, the potential trajectories are 396
confined by the global space in which the attractor unfolds, thereby achieving its global stability. 397
Within living systems, the fractal structures emerge from the ongoing interactions 398
between multiple subsystems on different temporal scales. For example, to execute rowing 399
strokes, several limbs need to exert movements in a precise sequence. The required changes 400
within the individual composite movements occurs on a smaller structural and temporal scale 401
than the entire rowing stroke (Den Hartigh, Cox, Gernigon, Van Yperen & Van Geert, 2015). 402
Each individual stroke is further embedded into the activity of rowing as several repetitions are 403
usually required. This means that several processes interact on different timescales to produce 404
the behavior. Behaviors that emerge from such fractal processes are referred to as metastable 405
(Rossi et al., 2023). Metastable behaviors can swiftly be adapted to avoid becoming stuck in 406
undesirable patterns and adopt new solutions to challenges within the environment (Hill, 407
Morison et al., 2024). This allows a system to operate at an optimal blend between rigidity and 408
flexibility allowing for various adaptational processes depending on the demands of the situation 409
(Den Hartigh, Otten, Gruszczynska, & Hill, 2021; Kiefer & Myer, 2015; Pincus & Metten, 410
2010). By being able to dissolve undesirable states (i.e., segregation tendencies) and form new, 411
more adaptive states (i.e., integration tendencies), a system can utilize a perturbation to form 412
more functional states. Thus, strange attractors allow for metastability of systems, which are 413
necessary for demonstrating growth. Note, however, that systems characterized by two stable 414
fixed-point attractors can also demonstrate metastability – as demonstrated by the Haken-Kelso-415
Bunz (1985) model – and thus do not exclusively occur in strange attractors. Nevertheless, 416
Dynamics of Resistance, Resilience, and Growth 19
strange attractors allow for more flexibility in the adaptational processes and may therefore be 417
considered as more functional than fixed-point attractors (Hill, Morison et al., 2024). 418
Because strange attractors are highly sensitive to changes in the starting condition, each 419
yet so small perturbation can initiate drastic changes in the unfolding trajectory (Grebogi, Ott, & 420
Yorke, 1987). That is, perturbations may be amplified or diversified, which lead to rather 421
unpredictable changes. Therefore, resistance to perturbations on a trajectory level is almost 422
impossible in strange attractors. Only if the attractor in its entirety is considered as the level of 423
analysis, it may be argued that a perturbation can be resisted. However, because the change in 424
the starting conditions has a significant impact on the shape of a strange attractor, its 425
periodization, and the space in which it unfolds (Tucker, 1999), resistance may only occur on the 426
level of the attractor as the same orbit or functionality may be maintained following a 427
perturbation. 428
Resilience may occur in strange attractors if the initial perturbation does not break its 429
global stability, or the recovery process yields similarly functional states (i.e., equifinality, Von 430
Bertalanffy, 1986). Even if the initial trajectory is perturbed, the fractal nature of the global 431
stability ensures that the system eventually revisits a similar state in future iterations. Thus, a 432
strange attractor allows for a perturbation to open multiple pathways for the system to return to a 433
similar state (i.e., demonstrating resilience). That is, the fractal nature of the strange attractors 434
within living systems allows the system to adopt various structural configurations that yield the 435
same functional output (i.e., degeneracy, Edelman & Gally, 2001). Note that in contrast fixed-436
point attractors and limit cycles typically allow for a very limited number of potential (or just a 437
single) recovery trajectories (Hill, Morison et al., 2024). Thus, the two key ingredients for 438
calculating resilience from time series according to Den Hartigh and Hill (2022) – perturbation 439
Dynamics of Resistance, Resilience, and Growth 20
strength and recovery time – can be explained by strange attractors in terms of the change in the 440
initial conditions caused by the perturbation. 441
To summarize, strange attractors are sensitive to initial conditions or perturbations. This 442
means that slight changes can have drastic changes on the emerging trajectories within the 443
attractor’s space. This feature of strange attractors limits the possibility for resistance against 444
perturbations. Instead, perturbations are marked by a return to a similar state in the future (i.e., 445
resilience) which can be achieved through multiple pathways. Moreover, the metastability of 446
strange attractors allows systems to operate at a state of constant criticality which enables quick 447
adaptations. This can result in increased functionality (i.e., growth) in response to perturbations. 448
DISCUSSION 449
The aim of this paper was to provide a theoretical foundation for the distinction between 450
resistance, resilience, and growth following stressors as proposed by Den Hartigh and Hill 451
(2022). Specifically, the authors argued that dynamical systems would present sound theoretical 452
approach to substantiate the proposition that these three are distinct processes. Therefore, we 453
discussed some key features of dynamical systems – interacting spatial and temporal scales as 454
well as attractor dynamics – to explain how systems respond to perturbations caused by a 455
stressor. Based on this discussion, we propose three key conclusions of how differences in 456
resistance, resilience, and growth may be explained within a dynamical systems framework. 457
First, to distinguish resistance, resilience, and growth, the perturbation caused by a stressor and 458
the subsequent response need to be observed at the adequate level of organization of the system. 459
Second, fixed point attractors and limit cycles will typically demonstrate resilience following a 460
perturbation but can also trigger global resistance and growth over longer time-frames through 461
changes in the basins of attraction. Third, strange attractors states may allow for resilience and 462
Dynamics of Resistance, Resilience, and Growth 21
growth in response to perturbations, while resistance should be rather unlikely to occur in these 463
attractors. 464
Practical Implications 465
Distinguishing resistance, resilience, and growth based on theoretical and methodological 466
approaches is not only relevant for research purposes but can also guide the way in which 467
interventions should be structured. For example, to enhance resilience from a fixed-point 468
attractor or limit cycle perspective, an intervention should be aimed at increasing the basin of 469
attraction. This means that the stability of the current state should be strengthened (see Schiepek, 470
Eckert, Aas, Wallot, & Wallot, 2016 for specific examples of such interventions). However, this 471
increased stability may not be useful for every situation. According to Pincus and Metten (2010) 472
systems need to demonstrate not only the ability to stabilize, but also to react flexibly depending 473
on the demands of a stressor (i.e., meta-flexibility). In the latter case, training flexible 474
adaptations rather than rigid response patterns would yield more desirable responses to 475
perturbations because they enable the possibility for improved functionality (i.e., growth). Such 476
flexible adaptations may be stimulated through the systematic exposure to stressors that foster re-477
organization processes within the system without risking harmful effects due to overdosing, 478
similar to the idea of toxoid vaccines (Hill, Kiefer et al., 2024; Kiefer & Martin, 2022). For 479
example, differential learning – inviting a person to continuously alter their movement pattern 480
during training – has been shown to diversify motor solutions and improved adaptations to 481
changing environments (e.g., e.g., Den Hartigh et al., 2021; Gray, 2020; Santos et al., 2018; 482
Savelsbergh, Kamper, Rabius, De Koning, & Schöllhorn, 2010; Schöllhorn, Hegen, & Davids, 483
2012; Schöllhorn, Mayer-Kress, Newell, & Michelbrink, 2009). Thus, it is imperative to 484
Dynamics of Resistance, Resilience, and Growth 22
understand what adaptational trajectories are associated with which attractor in order to tailor the 485
intervention content to the creation of the desirable patterns and avoid unintended side-effects. 486
Apart from the content of an intervention, another important implication is the match 487
between the perturbation and the according spatial and temporal scale for the evaluation. 488
Specifically, if the aim of a training is to foster resilience in response to a given perturbation, it 489
needs to be carefully considered which variables are expected to demonstrate either resistance, 490
resilience, or growth as well as on what timescale these processes are likely to unfold. For 491
example, emotion regulation training aims to increase the flexibility of emotion recognition and 492
adjustment on a micro scale thereby improving resilience. This improved resilience on the micro 493
scale fosters more stability at the macro scales, such as psychological well-being (Jenkins, 494
Hunter, Richardson, Conner, & Pressman, 2020). Thus, if only the well-being is considered in 495
the evaluation, it may be falsely concluded that the emotion regulation training induces 496
resistance against perturbations. 497
CONCLUSION 498
To achieve more conceptual clarity on what resilience is and what it is not, Den Hartigh 499
and Hill (2022) proposed a conceptual distinction between resistance, resilience, and growth. 500
This paper aimed to define dynamical systems principles that could provide a deeper theoretical 501
explanation underlying the distinction. Specifically, in dynamical systems what appears to be 502
resistance on macro scales may actually be classified as resilience or growth on micro scales. 503
Thus, resilience is a more likely response in different scales and conforms with attractor 504
dynamics that are commonly used in psychology and general human behavior. Growth can also 505
occur in dynamical systems, especially the underlying attractor dynamics allow for multiple 506
pathways of adaptation and swift changes between different states. Considering such theoretical 507
Dynamics of Resistance, Resilience, and Growth 23
distinctions can provide important clues for the development of theoretical models, empirical 508
measurements, and theory-driven interventions. 509
510
Dynamics of Resistance, Resilience, and Growth 24
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