| Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 20, Iss. 3, Jul, 2016, pp. 369-399 @2016 Society for Chaos Theory in Psychology & Life Sciences The Mathematics of Psychotherapy: A Nonlinear Model of Change Dynamics Abstract: Psychotherapy is a dynamic process produced by a
complex system of interacting variables. Even though there are
qualitative models of such systems the link between structure and function,
between network and network dynamics is still missing. The aim of this
study is to realize these links. The proposed model is composed of
five state variables (P: problem severity, S: success and therapeutic progress,
M: motivation to change, E: emotions, I: insight and new perspectives)
interconnected by 16 functions. The shape of each function is
modified by four parameters (a: capability to form a trustful working alliance,
c: mentalization and emotion regulation, r: behavioral resources and skills,
m: self-efficacy and reward expectation). Psychologically,
the parameters play the role of competencies or traits, which translate
into the concept of control parameters in synergetics.
The qualitative model was transferred into five coupled,
deterministic, nonlinear difference equations generating the dynamics of
each variable as a function of other variables.
The mathematical model is able to reproduce important features of
psychotherapy processes. Examples of parameter-dependent bifurcation diagrams are given.
Beyond the illustrated similarities between simulated and empirical dynamics,
the model has to be further developed, systematically tested by simulated experiments,
and compared to empirical data. Keywords: Mathematical modeling, psychotherapy, process research, common factors, computer simulation, nonlinear dynamics |