Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 20, Iss. 3, July, 2016, pp. 369-399
@2016 Society for Chaos Theory in Psychology & Life Sciences

 
 
 

The Mathematics of Psychotherapy: A Nonlinear Model of Change Dynamics

Gunter Schiepek, Paracelsus Medical University Salzburg, Austria and Ludwig Maximilians University Munich, Germany
Benjamin Aas, Paracelsus Medical University Salzburg, Austria and Ludwig Maximilians University Munich, Germany
Kathrin Viol, Paracelsus Medical University Salzburg, Austria and Ludwig Maximilians University Munich, Germany

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