Mathematical counseling for all who wonder why their relationship is like a sinus wave

The publication in Applied Mathematics and Computation is now available online.

In 1988, Steven Strogatz was the first to describe romantic relationships with mathematical dynamical systems. He constructed a two-dimensional model describing two hypothetical partners that interact emotionally. He used a well known example: the changes of Romeo's and Juliet's love (and hate) over time. His model became famous and inspired others to analyze (fictional) relationship case studies like Jack and Rose in the Titanic movie. However, the Strogatz model does not include delays in the partner's responses to one another. Therefore it is only a good start for fruitful studies on human emotions and relationships.

That is why Natalia Bielczyk adjusted Strogatz to a more life-like model by considering the time necessary for processing and forming the complex emotions in relationships. The reactivity in the relationship model is based on four parameters: both partners have a personal history (their 'past'), and a certain reactivity to their partner and his/her history. Depending on these parameters, different classes of relationships can be found: some seem doomed to break regardless of the partners promptness to one another while others are solid enough to always be stable. In the calculated models, stability occurs when both partners reach a stable level of satisfaction and the sinus wave disappears. The paper concludes that for a broad class of relationships, delays in reactivity can bring stability to couples that are originally unstable.

These results are pretty intuitive: too prompt or too delayed responses evoke trouble. Below a certain value, delays caused instability and above this value they caused stability, showing that some minimum level of sloth can be beneficial for a relationship. The fact that too fast emotional reactivity can lead to destabilization, shows that reflecting each other's moods is not enough for a stable relationship: a certain time range is necessary for compound emotions to form. Summarized, the publication offers mathematical justification for intuitive phenomena in social psychology. Working on good communication, studying each other's emotions and working out the right timing can improve your relationship, even without trying to change your partners traits (which is harder and takes more time).