Dynamic systems models describe how attraction, emotional exhaustion, and reciprocal responses can determine whether a relationship recovers from arguments and stresses or deteriorates. Mathematician Laurent Pijoux-Manjou emphasizes: This is about identifying trends, not a formula that predicts the fate of every couple.
Can the ups and downs of a relationship be described using equations? Mathematicians and behavioral researchers have been grappling with this question for decades. A new book by French mathematician Prof. Laurent Pijoux-Manjou presents attempts to describe Love וRelationship As dynamic systems – systems that change over time depending on the situation of both partners and how each of them responds to the other.
The book, Love! Valour! Equations! Mathematics of the Couple, combines Mathematics עם psychology, literature, cinema and everyday life. It does not offer a magic formula for finding a partner, but rather examines whether Mathematical models can explain why some relationships recover from crises while others gradually erode. The book is expected to be published by CRC Press in 2026.
Pijoux-Manjois is a professor of mathematics at the University of Lyon 1 and a member of the Camille Jourdain Institute. His work focuses on mathematical models of biological and medical systems, including degenerative diseases, blood cell development, and the effects of radiation. In recent years, he has also applied these tools to the study of human behavior and romantic relationships.
From rabbits and crazy people to relationships
The mathematical path to the study of relationships began in fields seemingly far removed from love. Models developed by Thomas Malthus in the 18th century were designed to describe population growth, while the Lotka-Volterra equations described fluctuations in the numbers of predators and their prey.
Other models describe what is known as the Alle effect: a population that is too small may have difficulty surviving and reproducing, but once it passes a certain threshold it is able to stabilize and grow. According to Pijoux-Manjouy, a similar idea can be used to describe a couple's bond: a sufficiently strong bond may survive an argument, a period of stress or external change, while an already weakened bond may have difficulty recovering from even a relatively minor disturbance.
“Just as you can describe population growth or the state of a species in danger of extinction, you can model the development of feelings within a couple,” he says. “We’re not trying to predict who you’ll fall in love with, but rather to understand why some couples remain stable despite life’s inevitable shocks, while others gradually drift apart.”
Three key variables
The model presented by Piaget-Manjou has three central components: the degree of mutual attraction between the partners, the natural rate of erosion of feelings over time, and the way in which each of them responds to the other's feelings and behavior.
One person may respond very warmly when their partner expresses affection, while another may react cautiously or withdraw when emotions become too strong. Some people react immediately to conflict, while others need time to process what has happened. The combination of the two partners' traits and reactions may create a stable equilibrium, a recurring oscillation between closeness and distance, or a continuous process of weakening the bond.
Previous researchers have tried to use dynamical systems to describe love. Mathematician Steven Strogatz presented simple models in which two people’s emotions influence each other. Physicist Sergio Rinaldi and his colleagues used equations to analyze relationships in literature and film, and researchers in Poland added to the models the factor of delay – the time that passes between an action by one partner and the reaction of the other.
Delay is not necessarily negative. An immediate response out of anger can escalate a conflict, while a short period of thought can defuse it. On the other hand, prolonged ignoring or repeatedly rejecting conversations can deepen the distance.
The threshold below which it is difficult to recover
One of the central ideas in the book is the existence of a critical threshold. According to the model, if the strength of the bond falls below a certain level and remains there, the system may gradually move towards a state of distance or separation.
This does not mean that every temporary drop in emotions heralds the end of a relationship. Many couples argue, deal with stress, or experience periods of distance. The important mathematical question is whether the system is able to return to equilibrium after the disruption.
“The important question is not whether the couple argues,” says Pijoux-Manjouy, “but whether they have enough resilience to return to equilibrium afterward.”
He compares it to the story of the Three Little Pigs. Life invites “wolves” to every relationship – illness, difficulties at work, raising children, emotional strain or financial problems. The goal is to understand what qualities make a relationship a brick house that withstands shocks, instead of a straw house that collapses when pressure is applied.
Not a system for predicting breakups
Mathematical models of relationships are not accurate predictors. They greatly simplify a reality that includes personality, memories, culture, economic status, family, health, and changing life circumstances. Existing models are based primarily on Western perceptions and patterns of relationships, and therefore there is no certainty that they are equally appropriate for other societies, religions, and family traditions.
The variables themselves are also difficult to measure. It is impossible to attach an exact number to human emotion the way one measures temperature or speed. Researchers are forced to rely on questionnaires, behavioral observations, personal reports, and assumptions about the relationship between external behavior and emotional state.
Therefore, the equations indicate tendencies and possible scenarios, not a predetermined fate. People can change their response patterns, seek treatment, learn to communicate differently, or make decisions that the model did not take into account.
An early warning system for relationships?
In the future, Peugeot-Manjou hopes to combine mathematical models with artificial intelligenceSuch a system could, at least theoretically, identify a continuous change in communication patterns and warn that the connection is approaching an area where it is more difficult to return to stability.
He compares the idea to a navigation system: it does not select the destination and does not require passengers to travel a specific route, but rather shows where they are, where the current road leads, and what alternative routes may be available.
But even such a system would raise serious questions about privacy, cultural bias, misinterpretation, and overreliance on algorithms. There is currently no agreed-upon mathematical tool that can determine whether a given couple should break up or stay together.
“Mathematics can illuminate patterns and provide direction,” concludes Pijoux-Manjouy, “but the success of a relationship ultimately depends on two people choosing every day to nurture their connection. The equations can show the way, but you still have to walk it together.”
Source: Love! Valour! Equations! Mathematics of the Couple, by Laurent Pujo-Menjouet, published by CRC Press, DOI: 10.1201/9781003758105.
Questions and Answers
Can an equation predict whether a particular couple will break up? No. The models describe general patterns and possible scenarios, but do not take into account all the psychological, cultural, and personal factors that affect people.
What is a “critical threshold” in a relationship? This is a mathematical concept that describes a minimum level of bond strength. When a bond falls below this level for an extended period of time, the model predicts that it will become increasingly difficult to return to a stable state.
What factors are included in the model? Among the main factors: mutual attraction, the rate of erosion of feelings, and the way each partner responds to the other's feelings and behavior.
Can the model be used in couples therapy? In the future, it may help identify patterns and provide early warning, but currently it is not a validated diagnostic or clinical treatment tool.
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