A view From the Bridge

The equations of love

Posted on behalf of Marten Scheffer

Gustav Klimt, The Kiss, 1907-8

The Kiss (Lovers) by Gustav Klimt, 1908.

Österreichische Galerie Belvedere/Google Cultural Institute

Few topics are as disparate as mathematics and love — or are they? Modeling Love Dynamics (World Scientific, 2016) by systems theorist Sergio Rinaldi and others playfully, but convincingly, makes the point that even amorous relationships cannot escape the fundamental laws of dynamical systems.

The argument propounded by Rinaldi and colleagues builds on the classical framework of coupled differential equations, which have proven so powerful in describing the essence of relationships in nature such as competition, cooperation and predation. The book’s cover illustration hints at the road ahead: it shows Gustav Klimt’s 1908 painting The Kiss (Lovers). A glance inside reveals that art is an essential part of the analysis of the drama of passion — a drama resulting in large part from the interplay of two strong forces, attraction and repulsion. Simple equations illustrated with elegant diagrams show how, depending on personalities, those forces can result in a transient affair, long-lasting stable equilibrium, or everlasting cycles of attraction and repulsion.

Portrait of Francesco Petrarca (Petrarch).

Portrait of Petrarch, whose Canzoniere can be linked to the  limit cycle.

Via Wikimedia Commons

Miniature of Laure de Novis, Petrarch's platonic love, 1463.

Miniature of Laure de Noves, who may have been Petrarch’s ‘Laura’.

Laurentian Library, Florence, 1463

The tales and poems chosen masterfully illustrate a range of mathematical features. The limit cycle, known for driving the oscillating dynamics of many economic or biological systems, is linked, for instance, to one of the greatest love stories in Western culture. That is, the cyclical 21-year platonic relationship between fourteenth-century Italian humanist and poet Francesco Petrarca (Petrarch) and the married Laura (possibly the Provençal noblewoman Laure de Noves), charted in Petrarch’s celebrated collection Il Canzoniere.

If three variables are mixed in the differential equations of passion, chaotic dynamics can arise. This is illustrated vividly in Henry-Pierre Roché’s semi-autobiographical 1953 novel Jules et Jim (which inspired François Truffaut’s 1962 film of the same name). Roché documents the love triangle between himself, the brilliant and charming journalist Helen Grund and her shy husband Franz Hessel, his best friend. As with the weather, the course of these dynamics is fundamentally unpredictable in the long run, as the smallest event can put things on a different trajectory. This phenomenon is also known as ‘the butterfly effect’, hinging on the idea that the flap of a butterfly’s wing may eventually lead to a hurricane in a distant place.

Other aspects of relationship dynamics generated by the models are illustrated by a range of classics. The complex tides of emotion between Rhett, Scarlett, Ashley and Melanie in Margaret Mitchell’s 1936 blockbuster Gone with the Wind (and the 1939 film directed by Victor Fleming) are tied to the mathematics of alternative basins of attraction. Gabrielle-Suzanne Barbot de Villeneuve’s eighteenth-century fairytale La Belle et La Bête (Beauty and the Beast) exemplifies a saddle-node bifurcation (often referred to as a tipping point) in Beauty’s slow, barely perceptible progress towards the transition from repulsion to attraction — a pattern also seen in the evolving relationship of Elizabeth Bennett and Darcy in Jane Austen’s 1813 novel Pride and Prejudice. The love triangles in Edmond Rostand’s 1897 play Cyrano de Bergerac meanwhile illustrate how temporary bluffing of one partner can sometimes make the difference needed to move the dynamical system from indifference to attraction for a stable love relationship.

When it comes to making mathematics easy, the book saves the best for last. The 40-page appendix is a complete primer on dynamical systems and their bifurcations. Starting with an example of a love model, a simple, lucid text illustrated by elegant drawings explains everything you always wanted to know but never dared to ask about attractors, repellors, saddles, torusses, strange attractors, tipping points and more.

Scientists and artists alike try to capture the essence of things, whether that is atomic structure, the psychological depths of a fictional character or the crystallization of emotion in music. Perhaps this is why the swirling dance between these two seemingly opposite endeavours works out surprisingly well in Modelling Love Dynamics. Clearly, the arts are superior when it comes to capturing the depths of love. Yet disarmingly easy maths powerfully captures the underlying drivers of stable alliances and transient dalliances. What topic is better suited to seduce a broad audience to play with equations?

Marten Scheffer​’s research focuses on complex systems and their adaptability. He is an ecologist and mathematical biologist at Wageningen University and Research Centre and is founder-director of the Synergy Program for Analysing Resilience and Critical Transitions (SparcS). His latest book is Critical Transitions in Nature and Society (Princeton University Press, 2009.)  He is also a multi-instrumentalist and composer.


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