Love Affairs Dynamics with One Delay in Losing Memory or Gaining Affection

A dynamic model of a love affair between two people is examined under different conditions. First the two-dimensional model is analyzed without time delays in the interaction of the lovers. Conditions are derived for the existence of a unique as well as for multiple steady states. The nonzero steady states are always stable and the stability of the zero steady state depends on model parameters. Then a delay is assumed in the mutual-reaction process called the Gaining-affection process. Similarly to the no-delay case, the nonzero steady states are always stable. The zero steady state is either always stable or always unstable or it is stable for small delays and at a certain threshold stability is lost in which case the steady state bifurcates to a limit cycle. When delay is introduced to the self-reaction process called the Losing-memory process, then the asymptotic behavior of the steady state becomes more complex. The stability of the nonzero steady state is lost at a certain value of the delay and bifurcates to a limit cycle, while the stability of the zero steady state depends on model parameters and there is the possibility of multiple stability switches with stability losses and regains. All stability conditions and stability switches are derived analytically, which are also verified and illustrated by using computer simulation.