dynamics of love and their relation to ode

the relation between ode andpde

1. Department of Mathematics Indian Institute of Technology , Guwahati  MSc Project 2024 Siraj Ahmed Shaikh 222123050 Supervisor : Prof Jiten C. Kalita Dynamics of Love

2. Story Let's Visualize this R J NOTE  R > 0 : Love R < 0 : Hate Love at first side Both in love R start disliking J still love him But R start hating J is NOT in love   R  hates J They hate each other J hates R And R is indifferent to J R begin to express love BUT J continue to dislike R Romeo is fickle lover . The more Juliet loves him the more he begin to dislike her. But when she loses interest his feelings for her warmup . She on the other hand tends to echo him.

3. Mathematical Modeling 1 Solution Conclusion Feelings of R and J are periodic in nature . The sad outcome of their affairs is of never-ending cycle of LOVE and HATE R J

4.  Mathematical modeling 2 a, b > 0           Eager-Beaver ( Romeo is encouraged both by his own feelings and by Juliet’s.) a> 0 , b< 0     Narcissistic ( Romeo is encouraged by his own feelings , but responds negatively to Juliet’s. ) a< 0 , b> 0     Cautious Lover ( Romeo is discouraged by his own feelings but responds positively to Juliet’s.) a, b< 0            Hermit ( Romeo is discouraged both by his own feelings as well as Juliet’s. ) Fire and Ice situation c=-b & d=-a        Never Ending situation a=d=0 & bc <0    

5. Case 1 : |a|>|b| Eigen values : Real unequal with different signs. Nature of critical point (0,0) : Saddle point Stability : unstable If a>b>0 (0,0) critical point = unstable emotional state Mathematical description = real-life emotional dynamics Romeo = Eiger Beaver, Juliet = Hermit Exponential solution = emotions align along V2 over time Phase portrait shows transition from love to dislike or hate to love, reflecting hyperbolic, unstable critical point, symbolizing dynamic relationship as " Fire and Ice ".

6. Case 1 : |a|>|b| Eigen values : Real unequal with different signs. Nature of critical point (0,0) : Saddle point Stability : unstable If a>0>b

7. Case 2 : |b|>|a| Eigen values : Purely Imaginary Nature of critical point (0,0) : Centre Stability : Stable BUT not asymptotically  If b>a>0 Solution: Rotated ellipse centered at (0,0) Initial emotional states don't matter Emotions cycle between affection and disdain Trajectory resembles a roller coaster ride Cyclic return to initial states perpetuates indefinitely, termed " Eternal Oscillation of Affection "

8. Case 2 : |b|>|a| Eigen values : Purely Imaginary Nature of critical point (0,0) : Centre Stability : Stable BUT not asymptotically  If b>0>a

9. Summary

10. Mathematical Modeling 2 The above model doesn’t consider the effect of learning , adaptation and synergism after living together . Hence emotion may vary; we assume that it proportional to RJ. Summary Model assumptions lead to positive linear behavior Model exhibits remarkable properties aligned with common understanding Individual behavior influences community structure Main conclusion: Individual appeal drives order within the community

11. Mathematical Modeling 4 The coefficient is time dependent

12. References Mathematics magazine Vol 61 , Feb 1988         Steven H. Strogatz ( Harward University) 2.   Journal of Humanistic mathematics Vol 2 , July 2019        Differential Equations of Love and Love of differential Equation by Issac Elishakoff   ( Flordia Atlantic   university) 3.   Dynamics of Love and happiness : A mathematical Analysis , June 2012        Dharna Satsangi and Arun k. Sinha  4.   Phase diagram: https://www.geogebra.org/m/fYxXgbsU

13. Thank You