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Love Dynamical Model with persepectives of Piecewise Differential Operators

Atul Kumar

TL;DR

This work develops a piecewise differential-dynamic framework for love dynamics that blends integer-order, fractional, and stochastic processes to capture crossover emotional patterns. It introduces two Owolabi models (linear and nonlinear) across interval-based memory kernels using Caputo, Atangana–Baleanu, and Caputo–Fabrizio derivatives, and proves existence and uniqueness for the Caputo-based piecewise system. The authors formulate Newton interpolation–based numerical schemes for the three derivative types and present simulations across varying fractional orders $\alpha$, revealing chaotic dynamics consistent with complex emotional behavior. The combination of analytical results and robust numerics demonstrates the model's flexibility and potential to describe rich, memory-driven, stochastic patterns in romantic relationships. These insights could inform broader applications of piecewise fractional calculus to multi-regime dynamical systems with uncertainty.

Abstract

For love dynamical models, a new idea combining piecewise concept for integer-order, stochastic, and fractional derivatives is presented in order to capture the chaos and several crossover emotional scenerios. Under the assumptions of linear growth and Lipschitz condition, the fixed-point theorem explain the uniqueness and existence to the models under the investigation. The piecewise derivatives were approximated utilising the Lagrange interpolation method, and the computer results were demonstrated numerically for several values of order $α$. It was observed that the recently presented new idea in love dynamical models can represent disordered emotional patterns in passionate loving partnerships.

Love Dynamical Model with persepectives of Piecewise Differential Operators

TL;DR

This work develops a piecewise differential-dynamic framework for love dynamics that blends integer-order, fractional, and stochastic processes to capture crossover emotional patterns. It introduces two Owolabi models (linear and nonlinear) across interval-based memory kernels using Caputo, Atangana–Baleanu, and Caputo–Fabrizio derivatives, and proves existence and uniqueness for the Caputo-based piecewise system. The authors formulate Newton interpolation–based numerical schemes for the three derivative types and present simulations across varying fractional orders , revealing chaotic dynamics consistent with complex emotional behavior. The combination of analytical results and robust numerics demonstrates the model's flexibility and potential to describe rich, memory-driven, stochastic patterns in romantic relationships. These insights could inform broader applications of piecewise fractional calculus to multi-regime dynamical systems with uncertainty.

Abstract

For love dynamical models, a new idea combining piecewise concept for integer-order, stochastic, and fractional derivatives is presented in order to capture the chaos and several crossover emotional scenerios. Under the assumptions of linear growth and Lipschitz condition, the fixed-point theorem explain the uniqueness and existence to the models under the investigation. The piecewise derivatives were approximated utilising the Lagrange interpolation method, and the computer results were demonstrated numerically for several values of order . It was observed that the recently presented new idea in love dynamical models can represent disordered emotional patterns in passionate loving partnerships.
Paper Structure (11 sections, 37 equations, 12 figures)

This paper contains 11 sections, 37 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Modeling of the love dynamical system
  4. First Mathematical ModelOwolabi
  5. Second Mathematical Model Owolabi
  6. Existence and uniqueness of Love Dynamical modelAtangana3
  7. Numerical algorithm
  8. Numerical algorithm in the aspect of Caputo derivative
  9. Numerical algorithm in the aspect of Atangana-Baleanu derivative:
  10. Numerical algorithm in the aspect of Caputo-Fabrizio derivative:
  11. Conclusion

Figures (12)

  • Figure 1: The Numerical simulations for first model \ref{['caputolinear']} with $t(0)=0$, $s(0)=1$, $p(0)=1$, $\rho_{1}=0.12$, $\rho_{2}=0.05$, $\psi_{1}=0.8$, $\psi_{2}=0.81$, $\gamma_{1}=0.5$, $\gamma_{2}=1.2$, $\sigma_{1}=0.02$, $\sigma_{2}=0.01$, $\omega_{1}=6.1$, $\omega_{2}=-1$.
  • Figure 2: Chaotic dynamics for the first model \ref{['caputolinear']} with $t(0)=0$, $s(0)=1$, $p(0)=1$, $\rho_{1}=0.12$, $\psi_{1}=0.8$, $\psi_{2}=0.81$, $\gamma_{1}=0.5$, $\gamma_{2}=1.2$, $\sigma_{1}=0.02$, $\sigma_{2}=0.01$, $\omega_{1}=6.1$, and $\alpha=0.95$..
  • Figure 3: The Numerical simulations for second model \ref{['caputononlinear']} with $t(0)=0$, $r(0)=1$, $s(0)=1$, $\rho_{1}=0.12$, $\rho_{2}=0.01$, $\phi_{1}=1$, $\phi_{2}=1$, $\sigma_{1}=0.01$, $\sigma_{2}=0.02$, $\omega_{1}=6.1$, $\omega_{2}=-1$.
  • Figure 4: Chaotic dynamics for the second model \ref{['caputononlinear']} with $t(0)=0$, $r(0)=1$, $s(0)=1$, $\rho_{1}=0.12$, $\rho_{2}=0.01$, $\phi_{1}=1$, $\phi_{2}=1$, $\sigma_{1}=0.01$, $\sigma_{2}=0.02$, $\omega_{1}=6.1$, $\omega_{2}=-1$.
  • Figure 5: The Numerical simulations for first model \ref{['ABlinear']} with $t(0)=0$, $s(0)=1$, $p(0)=1$, $\rho_{1}=0.12$, $\rho_{2}=0.05$, $\psi_{1}=0.8$, $\psi_{2}=0.81$, $\gamma_{1}=0.5$, $\gamma_{2}=1.2$, $\sigma_{1}=0.02$, $\sigma_{2}=0.01$, $\omega_{1}=6.1$, $\omega_{2}=-1$.
  • ...and 7 more figures