Solving Differential Equations Using Fourth-Order Runge-Kutta in The Context of Surrealism

This article presents a didactic application of the fourth-order Runge-Kutta method to nonlinear initial-value problems. The work continues and extends a previously reported teaching strategy based on reduction to first-order systems and a double-index notation for the RK4 stage constants. Four nonlinear cases are considered: the Lorenz system, the Duffing oscillator, the van der Pol oscillator, and a coupled strongly nonlinear system proposed by Cveticanin. For each case, the first three RK4 iterations are developed explicitly before the complete numerical simulation is carried out in Maple 2021. This makes the relation between the mathematical model, the stage constants, and the recursive update visible before software is used for the full trajectory. Numerical results are presented through time histories, phase portraits, and three-dimensional trajectories. In addition, surrealist visual compositions are integrated with the scientific graphs while the numerical curves, axes, scales, and data are preserved. From an educational perspective, this visual layer is explored as a way of broadening the conventional presentation of numerical results and of partially moving away from standardized forms associated with school mathematical discourse. To the best of the authors’ knowledge, this is the first educational numerical-methods study in which surrealist visual compositions are systematically integrated with RK4 simulations of nonlinear initial-value problems.