Insights into transferal to fractal space modeling: delayed forced Helmholtz–Duffing oscillator with the non-perturbative approach

Abstract The damped Helmholtz–Duffing oscillator is a topic of great interest in many different fields of study due to its complex dynamics. By transitioning from conventional continuous differential equations to their fractal counterparts, one gains insights into the system’s response under new mathematical frameworks. This paper presents a novel method for converting standard continuous differential equations into their fractal equivalents. This conversion occurs after the nonlinear system is transformed into its linear equivalent. Numerical analyses show that there are several resonance sites in the fractal system, which differ from the one resonance point found in the continuous system. One important finding is that the fractal system loses some of its stabilizing power when decaying behavior is transformed into a diffuse pattern. Interestingly, a decrease in the fractal order in resonance settings shows a stabilizing impact, highlighting the dynamics’ complexity inside fractal systems. This endeavor to convert to fractals is a revolutionary technique that is being employed for the first time.