Limiting multistability with fractional calculus

Department of Dynamics, Lodz University of Technology - Stefanowskiego 1/15, 90–537 Lodz, Poland

Received: 25 March 2026
Accepted: 18 May 2026

Abstract

In this paper we investigate the problem of limiting multistability in complex dynamical systems with fractional calculus. The study is based on the famous van der Pol-Duffing oscillator, which in the case of external excitation exhibits the co-existence of numerous attractors. We identify possible states of the system using the integer-order scheme and compare them to the ones obtained for fractional modelling. We show that depending on the order of derivatives, the number of solutions can be limited, leading to monostability of the system. The scenarios of attractors exclusion are discussed, including the study of their basin stabilities. Moreover, our results uncover the impact of fractional components on transient properties of the oscillator. Depending on the chosen order, the transient response can be majorly limited, allowing the trajectories to stabilize on the final state much earlier. We confirm our results in both periodic and chaotic dynamics, showing that the proposed methodology can be applied in various complex systems.