Strong chaotification and robust chaos in the Duffing oscillator induced by two-frequency excitation

• Published in: Nonlinear dynamics Vol. 103; no. 2; pp. 1955 - 1967
• Main Authors: Gusso, André, Ujevic, Sebastian, Viana, Ricardo L.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 2021; Springer Nature B.V
• Subjects: Automotive Engineering; Chaos theory; Classical Mechanics; Control; Duffing oscillators; Dynamical Systems; Engineering; Excitation; Initial conditions; Mechanical Engineering; Original Paper; Parameters; Robustness (mathematics); Tolerances; Vibration; Chaos; Duffing oscillator; Robust Chaos; Chaotification
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-020-06183-4

In this work, we demonstrate numerically that two-frequency excitation is an effective method to produce chaotification over very large regions of the parameter space for the Duffing oscillator with single- and double-well potentials. It is also shown that chaos is robust in the last case. Robust chaos is characterized by the existence of a single chaotic attractor which is not altered by changes in the system parameters. It is generally required for practical applications of chaos to prevent the effects of fabrication tolerances, external influences, and aging that can destroy chaos. After showing that very large and continuous regions in the parameter space develop a chaotic dynamics under two-frequency excitation for the double-well Duffing oscillator, we demonstrate that chaos is robust over these regions. The proof is based upon the observation of the monotonic changes in the statistical properties of the chaotic attractor when the system parameters are varied and by its uniqueness, demonstrated by changing the initial conditions. The effects of a second frequency in the single-well Duffing oscillator is also investigated. While a quite significant chaotification is observed, chaos is generally not robust in this case.