Duffing harmonic oscillator. Evolution of the attractor

Introduction. The paper investigates nonlinear oscillations of the Duffing harmonic oscillator, which are relevant for assessing the stability of mechanical and structural systems with elastic and damping components. The study’s relevance is linked to understanding the transition from regular to chaotic oscillations in seismic isolation and vibration control elements. The novelty lies in applying the Rosenstein method to quantitatively estimate the exponent-like quantity e1, replacing the qualitative approach used in Ueda’s classical studies (1979).

Materials and methods. The oscillator is described by the equation ẍ + kẋ + x3 = Bcos(ωt). Numerical integration was performed in MATLAB using the custom “+ueda” package. The dynamics were analyzed through Poincaré sections and by calculating the exponent-like value e1 using the Rosenstein algorithm, which tracks divergence of nearby trajectories in the reconstructed phase space.

Results. It is shown that increasing the excitation amplitude B (at k = 0.10) drives the system from periodic to chaotic motion via quasi-periodic states. Increasing the damping k (at B = 12.0) suppresses chaotic oscillations, restoring regular behaviour. The dependencies e1(B) and e1(k), refine the boundaries of chaos onset compared to Ueda’s results, identifying critical intervals Bc = 11.9–12.2 and kc = 0.26–0.30.

Conclusions. The results of the study confirmed the applicability of the Rosenstein method for the quantitative identification of chaotic modes in Duffing systems. The data obtained refine the boundaries of the transitions, in line with classical studies, and can be used in the design of seismic isolators and vibration-damping devices where it is necessary to avoid chaotic oscillations.

1. Nayfeh A.H., Mook D.T. Nonlinear Oscillations. Wiley, 1979.

2. Strogatz S.H. Nonlinear Dynamics and Chaos. CRC Press, 2018. DOI: 10.1201/9780429492563

3. Tanekou S.Ts., Ramadoss Ja., Kengne Ja., Kenmoe G.D., Rajagopal K. Coexistence of Periodic, Chaotic and Hyperchaotic Attractors in a System Consisting of a Duffing Oscillator Coupled to a van der Pol Oscillator. International Journal of Bifurcation and Chaos. 2023; 33(2). DOI: 10.1142/S0218127423300045. EDN XALVGC.

4. Kuznetsov S.V. Forced harmonic vibrations of duffing oscillator with cubic-quintic spring nonlinearity. Chaos, Solitons & Fractals. 2025; 199(1). DOI: 10.1016/j.chaos.2025.116699. EDN KKPYUH.

5. Da L., Soares I., Krol M.F., Rech P.C. Coexisting attractors and basins of attraction of an extended forced Duffing oscillator. The European Physical Journal B. 2024; 97(6):71. DOI: 10.1140/epjb/s10051-024-00709-0. EDN JWBVIF.

6. Moon F.C. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. Wiley, 1987.

7. Kovacic I., Brennan M.J. The Duffing Equation. Wiley, 2011. DOI: 10.1002/9780470977859

8. Thompson J.M.T., Stewart H.B. Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists. Wiley, 2002.

9. Rand R.H. Lecture Notes on Nonlinear Vibrations. Cornell University, 2005.

10. Chen G., Ueta T. Chaos in Circuits and Systems. World Scientific Series on Nonlinear Science Series B. 2002. DOI: 10.1142/4945

11. Guckenheimer J., Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences. 1983. DOI: 10.1007/978-1-4612-1140-2

12. Montenegro Joo J. Evolution of the Lyapunov Exponent during a Chaotic Event in the Nonlinear Damped and Forced Oscillator. Chaos and Complexity Letters. 2022; 16(1).

13. Abohamer Mk., Amer Ts., Arab A., Galal Aa. Analyzing the chaotic and stability behavior of a duffing oscillator excited by a sinusoidal external force. Journal of Low Frequency Noise, Vibration and Active Control. 2024; 44(2):969-986. DOI: 10.1177/14613484241298998. EDN PUDJES.

14. Ueda Y. Randomly transitional phenomena in the system governed by Duffing’s equation. Journal of Statistical Physics. 1979; 20(2):181-196. DOI: 10.1007/bf01011512. EDN WDFUCD.

15. Wolf A., Swift J.B., Swinney H.L., Vastano J.A. Determining Lyapunov exponents from a time series. Physica D: Nonlinear Phenomena. 1985; 16(3):285-317. DOI: 10.1016/0167-2789(85)90011-9

16. Kantz H., Schreiber T. Nonlinear Time Series Analysis. Cambridge University Press, 2004. DOI: 10.1017/cbo9780511755798

17. Rosenstein M.T., Collins J.J., De Luca C.J. A pra-ctical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena. 1993; 65(1-2):117-134. DOI: 10.1016/0167-2789(93)-90009-p

18. Gao J.B., Zheng Z., Sanjuán M.A.F. Chaos Theory and Complex Systems. Springer, 2020.

19. Luongo A., Zulli D. Nonlinear dynamic models for seismic isolation systems. International Journal of Non-Linear Mechanics. 2014; 58:156-166.

20. Casciati F., Rodellar J. Smart Structures: Innovative Systems for Seismic Response Control. WIT Press, 2011.