THERMAL SCIENCE
International Scientific Journal
BIFURCATION CHARACTERISTICS AND BURSTING OSCILLATION OF DUFFING-VAN DER POL OSCILLATOR
ABSTRACT
In this paper, we study the bifurcation characteristics and bursting oscillation of the Duffing-Van der Pol system with periodic excitation. Due to the different frequency scales between the excitation frequency and the natural frequency in the oscillator, when the periodic excitation changes slowly with time, the system is considered as a slow subsystem, and when it is fixed, the system is considered as a fast subsystem. We analyze the bifurcation characteristics of the fast subsystem and use the slowly varying parameter as the bifurcation parameter to show how the bursting oscillations are generated. Furthermore, the phase diagram and time-history diagram of fold-fold bursting oscillation, fold-subHopf bursting oscillation, supHopf-supHopf bursting oscillation, and homoclinic-homoclinic bursting oscillation are given by numerical simulation. Combined with the fig-ures, it is found that these four kinds of bursting oscillations with bifurcation delay phenomenon are symmetrical and further reveal the bifurcation mechanisms of these four kinds of bursting oscillations.
KEYWORDS
PAPER SUBMITTED: 2023-12-25
PAPER REVISED: 2024-07-06
PAPER ACCEPTED: 2024-07-06
PUBLISHED ONLINE: 2025-07-06
THERMAL SCIENCE YEAR
2025, VOLUME
29, ISSUE
Issue 3, PAGES [1849 - 1859]
- Prandtl, L., Motion of Fluids with Very Little Viscosity, Technical Memo-452, NACA, Washington, DC, 1928
- Yang, W., Towfighian, S., A Parametric Resonator with Low Threshold Excitation for Vibration Energy Harvesting, Journal of Sound and Vibration, 446 (2019), Apr., pp. 129-143
- Beims, M. W., Gallas, J. A. C., Predictability of the Onset of Spiking and Bursting in Complex Chemical Reactions, Physical Chemistry Chemical Physics, 20 (2018), 27, pp. 18539-18546
- Theodosiou, C., et al., Periodic Steady State Response of Large Scale Mechanical Models with Local Non-linearities,. International Journal of Solids and Structures, 46 (2009), 20, pp. 3565-3576
- Yang, S. C., Hong, H. P., Non-Linear Inelastic Responses of Transmission Tower-Line System Under Downburst Wind, Engineering Structures, 123 (2016), 15, pp. 490-500
- Zhang, L., et al., Chaos Control for the Periodically Excited Van der Pol-Duffing. Journal of Wenzhou University, Natural Sciences, 28 (2007), 2, pp. 11-14
- Ma, X. D., et al., Complex Mixed-Mode Vibration Types Triggered by the Pitchfork Bifurcation Delay in a Driven Van der Pol-Duffing Oscillator, Applied Mathematics and Computation, 411 (2021), 126522
- He, C. H., El-Dib, Y. O., A Heuristic Review on the Homotopy Perturbation Method for Non-conservative Oscillators, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 2, pp. 572-603
- He, C. H., et al., Controlling the Kinematics of a Spring-Pendulum System Using an Energy Harvesting Device, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 3, pp. 1234-1257
- Liu, Y. B., Chen, Y. S., Global Bifurcations for a Generalized Codimension-4 Duffing-Van der Pol Equation, Journal of Vibration and Shock, 30 (2011), 1, pp. 69-72
- He, C. H., et al., Hybrid Rayleigh-Van der Pol-Duffing Oscillator: Stability Analysis and Controller, Journal of Low Frequency Noise, Vibration & Active Control, 41 (2022), 1, pp. 244-268
- He, J.-H., El-Dib, Y. O., The Reducing Rank Method to Solve Third-Order Duffing Equation with the Homotopy Perturbation, Numerical Methods for Partial Differential Equations, 37 (2021), 2, pp. 1800-1808
- He, J.-H., et al., Homotopy Perturbation Method for fractal Duffing Oscillator with Arbitrary Conditions, Fractals, 30 (2022), 9, 22501651
- He, J.-H., et al., A Good Initial Guess for Approximating Non-Linear Oscillators by the Homotopy Perturbation Method, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 1, pp. 21-29
- Anjum, N., et al., Free Vibration of a Tapered Beam by the Aboodh Transform-Based Variational Iteration Method, Journal of Computational Applied Mechanics, 55 (2024), 3, pp. 440-450
- Tang, W., et al., Variational Iteration Method for the Nanobeams-Based N/MEMS System, MethodsX, 11 (2023), 102465
- He, J.-H., The Simplest Approach to Non-Linear Oscillators, Results in Physics, 15 (2019), 102546
- Zhang, J. G., et al., Application of He's Frequency Formula to Non-Linear Oscillators with Generalized Initial Conditions, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 4, pp. 701-712
- Kaid, N., et al., Enhancing Thermal Comfort in Building Innovations in Sustainable Cooling and Heating Systems Utilizing Geothermal Energy, Thermal Science, 27 (2023), 4B, pp. 3477-3486
- Yau, H. T., et al., Proximal Policy Optimization‐Based Controller for Chaotic Systems, International Journal of Robust and Non-linear Control, 34 (2024), 1, pp. 586-601
- Kuo, P.-H., et al., Novel Fractional-Order Convolutional Neural Network Based Chatter Diagnosis Approach in Turning Process with Chaos Error Mapping, Non-linear Dynamics, 111 (2023), 8, pp. 7547-7564
- Kuo, P.-H., et al., Machine Tool Chattering Monitoring by Chen-Lee Chaotic System-Based Deep Convolutional Generative Adversarial Nets, Structural Health Monitoring, 22 (2023), 6, pp. 3891-3907
- He, J.-H., et al., Pull-Down Instability of the Quadratic Non-Linear Oscillators, Facta Universitatis, Series: Mechanical Engineering, 21 (2023), 2, pp. 191-200
- He, J.-H., et al., Pull-in Stability of a Fractal MEMS System and Its Pull-In Plateau, Fractals, 30 (2022), 9, 22501857
- He, J.-H., et al., Piezoelectric Biosensor Based on Ultrasensitive MEMS System, Sensors and Actuators: A. Physical., 376 (2024), 115664
- Tian, D., et al., Fractal N/MEMS: From Pull-in Instability to Pull-in Stability, Fractals, 29 (2021), 2, 2150030
- Tian, D., He, C. H., A Fractal Micro-Electromechanical System and Its Pull-in Stability, Journal of Low Frequency Noise Vibration and Active Control, 40 (2021), 3, pp. 1380-1386
- He, C. H., A Variational Principle for a Fractal Nano/Microelectromechanical (N/MEMS) System, Int. J. Numer. Method. H., 33 (2023), 1, pp. 351-359
- Zhang, L., Wang, T., Qualitative Properties, Bifurcations and Chaos of a Discrete Predator-Prey System with Weak Allee Effect on the Predator, Chaos, Solitons & Fractals, 175 (2023), Oct., 113995
- Hastir, A., Muolo, R., A Generalized Routh-Hurwitz Criterion for the Stability Analysis of Polynomials with Complex Coefficients: Application to the PI-control of Vibrating Structures, IFAC Journal of Sys-tems and Control, 26 (2023), 100235
- Zhang, X. Y., et al., Bursting Oscillations Induced by Coexisted Cycles Separated by Fold Limit Cycle Bifurcation, Journal of Vibration Engineering & Technologies, 12 (2024), Suppl. 1, pp. S573-S583
- Hou, H. S., et al., On Fractional Ring Neural Networks with Multiple Time Delays: Stability and Hopf Bifurcation Analysis, Chinese Journal of Physics, 90 (2024), Aug., pp. 303-318
- Gjata, O., Zanolin, F., An Application of the Melnikov Method to a Piecewise Oscillator, Contemp. Math., 4 (2024), 2, pp. 249-269
- Colombo, G., et al., Inflated Deterministic Chaos and Smale's Horseshoe, Journal of Difference Equations and Applications, 18 (2012), 3, pp. 471-488