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<p>The chaos in bubble dynamics complicates the control of its behavior across various applications. Eliminating or reducing the chaotic oscillations of the bubble is of great importance due to its unpredictable and uncertain nature. This paper investigates the chaotic behavior of a single spherical bubble before stabilizing the bubble dynamics modeled by Rayleigh–Plesset (RP) equation through a periodic perturbation method. The system can transition from chaotic regions to periodic motion by applying a controller that is designed using Melnikov integral. The controller effect on dynamical behavior is analyzed through numerical methods, including Lyapunov exponent (LE) and bifurcation diagram. Simulation results demonstrate that the proposed method significantly reduces the level of chaotic oscillations in the system. Additionally, one of the control objectives achieved by the controller designed in this paper is the system’s robustness against uncertainties in bubble parameters. Despite the desired performance, this controller does not increase control effort.</p>

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Stabilization of chaotic spherical bubble oscillations: a Melnikov integral approach

  • 期刊论文

  • 发表日期：

    2025年4月16日

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  • Bahar Rahmatizadeh,
  • Masoumeh Azadegan,
  • Mohamad Taghi Hamidi Beheshti,
  • Mahmoud Najafi

摘要

The chaos in bubble dynamics complicates the control of its behavior across various applications. Eliminating or reducing the chaotic oscillations of the bubble is of great importance due to its unpredictable and uncertain nature. This paper investigates the chaotic behavior of a single spherical bubble before stabilizing the bubble dynamics modeled by Rayleigh–Plesset (RP) equation through a periodic perturbation method. The system can transition from chaotic regions to periodic motion by applying a controller that is designed using Melnikov integral. The controller effect on dynamical behavior is analyzed through numerical methods, including Lyapunov exponent (LE) and bifurcation diagram. Simulation results demonstrate that the proposed method significantly reduces the level of chaotic oscillations in the system. Additionally, one of the control objectives achieved by the controller designed in this paper is the system’s robustness against uncertainties in bubble parameters. Despite the desired performance, this controller does not increase control effort.