<p>Monitoring for the transition between different states of complex dynamic systems contributes to anticipating systemic catastrophic events, and any research that enriches the solutions to such problem is worthy of being encouraged. The two wings of the Lorenz attractor that exists in a classic simplified model of atmospheric convection are often regarded as two extreme climatic states. Therefore, research on the internal transition of the Lorenz attractor is conducive to addressing general transition problems. In this article, an analytical–numerical hybrid method is proposed, providing a new perspective for this problem. Three pairs of surfaces are analytically designed to segment the trajectory in the attractor into different segments representing different states. The criterion of this method is that the transition will undoubtedly occur if the trajectory is above one of the three surface pairs in certain spaces, while if not, the transition will not occur in the current evolution loop. The starting and ending positions of the transition can be marked by the intersection points between the remaining two pairs of surfaces and the trajectory, respectively. Throughout the entire process of monitoring transition, this method enables earlier detection of the transition compared with other methods, indicating that this method allows for more time to respond to catastrophic events. The effectiveness of this method for the Lorenz attractor has been confirmed, and there is no occurrence of missed or false detection, so that the robustness of this monitoring framework has been verified. Researchers engaged in condition monitoring for chaotic systems can gain new ideas from this work.</p>

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A novel geometric method for monitoring the transition between the wings of the Lorenz attractor

  • Honglei Wu,
  • Guilin Wen,
  • Jie Liu

摘要

Monitoring for the transition between different states of complex dynamic systems contributes to anticipating systemic catastrophic events, and any research that enriches the solutions to such problem is worthy of being encouraged. The two wings of the Lorenz attractor that exists in a classic simplified model of atmospheric convection are often regarded as two extreme climatic states. Therefore, research on the internal transition of the Lorenz attractor is conducive to addressing general transition problems. In this article, an analytical–numerical hybrid method is proposed, providing a new perspective for this problem. Three pairs of surfaces are analytically designed to segment the trajectory in the attractor into different segments representing different states. The criterion of this method is that the transition will undoubtedly occur if the trajectory is above one of the three surface pairs in certain spaces, while if not, the transition will not occur in the current evolution loop. The starting and ending positions of the transition can be marked by the intersection points between the remaining two pairs of surfaces and the trajectory, respectively. Throughout the entire process of monitoring transition, this method enables earlier detection of the transition compared with other methods, indicating that this method allows for more time to respond to catastrophic events. The effectiveness of this method for the Lorenz attractor has been confirmed, and there is no occurrence of missed or false detection, so that the robustness of this monitoring framework has been verified. Researchers engaged in condition monitoring for chaotic systems can gain new ideas from this work.