## 1 Introduction  
Due to the high directionality and high coherence of laser pulses, precise control can be achieved by adjusting laser parameters during laser-matter interactions. By designing parameters such as the polarization direction, chirp parameter, pulse width, beam waist radius, and repetition rate of the laser pulse, precise control over the interaction between the laser pulse and electrons can be achieved. Furthermore, controlling the initial position, energy, and motion direction of the electrons enables the interaction between the electron and laser pulse in different scenarios. Through simulations of experiments under various conditions, corresponding data for real-world situations can be obtained, thereby providing a reference for actual experiments. Laser-matter interactions have numerous application scenarios, making this field a subject of extensive research by many scholars [1, 2, 3]. Among these, nonlinear Thomson scattering has become a particularly popular research direction, and the nonlinear Thomson scattering of intense laser pulses in single electrons, particle beams, plasmas, and other materials has significant research value [4, 5, 6]. Thomson scattering refers to the elastic scattering of electromagnetic radiation by a free charged particle. During the scattering process, the frequency of the photon remains unchanged. When an electron is scattered by electromagnetic radiation, the electric field component causes the electron to undergo forced oscillations. According to classical electromagnetism, the oscillating electron radiates electromagnetic waves in all directions, which results in scattered X-rays, with the radiation primarily distributed in the forward direction of the incident electromagnetic radiation. On the other hand, nonlinear Thomson scattering occurs when electrons are driven by intense fields, causing their motion to become more complex. The equations of motion exhibit nonlinear characteristics under such conditions [7; 8; 9; 10]. In the late 1980s, [PERSON] et al. proposed the technique of Chirped Pulse Amplification (CPA) [11]. The advent of chirped pulses further enhanced the intensity and quasi-stability of laser pulses, enabling the generation of higher-quality pulses [12], and it has given rise to applications in various fields, including laser pulse amplifiers, electron dynamics, and more [13; 14; 15; 16; 17; 18]. Considering the high costs associated with the instruments, manpower, and material preparation required for conducting nonlinear Thomson scattering experiments, as well as the difficulty in performing large-scale experiments on individual electrons, simulations through computational experiments offer a more efficient approach. Such simulations allow for large-scale data acquisition and analysis in a relatively shorter time and at a lower cost. Therefore, the simulation of nonlinear Thomson scattering and the subsequent data analysis through simulations hold significant research value. In recent years, with the continuous development of ultrashort and ultra-intense chirped laser pulses, nonlinear Thomson scattering induced by electrons under relativistic laser pulse intensities has attracted significant attention. For example, [PERSON] et al. [19] investigated the spatial radiation characteristics of interactions between circularly polarized laser pulses and electrons. [PERSON] et al. [20] studied attosecond and zeptosecond X-ray pulses generated by nonlinear Thomson backscattering and found that variations in the laser pulse beam waist affect the pulse shape and monochromaticity of the process. Building upon the aforementioned studies, this paper analyzes the interaction between circularly polarized chirped pulses and electrons in a coaxial collision process, discovering a novel trajectory--an hourglass-spindle shape. The trajectory is defined and analyzed, and the principles and process of its formation are determined.