The study of systems with linear restoring forces and recurring nonlinear perturbations is crucial across physics. Resonance occurs when a system's natural oscillation frequencies and the frequency of nonlinear restoring forces satisfy specific algebraic relations. In accelerator physics, understanding resonances and nonlinear dynamics is essential to prevent particle loss. This paper focuses on a two-dimensional coupled resonance, a system with increased complexity compared to simpler one-dimensional systems. The Poincaré surface of section is a powerful tool for visualizing the dynamics of such systems, particularly in periodic systems. This method, developed by Henri Poincaré, allows for a simplified representation of the complex behavior. In two-degree-of-freedom systems, resonant dynamics manifest in a four-dimensional phase space, the topology of which is not always intuitive. When a nonlinear coupling force is present, the mixed coordinates can exhibit a surprising correlation, with the resonant dynamics only apparent in the mixed planes, not in individual (*q*, *p*) planes. The collection of points representing resonant particles across the Poincaré surface of section forms a four-dimensional closed curve termed a 'fixed line'. This work experimentally verifies the existence of these fixed lines, a phenomenon previously only observed in simulations or theoretical studies. The practical implications are significant, as the existence and behavior of fixed lines influence beam degradation and the dynamic aperture (DA), which directly impacts the design and operational efficiency of particle accelerators. Understanding fixed lines is vital for optimizing future high-energy colliders and improving high-intensity, high-brightness beams in lower-energy accelerators. Previous research on one-dimensional resonances has shown the importance of resonance crossing in halo formation and particle loss in high-intensity bunches. Recent suggestions linked fixed lines to asymmetric halo formation, underscoring the need for experimental verification. This research directly addresses this gap in experimental evidence.

The paper cites several studies highlighting the importance of understanding nonlinear dynamics and resonances in accelerator physics. The Large Hadron Collider (LHC) at CERN is presented as a prominent example, where the impact of nonlinearities from magnet imperfections required a larger-than-necessary dynamic aperture to ensure stable operation. The LHC's operational experience, along with subsequent systematic measurements and modelling, significantly improved the predictability of the dynamic aperture. Studies on one-dimensional resonances and their connection to space-charge-induced resonance crossing and halo formation are also discussed, leading to the hypothesis that fixed lines contribute to asymmetric halo formation. However, the experimental evidence for the existence of fixed lines has been lacking until now, motivating the current study.

The experiment was conducted at the CERN Super Proton Synchrotron (SPS). A proton beam was accelerated to 100 GeV/c, minimizing the influence of collective effects and enabling the beam to function effectively as a single particle probe. Transverse oscillations were induced using kicker magnets, and the effects of a third-order resonance excited by sextupole magnets were studied. The beam positions were recorded turn-by-turn using beam position monitors (BPMs). Four consecutive BPMs (two per plane) allowed the reconstruction of Courant-Snyder coordinates (*x*, *p_x*, *y*, *p_y*) at a specific location in the accelerator. Using an action-angle representation, the scaled Courant-Snyder coordinates and invariants (*ā_x*, *ā_y*, *φ_x*, *φ_y*, *Ω*) were extracted. This allowed visualization of the Poincaré surface of section and analysis of the resonant dynamics in (*Ω*, *ā_y*) space. The experiment faced significant challenges, including the inherent fragility of the fixed line effect, tune modulation due to power converter ripple, the presence of beta-beating (optics perturbation from quadrupole imperfections), and limitations in the available kicker magnets, which restricted the accessible fixed-line orientations. To address these challenges, the beam energy was increased to mitigate ripple effects, and careful machine tuning was performed. The third-order resonance was excited using two sextupoles, enabling variation of the fixed line orientation. A systematic measurement sequence varied sextupole strengths, distance from resonance, and kicker strengths to find optimal conditions. The Courant-Snyder coordinates were scaled to simplify the analysis and facilitate direct extraction from BPM data. Data analysis involved fitting the experimental data to the theoretical fixed line equation, providing a visual confirmation of the fixed line topology. The effect of BPM noise on the analysis was addressed by applying a moving average filter, and error bars were carefully estimated considering BPM fluctuations and beta-beating. Data selection criteria were established based on the stability of the beam oscillations and the closeness to the resonance. A drift parameter (D) was defined to quantify the stability of the machine parameters during measurements. Simulations using MAD-X were performed to validate the experimental results and to compare the Poincaré surface of section projections from simulation and experiment.

The experiment successfully measured fixed lines, providing the first experimental confirmation of their existence. The Poincaré surface of section projections from the experimental data clearly showed Lissajous patterns consistent with the predicted fixed line topology. The measured fixed line orientation agreed reasonably well with theoretical predictions, considering the unavoidable presence of beta-beating. The analysis demonstrated that the beam remained trapped around the fixed line even in the presence of machine parameter drifts, highlighting the robustness of the resonance. The study systematically explored the parameter space to identify suitable experimental conditions, showing a clear dependence on the strength of the sextupole excitation and the initial beam conditions. The experimental data closely matched simulation results generated using MAD-X, further validating the experimental setup and analysis methods. Quantitative measures of the stability of the beam and the deviations from the idealized fixed line were presented, showing small oscillations consistent with expected error sources.

The experimental observation of fixed lines directly confirms the theoretical predictions and provides compelling evidence for their role in nonlinear beam dynamics in particle accelerators. This significantly advances our understanding of resonant behavior in multi-dimensional systems. The ability to experimentally measure and characterize fixed lines is crucial for improving the design and operation of particle accelerators. The findings directly address the need for experimental verification of fixed lines and their relationship to halo formation, potentially leading to more precise models and improved beam control strategies. This has direct practical implications for optimizing dynamic aperture, reducing particle loss, and enabling higher beam intensities and brightness in both existing and future accelerators. The challenges addressed in the methodology emphasize the complexity of this experimental task, and the methods used demonstrate a thorough approach to controlling error sources and validating results.

This research provides the first experimental verification of fixed lines induced by nonlinear resonances in a particle accelerator. The findings confirm the theoretical predictions and highlight the importance of understanding fixed lines for optimizing accelerator performance. Future research could explore unstable fixed lines, more complex resonance structures, and the application of these findings to advanced accelerator designs. The detailed methodology developed can be applied to other accelerator facilities, paving the way for more comprehensive understanding of nonlinear dynamics.

The experimental setup had limitations regarding the range of accessible fixed-line orientations due to constraints in the kicker magnet system. The presence of beta-beating introduced uncertainty in the exact orientation of the fixed lines. Although efforts were made to minimize the impact of power converter ripple and machine parameter drifts, these factors introduced some uncertainties into the measurements. The study focused primarily on stable fixed lines; further investigation into unstable fixed lines would be valuable.