| Abstract: |
| Invariant algebraic surfaces are polynomial surfaces that remain fixed under the flow of a differential system, meaning solutions starting on the surface stay on the surface. They are crucial for understanding the global dynamics of a system, and their existence often indicates the presence of Darboux integrals, which can simplify the analysis of chaotic and nonchaotic behavior. In this talk, we propose a computational framework that combines Gr\obner bases and triangular sets. This framework first utilizes Gr\obner bases to compute elimination ideals and obtain parametric constraints for the existence of invariant algebraic surfaces, then employs triangular decomposition algorithms to derive explicit expressions, and finally performs reduction through variable substitution using the previously obtained constraints to yield concise expressions for invariant algebraic surfaces. The effectiveness of the proposed methods is validated through computations of invariant algebraic surfaces (curves) for the Belousov--Zhabotinsky reaction model and the Rayleigh--Li\`enard oscillation system. |
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