TORUS-BASED PHASE SPACE REPRESENTATIONS FOR COMPLEX ENVIRONMENTAL SYSTEMS.

Phase space descriptions play a major part in the study of complex dynamical systems; but traditional Euclideanformulations do not usually exhibit much of intrinsic periodicity, multi-frequency interactions and topologicalconstraints found in real-world systems. We propose a new analytical structure of the torus-based phase spacerepresentations in this work, which is aimed at offering a mathematically consistent and topologically faithfuldescription of the systems with cyclic and quasi-periodic dynamics. The suggested construction takes the form offormalization of the production of low-dimensional toroidal manifolds directly out of the variables of the system,allowing the retention of periodicity conditions of the boundaries and invariant structures that are being deformedunder the standard representations. The framework has analytical properties, such as stability features and invariantsets, which are obtained and analyzed. We use a set of numerical experiments with a set of representative nonlineardynamical systems, where the results of the computations are the toroidal orbits as compared to the Euclideancounterparts. The results of the simulations prove that the proposed approach provides better structural coherence,better interpretability of the system dynamics, and better resistance to changes in parameters and noise. These resultsdemonstrate the benefits of representations using torus when it comes to analytical understanding as well as towardscomputational modeling. The framework has a generalizable structure that can be applied to a large variety of complexsystems, providing a single methodology of how to combine topology-aware analysis with validation throughsimulations.