Exact Dimensional Reduction for Quasi-Linear ODE Ensembles
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913127859224576 |
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| author | Augustsson, Felix Martens, Erik Andreas Cestnik, Rok |
| author_facet | Augustsson, Felix Martens, Erik Andreas Cestnik, Rok |
| contents | We present an exact dimensional reduction for high-dimensional dynamical systems composed of $N$ identical dynamical units governed by quasi-linear ordinary differential equations (ODEs) of order $M$. In these systems, each unit follows a linear differential equation whose coefficients depend nonlinearly on the ensemble variables, such as a mean field variable. We derive $M+1$ closed-form macroscopic equations of order $M$ with variables that exactly capture the full microscopic dimensional dynamics and that allow reconstruction of individual trajectories from the reduced system. Our approach enables low-dimensional analysis of collective behavior in coupled oscillator networks and provides computationally efficient exact representations of large-scale dynamics. We illustrate the theory with examples, highlighting new families of solvable models relevant to physics, biology and engineering that are now amenable to simplified analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_18755 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exact Dimensional Reduction for Quasi-Linear ODE Ensembles Augustsson, Felix Martens, Erik Andreas Cestnik, Rok Adaptation and Self-Organizing Systems Exactly Solvable and Integrable Systems We present an exact dimensional reduction for high-dimensional dynamical systems composed of $N$ identical dynamical units governed by quasi-linear ordinary differential equations (ODEs) of order $M$. In these systems, each unit follows a linear differential equation whose coefficients depend nonlinearly on the ensemble variables, such as a mean field variable. We derive $M+1$ closed-form macroscopic equations of order $M$ with variables that exactly capture the full microscopic dimensional dynamics and that allow reconstruction of individual trajectories from the reduced system. Our approach enables low-dimensional analysis of collective behavior in coupled oscillator networks and provides computationally efficient exact representations of large-scale dynamics. We illustrate the theory with examples, highlighting new families of solvable models relevant to physics, biology and engineering that are now amenable to simplified analysis. |
| title | Exact Dimensional Reduction for Quasi-Linear ODE Ensembles |
| topic | Adaptation and Self-Organizing Systems Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2509.18755 |