动力系统：2025–2026 年度综述

An annual survey of the math.DS arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 132 papers and organised into 37 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Invariant Measures and Periodic Orbits in Smooth Dynamical Systems (39.9%) — This cluster focuses on proving existence and stability of invariant measures, periodic orbits, and local dynamics in smooth flows and maps on manifolds, often via normal forms and variational methods.
• Stochastic and Network-Enhanced Bifurcation Dynamics (28.8%) — This cluster investigates bifurcations, stochastic effects, and collective behavior in networked or higher-order dynamical systems, including time-scale models and geometric constraints.
• Nonlinear Stability and Control of Hybrid Dynamical Systems (11.9%) — This cluster addresses stability, control, and identification of nonlinear and hybrid dynamical systems, with applications ranging from neural networks to robotics and PDEs.
• Lyapunov Exponents and Thermodynamic Formalism (7.0%) — This cluster develops explicit bounds, Lyapunov exponents, and variational principles for cocycles and thermodynamic quantities, often with number-theoretic or algebraic flavor.
• Data-Driven Discovery and Neural Control of Dynamics (6.6%) — This cluster uses machine learning and data-driven methods to discover, control, or model dynamical systems, including symbolic regression and neural network-based control.
• Dimensions of Attractors and Iterated Function Systems (2.0%) — This cluster studies Hausdorff and mean dimensions of attractors, iterated function systems, and related fractal sets, often with number-theoretic or combinatorial constructions.
• Ergodic Averages, Mixing, and Topological Entropy (1.6%) — This cluster examines convergence of ergodic averages, mixing rates, entropy growth, and related topological or ergodic notions along sequences and in weighted settings.
• Normal Forms and Singularities of Vector Fields (1.3%) — This cluster focuses on analytic and geometric properties of vector fields, including singularities, normal forms, and pattern formation in smooth or piecewise-smooth systems.
• Billiards, Optimization, and Parameter Families (0.6%) — This cluster explores integrable billiards, parameter families of dynamical systems, and connections to optimization algorithms and boundary-value problems.
• Quantum and Categorical Extensions of Classical Dynamics (0.1%) — This cluster extends classical dynamical concepts to quantum or categorical settings, including cellular automata and noncommutative foliation invariants.
• Regularity and Sufficient Conditions for Slow-Fast Systems (0.1%) — This cluster investigates sufficient conditions for regularity and well-posedness in slow-fast, regularized, or descriptor dynamical systems.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.