normal form computation

This work addresses the challenge of distinguishing non-transverse intersecting motion branches in linkage mechanisms, which has hindered fine-grained analysis of their singularities and mobility. By leveraging the intrinsic geometric information of kinematic tangent cones—defined constructively for the first time—the study integrates local singularity analysis with computational algebraic geometry to extend existing algorithmic frameworks. The proposed computable method effectively identifies and separates non-transverse bifurcating motion branches at specific configurations. This approach overcomes the limitations of conventional local analyses and significantly enhances the understanding and modeling of complex kinematic structures.

This work addresses the complex training dynamics of time series models near bifurcation points, where rich features can be learned but optimization remains poorly understood. We propose the state-space Neural Tangent Kernel (sNTK), which integrates bifurcation theory and normal form analysis to reduce the learning geometry of high-dimensional recurrent systems to a low-rank description. We prove that near common codimension-one bifurcations, the sNTK collapses into a highly amplified rank-one operator, revealing the dominant learning direction and rendering the optimization landscape predictable. Leveraging this insight, we design a low-rank natural gradient method and demonstrate in teacher-student recurrent networks that the onset of bifurcation coincides precisely with a sharp drop in the effective rank of the sNTK and alignment along a dominant parameter direction, substantially improving training efficiency and stability.

This work addresses two key challenges in multiscale dynamical systems: the difficulty of initializing slow manifolds and the high computational cost of computing steady-state solutions for bifurcation diagrams. We propose a geometry-driven inverse modeling framework based on conditional score-based generative models (cSGMs). For the first time, conditional generative modeling is applied to slow manifold sampling and bifurcation diagram interpolation—enabling high-fidelity steady-state initial conditions to be generated directly from prescribed slow-variable values or new parameter configurations, without explicitly solving differential equations. The method integrates manifold learning, dynamical system dimensionality reduction, and generative inverse modeling to achieve label-controllable, mesh-free, data-driven slow manifold initialization and bifurcation diagram extrapolation and completion. Extensive validation across multiple ODE and PDE systems demonstrates substantial acceleration in steady-state acquisition while preserving accuracy, generalizability, and computational efficiency.

This paper addresses the theoretical equivalence between Lyapunov subcenter manifolds (LSMs) and eigenmanifolds: under what conditions can eigenmanifolds be rigorously characterized as LSMs exhibiting spatiotemporal symmetry? By integrating differential geometry, nonlinear dynamics, and Lie group analysis, we establish— for the first time—a rigorous proof that, in conservative, time-reversal-symmetric multibody systems, eigenmanifolds coincide precisely with the LSMs induced by the eigensubspaces of the linearized system. We further uncover their intrinsic relationship with Rosenberg manifolds and derive explicit sufficient conditions for local existence and uniqueness. Numerical experiments—including the double pendulum, five-link pendulum, and two variable-inertia systems—validate the theoretical framework. The results provide a rigid geometric foundation and verifiable design principles for periodic motion control in robotic systems.

This work addresses the rational realization problem for first-order differential-algebraic input-output equations, focusing on existence criteria and constructive methods under observability and real-coefficient constraints. Methodologically, it establishes an equivalence theorem between the existence of a rational realization and that of an observable rational realization for first-order systems; develops a decidable criterion for real rational realizability via differential-algebraic geometry and field extension theory; and devises a fully algorithmic, symbolically computed construction procedure that systematically generates both observable and real-coefficient rational realizations. Furthermore, several key results are extended to higher-order differential-algebraic equations. The contributions provide a rigorous yet practical algebraic framework for modeling, identification, and realization theory of linear and nonlinear dynamical systems, bridging theoretical algebra with applied system theory.

This work proposes a model-free method for detecting Hopf bifurcation points directly from time series without prior knowledge of the underlying dynamical equations. By leveraging Takens’ embedding to reconstruct the phase space and applying one-dimensional persistent homology, the approach introduces maximal persistence—a scalar topological functional with clear interpretability—as a bifurcation criterion, framing dynamical transitions as topological phase changes. Experimental validation on several canonical nonlinear systems demonstrates that the method reliably and accurately identifies Hopf bifurcations, confirming its effectiveness and broad applicability. This study thus offers a novel topological perspective for data-driven bifurcation analysis.

This work addresses data-driven modeling of physical systems—including autonomous and forced dynamical systems—by introducing a novel dynamics learning framework grounded in invariant foliation theory. Methodologically, it extends invariant foliation theory for the first time to parameter-dependent and volume-preserving forced systems; proposes a multi-neighborhood foliation fusion strategy to robustly reconstruct invariant manifolds; and integrates normal-form transformations to enable interpretable extraction of instantaneous frequency and damping characteristics. The contributions are threefold: (i) it effectively mitigates overfitting and underfitting, achieving high-fidelity reconstruction of invariant manifolds from sparse single- or multi-trajectory data; (ii) it delivers accurate long-term predictions across autonomous, periodically forced, quasiperiodically forced, and chaotic forced systems; and (iii) it offers strong interpretability and quantitative physical insight, establishing a new paradigm for low-data, high-accuracy, and physically interpretable dynamical modeling.

This work addresses the long-standing scarcity of large-scale annotated datasets in the domain of solving zero-dimensional nonlinear systems by introducing the largest benchmark dataset to date, specifically designed to support the evaluation of subdivision-based solvers and the learning of real root classification in parametric systems. Constructed through the integration of subdivision algorithms and nonlinear system theory, and informed by a comprehensive review of nearly two decades of relevant literature, the dataset provides a high-value resource for machine learning–driven real root classification and solver performance assessment. Experimental results demonstrate that the dataset effectively enables systematic comparisons among diverse solvers and facilitates the training of robust classification models, thereby filling a critical data gap in this research area.

This work addresses the challenge of constructing globally consistent Koopman eigenfunction representations for continuous-time dynamical systems exhibiting singularities—such as multistability, limit cycles, or separatrices—when only sparse, local observations are available. Conventional approaches struggle to achieve this efficiently. Leveraging the algebraic structure that non-zero Koopman eigenfunctions form a multiplicative group, the authors propose generating an expanded feature space via polynomial combinations of a small set of principal eigenfunctions. They further introduce a cross-singularity matching and continuation strategy that substantially enriches the repertoire of usable eigenfunctions. This framework enables high-fidelity, globally coherent modeling of dynamics from sparse data and significantly enhances the representation of key observables in complex systems.

This study investigates atypical bifurcation diagram phenomena arising in a class of one-dimensional discrete maps. Distinct from conventional bifurcation structures, these patterns exhibit complex dynamical behaviors rarely documented or entirely absent in standard bifurcation theory. By integrating dynamical systems theory, numerical simulations, and advanced visualization techniques, the work systematically uncovers and classifies a range of singular dynamical regimes within this map family. The findings not only reveal a rich repertoire of novel bifurcation scenarios but also deepen the understanding of complexity-generating mechanisms in nonlinear systems, offering fresh theoretical insights and empirical instances of nonstandard bifurcation phenomena.