Special Session 103: The integrability and bifurcation theory of dynamical systems and its applications

Elimination Methods for Computing Invariant Algebraic Surfaces of Dynamical Systems

Bo Huang Beihang University Peoples Rep of China Co-Author(s):    Changbo Chen, Bo Huang, Valery Romanovski and Linpeng Wang

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.

First Integrals and Invariants of Systems of ODEs

Abdul Jarrah American University of Sharjah United Arab Emirates Co-Author(s):    Mateja Grasic and Valery Romanovski

Abstract: In this talk, we study the relationship between monomial first integrals, polynomial invariants arising from certain group actions, and the Poincar\`e--Dulac normal forms of autonomous systems of ordinary differential equations with a diagonal linear part. Using tools from computational algebra, we develop algorithmic methods for identifying generators of the algebras of monomial and polynomial first integrals, including the general case where the eigenvalues of the linear part are algebraic complex numbers. The approach also provides a practical way to explore the structure of polynomial invariants and their connection to the Poincar\`e--Dulac normal forms of the underlying vector fields.

On the IFS which admits a specified attactor

Wenxia Li East China Normal University Peoples Rep of China Co-Author(s):    D.R Kong, W.X. Li, Z.Q.Wang, Y.Y. Yao, Y.X. Zhang

Abstract: An iterated function system (IFS) is a finite collection of contractive maps $\{f_i(x)=\beta _ix+\gamma _i: i=1,2,\cdots , n\}$ with $\beta _i, \gamma _i\in {\mathbb R}$ and $|\beta _i|

Support measures and complex dimensions in dynamics

Goran Radunovic University of Zagreb Croatia Co-Author(s):    Goran Radunovi\`c and Steffen Winter

Abstract: The general Steiner formula of Hug, Last and Weil describes the tube volume of any closed set in R^d, and the support measures arising from this formula encode its geometric properties. Recently, basic contents and support contents have been introduced as tools to extract fractal properties of a set from these measures. The original motivation was to extract the geometric meaning of the coefficients in fractal tube formulas that arise in the theory of complex dimensions by Lapidus, Radunovic, and Zubrinic. This is achieved by introducing appropriate zeta functions associated to each support measure, which turn out to be useful tools for computing basic contents and support contents. We will reflect on applications of these new functionals to analysis of orbits of dynamical systems. Based on joint work with Steffen Winter.

Rigidity of saddle loops

Maja Resman University of Zagreb Croatia Co-Author(s):    D. Panazzolo, L.Teyssier

Abstract: We define an abstract complex saddle loop in $\mathbb C^2$ as a pair $(\mathcal F,R)$ of a hyperbolic normalized saddle foliation $\mathcal F$ with a corner Dulac map $D$ and a regular map $R\in \mathrm{Diff}(\mathbb C,0)$. Up to an appropriate equivalence relation that corresponds to different determinations of complex Dulac and to transversal changes, the first return map is given by $F=RD$ on the universal cover of the standard quadratic domain. We show that such Poincar\` e maps are \emph{rigid}, in the sense that their non-ramified formal conjugacy implies the analytic conjugacy (in $\mathrm{Diff}(\mathbb C,0)$, lifted to the universal cover).

Research on Center-Related Problems of Generalized Kukles Systems

Yilei Tang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):

Abstract: In this talk we investigate center-related problems for generalized Kukles systems. We derive sufficient and necessary conditions for the origin of such systems with $\mathbb{Z}_2$-symmetry or weak $\mathbb{Z}_2$-symmetry to be a center. Moreover, we provide examples to illustrate the center conditions using our theoretical results and give a negative answer to a conjecture proposed in the literature. Moreover, we investigate the problem of integrability for general generalized Kukles systems when they admit a center.

Long-time behavior for some reaction-diffusion systems

Shi-Liang Wu xidian university Peoples Rep of China Co-Author(s):

Abstract: In this talk, we first review some existing results on the long-time behavior of bounded solutions to reaction-diffusion equations. We then present our recent works on the long-time dynamics of solutions to several classes of biological reaction-diffusion systems.

The stability and bifurcation of ecosystems within a defined energy landscape

Dongmei Xiao Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Tianwei Gan, Dongmei Xiao and Chenwan Zhou

Abstract: In this talk, we first assume that a defined energy landscape becomes the resource constraint in ecological community, then give three-dimensional Lotka-Volttera models within a defined energy landscape, and study bifurcations for a class of the three-dimensional Lotka-Volttera models as an example, discover the existence of strange bifurcations by using a conserved quantity as a bifurcation parameter, which reveals how resource constraints influence the coexistence of the species and their long-term dynamics.

Parametric normal forms and further simplification

Weinian Zhang Sichuan University Peoples Rep of China Co-Author(s):    Weinian Zhang

Abstract: Given a family of vector fields parametrized by $\xi$ in its linear part, one usually obtains its versal unfolding by the so-called two-step approach. It is also a common practice to suspend $\xi$ and calculate normal forms of the extended system. In this work we reformulate normal forms on modules of homogeneous polynomials over the ring of all continuous functions of $\xi$ and give a direct computation of versal unfolding. Our procedure enables us to determine coefficients of all terms of a certain degree in the normal form before we give a near-identity transformation of this degree. We can give all available near-identity transformations and choose an appropriate one to eliminate more terms of higher degree for a simpler normal form. We prove that the normal form reduced in our procedure is the simplest and unique. We illustrate our method with systems of linear centre and nilpotent linear parts separately.

Traveling wave trains and fronts in a reaction--diffusion model of seagrass meadows

Xiang Zhang Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Qi Qiao and Baodong Zhang

Abstract: n this talk we report our results on a model, posed by Moreno-Spiegelberg et al. [Proc. Natl. Acad. Sci. 2025], that incorporates positive feedback together with negative feedback mediated by an inhibitor, and successfully applied it to Posidonia oceanica meadows to explain the observed spatiotemporal phenomena. The original authors worked via numerical simulations and theoretical analysis and produced various spatiotemporal patterns, such as traveling pulses, wave trains, expanding rings, and spiral waves. Here our work rigorously establishes the existence of traveling pulses, traveling fronts, wave trains, as well as pulled and pushed fronts for this model. Our approach, based on geometric singular perturbation theory, allows us to construct traveling waves far from equilibrium, and can reveal more organized underlying structures. This provides a rigorous mathematical foundation for the wave phenomena, and contribute to a deeper understanding of the mechanisms generating complex spatiotemporal patterns in spatially extended ecological systems. This is a joint work with Qi Qiao and Baodong Zhang, which is based on our paper in J. London Math. Soc. (2) 2026; 113: e70428.