Special Session 16: The recent progress on qualitative theory of dynamical systems

Smooth conjugacy of center manifold dynamics

Shuang Chen Central China Normal University Peoples Rep of China Co-Author(s):

Abstract: Center manifolds are important invariant sets for dynamical systems which reduces high-dimensional systems to simplers ones on the center manifolds. This makes it possible to analyze bifurcations and stability that would otherwise be intractable. But their non-uniqueness poses a critical obstacle. In this talk, I will present our recent result on smooth conjugacy of center manifold dynamics, i.e., proving that the reduced dynamics are uniquely determined up to smooth equivalence, and then apply the developed theory to resolve some problems in the literature.

Global Dynamics of the Rosenzweig-MacArthur System with Allee Effect

Zhaosheng Feng University of Texas Rio Grande Valley USA Co-Author(s):

Abstract: In this talk, we investigate the global dynamics of a Rosenzweig-MacArthur system incorporating an Allee effect in the prey population. Our analysis is carried out within the first quadrant of the Poincare disc, which provides a compactification of the phase space and enables a complete characterization of the system behavior at infinity. The Allee effect introduces a critical threshold below which the prey population cannot persist, thereby fundamentally altering the system bifurcation structure and long-term dynamics. We rigorously characterize equilibria, limit cycles, and their stability, and complement the theoretical results with numerical simulations that illustrate biologically relevant scenarios. Implications for ecological resilience and species extinction thresholds are also discussed.

Stability of some monodromic singularities with two edges in the Newton diagram

Jaume Gine Universitat de Lleida Spain Co-Author(s):    Isaac A. Garcia, Victor Manosa

Abstract: This work focuses on the study of monodromic singularities in planar analytic families of vector fields whose Newton diagram consists of exactly two edges. We begin by analyzing the desingularization scheme of a \emph{minimal model} of polynomial vector fields, denoted by $\mathcal{X}$, which includes only the monomials corresponding to the vertices of the Newton diagram. We then extend this minimal model to the so-called \emph{Brunella-Miari vector fields} $\mathcal{X} \subset \mathcal{X}^{[1]}$, incorporating all monomials associated with points lying on the edges of the Newton diagram. As a second extension, we consider vector fields $\mathcal{X}^{[1]} \subset \mathcal{X}^{[2]}$ that include higher-order terms corresponding to points located above the polygonal line in the Newton diagram. The key point of our approach is to preserve the desingularization geometry at each extension step. We provide explicit desingularization procedures, which enables the computation of the linear part of the return map $\Pi$ in cases where the desingularized singularity is associated with a hyperbolic polycycle.

Bifurcation and Chaotic Dynamics in Predator-Prey Ecological Models

Kunlun Huang Beihang University Peoples Rep of China Co-Author(s):    Kunlun Huang, Xintian Jia, Yanqi Zhang, Ya li, Cuiping Li

Abstract: We study a discrete-time predator-prey system with strong Allee effect derived by the Euler discretization. Stability analysis shows that exceeding a critical predator mortality rate generates two interior equilibria and enables species coexistence. The system undergoes fold, flip, and Neimark-Sacker bifurcations with parameter variations. The strong Allee effect stabilizes the system, while weak Allee effect, high predator growth rate, and intensified environmental protection tend to induce periodic orbits and chaotic attractors, verified by Lyapunov exponents and numerical simulations.

Positive Solutions for a Class of Integral Boundary Value Problem of Fractional q-Difference Equations

Shugui Kang Shanxi Datong University Peoples Rep of China Co-Author(s):    Yunfang Zhang, Huiqin Chen and Wenying Feng

Abstract: In this talk, we studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green's function and obtain an important property of the function. The new property allows us to prove sufficient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a fixed point theorem on order intervals.

The number of zeros of Abelian integrals from Nilpotent Codimension 3 birfucation (Saddle and Elliptic cases)

Changjian Liu Sun Yat-sen university Peoples Rep of China Co-Author(s):

Abstract: Dumortier, Roussarie and Sotomayor gave a complete investigation of the unfolding of Bogdanov-Takens bifurcations of codimension $3$ based on the hypothesis that there are at most two limit cycles. From then on, this hypothesis becomes a longstanding open problem. According to Dumortier, there are three unsolved classes: elliptic, saddle, focus. For all these three cases, the problem are changed to estimate the number of zeros of some Abelian integrals. By giving new criteria, we prove that the associated Abelian integrals have at most two zeros for elliptic case and saddle case, that is, we give an affirmative answer of this hypothesis for these two cases.

Derivatives of the separation function of generalized saddle connections

Jordi Villadelprat Universitat Autonoma de Barcelona Spain Co-Author(s):    David Marin

Abstract: A classical formula shows that the breaking of a connection between two hyperbolic saddles $s_0^+$ and $s_0^-$ can be studied by means of a convergent improper integral that is often called the Melnikov integral. In this talk we will show that this formula extends to more general situations, for instance, when the singularities $s_0^\pm$ are semi-hyperbolic or even nilpotent. In some of these cases the improper integral is no longer convergent but nevertheless, under convenient hypothesis, there is a kind of residue that provides the desired information. Our main result expands the scope of situations in which we can study the breaking of homoclinic or heteroclinic connections. We show that this is indeed the case by analysing three different examples: a heteroclinic connection between nodes, a heteroclinic connection between semi-hyperbolic saddles at infinity and a homoclinic connection in a singularity at infinity. As an application we obtain a general result aimed at studying the breaking of hemicycles and we present several results to analyse the perturbation of unbounded polycycles within a quadratic unfolding that is versal.

Progress in Differentiable Linearization of Dynamical Systems

Wenmeng Zhang Chongqing Normal University Peoples Rep of China Co-Author(s):    Weinian Zhang, Kening Lu, Davor Dragicevic, Yonghui Xia, Weijie Lu

Abstract: This talk systematically reviews recent advances in differentiable linearization of dynamical systems, a refinement of the classical Hartman-Grobman Theorem. While the theorem guarantees topological conjugacy between a smooth diffeomorphism and its linear part at a hyperbolic fixed point, this conjugacy is generically non-smooth, motivating the core question of its differentiability at the fixed point. We are also concerned with the differentiable normal linearization for partially hyperbolic systems, achieving C^0 conjugacy that is C^1 on the center manifold.

Hyperbolicity of Algebraic Limit Cycles via Abelian Integrals

Yulin Zhao Sun Yat-sen University Peoples Rep of China Co-Author(s):

Abstract: This talk establishes a rigorous connection between the hyperbolicity of algebraic limit cycles and Abelian integrals for planar polynomial differential systems admitting invariant algebraic curves. Our results are non-perturbative, i.e., independent of any small parameter, and thus extend the classical Abelian-integral criterion beyond the perturbative framework. We prove that the vanishing of the associated Abelian integral $I(h)$ is a necessary condition for a periodic orbit to lie on an irreducible invariant algebraic curve. Moreover, for a class of systems under suitable constraints, we show that an algebraic limit cycle is hyperbolic if and only if $I`(h)\not=0$. As an application, we study Kukles systems of arbitrary degree and obtain existence results for algebraic (reversible) limit cycles. By analyzing the zeros of the relevant Abelian integrals, we further demonstrate the coexistence of the algebraic cycle with additional limit cycles, revealing rich global dynamics for this family.