This MATH-Amsud research project, Stability in Dynamical Systems: Invariant Structures, Hyperbolicity, and Singular Perturbations, explores key topics in the qualitative theory of dynamical systems. Aiming the understanding of global organization of phase spaces and the robustness of dynamical behaviors, we address local and global stability problems, the existence and bifurcation of invariant structures in both regular and singularly perturbed differential equations, and the properties of hyperbolicity and normal hyperbolicity of these structures.

The project seeks to strengthen scientific collaboration among research groups in Brazil, France, Chile, and Colombia by promoting researcher mobility, supporting joint investigations, and fostering transnational networks in Dynamical Systems. Expected outcomes include publishing cutting-edge research in high-impact journals, training graduate students, organizing scientific events, and expanding international collaborations.

The project is structured around five interconnected research axes. The synergy among them stems from their complementarity, with each axis addressing a distinct aspect of the broader theme:

Axis 1 – Homotopical Methods in Dynamical Systems: The dynamical behavior of continuous flows on their chain recurrent set R can be highly intricate and depends on the class of dynamical systems considered. In the differentiable setting, we focus on Smale flows; in the non-smooth context, on Gutierrez–Sotomayor flows. The invariant structure of R in each case will be analyzed using homotopical methods, many derived from Conley index theory and singularity theory. The continuation property of Conley index methods will be explored via a Morse-Smale type chain complex and its associated spectral sequence. This approach yields cancellation and bifurcation results and opens new directions for studying continuation behavior in one-parameter families of flows.

Axis 2 – Limit Tori in Higher Dimensional Differential Equations: The existence of compact invariant manifolds is a central problem in the qualitative theory of differential equations, as it reveals key aspects of the system’s dynamics. Understanding improves by analyzing the asymptotic behavior of trajectories near such manifolds. This axis focuses on invariant tori, which extend the role of limit cycles from planar systems to higher dimensions. The goal is to develop tools for identifying normally hyperbolic invariant tori in higher-dimensional systems and studying their dynamics. Key aims include using averaging theory to establish their existence from lower-dimensional tori in the averaged system, and investigating their regularity, convergence, and the behavior of nearby trajectories.

Axis 3 – Piecewise Smooth Dynamical Systems and Singular Perturbations: This axis is dedicated to the study of non-smooth dynamical systems modeled by piecewise-smooth vector fields, which arise in numerous applications. It employs the geometric theory of Singular Perturbations and singularity resolution as central tools to analyze the dynamics near discontinuities and describe Invariant Structures such as limit cycles and sliding regions.

Axis 4 – Markus-Yamabe Conjecture and Globally Asymptotically Stable Vector Fields: This axis aims to investigate the connection between Globally Asymptotically Stable (GAS) vector fields and the Markus-Yamabe Conjecture (MYC). Consider a C1 vector field F in Rn with the origin as its unique equilibrium. If the origin is locally asymptotically stable, there exists a neighborhood U where all trajectories converge to the origin. The basin of attraction is the largest open set with this property. F is GAS if the basin coincides with all of Rn. Determining this basin is crucial in the stability theory of ordinary differential equations. The GAS property is closely related to the Markus-Yamabe Conjecture, which asserts that certain conditions on the Jacobian matrix imply GAS. This research axis aims to explore this connection, combining analysis, geometry, and dynamical systems theory.

Axis 5 – Hyperbolicity and Control Systems on Lie Groups and Homogeneous Spaces: Uniform hyperbolicity, introduced by Smale, is a foundational concept in dynamical systems theory and has been extended to analyze various contexts, including control systems. This axis applies hyperbolic theory to study control and chain control sets of affine and linear control systems on Lie groups. Their relevance is underscored by the Jouan Equivalence Theorem, which states that certain control-affine systems on manifolds are diffeomorphic to affine systems on Lie groups or homogeneous spaces. Recent work has explored controllability on Lie groups. For linear systems, strong local controllability near the identity ensures a unique control set with nonempty interior, although such sets may also exist without this assumption. This axis seeks weaker conditions for the existence of control sets on Lie groups using hyperbolic techniques from dynamical systems theory.