2025/05/17 by Jaume Llibre, Llibre, Jaume, Gabriel Rondón +1
Mathematics · Physics and Astronomy · #34C05 #Advanced Differential Equations and Dynamical Systems #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2505.12047
openalex publication_date 2025/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the global dynamics of the Ehrhard-Müller differential system x = s(y - x), y = rx - xz - y + c, z = xy - z, where s, r and c are real parameters, and x, y, and z are real variables. We classify the invariant algebraic surfaces of degree 2 of this differential system. After we describe the phase portraits in the Poincaré ball of this differential system having one of this invariant algebraic surfaces. The Poincaré ball is the closed unit ball in ℝ3 whose interior has been identified with ℝ3, and his boundary, the 2-dimensional sphere \mathbbS2, has been identified with the infinity of ℝ3. Note that in the space ℝ3 we can go to infinity in as many as directions as points has the sphere \mathbbS2. A polynomial differential system as the Ehrhard-Müller system can be extended analytically to the Poincaré ball, in this way we can study its dynamics in a neigborhood of infinity. Providing these phase portraits in the Poincaré ball we are describing the dynamics of all orbits of the Ehrhard-Müller system having an invariant algebraic surface of degree 2.