The dynamics of the Ehrhard-Müller system with invariant algebraic surfaces

TL;DR

This work analyzes the global dynamics of the polynomial Ehrhard--Müller system in $\mathbb{R}^3$ with parameters $(s,r,c)$ by classifying invariant algebraic surfaces of degree $2$ and studying the induced phase portraits on the Poincaré ball. It identifies four invariant degree-$2$ surfaces, $f_1=x^2-z$, $f_2=y^2+z^2-cx$, $f_3=y^2+z^2$, and $f_4=y^2+z^2-rx^2$, and establishes corresponding Darboux invariants that constrain trajectory behavior. The paper then analyzes the dynamics on two key surfaces, $x^2-z=0$ and $y^2+z^2-cx=0$, using Poincaré compactification, vertical blow-ups, and discriminant analysis in the $(c,r)$-plane to describe $\,\alpha$- and $\omega$-limits and to show absence of limit cycles in these invariant sets. Together, these results extend Lorenz-type insights by providing a global, surface-restricted view of the system’s behavior, including infinity, and clarify how invariant algebraic structures govern the asymptotic dynamics of polynomial vector fields in three dimensions.

Abstract

In this paper we study the global dynamics of the Ehrhard-Müller differential system \[ \dot{x} = s(y - x), \quad \dot{y} = rx - xz - y + c, \quad \dot{z} = xy - z, \] 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 $\mathbb{R}^3$ whose interior has been identified with $\mathbb{R}^3$, and his boundary, the $2$-dimensional sphere $\mathbb{S}^2$, has been identified with the infinity of $\mathbb{R}^3$. Note that in the space $\mathbb{R}^3$ we can go to infinity in as many as directions as points has the sphere $\mathbb{S}^2$. 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 $2$.

Figures (7)

• Figure 1: Phase portrait of the Ehrhard-Müller system \ref{['main_eq_1']} at the infinity sphere.
• Figure 2: The topological phase portraits on the invariant algebraic surface $x^2-z=0$.
• Figure 3: The topological phase portraits on the invariant algebraic surface $y^2+z^2-cx=0$. The $Q$ point is the stable focus of system \ref{['main_eq_1']} on the algebraic surface $y^2+z^2-cx=0.$
• Figure 4: (a) The invariant algebraic surface $x^2-z=0$. (b) The interception of the invariant algebraic surface $x^2-z=0$ with the sphere $\mathbb{S}^2$ of the infinity.
• Figure 5: The blow up of the origin of the local chart $U_2$.

Theorems & Definitions (26)

• Theorem 1
• Proposition 2
• Proposition 3
• Theorem 4
• Theorem 5
• Lemma 6
• Proposition 7
• proof : Proof of Theorem $\ref{['prop_alg']}$
• proof : Proof of Proposition $\ref{['p1']}$
• proof : Proof of Proposition $\ref{['p2']}$