# On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals

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## Abstract

**:**

## 1. Introduction

## 2. Definitions and First Properties

- $\lambda \in \Lambda $, where $\Lambda $ is a connected open set of ${\mathbb{R}}^{m}$ ($m>0$),
- $x\in V$, where $V\subset {\mathbb{R}}^{n}$, is a compact submanifold of ${\mathbb{R}}^{n}$ of codimension zero, whose boundary $\partial V$ is a stratified manifold. The system is assumed to stay inside V, that is on $\partial V$, the vector field either points inward or is tangent to $\partial V$.
- There exist k smooth independent first integrals ${h}_{1},\cdots ,{h}_{k}$ $(k>0)$, independent of the parameter(s) $\lambda $, so that for every i, ${h}_{i}:{\mathbb{R}}^{n}\to \mathbb{R}$ is a smooth function. Here the independence of the functions ${h}_{1},\cdots ,{h}_{k}$ means that their gradients are linearly independent at every points.

**Proposition**

**1.**

**Proof.**

**Non-degeneracy**

**Conditions:**

**Remark**

**1.**

**Example**

**1.**

- 1.
- Consider the following vector field in ${\mathbb{R}}^{2}$: $f=\left(v\right(\lambda ,x,y),0)$, such that for all $\lambda \in \Lambda =\mathbb{R}$, $v(\lambda ,x,y)x\le 0$ when ${x}^{2}+{y}^{2}=1$, so that V can be taken to be the Euclidean unit ball of ${\mathbb{R}}^{2}$. Obviously, ${h}_{1}=y$ is a first integral and$$\frac{\partial f}{\partial x}=\left(\begin{array}{cc}\frac{\partial v}{\partial x}& \frac{\partial v}{\partial y}\\ 0& 0\end{array}\right).$$Of course, ${\left(\frac{\partial f}{\partial x}\right)}^{T}\nabla {h}_{1}=0$. A similar result is obtained if one consider $\frac{1}{2}{y}^{2}$ instead of y. The set of equilibrium points $E=\left\{\right(\lambda ,x,y\left)\right|v(\lambda ,x,y)=0\}$ has dimension 2.
- 2.
- Consider now a vector field f is ${\mathbb{R}}^{3}$, such that ${h}_{1}={x}^{2}+{y}^{2}+{z}^{2}$ and ${h}_{2}=4{x}^{2}+4{y}^{2}+{z}^{2}/4$ are first integrals. These functions are obviously differentially independent, outside the union of the plane $z=0$ and the line $x=y=0$. Take V to be the union of the points $\left\{\right(x,y,z\left)\right\}$ such that ${h}_{1}={R}_{1}$ and ${h}_{2}={R}_{2}$, with ${R}_{1},{R}_{2}\in [1,3]\times [5,15]$ and ${R}_{2}\ge 5{R}_{1}$.Then let us consider $f=(-\lambda y(z-x),\lambda x(z-x),0))$ with $\Lambda ={\mathbb{R}}_{+}^{*}$; it is straightforward to compute$$\frac{\partial f}{\partial x}=\left(\begin{array}{ccc}\lambda y& -\lambda (z-x)& -\lambda y\\ \lambda z-2\lambda x& 0& \lambda x\\ 0& 0& 0\end{array}\right)$$

**Proposition**

**2.**

**Proof.**

**We assume from now on that f is analytic and write $\mathit{\pi}:\mathit{E}\subset \mathbf{\Lambda}\times \mathit{V}\to \mathbf{\Lambda}$**. This is an analytic and proper map whose image we denote by ${\Lambda}_{0}\subset \Lambda $.

## 3. On the Nature of ${\Lambda}_{\mathbf{0}}$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 4. Fibers over the Parameter Space

**Proposition**

**4.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Proof.**

**Corollary**

**2.**

## 5. Parallel Transport of the Fibers

#### 5.1. Connection

#### 5.2. Transversality between Fibers and Integral Manifolds

**Proposition**

**7.**

#### 5.3. Lifting Curves

**Proposition**

**8.**

**Proof.**

**Theorem**

**1.**

**Proof.**

#### 5.4. The Natural Connection

**Lemma**

**3.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 6. The Natural Connection via a Riemannian Submersion

**Corollary**

**3.**

**Proof.**

## 7. The Eigenvalues along the Fibers

## 8. Examples and Applications

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Kaminski, Y.; Lochak, P.
On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals. *Mathematics* **2024**, *12*, 457.
https://doi.org/10.3390/math12030457

**AMA Style**

Kaminski Y, Lochak P.
On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals. *Mathematics*. 2024; 12(3):457.
https://doi.org/10.3390/math12030457

**Chicago/Turabian Style**

Kaminski, Yirmeyahu, and Pierre Lochak.
2024. "On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals" *Mathematics* 12, no. 3: 457.
https://doi.org/10.3390/math12030457