# Dynamics of $\theta$-invariant systems and normed actions

Danilo R Lima, Rosane Ushirobira, Denis Efimov

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Danilo R Lima, Rosane Ushirobira, Denis Efimov. Dynamics of $\theta$-invariant systems and normed actions. CDC 2025 - 64th IEEE Conference on Decision and Control, Dec 2025, Rio De Janeiro, Brazil. hal-05435682

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# Dynamics of $\theta$-invariant systems and normed actions

Danilo R. Lima<fnref n="1" />, Rosane Ushirobira<fnref n="1" />, Denis Efimov<fnref n="1" />

*Abstract*— This paper expands the dynamical analysis of $\theta$-invariant systems, with focus on autonomous systems containing equilibrium points. In particular, we present dynamical behavior that can occur only with $\theta$-invariant systems but not $G$-invariant systems. Those include finite-time stability and the existence of isolated periodic orbits. Furthermore, we show that many of the good dynamical properties present in homogeneous systems can be obtained in the more general context of invariance. This is done through the introduction of the concept of normed actions.

## I. INTRODUCTION

The understanding of the intrinsic symmetries of a dynamical system has been shown to play an important part in its dynamical analysis [7]. Various models are defined in terms of their particular symmetric properties, such as linear systems and homogeneous systems. Others are already extensively studied, and are shown to benefit from the symmetry approach, namely robotic models, population dynamics, and epidemiology models [3], [5]. This approach made it easier to infer properties about stability, observability, controllability, robustness and others, following the need imposed by their applications [8], [4], [2].

Some attention has also been given to the general theory of symmetries of differential equations. In particular, questions were raised concerning the search for all possible symmetries of a given system [7]. This theory has more recently had applications in observer design [4], [6].

A continuous-time system with input is represented by a differential equation of the form $\dot{x} = f(x, u)$. For such a system, an effective and simple way to represent symmetries of a system is through the action of Lie groups over the state and input space [4]. Systems equipped with such an action are called $G$-invariant systems. This setting can be limiting as it fails to include important examples such as weighted homogeneous systems. Despite this limitation, we have shown that a minor modification can be made to allow time-symmetry and similar methods can be applied in the context of observer design [6]. Systems possessing this extended symmetry are called $\theta$-invariant systems.

In this paper we turn our attention to autonomous systems, represented by a differential equation of the form $\dot{x} = f(x)$. We continue the analysis of $G$-invariant and $\theta$-invariant systems, presenting dynamical features that are exclusive to

one or the other type, together with features that are common to both. Those include symmetry for solutions, relations of local and global behaviors and implication of stability by attractivity. We also introduce the concept of invariant norm of a group action, inspired by the well-known homogeneous norm [9]. We then show that most of the hypotheses used in proving those dynamic features reduce to the existence of such a norm. To conclude we pose questions concerning conditions for the existence of invariant norms. Particular attention is paid to finite-time convergence in the systems with symmetries and the conditions of its existence.

This paper is organised as follows: in Section II, we will recall the concepts of $G$-invariant and $\theta$-invariant systems, together with some of their basic properties. In Section III, we discuss conditions for $\theta$-invariant systems to be finite-time stable. In Section IV, we introduce the concept of invariant norm and reproduce the results obtained previously under this unified hypothesis.

The notation used will be as follows:

* We denote $B_\delta(x_0) = \{x \in \mathbb{R}^n : \|x - x_0\| < \delta\}$, $B_\delta[x_0] = \{x \in \mathbb{R}^n : \|x - x_0\| \le \delta\}$ the open and closed balls of radius $\delta$ centered at $x_0$, respectively. The sphere of radius $\delta$ centered at $x_0$ is denoted $S_\delta(x_0) = \{x \in \mathbb{R}^n : \|x_0\| = \delta\}$.

* We denote by $\mathbb{R}_{\ge 0} = \{x \in \mathbb{R} : x \ge 0\}$ and $\mathbb{R}_{> 0} = \{x \in \mathbb{R} : x > 0\}$ the set of positive and strictly positive real numbers, respectively.

* We denote by $C^p(\mathbb{R}^n, \mathbb{R}^m)$ the set of functions $f: \mathbb{R}^n \to \mathbb{R}^m$ possessing continuous derivatives up to order $p$ and by $C^0(\mathbb{R}^n, \mathbb{R}^m)$ the set of continuous functions. For brevity, we will denote $C^p(\mathbb{R}^n, \mathbb{R})$ simply by $C^p(\mathbb{R}^n)$.

* We say that a function $\alpha: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is of class $\mathcal{K}$ if it is continuous, strictly increasing, and $\alpha(0) = 0$. If it additionally satisfies $\lim_{x \to \infty} \alpha(x) = +\infty$, we say that $\alpha$ is of class $\mathcal{K}_\infty$.

## II. INVARIANT SYSTEMS

The results in this paper apply to the system

$$ \dot{x}(t) = f(x(t)), \quad t \ge 0, \quad (1) $$

with $x(t) \in \mathbb{R}^n$, and $f: \mathbb{R}^n \to \mathbb{R}^n$ being a continuous vector field. We will denote by $x(t, x_0)$ the solution curve $x(t)$ for system (1) at time $t$ having initial condition $x(0) = x_0$. Further, for brevity we assume the the solutions of the system are well-defined for all $t \in \mathbb{R}_{\ge 0}$ for all $x_0 \in \mathbb{R}^n$ (i.e., the system (1) is forward complete).

We are mostly interested in equilibrium points of the system (1) and their stability, as defined below.

<footnote>*This work was partially supported by the Hauts-de-France Region, France and by ECOS NORD Project M20M04.</footnote>

<footnote n="1">¹Danilo R. Lima, Rosane Ushirobira and Denis Efimov are with Inria, Univ. Lille, CNRS, UMR 9189 - CRISTAL, F-59000 Lille, France {Danilo.Rodrigues-de-Lima,Rosane.Ushirobira, Denis.Efimov}@inria.fr</footnote>
**Definition II.1.** We say that a point $x_0 \in \mathbb{R}^n$ is an equilibrium of system (1) if $f(0) = 0$. An equilibrium is said to be

* Stable, if for every $\varepsilon > 0$, there exists $\delta > 0$ such that $\|x(t, x_0)\| < \varepsilon$ for every $x_0 \in B_\delta(0)$;

* Attractive, if there exists $\delta > 0$ such that $\lim_{t \to \infty} x(t, x_0) = 0$, for every $x_0 \in B_\delta(0)$;

* Asymptotically stable if it is attractive and stable;

* Finite-time stable (FTS) if there exists a so called time-settling function $T: \mathbb{R}^n \to \mathbb{R}_{\ge 0}$ such that $x(t, x_0) = 0$, for every $x_0 \in \mathbb{R}^n$ and every $t \ge T(x_0)$.

We will often assume that the equilibrium point is $x_0 = 0$. This is natural, since any other case can be obtained through translation of the variable, without significant change to the dynamics or the symmetries of the system.

Next we define important concepts of stability associated with subsets of $\mathbb{R}^n$ that will be useful to infer stability of equilibria.

**Definition II.2.** Let $\mathcal{H}_t: \mathbb{R}^n \to \mathbb{R}^n$ denote the time-t flow of the system (1) and let $K \subset \mathbb{R}^n$. We say that $K$ is

* positively invariant under $f$, if $\mathcal{H}_t(K) \subseteq K$ for every $t \ge 0$;

* strictly positively invariant under $f$, if $\mathcal{H}_t(K) \not\subseteq K$ for every $t \ge 0$;

* invariant under $f$, if $\mathcal{H}_t(K) = K$ for every $t \ge 0$;

We say that (1) is $\theta$-invariant [6] if there is a connected non-trivial Lie group $G$, an action $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ (also denoted $\phi_g: \mathbb{R}^n \to \mathbb{R}^n$, for a given $g \in G$) and a homomorphism $\theta: G \to \mathbb{R}_{>0}$, such that

$$f(\phi_g(x)) = \theta(g)D\phi_g(x)[f(x)], \quad (2)$$

for every $x \in \mathbb{R}^n$ and $g \in G$, where $D\phi_g(x)$ is the Jacobian of $\phi_g$ at the point $x \in \mathbb{R}^n$. We say that (1) is $G$-invariant if the property (2) is satisfied for $\theta \equiv 1$.

This class of systems demonstrates a very important phenomenon of scaling of trajectories.

**Proposition II.3.** [6] The $\theta$-invariance of (1) is equivalent to

$$x(t, \phi_g(x_0)) = \phi_g(x(\theta(g)t, x_0)), \quad (3)$$

for every $g \in G$ and every solution curve $x(t, x_0) \in \mathbb{R}^n$ with initial condition $x_0 \in \mathbb{R}^n$ and $t \ge 0$.

*Proof.* Suppose that property (2) holds and $x(t)$ is a trajectory of the system. We need to show that

$$\frac{d}{dt}\phi_g(x(\theta(g)t)) = f(\phi_g(x(\theta(g)t))) \quad (4)$$

also holds for every $t \in \mathbb{R}_{\ge 0}$. We compute the left-hand side of (4), obtaining:

$$\begin{aligned} \frac{d}{dt}\phi_g(x(\theta(g)t)) &= D\phi_g(x(\theta(g)t)) [\theta(g)\dot{x}(\theta(g)t)] \\ &= \theta(g)D\phi_g(x(\theta(g)t)) [f(x(\theta(g)t))], \end{aligned}$$

for every $t \in \mathbb{R}_{\ge 0}$. By invariance, the last term equals $f(\phi_g(x(\theta(g)t)))$, as desired.

Conversely, suppose that equation (4) holds. By the above calculation, we have

$$\begin{aligned} \theta(g)D\phi_g(x(\theta(g)t))[f(x(\theta(g)t))] \\ &= f(\phi_g(x(\theta(g)t))), \end{aligned}$$

for every curve $x(t)$ that is a trajectory of the system. Making $t=0$, we have that $x_0 = x(\theta(g)t, x_0)$, for any $g \in G$. Therefore, we can omit the argument $\theta(g)t$, and the above equation is equivalent to the time-independent equation (2). That proves the other side of the equivalence. $\square$

A natural question is whether $G$-invariant systems can be made $\theta$-invariant without changing its trajectories. To answer that question, we make a brief digression into the concept of full-rank condition and the moving frame [7] (see also [4]).

**Remark:** Since $G$ is a smooth manifold of dimension $r \ge 1$, up to composition with a chart, we can regard any function $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ as a smooth function between open sets of $\mathbb{R}^{r+n}$ and $\mathbb{R}^n$, written $\phi(g, x) = \phi(g_1, ..., g_r, x_1, ..., x_n)$. We can then consider $D_g\phi(g, x)$ as the transformation associated to the $n \times r$-matrix

$$\left(\frac{\partial \phi_i}{\partial g_j}\right)_{1 \le i \le n, 1 \le j \le r} \in \mathbb{R}^{n \times r} \quad (5)$$

which is a submatrix of $D\phi(g, x) \in \mathbb{R}^{n \times (r+n)}$. We say that the application $g \mapsto \phi_g(x)$ is full-rank in $x_0$ if the matrix $D_g\phi(e, x_0)$ has full rank, where $e \in G$ is the identity. In this case we say that $\phi$ satisfies the full-rank condition on $x_0$.

**Lemma II.4.** [4] Let $\phi: G \times \mathcal{X} \to \mathcal{X}$ be a group action and suppose $r \le n$. Let $x_0 \in \mathcal{X}$ be such that $g \mapsto \phi_g(x)$ has full-rank in $x_0$. Then there exists a neighborhood $\mathcal{O}$ of $x_0$ and a function $\gamma: \mathcal{O} \to G$ (called moving frame) such that

$$\gamma(\phi_g(x))g = \gamma(x) \quad (6)$$

for every $g \in G$ and $x \in \mathcal{O}$.

The moving frame allows us to write elements of $\mathbb{R}^n$ locally using group coordinates, and among its many applications is the following result:

**Proposition II.5.** Let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a such that $\dot{x} = f(x)$ is a $G$-invariant system for an action $\phi: G \to \mathbb{R}^n \to \mathbb{R}^n$ satisfying the full rank property in every $x \in \mathbb{R}^n$. Then the function $\gamma: \mathbb{R}^n \to G$ is defined everywhere and the system $\dot{x} = \theta(\gamma(x))^{-1}f(x)$ is $G$-invariant and has the same trajectories as $\dot{x} = f(x)$, up to time-scaling.

*Proof.* It is clear that if $t \mapsto x(t)$ is a solution of $\dot{x} = f(x)$, then $t \mapsto x(\theta(\gamma(x))^{-1})$ is a solution of $\dot{x} = \theta(\gamma(x))^{-1}f(x)$, and both curves have the same image.

Let us prove $\theta$-invariance. Define $\tilde{f}(x) = \theta(\gamma(x)f(x))$. Then

$$\tilde{f}(\phi_g(x)) = \theta(\gamma(\phi_g(x)))^{-1}f(\phi_g(x))$$

Using $G$-invariance of $f$, the property of the moving frame and the fact that $\theta$ is a homomorphism, we have

$$\theta(\gamma(\phi_g(x)))f(\phi_g(x)) = \theta(g^{-1})^{-1}\theta(\gamma(x))^{-1}D\phi_g(x)[f(x)]$$
$$= \theta(g)D\phi_g(x)[\theta(\gamma(x))^{-1}f(x)] = \theta(g)D\phi_g(x)[\tilde{f}(x)],$$

so property 2 holds for $\tilde{f}$. □

There are nonetheless challenges and limitations in the construction of an invariant frame. One of them is posed by the existence of equilibria.

**Proposition II.6.** If $x_0 \in \mathbb{R}^n$ is an isolated equilibrium point of a $\theta$-invariant system (1) with $\dim G \ge 1$, then the action $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ cannot satisfy the full-rank condition at $x_0$.

*Proof.* Let $x_0$ be an equilibrium point of (1). Consider the $G$-orbit of $x_0$, $\{\phi_g(x_0) : g \in G\}$. It is a connected set containing $x_0$ and made up entirely of equilibrium points (thanks to Lemma II.7). Then, since $x_0$ is isolated as an equilibrium point, such a $G$-orbit must be equal to the unitary set $\{x_0\}$. That means that the image of a neighborhood of the identity $e$ through the application $g \mapsto \phi_g(x_0)$ is a dimension zero set. Therefore, rank $(D_g\phi_g(x_0)) = 0 < \dim(G)$. □

Now we make some basic observations on existence of periodic solutions, as examples of features that cannot happen in $\theta$-invariant systems.

**Lemma II.7.** Suppose $x_0 \in \mathbb{R}^n$ generates a periodic solution for system (1) with period $\tau > 0$ (that is, $x(t, x_0) = x(t + \tau, x_0)$ for all $t \ge 0$) and let $g \in G$. Then the point $y_0 = \phi_g(x_0)$ generates a periodic solution with period $\theta(g)^{-1}\tau$.

*Proof.* A direct computation using the formula (3) for the scaling of trajectories gives us

$$\begin{aligned} x(\theta(g)^{-1}\tau, y_0) &= \phi_g(x(\theta(g)\theta(g)^{-1}\tau, x_0)) \\ &= \phi_g(x(\tau, x_0)) = \phi_g(x_0) = y_0. \end{aligned}$$

<figure><img src="image_2.png" /></figure>

**Proposition II.8.** System (1) cannot have a non-trivial isolated periodic orbit (that is, one that is different from an equilibrium point) if $\theta \ne 1$.

*Proof.* Suppose that there exists such orbit $\mathcal{O} \subset \mathbb{R}^n$, and let $x_0 \in \mathcal{O}$. Consider the $G$-orbit of $x_0$, that is $Gx_0 = \{\phi_g(x_0) : g \in G\}$. It is a connected set that contains $x_0$. Furthermore, by Lemma II.7, every point in it is periodic. Therefore, since $\mathcal{O}$ is isolated, we must have $Gx_0 \subseteq \mathcal{O}$. But that is a contradiction, since for any $g \in G$ such that $\theta(g) \ne 1$, the point $y_0 = \phi_g(x_0)$ has period $\theta(g)^{-1}\tau \ne \tau$. □

The propositions II.5 and II.8 show that some $G$-invariant systems can be easily turned into a $\theta$-invariant one possessing the same trajectories, while others cannot. It is still unclear what is the exact condition for a $G$-invariant system to be transformable into a $\theta$-invariant one, for non-trivial $\theta$. The latter type of invariance, as we will see in the next section, may be necessary to obtain accelerated rates of convergence. Consequently, pursuing that question might be a fruitful direction in the research of invariant systems.

### III. CONVERGENCE RATE OF INVARIANT SYSTEMS

In this section we will analyze the conditions permitting the invariant systems to demonstrate finite-time convergence rates. We will start our study by giving an easy-to-check algebraic condition that applies to the scalar case.

**Lemma III.1.** Consider the one-dimensional $G$-invariant system (1) with $f \in C^0(\mathbb{R}) \cap C^1(\mathbb{R} \setminus \{0\})$ and action $\phi \in C^2(\mathbb{R}_{>0} \times \mathbb{R})$. For each $x \in \mathbb{R} \setminus \{0\}$, define

$$\zeta_{\phi}(x) = x \left[ \frac{\partial}{\partial g \partial x} \phi(g, x) \cdot \frac{\partial}{\partial g} \phi(g, x)^{-1} \right]_{g=1} . \quad (7)$$

Let $L_{\phi}^{-} = \liminf_{x \to 0} |\zeta_{\phi}(x)|$ and $L_{\phi}^{+} = \limsup_{x \to 0} |\zeta_{\phi}(x)|$. If $L_{\phi}^{+} < 1$, then $f'(x)$ is unbounded around 0. If $L_{\phi}^{-} > 1$, then $f$ is differentiable at 0 and $f'(0) = 0$.

*Proof.* Since system (1) is invariant, we have

$$f(\phi_g(x)) = \phi'_g(x)[f(x)], \text{ for every } g \in \mathbb{R}_{>0} \text{ and } x \in \mathbb{R}.$$

Differentiation on the variable $g$ yields

$$f'(\phi_g(x)) \frac{\partial}{\partial g} \phi(g, x) = \frac{\partial}{\partial g \partial x} \phi(g, x) \cdot f(x),$$

for every $g \in \mathbb{R}_{>0}$ and $x \in \mathbb{R}$. Making $g = 1$ and rearranging the terms, we have

$$|f'(x)| = |\zeta_{\phi}(x)| \left| \frac{f(x)}{x} \right|, \text{ for every } x \in \mathbb{R}.$$

Let $x_0 > 0$. By the Mean Value Theorem, we can find $x_1 \in [0, x_0]$ such that

$$|f'(x_1)| = \frac{|f(x_0)|}{|x_0|} = \frac{1}{|\zeta_{\phi}(x_0)|} |f'(x_0)|.$$

Proceeding recursively, we can produce a decreasing sequence $x_n$ such that $\lim_{n \to \infty} x_n = 0$ (using transfinite induction if needed), such that

$$|f'(x_n)| = \prod_{k=0}^{n-1} \frac{1}{|\zeta_{\phi}(x_k)|} |f'(x_0)|.$$

If $L_{\phi}^{+} < 1$ and we choose $x_0$ small enough, then

$$|f'(x_n)| \ge \left( \frac{1}{L_{\phi}^{+}} \right)^n |f'(x_0)| \to \infty, \text{ as } n \to \infty,$$

proving that $f'$ is unbounded near 0.

Likewise, if $L_{\phi}^{-} > 1$ and we choose $x_0$ small enough, we have

$$|f'(x_n)| \le \left( \frac{1}{L_{\phi}^{-}} \right)^n |f'(x_0)| \to 0, \text{ as } n \to \infty.$$

Since $f \in C^1(\mathbb{R} \setminus \{0\}) \cap C^0(\mathbb{R})$, we have that $\lim_{x \to 0^+} f'(x) = 0$. The proof that $\lim_{x \to 0^-} f'(x) = 0$ is entirely analogous. □

It is well-known that finite-time stable smooth dynamical systems have right-hand sides with unbounded derivatives at the equilibrium. This result establishes when we can have
bounded derivatives purely from looking at the action $\phi$. This allows us to formulate the following result:

**Theorem III.2.** If a scalar system (1) is $G$-invariant for an action $\phi$ such that $L_{\phi}^{-} > 1$, then the origin cannot be FTS.

We hope to extend this result to systems in higher dimensions through differentiation in the direction of the group infinitesimal generators. Such an extension would provide easy-to-compute criteria to determine when a system can be FTS purely through its symmetries.

The next criteria also states that certain $G$-invariant systems cannot be FTS, under a topological hypothesis over $\phi$.

**Theorem III.3.** Suppose that 0 is FTS for the $\theta$-invariant system (1), and suppose that there exists a compact set $K \subset \mathbb{R}^n$ such that $0 \notin K$ and $G \cdot K = \mathbb{R}^n \setminus \{0\}$. Then $\theta$ cannot be the trivial homomorphism $\theta \equiv 1$.

Despite not depending on a direct computation, like in Lemma III.1, there is a considerably gain of generality in Theorem III.3. As we will see in next section, it still encompasses a wide range of symmetries. For further analysis we need the following result.

**Lemma III.4.** [[2], Lemma 2.2] Let system (1) be such that the origin is an attractive equilibrium point. Then for every compact set $K \subset \mathbb{R}^n$, there is $M_K > 0$ such that $\|x(t, x_0)\| \le M_K$, for every $t \in \mathbb{R}_{\ge 0}$ and every $x_0 \in K$.

Now we turn our attention to attractive systems. In particular, we are interested in whether stability is inherited by attractiveness. In order to formulate those stability conditions we need to introduce some extra topological hypotheses on $\phi$.

**Theorem III.5.** Let system (1) be such that the origin is an attractive equilibrium point. Suppose also that $f$ is $\theta$-invariant for an action $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ and a homomorphism $\theta: G \to \mathbb{R}_{>0}$ satisfying the following properties:

1) There exists $\delta_0, \delta_1 > 0$ such that for every $x \neq 0$ with $\|x\| < \delta_1$, there is $g \in G$ such that $\|\phi_g(x)\| \ge \delta_0$;

2) For every $M > 0$ and every $\delta > 0$, there exists a class $\mathcal{K}$ function $\alpha_{M,\delta}: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ such that, for every $x \in S_{\delta}$

$$ \phi_g(B_M[0]) \subset B_{\alpha_{M,\delta}(\|\phi_g(x)\|)}[0]. \quad (8) $$

Then, the origin is a stable equilibrium.

## IV. NORMED ACTIONS

The conditions imposed on the action $\phi$ in Theorems III.3 and III.5 are rather sophisticated, and they can be simplified by introducing an associated norm.

**Definition IV.1.** Let $\|\cdot\|: \mathbb{R}^n \to \mathbb{R}_{>0}$ denote the usual Euclidean norm. We say that a function $\|\cdot\|_{inv}: \mathbb{R}^n \to \mathbb{R}_{\ge 0}$ is an invariant norm for the action $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ if there exists a non-constant function $\xi: G \to \mathbb{R}_{>0}$ and $\alpha_1, \alpha_2 \in \mathcal{R}_{\infty}$ such that

$$ \bullet \quad \alpha_1(\|x\|) \le \|x\|_{inv} \le \alpha_2(\|x\|), \text{ for every } x \in \mathbb{R}^n; $$

$$ \bullet \quad \|\phi_g(x)\|_{inv} = \xi(g)\|x\|_{inv}, \text{ for every } g \in G \text{ and } x \in \mathbb{R}^n $$

An action possessing an invariant norm will be called a normed action.

As we will demonstrate below, introducing the hypothesis that the action $\phi$ is normed allows us to get all the properties used in the proof of Theorems III.3 and III.5. Moreover, existence of such a norm is standard for the homogeneous systems [9], which form an important class of examples for $\theta$-invariant dynamics.

The invariant norm may not always satisfy the condition $\|\lambda x\| = |\lambda|\|x\|$ for every $\lambda \in \mathbb{R}$ and $x \in \mathbb{R}^n$ associated to the usual definition of norm. However, we show in the next Lemma that the function $\xi$ from Definition IV.1 ranges over all $\mathbb{R}_{>0}$. That means that large initial conditions can always be obtained from small ones through the group symmetry. This is crucial to understand global behavior.

**Lemma IV.2.** Let $G$ be a connected Lie group, $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ a normed action and $\xi: G \to \mathbb{R}_{>0}$ be such that $\|\phi_g(x)\|_{inv} = \xi(g)\|x\|_{inv}$. Then $\xi$ is a surjective group homomorphism.

*Proof.* Let us first show that $\xi$ is a homomorphism. We clearly have $\xi(e) = 1$, since $\phi_e$ is the identity transformation. If $g, h \in G$ and $x \in \mathbb{R}^n$, then

$$ \|\phi_g(\phi_h(x))\|_{inv} = \xi(g)\|\phi_h(x)\|_{inv} = \xi(g)\xi(h)\|x\|_{inv} $$

But since $\phi_g \circ \phi_h = \phi_{gh}$, we have

$$ \|\phi_g(\phi_h(x))\|_{inv} = \|\phi_{gh}(x)\|_{inv} = \xi(gh)\|x\|_{inv} $$

Therefore, the right-hand side of last two equation are the same and, choosing any $x \neq 0$, we have the equality $\xi(gh) = \xi(g)\xi(h)$, showing that $\xi$ is a homomorphism.

Since $\xi$ is non-constant by hypothesis, there exists $g \in G$ such that $\xi(g) \neq 1$. Then, $\xi(g^n) = \xi(g)^n$ for every $n \in \mathbb{Z}$ and the set $\{\xi(g^n) : n \in \mathbb{Z}\}$ is contained in $\xi(G)$ and unbounded. Therefore, the set $\xi(G)$ is a connected unbounded subgroup of $(\mathbb{R}_{>0}, \cdot)$. This is only possible if $\xi(G) = \mathbb{R}_{>0}$. $\square$

Now we reproduce Theorem III.3 under the much simpler hypothesis of the existence of an invariant norm.

**Theorem IV.3.** Suppose that 0 is FTS for the $\theta$-invariant system (1), and suppose that the action $\phi: G \times \mathbb{R}^n \to \mathbb{R}^n$ of equation (3) is normed. Then $\theta$ cannot be the trivial homomorphism $\theta \equiv 1$.

As an application, we show that we cannot represent any dilation symmetry of a homogeneous system of negative degree using only an action on the state space. Thus, we prove rigorously that $\theta$-invariance is indeed necessary, as suggested informally in [6].

Recall that, given $r = (r_1, ..., r_n) \in \mathbb{R}^n$, $r_i > 0$ and $\nu \in \mathbb{R}$, we denote $\Lambda_{\varepsilon}^r = \text{diag}(\varepsilon^{r_1}, ..., \varepsilon^{r_n})$. We define a vector field $f: \mathbb{R}^n \to \mathbb{R}^n$ to be $r$-homogeneous of degree $\nu$ if it satisfies the equation

$$ f(\Lambda_{\varepsilon}^{r} x) = \varepsilon^{\nu} \Lambda_{\varepsilon}^{r} f(x), $$
for every $\varepsilon > 0$ and $x \in \mathbb{R}^n$. We also say that the associated system $\dot{x} = f(x)$ is $r$-homogeneous of degree $\nu$ in this case.

To makes matters more precise, we define what we mean by a dilation symmetry.

**Definition IV.4.** Let $G$ be a connected Lie group. We call an action $\phi : G \times \mathbb{R}^n \to \mathbb{R}^n$ a dilation if for every $x \in \mathbb{R}^n$ there exists a curve $\gamma : \mathbb{R}_{>0} \to G$ such that $\lim_{t \to +\infty} \|\phi_{\gamma(t)}(x)\| = +\infty$ and $\lim_{t \to 0^+} \phi_{\gamma(t)}(x) = 0$

**Corollary IV.5.** Let system (1) be $r$-homogeneous of degree $\nu < 0$. Then there is no dilation $\phi : G \times \mathbb{R}^n \to \mathbb{R}^n$ that makes it $G$-invariant.

*Proof.* It has been shown that homogeneous systems of negative degree are finite-time stable [1].

It is also clear that for a dilation $\phi$, the set $K = \{x \in \mathbb{R}^n : \|x\|_{inv} = 1\}$ has the property $G \cdot K = \mathbb{R}^n \setminus \{0\}$. The result then follows by Theorem III.3. $\square$

In the case of homogeneous systems we also have an associated homogeneous norm, given by

$$ \|x\|_{r,\nu} = (|x_1|^{\frac{\nu}{r_1}} + \dots + |x_n|^{\frac{\nu}{r_n}})^{\frac{1}{\nu}} $$

and first introduced in [9]. This norm fits the definition of invariant norm, since it satisfies

$$ \|\Lambda_{\varepsilon}^{r} x\|_{r,\nu} = \varepsilon \|x\|_{r,\nu}, $$

for every $\varepsilon > 0$ and $x \in \mathbb{R}^n$.

Now we reproduce Theorem III.5 using again the hypothesis of existence of an invariant norm.

**Theorem IV.6.** Let system (1) be such that the origin is an attractive equilibrium point. Suppose also that $f$ is $G$-invariant for a normed action $\phi : G \times \mathbb{R}^n \to \mathbb{R}^n$. Then the origin is a stable equilibrium point.

Next result shows that we can infer global behavior from local, following what should intuitively extend from the theory of linear and homogeneous systems.

**Corollary IV.7.** Let system (1) be $\theta$-invariant for a normed action $\phi : G \times \mathbb{R}^n \to \mathbb{R}^n$. If the origin is a locally attractive equilibrium point, then it is also globally asymptotically stable.

*Proof.* By Theorem IV.6, the origin is stable. So all we need to prove is that $\lim_{t \to \infty} x(t, x_0) = 0$, for every $x_0 \in \mathbb{R}^n$.

Let $x_0 \in \mathbb{R}^n \setminus \{0\}$ and $\delta > 0$ be such that $x(t, y_0) \to 0$, for every $y_0 \in B_{\delta}^{inv}(0)$. If we chose $g \in G$ such that $\xi(g) < \delta/\|x_0\|_{inv}$ and define $y_0 = \phi_g(x_0)$, then $\|y_0\|_{inv} = \|\phi_g(x_0)\|_{inv} < \delta$, therefore $x(t, y_0) \to 0$. But, by scaling of trajectories, we have

$$ x(t, x_0) = \phi_{g^{-1}}(x(\theta(g)t, y_0) $$

Since $\theta(g) \neq 0$ and $\phi$ is continuous, we have that the right-hand side tends to 0, showing that $x(t, x_0) \to 0$ for an arbitrary $x_0 \in \mathbb{R}^n$. $\square$

Another useful condition for the asymptotically stability of an equilibrium point is given in terms of the existence

of invariant sets. The next Theorem is a generalization of Theorem 6.1 of [1] for $\theta$-invariant systems with normed actions.

**Theorem IV.8.** Let system (1) be $\theta$-invariant for a normed action $\phi : G \times \mathbb{R}^n \to \mathbb{R}^n$ and such that the origin is an equilibrium point. Suppose that the system possesses a strictly positively invariant compact set $A$. Then the origin is globally asymptotically stable.

## V. CONCLUSION

In this paper we made a dynamical analysis of $\theta$-invariant autonomous systems with equilibrium solutions. We compared its dynamical features with those of $G$-invariant systems. We then established stability properties that extend naturally from similar ones encountered in homogeneous systems. Finally, we introduced the concept of invariant norm, which brings together the conditions necessary to establish those stability results.

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