# The Differential Stability Problem. Part II: the Dynamical Spectral Decomposition

Mario Spirito

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Mario Spirito. The Differential Stability Problem. Part II: the Dynamical Spectral Decomposition. 2026. hal-05528163

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# The Differential Stability Problem. Part II: the Dynamical Spectral Decomposition

Mario Spirito

**Abstract**— This work develops a unified framework for the analysis of stability and contractivity of nonlinear time-varying dynamical systems. After introducing the eigen-problems in the general settings of operators, we formalize system dynamics on manifolds with differentiable time-varying vector fields and their associates variational and prolonged dynamics in this general setting.

Motivated by the limitations of standard eigenvalue approach to study the stability analysis of nonlinear systems, we propose a dynamical spectral decomposition based on an operator eigenproblem, generalizing the classical Jacobian-based analysis for equilibrium points and enabling the treatment of systems that are non-diagonalizable. The resulting spectral framework yields criteria for differential asymptotic and differential exponential stability and provides constructive methods for designing contractive metrics on prescribed regions of the state space. Several examples illustrate the applicability of the approach to linear time-varying, linear time-invariant, and nonlinear systems.

Overall, the proposed framework bridges operator-theoretic and spectral perspectives on stability, offering new tools for the study of a broad class of nonlinear time-varying dynamical systems.

***Index Terms***— Contraction theory, Nonlinear systems, Operator theory, Time-varying systems, Spectral decomposition, Stability analysis.

## I. INTRODUCTION

STABILITY analysis is a foundational theme in systems and control theory, underpinning both the qualitative understanding of dynamical behavior and the design of controllers and observers. In the nonlinear autonomous setting, stability of an equilibrium point is traditionally established through the construction of a Lyapunov function whose time derivative remains strictly negative away from the equilibrium [1]. For linear time-invariant (LTI) systems, the problem reduces to evaluating the real parts of the eigenvalues of the state matrix, a criterion that has long served as the cornerstone for assessing asymptotic stability. Extending such spectral ideas to nonlinear systems build on early notions of dynamical eigenvectors for linear time-varying (LTV) systems [2], subsequent formalizations of nonlinear eigenvalues for affine and nonlinear systems [3], and the emergence of a nonlinear spectral theory tailored to stability and contraction analysis [4]–[6]. Further developments, applications, and perspectives on these nonlinear spectral ideas can be found in [7]. Related

concepts such as dynamical eigenvectors have also been used to characterize the analytic features of solutions of linear time-varying systems, as discussed in [8] and the references therein.

Linearization around an equilibrium yields an LTI model, whereas linearization along a general time-varying trajectory gives rise to LTV system, which constitutes the natural local approximation in time-dependent settings.

Although the eigenvalue test produces a complete and definitive characterization of stability for LTI systems (owing to the direct correspondence between the spectrum of the system matrix and the asymptotic behavior of solutions [9], [10]), the situation is fundamentally different for LTV systems. In the time-dependent setting, pointwise spectral information is generally neither necessary nor sufficient to infer stability properties. In particular, it is well known that an LTV system may exhibit instability even when all instantaneous eigenvalues of the system matrix possess strictly negative real parts at every time instant [1], [9]–[16].

This fundamental limitation stems from the fact that stability in LTV systems is not governed by local-in-time spectral properties, but rather by the dynamical effects of the time-varying linear transformation. As a result, evolving eigenspace bases can mask eigenvalues with positive real parts, or their inherent growth may lead to divergence despite a lack of instantaneous instability. These effects are captured not by the eigenvalues of the system matrix at a fixed time, but by the explicit solution of the dynamics and, in particular, by the state-transition matrix, which encodes the global-in-time evolution of the linear dynamics.

For LTI systems, this object has a closed-form expression via the matrix exponential, but for generic LTV systems its computation requires more elaborate machinery. Classical approaches include the Peano-Baker series [10], [13], [16], which provides a formal representation that is rarely tractable for analysis. Closed-form solutions exist only under restrictive commutativity conditions, as shown in [10]. Beyond these cases, several contributions have attempted to construct explicit or semi-explicit representations: the early closed-form approach of [17], later refined in [18] and extended in [19], [20]; the Taylor-series-based operator method proposed in [21]; constructions based on commutativity such as those in [22]; and the more recent developments and corrections surrounding [23], [24]. Stability criteria for LTV systems frequently exploit properties of the state-transition matrix, as in [25]–[27].

Meanwhile, the stability of a nonlinear system can be characterized through its variational dynamics, which takes
the form of a state-dependent LTV system. Building on this representation, contraction theory has emerged as a powerful and unifying framework for analyzing the stability, convergence, synchronization, and robustness properties of nonlinear dynamical systems. In contrast to classical Lyapunov theory, contraction theory provides a differential, trajectory-based viewpoint in which stability is characterized through the behavior along the system's solutions. The early precursors of the contraction paradigm can be traced to the works of Demidovich [28] and Willems [29], which established sufficient conditions for global convergence using constant quadratic metrics. These ideas were later revisited and reformulated in the context of nonlinear systems and observer design in [30]–[33], giving rise to the modern concepts of convergent systems and incremental stability [34]–[37], recently revised in [38].

A persistent challenge in contraction analysis is the construction of an appropriate contracting metric. Analytical derivation is often feasible only for low-dimensional or structurally simple systems [39]. To address this, several computational approaches have been developed, including convex optimization and LMI-based techniques for constructing control contraction metrics [40]–[42], pseudospectral and meshfree methods to compute metrics for equilibria and periodic orbits [43] and separability-based methods for distributed control [44]. Neural networks have recently been proposed to parameterize and learn contraction metrics for complex, high-dimensional systems, providing a flexible tool for trajectory generation and stability verification [45]–[47]. Data-driven and learning-based approaches have been proposed, including deep-learning techniques to discover contracting metrics [48], [49].

On the analytical side, recent efforts have sought to relate the construction of contracting metrics to nonlinear eigenvalues, nonlinear eigenvectors, and operator-theoretic formulations. These developments are based on early notions of nonlinear eigenvalues and eigenvectors for nonlinear systems [3], and the emergence of a nonlinear spectral theory tailored to stability and contraction analysis [4]–[6]. Within this framework, contraction metrics are characterized through differential Riccati-type equations and geometric operator formulations, establishing deep connections with differential Hamiltonian and Riccati structures [50]. Koopman operator and its associated eigenfunctions have been used as stability tools in [51], and a comparison between the Koopman operator approach and contraction is found in [52]. These analytical tools provide principled mechanisms to approximate or compute contractive Riemannian metrics for general nonlinear systems, thereby extending classical spectral methods beyond the linear time-invariant setting. Nevertheless, despite substantial progress, the systematic and constructive derivation of contractive Riemannian metrics remains a technically demanding and largely open problem, as emphasized in recent comprehensive surveys [53].

This rich landscape of time-varying nonlinear stability theory motivates the developments in this work, which aims to unify the tools to infer stability on nonlinear systems through the variational dynamics. Using operator-theoretic tools, we generalize classical spectral methods and show how to construct an associated contractive metric.

### A. Paper contribution

The present paper advances contraction and differential stability analysis. The central contribution of this paper is the introduction of a novel operator-theoretic formulation whose spectral structure allows to explicitly construct the state-transition matrix associated with the variational dynamics. This generalizes the classical eigenvalue-based approach used in the analysis of linear time-invariant systems, providing a dynamical spectral decomposition applicable to nonlinear time-varying settings. We further show that a weaker condition also yields the state transition matrix and facilitates the stability analysis of the variational dynamics. Finally, we leverage the resulting transformation matrix to construct a contractive metric, thereby offering a systematic, alternative, and constructive approach to contraction analysis.

### B. Structure of the paper

The remainder of the paper is organized as follows. We begin in Section II with the mathematical foundation required for our developments. This involves linear operators with their eigenstructure (Section II-A), and the concepts of nonlinear eigenvectors and eigenvalues (Section II-B).

Next, Section III formalizes the system-theoretic framework. We consider dynamics on manifolds under a time-varying vector field, discuss the associated flow map, recall the notion of variational state and variational dynamics, the related state transition matrix, and introduce the prolonged dynamics. In Section IV we recall the differential stability and contractive dynamics notions used throughout the document.

The main result on the differential stability analysis is addressed in Section V. Specifically, in Section V-A we introduce a dynamical spectral decomposition based on the eigenproblem of a novel defined operator and compare it with classical Jacobian-based eigenanalysis. In Section V-B we present a more general transformation that removes diagonalizability constraints on the vector field's Jacobian. This leads in Section V-C to differential asymptotic and exponential stability criteria. Finally, Section V-D shows how the eigenstructure of this operator is used to construct regional contractive metrics.

The effectiveness of the proposed framework is illustrated in Section VI. For LTV systems, in Section VI-A.1 we recover the transition matrix of a classical example, while Section VI-A.2 treats an LTI system with a non-diagonalizable state matrix, in the Jordan sense. A nonlinear contractive system is analyzed in Section VI-B, where the operator eigenproblem is solved to construct a contractive metric and assess the regional exponential stability of the variational dynamics.

The paper concludes in Section VII with a summary of the main contributions and a discussion of future research directions.

### C. Notation:

The set of real (complex) numbers is denoted by $\mathbb{R}$ ($\mathbb{C}$), and $\mathbb{N}$ denotes the set of natural numbers. Given $n \in \mathbb{N}$, vectors are assumed to belong to $\mathbb{R}^n$ and matrices to $\mathbb{R}^{n \times n}$. The symbol
$\| \cdot \|$ is used as the standard Euclidean norm for vectors and the induced 2-norm for matrices, unless stated otherwise.

$C^k(\mathcal{X})$ denotes the class of continuously $k$ times differentiable functions on $\mathcal{X}$.

Given a sufficiently smooth function $f: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$, the Jacobian matrix with respect to the state variable is denoted by $\frac{\partial f(x,t)}{\partial x}\bigg|_{(x,t)}$.

For a matrix $M \in \mathbb{R}^{n \times n}$, $\sigma(M)$ denotes its spectrum, $\lambda_i(M)$ its $i$-th eigenvalue, and $\sigma_i(M)$ its $i$-th singular value, defined as $\sigma_i(M) = \sqrt{\lambda_i(M^\top M)}$. The minimum and maximum singular values are denoted by $\sigma_{\min}(M)$ and $\sigma_{\max}(M)$, respectively, and the condition number of $M$ is $\mu(M) = \frac{\sigma_{\max}(M)}{\sigma_{\min}(M)}$. The 2-norm for matrices comes then directly as $\|M\| = \sigma_{\max}(M)$ and $\|M^{-1}\| = \sigma_{\min}(M)^{-1}$.

For any matrix $M \in \mathbb{R}^{n \times n}$, its skew-symmetric and symmetric parts are defined as skew($M$) = $\frac{1}{2}(M - M^\top)$ and sym($M$) = $\frac{1}{2}(M + M^\top)$. A matrix $P$ is said to be symmetric if $P = P^\top$. A symmetric matrix $P \in \mathbb{R}^{n \times n}$ is positive definite, denoted by $P > 0$, if $x^\top P x > 0$, for all $x \in \mathbb{R}^n \setminus \{0\}$.

## II. PRELIMINARIES

We begin by recalling the eigenproblem in the general settings of operators and the nonlinear eigenstructure introduced in [5]. These concepts provide the basic mathematical setting used throughout the paper.

### A. Operators, eigenvalues, and eigenfunctions

In this subsection, we extend the classical notions of eigenvalues and eigenvectors from linear algebra to a more general functional setting. Here, operators act on function spaces rather than finite-dimensional vector spaces, and eigenvalues may themselves be scalar-valued functions of the independent variables. This framework is fundamental for analyzing systems whose spectral properties depend on the state $x \in \mathcal{X} \subseteq \mathbb{R}^m$.

**Definition II-A.1 (Operator).** Let $\mathcal{X} \subseteq \mathbb{R}^m$ (or $\mathbb{C}^m$) and let $\mathcal{F}$ be a suitable space of functions $f: \mathcal{X} \to \mathbb{R}^n$ (or $\mathbb{C}^n$). An operator is a mapping $\mathcal{O}: \mathcal{F} \to \mathcal{F}$, $f \mapsto \mathcal{O}[f]$, which assigns to each function $f \in \mathcal{F}$ a new function $\mathcal{O}[f] \in \mathcal{F}$ defined on $\mathcal{X}$.

**Definition II-A.2 (Eigenvalue and eigenfunction).** Let $\mathcal{O}: \mathcal{F} \to \mathcal{F}$ be an operator. A nonzero function $T_j \in \mathcal{F}$ is called an eigenfunction of $\mathcal{O}$ if there exists a map $\lambda_j: \mathcal{X} \to \mathbb{R}$ (or $\mathbb{C}$) such that

$$ \mathcal{O}[T_j](x) = \lambda_j(x) T_j(x), \quad \forall x \in \mathcal{X}. $$

The map $\lambda_j$ is called the eigenvalue function associated with the eigenfunction $T_j$.

**Definition II-A.3 (Eigenspace).** The set of all eigenfunctions associated with a given eigenvalue function $\lambda_j$ (repeated eigenvalues are considered as one), together with the zero function,

$$ \mathcal{E}_{\lambda_j} = \{T_j \in \mathcal{F} \setminus \{0\} : \\ \mathcal{O}[T_j](x) = \lambda_j(x) T_j(x), \forall x \in \mathcal{X}\} \cup \{0\}, $$

is called the eigenspace of $T_j$ associated with $\lambda_j$.

**Remark II.1** The standard definition of eigenvalues and left (right) eigenvectors of a matrix $A \in \mathbb{R}^{n \times n}$, i.e., $TA = \lambda T$ ($AT = \lambda T$), with $\lambda \in \mathbb{C}$, and $T \in \mathbb{C}^n \setminus \{0\}$, is recovered as a particular case in which the operator, i.e. $\mathcal{O} = A$, is linear and finite-dimensional.

In the next sections, we will use the following notion.

**Definition II-A.4 (Pointwise linear independence).** Let $\mathcal{O}: \mathcal{F} \to \mathcal{F}$ be an operator, where $\mathcal{F}$ is a suitable function space of mappings $f: \mathcal{X} \to \mathbb{R}^n$ (or $\mathbb{C}^n$), and $\mathcal{X} \subseteq \mathbb{R}^m$ (or $\mathbb{C}^m$). A set of $n$ eigenfunctions $T_j, j = 1, \dots, n$, is said to be pointwise linearly independent or everywhere distinct if, for every $x \in \mathcal{X}$, the family of vectors

$$ \{T_1(x), \dots, T_n(x)\} \subset \mathbb{C}^n $$

is linearly independent. In particular, these vectors form a basis of $\mathbb{R}^n$ at each point $x \in \mathcal{X}$.

**Remark II.2** Pointwise linear independence of the eigenfunctions is equivalent to stating that the distribution of

$$ (T_1, \dots, T_n) $$

spans $\mathbb{C}^n$ at each point of $\mathcal{X}$. Equivalently, for column-vector eigenfunctions $T_j, j = 1, \dots, n$, the matrix

$$ T(x) := [T_1(x), \dots, T_n(x)] \in \mathbb{C}^{n \times n} $$

is invertible at every point of the domain, i.e.,

$$ \det(T(x)) \neq 0, \quad \forall x \in \mathcal{X}. $$

### B. Nonlinear eigenvectors and eigenvalues

We now recall the nonlinear system eigenstructure framework developed in [5]. The authors analyze the conditions under which a nonlinear system $\dot{x} = f(x)$ admits a diffeomorphic change of coordinates that diagonalized its dynamics. Specifically, a system $\dot{x} = f(x)$ is said to be diagonalizable on a compact set $X_d \subseteq \mathcal{X}$ if there exists a diffeomorphism $\phi: X_d \to \mathbb{C}^n$ such that, in the new coordinates $z = \phi(x)$, the dynamics simplifies to the decoupled form $\dot{z}_i = g_i(z_i)$, $i = 1, \dots, n$, where each state $z_i: \mathbb{R} \to \mathbb{C}$ and its dynamics $g_i: \mathbb{C} \to \mathbb{C}$ is analytic. In this context, the nonlinear eigenvalue and eigenvectors of the system are defined as follows.

**Definition II-B.1 (Def. 2 in [5]).** Let $\lambda: X_d \to \mathbb{C}$ and $v: X_d \to \mathbb{C}^n$ be complex analytic functions, such that $v(x^*) \neq 0$ at any equilibrium points $x = x^*$. They are called, respectively, a nonlinear eigenvalue and nonlinear left eigenvector of system $\dot{x} = f(x)$ if, for all $x \in X_d$,

$$ v^\top(x) \frac{\partial f(x)}{\partial x} + \left( \frac{\partial v(x)}{\partial x} f(x) \right)^\top = \lambda(x) v^\top(x). \quad (1) $$

At any equilibrium point $x = x^*$, identity (1) reduces to the classical eigenvalue relation for the vector field's Jacobian evaluated at that point, i.e.,

$$ v^\top(x^*) \frac{\partial f(x^*)}{\partial x} = \lambda(x^*) v^\top(x^*). \quad (2) $$
The authors further provide necessary and sufficient conditions for the diagonalizability of the nonlinear system, summarized below.

**Theorem II-B.1** (Th.1 in [5]). Let $\lambda_i : \mathcal{X}_d \to \mathbb{C}$ and $v_i : \mathcal{X}_d \to \mathbb{C}^n$, denote the nonlinear eigenvalues and their associated left eigenvectors of the system $\dot{x} = f(x)$. The system is diagonalizable on $\mathcal{X}_d \subset \mathbb{R}^n$ if and only if

a) the distribution of $(v_1(x), \dots, v_n(x))$ spans $\mathbb{R}^n$ for every $x \in \mathcal{X}_d$;

b) there exists complex analytic functions $\psi_i : \mathcal{X}_d \to \mathbb{C}^n$ such that $v_i^\top(x)dx = d\psi$ on $\mathcal{X}_d$, $i = 1, \dots, n.$ ▼

The first condition establishes pointwise linear independence of the nonlinear eigenvectors, ensuring that they form a basis of the tangent space at each $x \in \mathcal{X}_d$. The second condition is an integrability requirement, which guarantees that each left eigenvector field is locally exact and therefore admits a scalar potential function. This property enables the explicit construction of decoupling coordinates $z = \phi(x)$ through analytic integration, allowing the change of coordinates defined in the *lifted* variational space to be consistently *pulled down* to the underlying state space.

**Remark II.3** From this point onward, for notational simplicity, we identify the smooth manifold, denoted by $\mathcal{X}$, with its image in $\mathbb{R}^n$. Accordingly, the dynamical system $\dot{x} = f(x, t)$ is understood to be defined on $\mathcal{X} \subseteq \mathbb{R}^n$, while implicitly assuming that $\mathcal{X}$ is a smooth immersed submanifold of $\mathbb{R}^n$. ▼

### III. THE DYNAMICAL SYSTEM CONTEXT

In this work, we consider a *forward complete*<fnref n="1" /> dynamical system formulated as the following initial value problem (IVP). Let $\mathcal{X} \subseteq \mathbb{R}^n$ be a smooth manifold and let $f : \mathcal{X} \times \mathbb{R} \to \mathcal{T}\mathcal{X}$ be a $C^1$ map in $(x, t)$. The system under consideration is

$$ \Sigma: \begin{cases} \dot{x}(t) = f(x(t), t), \\ x(t_0) = x_0, \end{cases} \quad (3) $$

where $t \in [t_0, \infty) \subseteq \mathbb{R}$ and $x: [t_0, \infty) \to \mathcal{X} \subseteq \mathbb{R}^n$. The map $f$ is the system's vector field, and $x(t)$ denotes a local coordinate representation of a trajectory. System (3) is called *inhomogeneous* (or non-autonomous). If the dynamics does not depend explicitly on time, i.e.,

$$ \dot{x} = f(x), \quad x(t_0) = x_0, \quad (4) $$

then the system is referred to as *homogeneous* (or autonomous).

Given the IVP (3), we denote by $\varphi(x_0, t; t_0) \in \mathcal{X}$ the unique solution $x(t) = \varphi(x_0, t; t_0)$, initialized at $x(t_0) = x_0 \in \mathcal{X}$. The map $\varphi: \mathcal{X} \times [t_0, \infty) \to \mathcal{X}$ is referred to as the system's flow.

The flow map has the following fundamental properties:

a) **Identity:** $\varphi(x_0, t_0; t_0) = x_0$ for all $x_0 \in \mathcal{X}$.

b) **Semigroup property:** for all $t_0, t_1, t_2 \in \mathbb{R}$, such that $t_0 \le t_1 \le t_2$,

$$ \varphi(x_0, t_2; t_0) = \varphi(\varphi(x_0, t_1; t_0), t_2; t_1). $$

c) **Differentiability:** if $f(x, t)$ is $C^k$ in $x$, then $\varphi(x_0, t; t_0)$ is $C^k$ with respect to $x_0$.

The definition of flow allows to describe the (positively) invariant property of a subset.

**Definition III.-1** (Invariant and positively invariant subset). A subset $\bar{\mathcal{X}} \subseteq \mathcal{X} \subseteq \mathbb{R}^n$ is said to be *invariant* under the flow $\varphi$ of system (3) if, for every point $x_0 \in \bar{\mathcal{X}}$, the trajectory remains in $\bar{\mathcal{X}}$ for all backward and forward times, i.e.,

$$ \varphi(x_0, t; t_0) \in \bar{\mathcal{X}}, \quad \forall t \in \mathbb{R}. $$

Similarly, the set $\bar{\mathcal{X}}$ is said to be *positively invariant* if

$$ \varphi(x_0, t; t_0) \in \bar{\mathcal{X}}, \quad \forall t \ge t_0. $$

In other words, trajectories starting in a positively invariant set remain inside it for all future times. ▼

In the following, $\mathcal{X}$ is regarded as forward invariant.

a) **Variational State:** To study the sensitivity of trajectories with respect to perturbations in the initial conditions, we introduce a smooth family of solutions. In particular, for a reference initial state $x_0 \in \mathcal{X}$, consider a family of parameterized trajectories $\chi: (-\delta_\chi, \delta_\chi) \times [t_0, \infty) \to \mathcal{X}$ such that, for each $\varepsilon \in (-\delta_\chi, \delta_\chi) \subseteq \mathbb{R}$, with $\delta_\chi > 0$ sufficiently small, the curve $t \mapsto \chi(\varepsilon, t)$ satisfies

$$ \dot{\chi}(\varepsilon, t) = f(\chi(\varepsilon, t), t), \quad \chi(\varepsilon, t_0) = x_0 + \varepsilon\nu, \quad (5) $$

where $\nu \in \mathbb{R}^n$ is an initial perturbation direction. The nominal trajectory is recovered for $\varepsilon = 0$, i.e., $\chi(0, t) = \varphi(x_0, t; t_0) = x(t)$.

We can now introduce the notion of variational state and the associated variational dynamics. Starting from the parametrized family of trajectories $\chi: (-\delta_\chi, \delta_\chi) \times [t_0, \infty) \to \mathcal{X}$, where $\chi(\varepsilon, t)$ denotes the solution of (5) corresponding to the perturbed initial condition $x_0 + \varepsilon\nu$, we define the *variational state* as

$$ \delta x^{\pm}(t) := \lim_{\varepsilon \to 0^{\pm}} \left. \frac{\partial \chi(\varepsilon, t)}{\partial \varepsilon} \right|_{\varepsilon=0^{\pm}} = \frac{\partial \chi(\varepsilon, t)}{\partial \varepsilon} \quad (6) $$

Figure 1 provides a graphical illustration of the variational state $\delta x^{\pm}$, highlighting its role within the family of parameterized trajectories.

b) **Variational and prolonged dynamics:** The evolution of $\delta x(t)^2$ is governed by a linear time-varying system known as the *variational dynamics*, which captures system (3) infinitesimal behavior around the nominal solution $\varphi(x_0, t; t_0)$, i.e.,

$$ D\Sigma: \begin{cases} \dot{\delta x}(t) = \frac{\partial f}{\partial x}(\varphi(x_0, t; t_0), t) \delta x(t), \\ \delta x(t_0) = \nu, \end{cases} \quad (7) $$

where $\nu \in \mathcal{T}_{x_0}\mathcal{X} \subseteq \mathbb{R}^n$ denotes an admissible<fnref n="3" /> perturbation direction at time $t_0$. Classically, coming directly from the definition of variational state (6), this LTV state-dependent system characterizes the evolution of infinitesimal deviations from the reference trajectory $\varphi(x_0, t; t_0)$ and plays a fundamental role

<footnote n="1">¹A system is called forward complete if, for every initial condition, the corresponding solution is defined for all forward times [54].</footnote>

<footnote n="2">²In the following, the notation $\pm$ is omitted, and we simply consider the limiting case $\varepsilon \to 0$.</footnote>

<footnote n="3">³That is, lying on the tangent space of $\mathcal{X}$ at $x_0$.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1: Parametrized trajectories $\chi(\varepsilon, t)$ in the neighborhood of the nominal trajectory $x(t) = \chi(0, t)$. In both cases, the time evolution of $x(t)$ is shown together with two perturbed trajectories, $\chi(\delta_{\chi}, t)$ and $\chi(-\delta_{\chi}, t)$. The variational displacements $\delta x^{\pm}(t)$ are depicted as vectors orthogonal to the time axis, representing infinitesimal deviations along the family of trajectories.</figcaption></figure>

in the stability and local sensitivity analysis of dynamical systems [55]–[57]. As any LTV system, the evolution of variational state is described by the state transition matrix, $\Phi: \mathbb{R}^2 \to \mathbb{R}^{n \times n}$ defined such that $\delta x(t) = \Phi(t, t_0)\delta x(t_0)$, for all $t, t_0 \in \mathbb{R}$. In complete analogy with the flow $\varphi$, the state transition matrix $\Phi(t, t_0)$ inherits several important properties from the linear variational dynamics (7). Specifically, for all times $t, \tau, t_0 \in \mathbb{R}$, it satisfies:

a) **Identity at the initial time:** for all $t_0 \in \mathbb{R}$, $\Phi(t_0, t_0) = I_n$,

b) **Composition property:** for all $t, \tau, t_0 \in \mathbb{R}$, $\Phi(t, \tau) \Phi(\tau, t_0) = \Phi(t, t_0)$,

c) **Invertibility:** For all $t, t_0 \in \mathbb{R}$, $\Phi(t, t_0)^{-1} = \Phi(t_0, t)$.

d) **Differential equation:** The matrix $\Phi(t, t_0)$ satisfies the LTV matrix IVP

$$ \dot{\Phi}(t, t_0) = \frac{\partial f}{\partial x}(\varphi(x_0, t; t_0), t) \Phi(t, t_0), \Phi(t_0, t_0) = I_n. $$

These properties make the transition matrix $\Phi(t, t_0)$ a central object in the analysis of contraction properties.

The variational dynamics (7) can only be defined consistently together with the notion of the flow $\varphi$. In the general case, we consider the parallel configuration of systems (3) and (7), commonly referred to as the prolonged dynamics [58], [59]. The prolonged (also termed “extended”, or “lifted” as in [60]) system associated with $\Sigma$ is the augmented 2n-

dimensional system defined on the tangent bundle $T\mathcal{X}$, consisting of $\Sigma$ and its variational dynamics $\mathcal{D}\Sigma$ in parallel. More precisely, the prolonged system dynamics is given by

$$ T\Sigma: \left\{ \begin{array}{l} \dot{x} = f(x, t), \quad x(0) = x_0, \\ \delta x(t) = \frac{\partial f}{\partial x}(\varphi(x_0, t; t_0), t) \delta x(t), \quad \delta x(0) = \nu. \end{array} \right. $$

To complete this section, from [38], the family of parametrized trajectories $\chi(\varepsilon, t)$ can be built based on the notion of system flow, i.e., given an initial direction $\nu$, $\chi(\varepsilon, t) = \varphi(x_0 + \varepsilon\nu, t; t_0)$. This direct link provides the closed form expression for the transition matrix $\Phi(t, t_0)$. Specifically, from (6) we obtain

$$ \begin{aligned} \delta x(t) &= \frac{\partial \varphi}{\partial x_0}(x_0, t; t_0) \delta x(t_0) \\ &= \Phi(t, t_0) \delta x(t_0). \end{aligned} \quad (8) $$

All subsequent analyses and stability properties should be, in principle, formulated with respect to the prolonged system $T\Sigma$, since the variational dynamics (7) is well defined only along trajectories solution of the nonlinear system. However, in the remainder, it is always assumed that the variational evolution of the reference trajectory $x(t) = \varphi(x_0, t; t_0)$ is implicitly governed by (3). Accordingly, for notational simplicity, we will refer exclusively to the variational dynamics with the understanding that all quantities are evaluated along solutions of the underlying nonlinear system.

## IV. DIFFERENTIAL STABILITY AND CONTRACTIVE DYNAMICS

This paper focuses mainly on the differential stability concepts and their link to the contraction property. For completeness, in this section we recall these notions. A more detailed overview of different notions of stability is summarized in [38].

### A. Differential stability

The concept of differential stability originates from the analysis of the variational dynamics associated with the nonlinear system (3). The variational system (7) is a time-varying and trajectory-dependent linear system whose stability properties depend simultaneously on the state trajectory $x(t)$ and on the time variable $t$. Consequently, any stability property defined for this system can be equivalently interpreted as a differential stability property of the nonlinear dynamics (3).

**Definition IV-A.1 (differential Regional Stability (dRS)).** System (3) is said to be differentially (regionally) stable in a forward invariant region $\mathcal{X} \subseteq \mathcal{X}$ if, for any trajectory $\varphi(x_0, t; t_0)$ with $x_0 \in \mathcal{X}$, the solution $\delta x(t)$ of the associated variational dynamics remains bound for all forward time, i.e., for every $\varepsilon > 0$ there exists a $\delta = \delta_{\varepsilon, x_0, t_0} > 0$ such that, any initial infinitesimal displacement $\delta x(t_0) \in T_{x_0}\mathcal{X}$ satisfying

$$ \|\delta x(t_0)\| < \delta \implies \|\delta x(t)\| < \varepsilon, \forall t \ge t_0. \quad (9) $$

**Definition IV-A.2 (differential Regional Asymptotic Stability (dRAS)).** System (3) is said to be differentially (regionally)
asymptotically stable in a forward invariant region $\bar{\mathcal{X}}$ if, for any initial condition $x_0 \in \bar{\mathcal{X}}$, it is differentially (regionally) stable and the solution $\delta x(t)$ of the variational system satisfies

$$ \lim_{t \to \infty} \|\delta x(t)\| = 0, \quad \forall x_0 \in \bar{\mathcal{X}}. \quad \llap{\text{a}} $$

**Definition IV-A.3 (differential Regional Exponential Stability (dRES)).** System (3) is differentially (regionally) exponentially stable in a forward invariant region $\bar{\mathcal{X}}$ if there exist real constants $\kappa = \kappa_{x_0, t_0} > 0$ and $\alpha = \alpha_{x_0, t_0} > 0$ such that, for all initial conditions $x_0 \in \bar{\mathcal{X}}$ and $\delta x(t_0) \in \mathcal{T}_{x_0} \mathcal{X}$,

$$ \|\delta x(t)\| \le \kappa e^{-\alpha(t-t_0)} \|\delta x(t_0)\|, \quad \forall t \ge t_0. \quad (10) \llap{\text{b}} $$

In this framework, the classical notions of stability for the nonlinear system (3) naturally extend to its variational dynamics. Classically, the resulting differential stability properties capture the local, infinitesimal behavior of the system along its trajectories. This viewpoint forms the foundation for the forthcoming concepts of *contractive dynamics* and *differential Lyapunov function*.

## B. Contractive dynamics

The contraction approach was originally introduced as a tool for analyzing the stability properties of nonlinear systems, serving as a differential counterpart to the classical Lyapunov framework [32]. The theory was later extended to systems endowed with nonlinear, state-dependent metrics in [60], and further developed in [30], [34], [38].

**Definition IV-B.1 (Contractive time-varying system).** An inhomogeneous system (3) on a smooth manifold $\mathcal{X}$ is said to be contractive in a region $\mathcal{X}_* \subseteq \mathcal{X}$ if there exists a $C^1$ matrix-valued function $P: \mathcal{X} \times [t_0, +\infty) \to \mathbb{R}^{n \times n}$ with a well-defined time derivative, and two strictly positive real numbers $\underline{p}$ and $\bar{p}$ such that $P$ satisfies $\forall x \in \mathcal{X}_* \setminus \{x^*\}$ and $t \ge t_0$,

$$ \underline{p}I \le P(x, t) \le \bar{p}I, \quad (11) $$

$$ \frac{dP}{dt}(x,t) + P(x,t) \frac{\partial f}{\partial x}(x,t) + \frac{\partial f^\top}{\partial x}(x,t) P(x,t) < 0. \quad (11) $$

In such a case, $P$ is classically called a *contractive metric*. $\llap{\text{c}}$

The contractive metric $P$ naturally induced the differential Lyapunov function $V: \mathcal{T}\mathcal{X} \times \mathcal{X} \times [t_0, \infty) \to \mathbb{R}$, with $V(\delta x, x, t) := \delta x^\top P(x, t) \delta x$.

Using definitions IV-A.3 (with $\bar{\mathcal{X}} \equiv \mathcal{X}$) and IV-B.1, we can state the following.

**Lemma IV-B.1.** A *contractive dynamics* (3) whose associated metric $P(x)$ satisfies, for all $x \in \mathcal{X}$ and $t \ge t_0$

$$ 0 < \underline{p}I \le P(x, t) \le \bar{p}I, \quad \underline{p}, \bar{p} \in \mathbb{R} \quad (12) $$

$$ \frac{dP}{dt}(x,t) + P(x,t) \frac{\partial f}{\partial x}(x,t) + \frac{\partial f}{\partial x}(x,t) P(x,t)^\top \le -2\alpha P(x,t) \quad (12) $$

with $\alpha > 0$, is differentially *Globally Exponentially Stable* with $\kappa = \sqrt{\underline{p}/\bar{p}}$. $\llap{\text{d}}$

A proof of the lemma and a geometrical interpretation of Lyapunov functions can be found in [38]. However, it is worth noting that (12) is a stronger condition than (11).

## V. DIFFERENTIAL STABILITY ANALYSIS

To analyze the stability properties of nonlinear systems from a differential viewpoint, we study the evolution of the variational state along the system trajectory. In the following, we show that, under mild commutativity and invertibility conditions, the variational dynamics can be expressed in transformed coordinates through a matrix pair $(\mathcal{T}, \mathcal{A})$, yielding a closed-form representation of the state transition matrix. This result provides a natural basis for a differential stability analysis. The next subsection develops this idea, introducing the dynamical spectral decomposition and its role in characterizing the differential stability properties of the flow.

### A. The dynamical spectral decomposition

We now present our first main result of the paper.

**Theorem V-A.1.** Consider system (3), with $f \in C^1(\mathcal{X} \times [t_0, \infty))$ on the forward invariant manifold $\mathcal{X}$ and its variational dynamics (7) on the tangent bundle $\mathcal{T}\mathcal{X}$. Define $\mathcal{F}_{\mathcal{C}}$ as the set of complex functions in $\mathbb{C}^n$, i.e.,

$$ \mathcal{F}_{\mathcal{C}} := \{g \in C^1(\mathcal{X} \times [t_0, \infty)) \text{ s.t. } g: \mathcal{X} \times [t_0, \infty) \to \mathbb{C}^n\}. $$

Given the vector field $f$, define the operator $\mathcal{A}: \mathcal{F}_{\mathcal{C}} \to \mathcal{F}_{\mathcal{C}}$ as

$$ \mathcal{A}[\cdot] := \frac{d[\cdot]}{dt} + [\cdot] \frac{\partial f}{\partial x}(x,t). \quad (13) $$

Assume there exist $n$ eigenfunctions $T_j \in \mathcal{F}_{\mathcal{C}}$ of $\mathcal{A}$, $j = 1, \dots, n$, pointwise linearly independent, associated with (possibly repeated) eigenvalues $\lambda_j \in C^0(\mathcal{X} \times [t_0, \infty))$ with $\lambda_j: \mathcal{X} \times [t_0, \infty) \to \mathbb{C}$, $j = 1, \dots, n$, defined according to Def. II-A.2. That is, $T_j$ and $\lambda_j$ satisfy, for $j = 1, \dots, n$,

$$ \mathcal{A}[T_j](x,t) = \lambda_j(x,t) T_j(x,t), \quad \forall (x,t) \in \mathcal{X} \times [t_0, \infty). $$

Then the explicit solution of the variational state $\delta x$, initialized at $\delta x(t_0) = \delta x_0$ is given by

$$ \delta x(t) = T(x,t)^{-1} \exp \left( \int_{t_0}^{t} \Lambda(x(s),s) ds \right) T(x_0, t_0) \delta x_0 \quad (14) $$

where $\Lambda(x,t) = \text{diag}(\lambda_1(x,t), \dots, \lambda_n(x,t)) \in \mathbb{C}^{n \times n}$ and $T^\top = [T_1^\top | \dots | T_n^\top] \in \mathbb{C}^{n \times n}$. $\llap{\text{e}}$

**Proof.** The proof relies on the diagonalization of the variational dynamics (7) via the change of coordinates $\delta z_j: \mathbb{R} \to \mathbb{C}$, with $\delta z_j := T_j(x,t)\delta x(t)$, $j = 1, \dots, n$. By definition, $T_j$ is the left eigenfunction of the operator $\mathcal{A}$ associated with the eigenvalue $\lambda_j$ and thus satisfies, $\forall (x,t) \in \mathcal{X} \times [t_0, \infty)$ and for $j = 1, \dots, n$,

$$ \begin{aligned} \mathcal{A}[T_j](x,t) &= \lambda_j(x,t) T_j(x,t) \\ &= \frac{dT_j}{dt}(x,t) + T_j(x,t) \frac{\partial f}{\partial x}(x,t). \end{aligned} $$
Taking the time derivative of $\delta z_j(t)$ gives, for $j = 1, \dots, n$,

$$ 
\begin{aligned}
\dot{\delta z}_j(t) &= \dot{T}_j(x, t) \delta x + T_j(x, t) \frac{\partial f}{\partial x}(x, t) \delta x \\
&= \lambda_j(x, t) T_j(x, t) \delta x \\
&= \lambda_j(x, t) \delta z_j(t).
\end{aligned}
$$

Since, in the new coordinates, the system is diagonal (each state evolution is independent from one another), linear, and time-varying, its closed form solution is

$$ \delta z_j(t) = \exp \left( \int_{t_0}^{t} \lambda_j(x(\tau), \tau) d\tau \right) \delta z_j(t_0) $$

with $\delta z_j(t_0) = T_j(x_0, t_0) \delta x_0$, for $j = 1, \dots, n$.

Collecting the scalar equations into a vectorial form $\delta z(t) := (\delta z_1(t), \dots, \delta z_n(t)) \in \mathbb{C}^n$ yields

$$ \delta z(t) = \exp \left( \int_{t_0}^{t} \Lambda(x(\tau), \tau) d\tau \right) \delta z(t_0) $$

with $\delta z(t_0) = T(x_0, t_0) \delta x_0$. Pointwise linear independence of the eigenfunctions $T_j$, $j = 1, \dots, n$, ensures that the coordinate transformation

$$ \delta z(t) := T(x(t), t) \delta x(t) $$

is well-defined and invertible for all $t$. Therefore,

$$ 
\begin{aligned}
\delta x(t) &= T(x, t)^{-1} \delta z(t) \\
&= T(x, t)^{-1} \exp \left( \int_{t_0}^{t} \Lambda(x(\tau), \tau) d\tau \right) T(x_0, t_0) \delta x_0
\end{aligned}
$$

which concludes the proof. $\square$

<figure><img src="image_3.png" /></figure>

**Remark V.1** From an operator-theoretic point of view, the pointwise spanning property ensures that the set of eigenfunctions $\{T_1, \dots, T_n\}$ forms a complete basis of left eigenvectors of the operator $\mathcal{A}$ at each point of $\mathcal{X} \times [t_0, \infty)$, therefore allowing the spectral decomposition of the variational state. $\lrcorner$

The result in the theorem has a direct connection with the closed form expression of the variational state expressed via the system flow $\varphi$. Specifically, comparing (8) with the expression for $\delta x(t)$ in Th. V-A.1, we obtain an explicit closed form expression for the state transition matrix of the variational dynamics,

$$ 
\begin{aligned}
\Phi(t, t_0) &= \frac{\partial \varphi}{\partial x_0}(x_0, t; t_0) \\
&= T(x, t)^{-1} \exp \left( \int_{t_0}^{t} \Lambda(x(s), s) ds \right) T(x_0, t_0) \quad (15)
\end{aligned}
$$

and, because of the uniqueness of the flow $\varphi$, the transition matrix is also unique, independently of the choice of pairs $(\lambda_j, T_j)$, $j = 1, \dots, n$, provided they are valid dynamical eigenpairs of the variational state matrix.

### Comparison with the standard eigenproblem

Let $x(t) = \varphi(x_0, t; t_0)$ be a trajectory of system (3), and set, for notational simplicity, $A(t) := \frac{\partial f}{\partial x}(x(t), t) \in \mathbb{R}^{n \times n}$.

Writing explicitly the operator eigenproblem $\mathcal{A}[T_j] = \lambda_j T_j$, yields the left-eigenfunction relation

$$ \frac{dT_j}{dt}(t) + T_j(t) A(t) = \lambda_j(t) T_j(t), \quad \forall t \ge t_0 \quad (16) $$

where we use the shorthand $T_j(t) = T_j(x(t), t) \in \mathbb{C}^{1 \times n}$ and $\lambda_j(t) = \lambda_j(x(t), t) \in \mathbb{C}$.

We refer to the pair $(\lambda_j, T_j)$ solving (16) as the dynamical eigenpair$^4$. Furthermore, let $\mu_j \in \mathbb{C}^1$ and $v_j \in \mathbb{C}^1$, and let $(\mu_j(t), v_j(t)) \in \mathbb{C} \times \mathbb{C}^{1 \times n}$ denote a classical left-eigenpair of $A(t)$, i.e., a solution of the standard eigenproblem

$$ v_j(t) A(t) - \mu_j(t) v_j(t) = 0, \quad \forall t \ge t_0. \quad (17) $$

Comparing the two expressions (16) and (17), we see that the standard left eigenpair $(\mu_j, v_j)$ does not generally solve the operator eigenproblem $\mathcal{A}[T_j] = \lambda_j T_j$ unless $\dot{T}_j$ is identically zero, i.e., unless $v_j = T_j$ is a constant vector (as in the LTI systems case). Rewriting (16) as

$$ \dot{T}_j(t) = T_j(t) (\lambda_j(t) I - A(t)) \quad (18) $$

we observe that, for a prescribed function $\lambda_j$, the map $T_j$ satisfies an LTV ordinary differential equation (ODE). Its evolution can therefore be analyzed using the associated state transition matrix as is standard for LTV systems [1].

If $T_j$ happens to be a classical left eigenfunction of $A(t)$, i.e., if there exists a map $\mu_j$ such that $(\mu_j, T_j)$ is a standard left-eigenpair of $A(t)$ and a map $\lambda_j$ such that $(\lambda_j, T_j)$ is a dynamical left-eigenpair of $A(t)$, where $\mu_j$ does not necessarily agree with $\lambda_j$, i.e., $\mu_j(t) \neq \lambda_j(t)$ for a.a. $t \ge t_0$, then (18) reads as

$$ \dot{T}_j(t) = (\lambda_j(t) - \mu_j(t)) T_j(t) \quad (19) $$

and, for any given initial condition $T_j(t_0) \in \mathbb{C}^n$, the eigenfunction $T_j$, $j = 1, \dots, n$, has a closed form solution

$$ T_j(t) = \exp \left( \int_{t_0}^{t} \lambda_j(\tau) - \mu_j(\tau) d\tau \right) T_j(t_0). \quad (20) $$

However, even if one selects $T_j$ as a standard left eigenfunction of $A(t)$, there is no guarantee that there exists a map $\lambda_j$ making $(\lambda_j, T_j)$ a dynamical eigenpair of $A(t)$. Conversely, prescribing the map $\lambda_j$ to be a classical eigenvalue of $A(t)$ does not ensure that a corresponding dynamical eigenfunction $T_j$ exists.

To obtain the analytic expression of $T_j$, one must be able to solve the first-order linear partial differential equation (PDE) (16), although there are specific situations where the analytic expression can be derived without directly solving the PDE (16). This line of research is currently under development and will be the subject of a future work.

It is important to emphasize the distinction between the dynamical and the standard eigenfunctions:

* the dynamical eigenfunctions $T_j$ describe, in local coordinates, how the directions of the tangent space evolve in time along the system's trajectory for $t \ge t_0$;

<footnote>$^4$ $\lambda_j$ and $T_j$ are referred to as dynamical eigenvalue and dynamical eigenfunction, respectively.</footnote>
• the standard eigenfunctions $v_j$ describe the directions of the tangent space at a fixed point in the space $x \in \mathcal{X}$ at a fixed time $t \in \mathbb{R}$.

Similarly, the distinction between the operators $\mathcal{A}$ and $\mathcal{A}$ is the following:

• $\mathcal{A}$ describes the infinitesimal dynamical evolution of the variational dynamics on the tangent bundle along the system's trajectory $x(t) \in \mathcal{X}$, generalizing the classical analysis around equilibrium points.

• $\mathcal{A}(t)$ describes the instantaneous quasi-static linearization of the dynamics around a fixed point $x \in \mathcal{X}$ for a given time $t \in \mathbb{R}$. This linearization is exact only around equilibrium points $x = \bar{x} \in \mathcal{X}$ such that $f(\bar{x}, t) = 0$, for all $t \ge t_0$.

The set of eigenpairs $(\lambda_j, T_j)$, $j = 1, \dots, n$, is not unique. There may exist infinitely many valid combinations of solutions. As a counterexample to uniqueness, see the case described in Sec. VI-A.2 where a non-dialogizable matrix, in the sense of Jordan, is brought into diagonal form through a dynamical change of coordinates.

To conclude, once the eigenvalue function $\lambda_j$ is fixed, the analytic solution of $T_j$ relies on the solvability of the PDE (16), and not on the classical diagonalizability property of $\mathcal{A}(t)$. Instead, the existence and uniqueness of such a solution are tied to the properties of the LTV system (18), defined along the system trajectories $x(t) = \varphi(x_0, t; t_0)$ for any initial condition $T_j(x_0, t_0)$. In general, the eigenfunctions $T_j$ may not admits an analytic solution, but they can be obtained by integrating the linear ODE (18). The collection of the eigenvalue functions forms a time-varying basis for the tangent space at each point of $\mathcal{X} \times \mathbb{R}$, enabling the explicit diagonalization of the variational dynamics even when $\mathcal{A}(t)$ has repeated eigenvalues.

## B. A more general transformation

The full eigenproblem associated with the operator $\mathcal{A}$ defined in (13), i.e., finding $n$ eigenpairs $(\lambda_j, T_j)$ such that the left eigenfunction $T_j$, $j = 1, \dots, n$, are pointwise linearly independent, can be expressed in matrix form using the matrices $\Lambda, T \in \mathbb{C}^{n \times n}$ as in Th.V-A.1. Specifically, define

$$ \Lambda(x, t) := \operatorname{diag}(\lambda_1(x, t), \dots, \lambda_n(x, t)) $$

and

$$ T(x, t) := \begin{bmatrix} T_1(x, t) \\ \vdots \\ T_n(x, t) \end{bmatrix}. $$

Then the complete left-eigenproblem associated with the operator $\mathcal{A}$ is solved if the matrix pair $(T, \Lambda)$ satisfies the matrix ODE, $\forall t \ge t_0$,

$$ \dot{T}(x, t) + T(x, t) \frac{\partial f}{\partial x}(x, t) = \Lambda(x, t)T(x, t), \quad (21) $$

with the matrix $T(x, t) \in \mathbb{C}^1$ pointwise invertible and $\Lambda \in \mathbb{C}^0$.

Although (21) does not simplify the problem itself, it provides a compact complex-valued matrix representation of the eigenproblem. Moreover, for the stability analysis of the variational dynamics or for obtaining a closed-form expression

of the variational state transition matrix $\Phi(t, t_0)$ such that $\delta x(t) = \Phi(t, t_0) \delta x_0$, it is not necessary for the new coordinates $\delta z = T \delta x$ to be diagonal.

**Theorem V-B.1.** Consider system (3), with $f \in C^1(\mathcal{X} \times [t_0, \infty))$ on the forward invariant manifold $\mathcal{X}$ and its variational dynamics (7) on the tangent bundle $\mathcal{T}\mathcal{X}$. Define real-valued matrix functions $\mathcal{T}, \mathcal{A} : \mathcal{X} \times [t_0, \infty) \to \mathbb{R}^{n \times n}$, with $\mathcal{T} \in C^1(\mathcal{X}, \mathbb{R})$ and $\mathcal{A} \in C^0(\mathcal{X}, \mathbb{R})$. Assume that, for any $\sigma, \tau \in \mathbb{R}$, $\mathcal{A}$ satisfies the commutativity condition

$$ \mathcal{A}(x(\sigma), \sigma)\mathcal{A}(x(\tau), \tau) = \mathcal{A}(x(\tau), \sigma)\mathcal{A}(x(\sigma), \sigma) \quad (22) $$

and that $(\mathcal{T}, \mathcal{A})$ solves the matrix ODE

$$ \dot{\mathcal{T}}(x, t) + \mathcal{T}(x, t) \frac{\partial f}{\partial x}(x, t) = \mathcal{A}(x, t)\mathcal{T}(x, t), \quad \forall t \ge t_0 \quad (23) $$

with $\mathcal{T}(x, t)$ pointwise invertible on $\mathcal{X} \times [t_0, \infty)$. Then the explicit solution of the variational state $\delta x$, for all $t \ge t_0$, is

$$ \delta x(t) = \mathcal{T}(x, t)^{-1} \exp \left( \int_{t_0}^{t} \mathcal{A}(x(\tau), \tau) ds \right) \mathcal{T}(x_0, t_0) \delta x(t_0) \quad (24) $$

*Proof.* The solution pair $(\mathcal{T}, \mathcal{A})$, with $\mathcal{T}(x, t)$ everywhere invertible, allows us to define the change of coordinates $\delta z = \mathcal{T}(x, t)\delta x$, whose dynamics is

$$ \dot{\delta z} = \mathcal{A}(x, t)\delta z, \quad \delta z(t_0) = \mathcal{T}(x_0, t_0)\delta x_0. \quad (25) $$

Thanks to the commutativity assumption (22), the linear time-varying system (25) admits a closed form solution

$$ \delta z(t) = \exp \left( \int_{t_0}^{t} \mathcal{A}(x(\tau), \tau) \, d\tau \right) \delta z(t_0) $$

from which

$$ \delta x(t) = \mathcal{T}(x, t)^{-1} \exp \left( \int_{t_0}^{t} \mathcal{A}(x(\tau), \tau) \, d\tau \right) \mathcal{T}(x_0, t_0) \delta x_0, $$

which proves the claim. $\square$

Equation (24) provides another closed form expression for the state transition matrix of the variational dynamics

$$ \begin{aligned} \Phi(t, t_0) &= \frac{\partial \varphi}{\partial x_0}(x_0, t; t_0) \\ &= \mathcal{T}(x, t)^{-1} \exp \left( \int_{t_0}^{t} \mathcal{A}(x(\tau), \tau) d\tau \right) \mathcal{T}(x_0, t_0), \end{aligned} \quad (26) $$

which is equivalent to (15) by the uniqueness property of $\varphi$. *Remark V.2* Equation (26) shows that the time evolution of $\delta x(t)$ can be decomposed into contributions from $\mathcal{T}(x, t)$ and from $\mathcal{A}(x, t)$, which are dynamically interconnected with the Jacobian of $f(x, t)$ through the matrix equation (23). However, this decomposition reduces to the standard eigenstructure representation since it is well known that constant eigenvectors always provide an admissible solution to the Jordan eigenproblem associated with $\mathcal{A}$, for LTI systems. $\square$

Solving (23) with (22) is less restrictive than solving the full dynamical left-eigenproblem (21) because:
* it removes the constraint that the variational dynamics must be diagonal in the new coordinates;

* it allows an equivalent closed form expression of the variational state evolution thanks to the properties of invertibility of $\mathbb{T}$ and commutativity of $\mathbb{A}$;

* for time invariant vector fields, if $\mathbb{A}(x)$ is diagonal$^5$, the $i$-th row of $\mathbb{T}(x)$, i.e., $\mathbb{T}_i(x)$, associated with the $i$-th diagonal term of $\mathbb{A}$, for $i = 1, \dots, n$, it requires a weaker differentiability condition than the nonlinear left eigenfunctions in (1). Only differentiability in time is needed; see the discussion on the solution of the regulator equations in [61];

* it does not directly require the integrability property in Theorem II-B.1.

It is worth highlighting that the commutativity property (22) is satisfied by

* constant matrices;

* diagonal state-dependent and/or time-varying matrices;

* Jordan block representations (real or complex);

* matrices sharing the same eigenvector basis [62];

* polynomials or analytic functions of the same matrix [63],

* families of matrices that are simultaneously diagonalizable or triangularizable (for instance, commuting symmetric or commuting normal matrices) [62];

* circulant matrices [64];

* and, more generally, elements of the same commutative subalgebra [65].

### C. Differential asymptotic stability criteria

Under the assumption that system trajectories exist for all $t \in [t_0, \infty)$, which is ensured by the forward invariance of the manifold $\mathcal{X}$ under the flow $\varphi$, the following result characterizes the differential regional asymptotic stability of the nonlinear time-varying system (3).

**Proposition V-C.1 (differential Regional Asymptotic Stability).** Let $\mathcal{X}$ be the forward invariant manifold under the flow $\varphi$ of system (3). Assume that the following conditions hold:

a) **Vector field regularity:** the vector field $f(x, t)$ is of class $C^1$ on $\mathcal{X} \times [t_0, \infty)$;

b) **Existence of transformation matrices:** there exist matrix-valued mappings $\mathbb{T}, \mathbb{A}: \mathcal{X} \times [t_0, \infty) \to \mathbb{R}^{n \times n}$, with $\mathbb{T} \in C^1(\mathcal{X} \times [t_0, \infty))$ and $\mathbb{A} \in C^0(\mathcal{X} \times [t_0, \infty))$, satisfying the relation (23) and the commutativity condition (22), where $\mathbb{T}(x, t)$ is pointwise invertible for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$;

c) **Integral asymptotic attractivity:**

$$ \lim_{t \to \infty} \int_{t_0}^{t} \beta_{\mathbb{A}}(x(\tau), \tau) + \frac{d}{d\tau} \ln \left[ \sigma_{\min}(\mathbb{T}(x(\tau), \tau)) \right] d\tau = \infty \quad (27) $$

where the function $\beta_{\mathbb{A}}: \mathcal{X} \times [t_0, \infty) \to \mathbb{R}_{\ge 0}$ is defined as $\beta_{\mathbb{A}}(x, t) = -\lambda_{\max}(\text{sym}(\mathbb{A}(x, t)))$.

Then, system (3) is differentially Regionally Asymptotically Stable.

*Proof.* Since the manifold $\mathcal{X}$ is forward invariant under the flow $\varphi$, the solution $\varphi(x_0, t; t_0)$ exists for all $t \ge t_0$. Moreover, because the vector field $f$ is of class $C^1$, the flow map is continuously differentiable with respect to the initial condition. Consequently, the state-transition matrix $\Phi(t, t_0) = \frac{\partial \varphi}{\partial x_0}(x_0, t; t_0)$ is continuous in time.

Using the transformation induced by $\mathbb{T}$ and the associated representation of the variational dynamics, we obtain the bound

$$ \begin{aligned} \|\Phi(t, t_0)\| & \le \|\mathbb{T}(x, t)^{-1}\| \exp \left( \int_{t_0}^{t} \mathbb{A}(x(\tau), \tau) d\tau \right) \|\mathbb{T}(x_0, t_0)\| \\ & \le \frac{\sigma_{\max}(\mathbb{T}(x_0, t_0))}{\sigma_{\min}(\mathbb{T}(x(t), t))} \exp \left( -\int_{t_0}^{t} \beta_{\mathbb{A}}(x(\tau), \tau) d\tau \right), \end{aligned} $$

where $\beta_{\mathbb{A}}(x, t) = -\lambda_{\max}(\text{sym}(\mathbb{A}(x, t)))$.

Taking the natural logarithm of both sides allows us to separate the contributions of the exponential decay induced by $\mathbb{A}$ and the time variation of the transformation $\mathbb{T}$, yielding

$$ \begin{aligned} \ln \|\Phi(t, t_0)\| & \le \ln \sigma_{\max}(\mathbb{T}(x_0, t_0)) - \ln \sigma_{\min}(\mathbb{T}(x_0, t_0)) \\ & \quad - \int_{t_0}^{t} \beta_{\mathbb{A}}(x(\tau), \tau) + \frac{d}{d\tau} \ln [\sigma_{\min}(\mathbb{T}(x(\tau), \tau))] d\tau. \end{aligned} $$

Exponentiating both sides, we recover the bound

$$ \begin{aligned} \|\Phi(t, t_0)\| & \le \frac{\sigma_{\max}(\mathbb{T}(x_0, t_0))}{\sigma_{\min}(\mathbb{T}(x_0, t_0))} \\ & \exp \left( - \int_{t_0}^{t} \beta_{\mathbb{A}}(x(\tau), \tau) + \frac{d}{d\tau} \ln [\sigma_{\min}(\mathbb{T}(x(\tau), \tau))] d\tau \right). \end{aligned} $$

By assumption (27), the integral in the exponent diverges to $+\infty$ as $t \to \infty$, implying that $\|\Phi(t, t_0)\| \to 0$. Finally, since $\delta x(t) = \Phi(t, t_0) \delta x_0$, it follows that $\|\delta x(t)\| \le \|\Phi(t, t_0)\| \|\delta x_0\| \to 0$, which establishes differential regional asymptotic stability. $\square$

Assumption (27) is sufficient condition that guarantees asymptotic convergence of the variational state, without requiring an exponential decay.

In particular, the attractivity assumption (27) does not impose uniformity on the decay of $\|\Phi(t, t_0)\|$ with respect to time. Such uniformity becomes essential when characterizing differential regional exponential stability, which is addressed in the following result.

**Corollary V-C.1 (differential Regional Exponential Stability).** Let $\mathcal{X}$ be the forward invariant manifold under the flow $\varphi$ of system (3). Assume that:

a) **Vector field smoothness:** the vector field $f(x, t)$ is $C^1$ over $\mathcal{X} \times [t_0, \infty)$;

b) **Transformation matrices:** there exist mappings $\mathbb{T}, \mathbb{A}: \mathcal{X} \times [t_0, \infty) \to \mathbb{R}^{n \times n}$, with $\mathbb{T} \in C^1(\mathcal{X} \times [t_0, \infty))$ and $\mathbb{A} \in C^0(\mathcal{X} \times [t_0, \infty))$ satisfying (23) and (22), and such that $\mathbb{T}(x, t)$ pointwise invertible for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$;

<footnote n="5">⁵By allowing complex-valued solutions $(\mathbb{T}, \mathbb{A})$, i.e., $\mathbb{T}, \mathbb{A}: \mathcal{X} \to \mathbb{C}^{n \times n}$.</footnote>
c) Exponential Attractivity: for all $t_0 \in \mathbb{R}$, there exist positive real constant $\alpha_{t_0}$ such that, $\forall (x, t) \in \mathcal{X} \times [t_0, \infty)$,

$$ \int_{t_0}^{t} \beta_A(x(\tau), \tau) + \frac{d}{d\tau} \ln[\sigma_{\min}(\mathbb{T}(x(\tau), \tau))] d\tau \geq \alpha_{t_0} \cdot (t - t_0). \quad (28) $$

Then system (3) is differentially Regionally Exponentially Stable. Furthermore, if there exist $\mu_T > 0$ and $\alpha > 0$ such that, for all $t_0 \in \mathbb{R}$, $\mu(\mathbb{T}(x_0, t_0)) = \mu_T$ and (28) holds with $\alpha_{t_0} = \alpha$, then system (3) is uniformly differentially Regionally Exponentially Stable. ▼

*Proof.* The proof follows directly from Proposition V-C.1. In particular, for all $t \ge t_0$, the state transition matrix of the variational dynamics satisfies

$$ \begin{aligned} \|\Phi(t, t_0)\| &\le \frac{\sigma_{\max}(\mathbb{T}(x_0, t_0))}{\sigma_{\min}(\mathbb{T}(x_0, t_0))}. \\ \exp\left(-\int_{t_0}^{t} \beta_A(x(\tau), \tau) + \frac{d}{d\tau} \ln[\sigma_{\min}(\mathbb{T}(x(\tau), \tau))] d\tau\right) &< \infty. \end{aligned} $$

Invoking condition (28), this bound reduces to

$$ \|\Phi(t, t_0)\| \le \frac{\sigma_{\max}(\mathbb{T}(x_0, t_0))}{\sigma_{\min}(\mathbb{T}(x_0, t_0))} \cdot \exp(-\alpha_{t_0} \cdot (t - t_0)). $$

Consequently, for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$,

$$ \|\delta x(t)\| \le \kappa \exp(-\alpha_{t_0} (t - t_0)) \|\delta x_0\|, $$

where $\kappa = \mu(\mathbb{T}(x_0, t_0))$. This establishes differential regional exponential stability. The uniform result follows immediately when both $\kappa = \mu_T$ and $\alpha_{t_0} = \alpha$ are independent of the initial time $t_0$. □

### D. Constructing a contractive metric P

Solving (23) with (22) and ensuring that $\mathbb{T}$ is pointwise invertible is a more general problem than obtaining the nonlinear left eigenvectors of $\mathcal{A}$, i.e., finding the pair $(T, \Lambda)$ such that (21) holds and $T$ is pointwise invertible. This formulation is also more general than the nonlinear eigenvalues defined in [5], i.e., the solutions of the eigenproblem in (1).

Within the general framework of (23), one could, in principle, fix a generic matrix $\mathcal{A}$ and then search for a (possibly difficult to find) pointwise invertible matrix $\mathbb{T}$. In particular, if $\mathcal{A}$ is chosen to have a positive definite symmetric part and a pointwise invertible solution $\mathbb{T}$ exists, one can construct a contractive metric $P(x, t)$ for system (3).

**Lemma V-D.1 (Contractive Differential Regional Asymptotically Stable System).** Let $\mathcal{X}$ be a forward invariant manifold under the flow $\varphi$ of system (3). Assume the following:

a) Vector field smoothness: the vector field $f(x, t)$ is $C^1$ over $\mathcal{X} \times [t_0, \infty)$;

b) Transformation matrices: there exist $\mathbb{T}, \mathcal{A} : \mathcal{X} \times [t_0, \infty) \to \mathbb{R}^{n \times n}$, with $\mathbb{T} \in \mathbb{C}^1(\mathcal{X} \times [t_0, \infty))$ and $\mathcal{A} \in \mathbb{C}^0(\mathcal{X} \times) $ satisfying (23), with $\mathbb{T}$ pointwise invertible;

c) Smoothness of the transformation: let $\sigma_{\min}(\mathbb{T}), \sigma_{\max}(\mathbb{T}) \in \mathbb{C}^1(\mathcal{X} \times [t_0, \infty))$. Then, for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$,

$$ \sigma_{\min}(\mathbb{T}(x, t)) \le \|\mathbb{T}(x, t)\| \le \sigma_{\max}(\mathbb{T}(x, t)); \quad (29) $$

d) Asymptotic attractivity:

$$ \lim_{t \to \infty} \int_{t_0}^{t} \beta_A(x(\tau), \tau) + \frac{d}{d\tau} \ln[\sigma_{\min}(\mathbb{T}(x(\tau), \tau))] d\tau = \infty \quad (30) $$

where $\beta_A: \mathcal{X} \times [t_0, \infty) \to \mathbb{R}_{\ge 0}$ is given by $\beta_A(x, t) = -\lambda_{\max}(\text{sym}(\mathcal{A}(x, t)))$.

Then, system (3) has a contractive vector field with respect to the metric $P = \mathbb{T}^\top\mathbb{T}$ and is differentially Regionally Asymptotically Stable (dRAS). ▼

*Proof.* For notational convenience, we eliminate the explicit dependence on $(x, t)$. Define $P = \mathbb{T}^\top\mathbb{T}$, whose time derivative is given by

$$ \dot{P} = 2\mathbb{T}^{\top}\dot{\mathbb{T}} = 2\mathbb{T}^{\top}\left(\mathbb{A}\mathbb{T} - \mathbb{T}\frac{\partial f}{\partial x}\right). $$

Then, for all $(x, t) \in \mathcal{X} \times [t_0, \infty)$,

$$ \begin{aligned} \dot{P} + \mathbb{T}^{\top}\mathbb{T}\frac{\partial f}{\partial x} + \frac{\partial f^{\top}}{\partial x}\mathbb{T}^{\top}\mathbb{T} &= 2\mathbb{T}^{\top}\mathbb{A}\mathbb{T} \\ \dot{P} + P\frac{\partial f}{\partial x} + \frac{\partial f^{\top}}{\partial x}P &= 2\mathbb{T}^{\top}\mathbb{A}\mathbb{T} \le -2\beta_{\mathcal{A}}P. \end{aligned} $$

The last inequality provides the contractive property of the metric $P$, whose smoothness follows from (29). In particular, for each $(x, t) \in \mathcal{X} \times [t_0, \infty)$, we have

$$ 0 < \sigma_{\min}(\mathbb{T}(x, t))^2 I \le P(x, t) \le \sigma_{\max}(\mathbb{T}(x, t))^2 I. $$

Defining the differential Lyapunov function $V = \delta x^\top \mathbb{T}^\top \mathbb{T} \delta x$, we obtain

$$ \dot{V} = 2\delta x^\top \mathbb{T}^\top \mathbb{A} \delta x^\top \le -2\beta_{\mathcal{A}}V, $$

which implies

$$ V(\delta x, x, t) \le \exp\left(-2 \int_{t_0}^{t} \beta_A(x(\tau), \tau) d\tau\right) V(\delta x_0, x_0, t_0). $$

Using the bounds

$$ \sigma_{\min}(\mathbb{T})^2 \|\delta x\|^2 \le V \le \sigma_{\max}(\mathbb{T})^2 \|\delta x\|^2, $$

we obtain

$$ \|\delta x\| \le \frac{\sigma_{\max}(\mathbb{T}(x_0, t_0))}{\sigma_{\min}(\mathbb{T}(x, t))} \exp\left(-\int_{t_0}^{t} \beta_A(x(\tau), \tau) d\tau\right) \|\delta x_0\|. \quad (31) $$

This retrieves the same conditions as in Proposition V-C.1. In particular, under (30), we conclude that $\|\delta x\| \underset{t \to \infty}{\to} 0$, which proves the result. □

The proof is constructive, as it explicitly provides a method to build a contractive metric $P = \mathbb{T}^\top\mathbb{T}$ from the transformation matrix $\mathbb{T}$.
Furthermore, if the transformation $\mathcal{T}$ and the generalized eigenvalue matrix $\mathcal{A}$ admit fixed bounds on $\mathcal{X} \times [t_0, \infty)$, the contractive metric constructed in Lemma V-D.1 directly yields differential Regional Exponential Stability. This is formalized in the following.

**Corollary V-D.1 (Contractive and Differentially Regionally Exponentially Stable System).** Let $\mathcal{X}$ be a forward invariant manifold under the flow $\varphi$ of system (3). Suppose the following conditions hold:

a) *Vector field smoothness: the vector field $f(x, t)$ is $C^1$ over $\mathcal{X} \times [t_0, \infty)$;

b) *Transformation matrices: there exist $\mathcal{T}, \mathcal{A} : \mathcal{X} \times [t_0, \infty) \to \mathbb{R}^{n \times n}$, with $\mathcal{T} \in \mathbb{C}^1(\mathcal{X} \times [t_0, \infty))$ and $\mathcal{A} \in \mathbb{C}^0(\mathcal{X} \times) $ satisfying (23), with $\mathcal{T}$ pointwise invertible*;

c) *Smoothness of the transformation: for all $t_0 \in \mathbb{R}$, there exist positive real constant $\underline{\sigma}_{t_0}$, such that, for each $(x, t) \in \mathcal{X} \times [t_0, \infty)$,

$$ \underline{\sigma}_{t_0} \le \sigma_{\min} \mathcal{T}(x, t) \quad (32) $$

d) *Negative definiteness: there exists a positive real value $\beta_{\mathcal{A}}$ such that*

$$ \operatorname{sym}(\mathcal{A}(x, t)) \le -\beta_{\mathcal{A}} I, \forall (x, t) \in \mathcal{X} \times [t_0, \infty). \quad (33) $$

Then, system (3) has a contractive vector field with respect to the metric $P = \mathcal{T}^\top \mathcal{T}$ and is differentially Regionally Exponentially Stable (dRES).

**Proof.** All the assumptions of this corollary imply those of Lemma V-D.1. In particular, inequality (31), together with the uniform bound $\operatorname{sym}(\mathcal{A}) \le -\beta_{\mathcal{A}} I$, yields, for all $t \ge t_0$

$$ \|\delta x(t)\| \le \frac{\sigma_{\max} \mathcal{T}(x_0, t_0)}{\underline{\sigma}_{t_0}} \exp(-\beta_{\mathcal{A}}(t-t_0)) \|\delta x_0\|. $$

This proves differential Regional Exponential Stability with respect to the contractive metric $P = \mathcal{T}^\top \mathcal{T}$. $\square$

The conditions of Corollary V-D.1 are stronger than those of Corollary V-C.1, as they impose fixed bounds both on the contraction rate and on the conditioning of the induced metric. As in Corollary V-C.1, *uniform* differential Regional Exponential Stability follows immediately whenever these bounds are independent of the initial time $t_0$. Moreover, a weaker (possibly non-uniform) exponential stability property can be recovered by relaxing the assumptions (32) and (33), requiring (28) to hold. The corresponding arguments are omitted to avoid repetition.

**Remark V.3** If $\mathcal{T}$ is constant and full rank, and $\operatorname{sym}(\mathcal{A})$ satisfies (33) globally on $\mathbb{R}^n \times \mathbb{R}$, then the induced metric $P = \mathcal{T}^\top \mathcal{T}$ is constant and uniformly positive definite. In this case, the system is uniformly differentially globally exponentially stable, recovering the classical convergence dynamics framework [35], and the standard LTI system arguments. $\square$

## VI. EXAMPLES

In this section, we provide examples related to the different topics of the paper, illustrating the properties of the transformations $\mathcal{T}$ for LTV, LTI and nonlinear systems.

### A. Linear systems

1) **Example 4.22 in Khalil's book:** Consider the LTV system $\dot{x}(t) = A(t)x(t)$, initialized at $x(t_0) = x_0 \in \mathbb{R}^2$, where

$$ A(t) = \begin{bmatrix} -1 + 1.5 \cos(t)^2 & 1 - 1.5 \sin(t) \cos(t) \\ -1 - 1.5 \sin(t) \cos(t) & -1 + 1.5 \sin(t)^2 \end{bmatrix} $$

whose standard eigenvalues are constant and with negative real part, namely,

$$ \mu_{1,2} = -0.25 \pm i \frac{\sqrt{7}}{4}. $$

Despite the negative real part of $\mu_{1,2}$, the stability properties of the LTV system cannot be inferred from the pointwise eigenvalues of $A(t)$; see [1, Ex. 4.22].

However, the stability property can be determined through the dynamical eigenpair $(\mathcal{T}, \mathcal{A})$, which solves (23) with $\mathcal{T}$ pointwise invertible. We dropped the state-dependence as this is a purely time-varying example. A particular solution of the eigenproblem is given by

$$ \mathcal{T}(t) = \begin{bmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{bmatrix}, \quad \mathcal{A}(t) = \begin{bmatrix} \frac{1}{2} & 0 \\ 0 & -1 \end{bmatrix}. $$

In fact,

$$ \dot{\mathcal{T}} = \begin{bmatrix} -\sin t & -\cos t \\ \cos t & -\sin t \end{bmatrix}, \quad \mathcal{A} \mathcal{T} = \begin{bmatrix} \frac{1}{2} \cos t & -\frac{1}{2} \sin t \\ -\sin t & -\cos t \end{bmatrix}, $$

and, by direct computation,

$$ \begin{aligned} \mathcal{T} \mathcal{A} = & \begin{bmatrix} \cos t & -\sin t \\ \sin t & \cos t \end{bmatrix} \begin{bmatrix} -1 + \frac{3}{2} \cos^2 t & 1 - \frac{3}{2} \sin t \cos t \\ -1 - \frac{3}{2} \sin t \cos t & -1 + \frac{3}{2} \sin^2 t \end{bmatrix} \\ = & \begin{bmatrix} \sin t + \frac{1}{2} \cos t & \cos t - \frac{1}{2} \sin t \\ -\sin t - \cos t & \sin t - \cos t \end{bmatrix}. \end{aligned} $$

thus verifying that

$$ \dot{\mathcal{T}}(t) + \mathcal{T}(t)A(t) - \mathcal{A}(t)\mathcal{T}(t) = \mathbf{0}_{2 \times 2}. $$

Since $\mathcal{T}$ is orthonormal, the matrix pair $(\mathcal{T}, \mathcal{A})$ yields the transition matrix $\Phi(t, t_0)$ via (26). Therefore, $x(t) = \Phi(t, t_0)x_0$, where $\Phi(t, t_0) = \mathcal{T}(t)^\top \exp(\mathcal{A}(t - t_0)) \mathcal{T}(t_0)$, whose entries are

$$ \Phi_{11}(t, t_0) = e^{\frac{1}{2}(t-t_0)} \cos t \cos t_0 + e^{-(t-t_0)} \sin t \sin t_0 $$

$$ \Phi_{12}(t, t_0) = -e^{\frac{1}{2}(t-t_0)} \cos t \sin t_0 + e^{-(t-t_0)} \sin t \cos t_0 $$

$$ \Phi_{21}(t, t_0) = -e^{\frac{1}{2}(t-t_0)} \sin t \cos t_0 + e^{-(t-t_0)} \cos t \sin t_0 $$

$$ \Phi_{22}(t, t_0) = e^{\frac{1}{2}(t-t_0)} \sin t \sin t_0 + e^{-(t-t_0)} \cos t \cos t_0. $$

For $t_0 = 0$, this simplifies to

$$ \begin{aligned} \Phi(t, 0) &= \mathcal{T}(t)^\top \exp(\mathcal{A}t) \mathcal{T}(0) \\ &= \begin{bmatrix} e^{\frac{1}{2}t} \cos t & e^{-t} \sin t \\ -e^{\frac{1}{2}t} \sin t & e^{-t} \cos t \end{bmatrix}, \end{aligned} $$

which coincides with the expression reported in Example 4.22 in [1].
2) LTI system with non diagonalizable state matrix: Consider an LTI system $\dot{x} = Ax$, with $t_0 = 0$ and $x(0) = x_0 \in \mathbb{R}^2$, where A has two repeated eigenvalues $\lambda \in \mathbb{R}$ with geometric multiplicity $g = n = 2$. Without loss of generality, we take A in its Jordan block form, i.e.,

$$A = \begin{bmatrix} \lambda & 1 \\ 0 & \lambda \end{bmatrix},$$

and we search for a (dynamical) invertible transformation matrix

$$T(t) = \begin{bmatrix} a(t) & b(t) \\ c(t) & d(t) \end{bmatrix}, \quad \text{with} \quad T(0) = \begin{bmatrix} a_0 & b_0 \\ c_0 & d_0 \end{bmatrix}$$

that solves (23) and diagonalizes the state matrix. To this end, we fix $\mathbb{A} = \text{diag}(\lambda', \lambda')$ for some $\lambda' \in \mathbb{R}$, not necessarily equal to $\lambda$.

From (23), we write

$$\dot{T}(t) = \begin{bmatrix} (\lambda' - \lambda) a(t) & a(t) + (\lambda' - \lambda) b(t) \\ (\lambda' - \lambda) c(t) & c(t) + (\lambda' - \lambda) d(t) \end{bmatrix} \quad (34)$$

whose solution is $T(t) = \exp((\lambda' - \lambda)t)T_0(t)$, where

$$T_0(t) = \begin{bmatrix} a_0 & b_0 + t a_0 \\ c_0 & d_0 + t c_0 \end{bmatrix}.$$

Thus, the state transition matrix $\Phi(t, 0)$ reads as

$$\begin{aligned} \Phi(t, 0) &= T(t)^{-1} \exp(\lambda' t) T(0) \\ &= T_0(t)^{-1} \exp((\lambda - \lambda')t + \lambda' t) T(0) \\ &= T_0(t)^{-1} \exp(\lambda t) T(0). \end{aligned}$$

Taking $T(0)$ as the identity matrix, the singular values of $T_0(t)$ will read as

$$\begin{aligned} \sigma_{\min}(T_0) &= \frac{t^2 + 2 - \sqrt{t^2(t^2 - 4)}}{2} \\ \sigma_{\max}(T_0) &= \frac{t^2 + 2 + \sqrt{t^2(t^2 - 4)}}{2}. \end{aligned}$$

The integral in (27) reads as

$$-\lambda t + \ln \left( \frac{t^2 + 2 - \sqrt{t^2(t^2 - 4)}}{2} \right)$$

whose limit, for $t \to \infty$, tends to infinity if the sign of $\lambda$ is negative, recovering the classical asymptotic stability condition for LTI systems.

## B. A nonlinear contractive system

Consider the nonlinear homogeneous system in [5, ex.8], with dynamics

$$\begin{aligned} \dot{x}_1 &= -x_2(-1 + 4x_1 + 4x_2 - 4x_2^2), \\ \dot{x}_2 &= -2(x_1 + x_2 - x_2^2) \end{aligned} \quad (35)$$

and Jacobian matrix given by

$$\frac{\partial f}{\partial x}(x) = \begin{bmatrix} -4x_2 & 1 - 8x_2 + 12x_2^2 - 4x_1 \\ -2 & -2 + 4x_2 \end{bmatrix}.$$

In our case, by applying the method proposed in Sec. V-B, we obtain a real-valued solution pair $(T, \mathbb{A})$ to (23) given by

$$T(x) = \begin{bmatrix} 0 & \frac{1}{2} \\ 1 & \frac{1}{2} - 2x_2 \end{bmatrix}, \quad \mathbb{A} = \begin{bmatrix} -1 & -1 \\ 1 & -1 \end{bmatrix}.$$

It is worth noticing that, in this case, $\mathbb{A}$ is not diagonal but constant and, therefore, it satisfies the commutativity condition (22). Furthermore, its symmetric part is everywhere negative definite, which allows us to apply the analysis described in Section V-D. In particular, by defining the metric

$$P(x) = T^{\top}(x)T(x) = \begin{bmatrix} 1 & \frac{1}{2} - 2x_2 \\ \frac{1}{2} - 2x_2 & \frac{1}{4} + (2x_2 - \frac{1}{2})^2 \end{bmatrix}$$

whose principle minor is 1 and its determinant $\det(P) = 1/4 > 0$, we verify that $P(x)$ is everywhere positive definite. For completeness, its inverse is given by

$$P^{-1}(x) = \begin{bmatrix} 16x_2^2 - 8x_2 + 2 & 8x_2 - 2 \\ 8x_2 - 2 & 4 \end{bmatrix}.$$

With the constructed metric, we write the time derivative of the differential Lyapunov function

$$V(\delta x, x) := \delta x^{\top} T^{\top}(x) T(x) \delta x = \delta x^{\top} P(x) \delta x$$

as

$$\begin{aligned} \dot{V} &= 2\delta x^{\top} T^{\top} \left( \dot{T} + T \frac{\partial f(x)}{\partial x} \right) \delta x = 2\delta x^{\top} T^{\top} \mathbb{A} T \delta x \\ &= -2\delta x^{\top} T^{\top} T \delta x = -2\delta x^{\top} P(x) \delta x = -2V, \end{aligned}$$

resulting in $V(\delta x, x) = \exp(-2t)V(\delta x_0, x_0)$. The metric $P$ is lower and upper bound by its eigenvalues $\lambda_{\min}(x)$ and $\lambda_{\max}(x)$, so we have

$$\begin{aligned} \lambda_{\min}(x) \|\delta x(t)\|^2 &\le V(\delta x, x) = \exp(-2t)V(\delta x_0, x_0) \\ &\le \exp(-2t)\lambda_{\max}(x_0) \|\delta x_0\|^2 \end{aligned}$$

and thus

$$\|\delta x(t)\|^2 \le \frac{\lambda_{\max}(x_0)}{\lambda_{\min}(x)} \exp(-2t) \|\delta x_0\|^2.$$

The eigenvalues of $P(x)$ are given by

$$\begin{aligned} \lambda_{\min}(x) &= \frac{1 + 2(2x_2 - \frac{1}{2})^2 - \sqrt{(1 + 2(2x_2 - \frac{1}{2})^2)^2 - 1}}{2}, \\ \lambda_{\max}(x) &= \frac{1 + 2(2x_2 - \frac{1}{2})^2 + \sqrt{(1 + 2(2x_2 - \frac{1}{2})^2)^2 - 1}}{2}. \end{aligned}$$

A straightforward function analysis shows that $\lambda_{\min}(x) \underset{|x_2| \to \infty}{\longrightarrow} 0^+$, implying $\inf_{x \in \mathbb{R}^2} \lambda_{\min}(x) = 0$. Moreover, its maximum is achieved at the points $(x_1^*, x_2^*)$ with $x_2^* = \pm 1/2$, yielding $\max_{x \in \mathbb{R}^2} \lambda_{\min}(x) = 1/2$. However, $\lambda_{\min}(x)$ does not admit a uniform positive lower bound, but can be lower bounded based on the system's initial condition or on the forward invariant region of interest $\mathcal{X} \subset \mathbb{R}^2$. To this end, consider a positive real valued lower bound map $\lambda(x_0)$ defined for any $x_0 \in \mathcal{X}$, such that $\lambda(x_0) \le \lambda_{\min}(x(t))$, for all $t \ge t_0$, then

$$\|\delta x(t)\| \le \sqrt{\frac{\lambda_{\max}(x_0)}{\lambda(x_0)}} \exp(-t) \|\delta x(0)\|.$$

Since $\lambda_{\max}(x_0)$ is upper bounded by a real scalar at any $x_0 \in \mathbb{R}^2$, we conclude that the system (35) is differentially
semi-globally exponentially stable, with a non-uniform scaling factor $\kappa$. Equivalently, this established differential regional exponential stability with fixed bounds. Note that Kawano et al. in [5] proved the global asymptotic stability of the system (35).

# VII. CONCLUSIONS AND FUTURE WORKS

## A. Conclusions

In this work, we developed a comprehensive framework for the analysis of stability and contractivity of nonlinear time-varying dynamical systems. A minimal mathematical background was provided, discussing the operator eigenproblem that plays a central role in the differential stability analysis of nonlinear systems.

We then examined the differential stability properties through the lens of a dynamical spectral decomposition based on an operator formulation. This enabled a comparison with the classical Jacobian-based approach and motivated a more general transformation that relaxes the need for diagonalizability of the state matrix. Using this transformation, we derived criteria for differential asymptotic and exponential stability and constructed contractive metrics valid on regions of the state space. These results pave the way for the systematic design of a metric $P$ that ensures (regional) contractivity, even in systems where Jacobian-based techniques would be insufficient.

The examples provided, including an LTV system, an LTI system with a non-diagonalizable matrix, and a nonlinear system equipped with a constructed contractive metric, illustrate the versatility of the proposed framework and its applicability to a broad class of systems.

Overall, this work lays the foundations for a unified differential approach to stability and contractivity, offering new tools for the study of nonlinear, time-varying, and non-diagonalizable dynamical systems, and opening several promising avenues for future research.

## B. Future Works

Several research directions naturally emerge from the theoretical and methodological developments presented in this work. A first open question concerns the regularity of the spectral elements, namely the smoothness properties of the eigenfunctions $T_j$ and eigenvalues $\lambda_j$ inherited from the operator $\mathcal{A}$. Understanding how the differentiability or continuity of $\frac{\partial f}{\partial x}$ propagates to its dynamical eigenpair $(T_j, \lambda_j)$ is central to establishing stronger stability guarantees and allowing a more refined differential analysis.

A related challenge is the existence and well-posedness of solutions to the associated partial differential equation defining the transformation. This raises a “chicken-egg” dilemma: one may attempt to fix $\lambda_j$ and solve for $T_j$, or fix $T_j$ and solve for $\lambda_j$, but the interplay between them remains subtle. A deeper investigation of this coupled structure is required to determine when consistent solutions exist and to formalize any uniqueness properties.

From a computational standpoint, an important avenue of research is the numerical solution of the transformation problem,

in analogy with the methodology developed for linear time-varying systems. Developing robust numerical and symbolic schemes would significantly broaden the applicability of the framework, enabling automated construction of contractive metrics for nonlinear systems.

Concurrently, the developments in the differential stability presented in Sec. V-C provide a rigorous framework for investigating necessary and sufficient conditions for system instability, complementing traditional approaches based on the state transition matrix and offering a unified, trajectory-centric perspective on stability analysis.

Finally, an ambitious but natural direction is the extension to infinite-dimensional systems, including distributed parameter systems and systems governed by partial differential equations. The operator-based spectral decomposition developed here may offer a promising pathway to unify differential stability analyses in both finite and infinite-dimensional settings.

Overall, these directions highlight the richness of the proposed framework and point toward a broad range of theoretical, computational, and applied developments that can further advance the understanding of differential stability and contractive dynamics.

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**Mario Spirito** Mario Spirito received the B.S. and M.S. (Laurea) degrees cum laude in Automation Engineering from the University of Bologna, Bologna, Italy, in 2016 and 2018, respectively. He earned the Ph.D. degree in Systems Engineering (Automatic Control) from the same institution in 2022. Throughout 2022, he was a Research Fellow with the ACTEMA research group at the University of Bologna. From January 2023 to December 2024, he served as a Postdoctoral Researcher at Université Claude Bernard Lyon 1, Lyon, France, where he was funded by the ANR project IMPACT. Since March 2025, he has been a Postdoctoral Researcher with the Rehab Technologies Lab at the Italian Institute of Technology (IIT), Genoa, Italy. His current work involves control-oriented models for human-robot interaction systems, soft exoskeletons, and upper-limb prostheses.

His research interests include nonlinear control theory, output regulation, and observer design, including nonlinear identity observers and KKL observers. He also focuses on the control of non-minimum phase systems, energy-based modeling of complex multi-physics dynamics (such as port-Hamiltonian systems and SMA), and internal models in neuroscience and biological systems.