Autonomous Linear Dynamical Systems

• Autonomous linear dynamical systems are models defined by intrinsic state evolution, described by linear time-invariant or time-varying equations.
• They play a central role in control theory, signal processing, and system identification by leveraging spectral properties and invariant subspaces for stability analysis.
• Recent advances integrate data-driven learning, adaptive mechanisms, and numerical optimization to enhance system identification, simulation accuracy, and control policy tuning.

Autonomous linear dynamical systems are systems described by linear time-invariant (LTI) or linear time-varying dynamics in which the evolution of the system state proceeds without explicit driving inputs, typically governed by equations such as $\dot{x} = A x$ (continuous time) or $x_{k+1} = A x_k$ (discrete time), where $A$ is a matrix of system coefficients. These systems form the canonical backbone of modern control, signal processing, system identification, and theory of stability, and they serve as the starting point for understanding both natural and engineered processes in mathematics, engineering, and physics. The paper of autonomous linear systems has expanded to embrace not only theoretical analysis but also data-driven learning, optimization under autonomy, and biologically motivated adaptation mechanisms.

1. Mathematical Structure and Theory

An autonomous linear dynamical system evolves according to

$\frac{d}{dt} x(t) = A x(t)$

or, in discrete time,

$x_{k+1} = A x_k,$

with $A \in \mathbb{R}^{n \times n}$. The solution is governed by the spectral properties of $A$. For continuous systems,

$x(t) = e^{A t} x(0),$

and for discrete systems,

$x_k = A^k x_0.$

The local and global behavior—including stability, periodicity, and divergence—is dictated by the spectrum of $A$. For example, if all eigenvalues have negative real part (continuous) or modulus less than one (discrete), the origin is globally asymptotically stable.

For high-order systems (e.g., third order), the structure of phase trajectories, invariant lines, and invariant planes is directly dictated by the eigenstructure. Explicit construction techniques employ orthogonalization (e.g., Gram–Schmidt) to align the coordinate system with invariant manifolds, facilitating analytic and geometric understanding of the phase portrait, especially in the vicinity of a singular point (Puntus et al., 2022).

Extensions to more general settings model the system as a collection of linear maps arranged along a multigraph, with each node $g_i$ associated to a space $L_i$ and each edge a map $A_{ji}$, thereby introducing constraints on allowed sequences of state transitions. The dynamical evolution then depends on products along admissible paths, yielding a behavior characterized by the joint spectral radius (Cicone et al., 2016).

2. Stability, Invariant Structure, and Graph-Based Generalization

The asymptotic behavior and stability of autonomous linear systems is characterized by essential spectral and geometric quantities:

• Spectral radius: For a time-invariant $A$, the largest modulus of any eigenvalue determines stability.
• Joint spectral radius (JSR): In switching or graph-constrained settings, the JSR $$\rho(\xi) = \lim_{k\to\infty} \sup_{|\alpha|=k} \|\Pi_\alpha\|^{1/k}$$ quantifies maximal growth over all allowed trajectories, where $\Pi_\alpha$ denotes a product of matrices along path $\alpha$ (Cicone et al., 2016).
• Invariant (Barabanov) multinorms: For systems on graphs, an extremal multinorm $\{\|\cdot\|_i\}$ satisfies $\max_{A_{ji}} \|A_{ji}x\|_j \leq \rho(\xi)\|x\|_i$, and is used to characterize boundedness and to realize explicit growth bounds.

In traditional settings, the analysis of singular points (e.g., fixed points at the origin) and phase trajectories is based on identifying invariant subspaces and constructing explicit expressions for trajectories in terms of eigenvectors and exponentials. These approaches generalize naturally to block-diagonal or irreducible components under suitable coordinate changes, and invariant subspaces yield a reduction of the analysis to lower-dimensional, indecomposable factors (Cicone et al., 2016, Puntus et al., 2022).

3. Autonomous Learning and Adaptive Mechanisms

Recent frameworks embed slow, time-delayed feedback adaptation into autonomous linear systems, yielding dynamics of the form: $\frac{d}{dt} x = A(w) x,$

$$\tau \frac{d}{dt} w = - [\varepsilon(t) - \varepsilon(t-\Delta)][w(t) - w(t-\Delta)] + S \varepsilon(t) \xi(t),$$

where $w$ are adaptively updated parameters, $\varepsilon(t)$ measures deviation from a target, and parameters are steered by comparing error changes across a delay $\Delta$ (Kaluza et al., 2014). This structure enables the system to autonomously adapt internal parameters in order to achieve prescribed behaviors (e.g., synchronization levels, output function matching), independent of external supervision. The error-driven adaptation via time-delayed feedback provides a form of "memory" of changes in performance, and the inclusion of noise (scaled by error) promotes escape from local minima.

Discrete-time extensions address systems where the error is only intermittently available due to intrinsic temporal processing (e.g., feed-forward neural networks, oscillatory networks). In these cases, parameter updates occur at the end of each processing window, further reconciled with the continuous-time scheme. The update rule takes the form: $$w(n+1) = w(n) - K_T \delta w(n) \delta \varepsilon(n) + \varepsilon(n) S \xi(n),$$ aligning the direction of weight updates with past improvements in performance and including a noise term to promote exploration (Bilen et al., 2016).

4. System Identification and Data-Driven Learning

Reliable identification of autonomous linear systems—estimating $A$ from data—has advanced from analyses based on long, steady-state trajectories to more robust approaches using multiple short, independent trajectories, enabling identification even for unstable $A$ (i.e., when the system does not reach steady state). Subspace identification strategies employ a regression approach: $\hat{G} = Y_+ Y_-^* (Y_- Y_-^*)^{-1},$ where $Y_-$ and $Y_+$ are block vectors of past and future outputs, and $\hat{G}$ encodes the extended observability and controllability properties (Xin et al., 2022). Subsequent balanced realization (e.g., via SVD) recovers estimated system matrices up to similarity. Under Gaussian i.i.d. noise and zero-mean initial states, the estimation error converges as $\mathcal{O}(1/\sqrt{N})$ (number of trajectories $N$), regardless of stability or instability, and can be controlled logarithmically for nonzero-mean initial states by adjusting trajectory length.

Polynomial-time, method-of-moments approaches further stabilize the estimation procedure under minimal assumptions (observability, controllability, marginal stability), leveraging stabilized linear combinations of outputs to control variance and accurately estimate Markov parameters from a single long trajectory (Bakshi et al., 2023). Lower bounds demonstrate the necessity of nondegenerate observability and controllability; otherwise, the system is information-theoretically unidentifiable from reasonable sample sizes.

5. Optimization and Autonomous Policy Tuning

Optimizing autonomous dynamical systems—whether for control, system identification, or behavioral cloning—can be unified through a framework in which all policy and model parameters are embedded into the state evolution, so that the dynamics are given by

$x_{t+1} \sim P(x'|x,\theta),\quad L(x,\theta).$

The standard BeLLMan equation reads

$$V(x, \theta) = L(x, \theta) + \gamma \int P(x'|x, \theta) V(x', \theta) dx',$$

and the objective $J(\theta) = \mathbb{E}_{x \sim P_0}[V(x, \theta)]$ is differentiated with respect to all parameters $\theta$. The DSO (dynamical system optimization) gradient generalizes policy gradients, incorporating both direct parameter effects on the immediate loss and indirect effects via state transition probabilities (2506.08340): $$\nabla_{\theta} J(\theta) = \mathbb{E}_{x \sim \rho(\cdot, \theta)}\left[\nabla_{\theta} L(x, \theta) + \gamma\,\mathbb{E}_{x' \sim P(\cdot|x, \theta)}\left[\nabla_{\theta} \log P(x'|x, \theta) V(x', \theta)\right]\right].$$ This formulation directly recovers standard policy gradients, deterministic gradients, and, when further differentiated, supports Hessian and natural gradient (Fisher metric) methods. Proximal and off-policy learning analogs are also available, unifying a range of learning and optimization procedures including system identification, behavioral cloning, and mechanism design within a single autonomous framework.

6. Numerical Methods and Structure-Preserving Simulation

Numerical integration of autonomous linear systems must balance accuracy, preservation of invariants, and the maintenance of critical dynamical properties such as positivity and correct asymptotic stability. Explicit nonstandard Runge–Kutta (ENRK) schemes replace standard step sizes by functions $\varphi(h)$ satisfying $\varphi(h) = h + \mathcal{O}(h^{p+1})$ to enforce stability and positivity regardless of step size: $$\frac{y_{k+1} - y_k}{\varphi(h)} = b_1 K_1 + b_2 K_2 + \dots + b_s K_s,$$ with stages $$K_i = f(y_k + \varphi(h) \sum_{j=1}^{i-1} a_{ij} K_j)$$. The approach ensures that for suitably chosen $\varphi(h)$,

• the numerical solution retains the original order of accuracy (order $p$),
• positivity of solution components is preserved for all $h$,
• elementary (linear) stability regions include the negative real axis.

Applications to population dynamics, epidemiology, and metapopulation models demonstrate the practical importance of stability-preserving integration for autonomous systems (Dang et al., 2017).

7. Applications, Extensions, and Broader Impact

Autonomous linear dynamical systems underpin a wide spectrum of applications—ranging from modeling biological networks and chemical kinetics (log-linear dynamics (Diamond, 2020)), to digital twins and real-time Bayesian inference for high-dimensional PDE-based models of tsunami propagation (Henneking et al., 24 Jan 2025).

Advanced visualization techniques, such as line integral convolution (LIC) and grid-based critical point extraction, enhance qualitative understanding and education in two-dimensional autonomous systems by revealing subtle flow features, critical points, and separatrices otherwise hidden in sparse vector plots (Müller et al., 2015).

Geometric methods for constructing higher-order first integrals of motion—based on Killing tensors and differential geometric collineations—connect the integrability and superintegrability of autonomous mechanical systems to the classical theory of invariants and symmetries, revealing conserved quantities in both linear and nonlinear settings (Mitsopoulos et al., 2021).

Methods for identification and reduction of coupled linear systems with quadratic outputs leverage advanced model reduction and barycentric approximation frameworks (e.g., AAA algorithm) to match both linear and quadratic input-output behaviors, yielding structurally faithful reduced models with rigorous interpolation guarantees (Gosea et al., 2020).

Modern approaches extend to data-driven learning using recurrent switching linear dynamical systems, variational Bayes filtering, and deep neural network surrogates where the network architecture is explicitly derived from the autonomous system matrices, promoting both rigorous approximation and interpretable simulation (Linderman et al., 2016, Becker-Ehmck et al., 2019, Datar et al., 24 Mar 2024).

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In sum, autonomous linear dynamical systems serve as both a canonical theoretical construct and as a fertile ground for innovations in adaptation, learning, control, modeling, and computational methodology. Their paper continues to yield both foundational principles and practical tools for analyzing, simulating, identifying, and optimizing dynamical phenomena in diverse domains.