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Symbolic Reduction for Formal Synthesis of Global Lyapunov Functions
Jun Liu, Maxwell Fitzsimmons
https://arxiv.org/abs/2506.18171 https://

Symbolic Reduction for Formal Synthesis of Global Lyapunov Functions
We investigate the formal synthesis of global polynomial Lyapunov functions for polynomial vector fields. We establish that a sign-definite polynomial must satisfy specific algebraic constraints, which we leverage to develop a set of straightforward symbolic reduction rules. These rules can be recursively applied to symbolically simplify the Lyapunov candidate, enabling more efficient and robust discovery of Lyapunov functions via optimization or satisfiability modulo theories (SMT) solving. In…

On fast Lyapunov spectra for Markov-R\'{e}nyi maps
Lulu Fang, Carlos Gustavo Moreira, Zhichao Wang, Yiwei Zhang
https://arxiv.org/abs/2506.16291 https:…

On fast Lyapunov spectra for Markov-Rényi maps
In this paper, we study the multifractal analysis for Markov-Rényi maps, which form a canonical class of piecewise differentiable interval maps, with countably many branches and may contain a parabolic fixed point simultaneously, and do not assume any distortion hypotheses. We develop a geometric approach, independent of thermodynamic formalism, to study the fast Lyapunov spectrum for Markov-Rényi maps. Our study can be regarded as a refinement of the Lyapunov spectrum at infinity. We demonst…

A Hybrid Zernike-Lyapunov Framework for Aberration-Based Statistical Wavefront Reconstruction of Chaotic Optical Surfaces
Netzer Moriya
https://arxiv.org/abs/2506.16847

A Hybrid Zernike-Lyapunov Framework for Aberration-Based Statistical Wavefront Reconstruction of Chaotic Optical Surfaces
We present a comprehensive theoretical framework that unifies chaotic wavefront dynamics with classical aberration theory through a Statistical Wavefront Reconstruction Framework (SWRF) formalism. By establishing rigorous connections between ray trajectory deflections and wave-optical phase perturbations through the eikonal equation, we decompose chaotic wavefront perturbations into modified Zernike-Lyapunov hybrid expansions, establishing mathematical equivalences between Lyapunov exponents, f…

Polynomial Eigenfunctions and Matrix Lyapunov Equations from Energy Balance Integrals
Netzer Moriya
https://arxiv.org/abs/2506.15288 https://

Polynomial Eigenfunctions and Matrix Lyapunov Equations from Energy Balance Integrals
We establish a unified theoretical framework that connects classical orthogonal polynomial systems to matrix Lyapunov equations through the fundamental physics of energy dissipation in stochastic dynamical systems. Starting from the energy balance principle in infinite-dimensional Hilbert spaces, we derive a master integral representation that naturally encompasses both spectral geometry and covariance dynamics. The theory reveals that established orthogonal polynomials (Zernike, Hermite, spher…

Replaced article(s) found for eess.SY. https://arxiv.org/list/eess.SY/new
[2/2]:
- Observations on robust diffusive stability and common Lyapunov functions
Blake McGrane-Corrigan, Rafael de Andrade Moral, Oliver Mason

Quantum Fisher information from tensor network integration of Lyapunov equation
Gabriela W\'ojtowicz, Susana F. Huelga, Marek M. Rams, Martin B. Plenio
https://arxiv.org/abs/2506.11330

Quantum Fisher information from tensor network integration of Lyapunov equation
The Quantum Fisher Information (QFI) is a geometric measure of state deformation calculated along the trajectory parameterizing an ensemble of quantum states. It serves as a key concept in quantum metrology, where it is linked to the fundamental limit on the precision of the parameter that we estimate. However, the QFI is notoriously difficult to calculate due to its non-linear mathematical form. For mixed states, standard numerical procedures based on eigendecomposition quickly become impracti…

Dense Phenomena for Ergodic Schr\"odinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent
Artur Avila (Universit\"at Z\"urich,IMPA), David Damanik (Rice University)
https://arxiv.org/abs/2506.14259

Dense Phenomena for Ergodic Schrödinger Operators: I. Spectrum, Integrated Density of States, and Lyapunov Exponent
We consider Schrödinger operators in $\ell^2(\Z)$ whose potentials are defined via continuous sampling along the orbits of a homeomorphism on a compact metric space. We show that for each non-atomic ergodic measure $μ$, there is a dense set of sampling functions such that the associated almost sure spectrum has finitely many gaps, the integrated density is smooth, and the Lyapunov exponent is smooth and positive. As a byproduct we answer a question of Walters about the existence of non-unifor…

Lyapunov analysis for FISTA under strong convexity
Luis M. Brice\~no-Arias
https://arxiv.org/abs/2506.11785 https://arxiv.org/pdf/250…

Lyapunov analysis for FISTA under strong convexity
In this paper, we conduct a theoretical and numerical study of the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) under strong convexity assumptions. We propose an autonomous Lyapunov function that reflects the strong convexity of the objective function, whether it arises from the smooth or non-smooth component. This Lyapunov function decreases monotonically at a linear rate along the iterations of the algorithm for a fixed inertial parameter. Our analysis demonstrates that the best th…

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Some remarks on stochastic converse Lyapunov theorems
Pavel Osinenko, Grigory Yaremenko
https://arxiv.org/abs/2506.06053 https://arxi…

Some remarks on stochastic converse Lyapunov theorems
In this brief note, we investigate some constructions of Lyapunov functions for stochastic discrete-time stabilizable dynamical systems, in other words, controlled Markov chains. The main question here is whether a Lyapunov function in some statistical sense exists if the respective controlled Markov chain admits a stabilizing policy. We demonstrate some constructions extending on the classical results for deterministic systems. Some limitations of the constructed Lyapunov functions for stabili…