arXiv subjects

Swapnil Tripathi

Publications and source records attributed to Swapnil Tripathi.

11 recordsLinked to original sources

arXiv2026

Construction of Lyapunov density for nonautonomous dynamical systems on hypertorus

We present a semidefinite programming framework for constructing time-varying Lyapunov densities for nonautonomous dynamical systems on a hypertorus. The formulation leverages Gram matrix representations of hybrid (real-trigonometric) polynomials. In addition, we introduce a novel block decomposition of these Gram representations to confine the blow-up of the resulting density to a prescribed set. The results are then applied to establish the almost global synchronization of a time-varying Kuramoto model and the robust almost-global stability of a parameter-varying nonautonomous system. These examples demonstrate the applicability of the proposed method and validate the theoretical results. All computational results are obtained using an open-source MATLAB implementation, as referenced in the text, thereby facilitating reproducibility of the reported examples.

math.DS↗

arXiv2026

Dual Lyapunov-based Synchronization Control of R\"ossler System

This paper proposes a novel approach for the synchronization problem of nonlinear dynamical systems, integrating dual Lyapunov stability analysis with polynomial optimization. A comprehensive review of the relevant scientific literature on synchronization methods is conducted, with a particular focus on classical Lyapunov-based methods for chaotic systems. In this study, the R\"ossler system is synchronized by employing dual Lyapunov-based closed-loop synchronization method. This method uses semidefinite programming and sum-of-squares polynomials to compute a nonlinear state feedback function which synchronize a chaotic system to a selected reference model. It is aimed that chaotic behavior is destroyed and, instead, a limit cycle becomes attracting. Simulation works are performed for randomly selected 100 different initial conditions to show that synchronization process is successfully performed. Furthermore, bifurcation diagrams and phase portraits are evaluated to analyze the system dynamics. The paper discusses results and how new constraints should be employed and adapted to more complex systems.

math.DS↗

arXiv2026

Semidefinite Programming Certificates for Synchronization of Kuramoto Oscillators on Arcs

A class of Kuramoto models with a general coupling function that can be expressed in terms of a finite number of harmonics, each comprising sinusoidal terms, is studied. We propose a novel approach for certifying local phase synchronization in this class for all initial conditions lying on an arc. The trace parametrization property and Gram matrix representation of a trigonometric polynomial are utilized along with Putinar's Positivstellensatz to obtain semidefinite programming certificates for the stability of the phase-difference system, which in turn implies synchronization of the original system. The results can be extended to any system of coupled oscillators where the forward-invariance on arcs can be established.

math.DS↗

arXiv2023

Stability of Reset and Impulsive Continuous-time Linear Switched Systems

We study stability issue of reset and impulsive switched systems. We find time constraints (dwell time and flee time) on switching signals which stabilize a given reset switched system. For a given collection of matrices, we find an assignment of resets and time constraints on switching signals which guarantee stability of the reset switched system. Similar results are obtained for impulsive switched systems as well. Two techniques, namely, analysis of flow of the system and the multiple Lyapunov function approach is used to obtain the results. The results are later generalized to obtain mode-dependent time constraints for stability of these systems.

math.DS↗

arXiv2022

Stability of Bimodal Planar Switched Linear Systems with Both Stable and Unstable Subsystems

We study dynamics of a bimodal planar linear switched system with a Hurwitz stable and an unstable subsystem. For given \textit{flee time} from the unstable subsystem, the goal is to find corresponding \textit{dwell time} in the Hurwitz stable subsystem so that the switched system is asymptotically stable. The dwell-flee relations obtained are in terms of certain smooth functions of the eigenvalues and (generalized) eigenvectors of the subsystem matrices. The results are extended to a special class of symmetric bilinear systems. The results are also extended to a multimodal planar linear switched system in which the switching is governed by an undirected star graph, where the internal node corresponds to a Hurwitz stable (unstable, respectively) subsystem and all the leaves correspond to unstable (Hurwitz stable, respectively) subsystems. For this multimodal system, dwell-flee relations are obtained as solutions of a minimax optimization problem.

math.DS↗

arXiv2021

Stability of bimodal planar linear switched systems

We consider bimodal planar switched linear systems and obtain dwell time bounds which guarantee their asymptotic stability. The dwell time bound obtained is a smooth function of the eigenvectors and eigenvalues of the subsystem matrices. An optimal scaling of the eigenvectors is used to strengthen the dwell time bound. A comparison of our bounds with the dwell time bounds in the existing literature is also presented.

math.DS↗

arXiv2012

Stellar Structure Equations in Extended Palatini Gravity

We consider static spherically symmetric stellar configurations in Palatini theories of gravity in which the Lagrangian is an unspecified function of the form $f(R,R_{μν}R^{μν})$. We obtain the Tolman-Oppenheimer-Volkov equations corresponding to this class of theories and show that they recover those of $f(R)$ theories and General Relativity in the appropriate limits. We show that the exterior vacuum solutions are of Schwarzschild-de Sitter type and comment on the possible expected modifications, as compared to GR, of the interior solutions.

gr-qc↗

arXiv2012

Static Spherically Symmetric Solutions in Extended Palatini Gravity

We consider static spherically symmetric stellar configurations in Palatini theories of gravity in which the Lagrangian is an unspecified function of the form f(R,R_{μν}R^{μν}). We obtain the Tolman-Oppenheimer-Volkov equations corresponding to this class of theories and show that they recover those of f(R) theories and General Relativity in the appropriate limits. We compute exterior vacuum solutions and comment on the possible expected modifications, as compared to GR, of the interior solutions.

gr-qc↗

arXiv2010

Enriched Phenomenology in Extended Palatini Theories

We show that extended theories of gravity with Lagrangian f(R,R_{μν}R^{μν}) in the Palatini formulation possess a phenomenology much richer than the simpler f(R) or f(R_{μν}R^{μν}) theories. In fact, we find that the scalars R and Q=R_{μν}R^{μν} can be written as algebraic functions of the energy density and pressure of the energy momentum tensor. In the simpler cases of f(R) or f(R_{μν}R^{μν}) theories, R and Q are just functions of the trace T of the energy-momentum tensor. As a result, in radiation dominated universes f(R) and f(R_{μν}R^{μν}) theories exhibit the same dynamics as general relativity with an effective cosmological constant. This is not the case of f(R,R_{μν}R^{μν}) models, in which R=R(ρ,P) and Q=Q(ρ,P) and, therefore, modified dynamics exists even for traceless sources.

gr-qc↗

arXiv2009

Dynamical Aspects of Generalized Palatini Theories of Gravity

We study the field equations of modified theories of gravity in which the lagrangian is a general function of the Ricci scalar and Ricci-squared terms in Palatini formalism. We show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary metric which, in particular cases of interest, is related with the physical metric by means of a disformal transformation. This relation between physical and auxiliary metric boils down to a conformal transformation in the case of f(R) theories. We also show with explicit models that the inclusion of Ricci squared terms in the action can impose upper bounds on the accessible values of pressure and density, which might have important consequences for the early time cosmology and black hole formation scenarios. Our results indicate that the phenomenology of f(R_{ab}R^{ab}) theories is much richer than that of f(R) and f(R_{ab}R^{ab}) theories and that they also share some similarities with Bekenstein's relativistic theory of MOND.

gr-qc↗

arXiv2009

Examining the Field of Static Point Scalar Charge in Schwarzschild Spacetime

In this paper we examine some aspects of the field of a scalar point charge in curved spacetimes. First we find the closed form solution for the scalar field due to a point charge in Schwarzschild spacetime. Then we expand it locally in powers of r (coordinate distance from the charge) and compare it to Quinn's local expansion for the field of scalar charge. We show a term by term match, except for a mismatch in one term arising due to an error in a Riemann tensor term in Quinn's expression. We show the correct expression, the detailed derivation of which will be subject of a later paper.

gr-qc↗