Monodromy Operator Research Articles

Stability and Bifurcations in Time-Delay Systems: A Novel and Efficient Blend of Methods

The goal of this paper is to introduce a hybrid technique, combining two existing methods conveniently, to analyze the stability and bifurcations of equilibrium points and periodic orbits in delay-differential equations of retarded type. One method is called the frequency-domain approach, an analytical tool based on control theory allowing to detect and represent periodic solutions emerging from the Hopf bifurcations, via Fourier series and harmonic balances. The second one, the so-called semi-discretization method, provides a numerical scheme to approximate the monodromy operator of a periodic solution, thus permitting to establish the stability and bifurcations of limit cycles as well as analyzing the equilibrium points. It is shown that a proper combination of these methods provides a straightforward strategy to study both local and global bifurcations in time-delay systems. The usefulness of the proposed approach is shown by three examples, where the results are compared with those given by the software DDE-BIFTOOL.

On integrability of the one-dimensional Hubbard model

We find a family of solutions to Zamolodchikov's tetrahedral algebra corresponding to the fermionic R-operator for the free fermion model of the difference type in one of the spectral parameters, construct an extension of the R-operator for a system of two spins satisfying the Yang-Baxter equation, and find the local charges. We also construct a twisted monodromy operator, which leads to the one-dimensional Hubbard model.

Spectra of evolution operators of a class of neutral renewal equations: Theoretical and numerical aspects

In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear periodic equation with one discrete delay and fully characterize the spectrum of its monodromy operator. We perform numerical experiments discretizing the evolution operators via pseudospectral collocation, confirming the theoretical results and giving perspectives on the generalization to systems and to multiple delays. Although we do not attempt to perform a rigorous numerical analysis of the method, we give some considerations on a possible approach to the problem.

Non-linear instability of periodic orbits of suspensions of thin fibers in fluids

This paper is concerned with difficulties encountered by engineers when they attempt to predict the orientation of fibers in the creation of injection molded plastic parts. It is known that Jeffery's equation, which was designed to model a single fiber in an infinite fluid, breaks down very badly when applied, with no modifications, to this situation. In a previous paper, the author described how interactions between the fiber orientation and the viscosity of the suspension might cause instability, which could result in the simple predictions from Jeffery's equation being badly wrong. In this paper, we give some rigorous proofs of instability using Floquet Theory. We also give an example where the spectal radius of the monodromy operator is one, but there is still non-linear instability.

Complete Description of Bounded Solutions for a Duffing-Type Equation with a Periodic Piecewise Constant Coefficient

We consider the equation $u_{xx}^{}-u+W(x)u^3=0$ where $W(x)$ is a periodic alternating piecewise constant function. It is proved that under certain conditions for $W(x)$ solutions of this equation, which are bounded on $\mathbb{R}$, $|u(x)|<\xi$, can be put in one-to-one correspondence with bi-infinite sequences of numbers $n\in \{-N,\,\ldots,\,N\}$ (called ``codes'' of the solutions). The number $N$ depends on the bounding constant $\xi$ and the characteristics of the function $W(x)$. The proof makes use of the fact that, if $W(x)$ changes sign, then a ``great part'' of the solutions are singular, i.e., they tend to infinity at a finite point of the real axis. The nonsingular solutions correspond to a fractal set of initial data for the Cauchy problem in the plane $(u,\,u_x^{})$. They can be described in terms of symbolic dynamics conjugated with the map-over-period (monodromy operator) for this equation. Finally, we describe an algorithm that allows one to sketch plots of solutions by its codes.

The local Langlands correspondence for GLn over function fields

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze, we give a new proof of the local Langlands correspondence for $\GL_n$ over $F$. More specifically, we construct $\ell$-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map $\DeclareMathOperator{\rec}{rec}\pi\mapsto\rec(\pi)$ from isomorphism classes of irreducible smooth representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$. Our map $\rec$ is characterized in terms of a local compatibility condition on traces of a certain test function $f_{\tau,h}$, and we prove that $\rec$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).

The essential spectrum of periodically stationary pulses in lumped models of short‐pulse fiber lasers

AbstractIn modern short‐pulse fiber lasers, there is significant pulse breathing over each round trip of the laser loop. Consequently, averaged models cannot be used for quantitative modeling and design. Instead, lumped models, which are obtained by concatenating models for the various components of the laser, are required. As the pulses in lumped models are periodic rather than stationary, their linear stability is evaluated with the aid of the monodromy operator obtained by linearizing the round‐trip operator about the periodic pulse. Conditions are given on the smoothness and decay of the periodic pulse that ensure that the monodromy operator exists on an appropriate Lebesgue function space. A formula for the essential spectrum of the monodromy operator is given, which can be used to quantify the growth rate of continuous wave perturbations. This formula is established by showing that the essential spectrum of the monodromy operator equals that of an associated asymptotic operator. Since the asymptotic monodromy operator acts as a multiplication operator in the Fourier domain, it is possible to derive a formula for its spectrum. Although the main results are stated for a particular experimental stretched pulse laser, the analysis shows that they can be readily adapted to a wide range of lumped laser models.

Universality of Descendent Integrals over Moduli Spaces of Stable Sheaves on K3 Surfaces

We interprete results of Markman on monodromy operators as a universality statement for descendent integrals over moduli spaces of stable sheaves on K3 surfaces. This yields effective methods to reduce these descendent integrals to integrals over the punctual Hilbert scheme of the K3 surface. As an application we establish the higher rank Segre-Verlinde correspondence for K3 surfaces as conjectured by Göttsche and Kool.

Spectral criteria to stability and observability of mean-field stochastic periodic systems

This paper addresses stability and observability of discrete-time mean-field stochastic systems with periodic coefficients and multiplicative noise. By constructing a monodromy operator associated to the periodic coefficients of the considered dynamics, spectral criterion and Lyapunov-type criterion are presented for asymptotic mean square stability and weak stability, respectively. Based on the Lyapunov criterion, a necessary and sufficient condition is supplied for regional stabilizability. Furthermore, Popov–Belevitch–Hautus (PBH) criterion is obtained for observability of mean-field stochastic periodic systems. By the proposed spectral criterion of stability/observability, the intrinsic relationships are clarified for stability/observability among deterministic periodic systems, stochastic periodic systems and mean-field stochastic periodic systems. Finally, as an application of PBH criterion, a Barbashin–Krasovskii stability theorem is established to indicate the connection between observability and stability of the considered systems.

On the Branching of a Large Periodic Solution of a System of Differential Equations with a Parameter

We study a normal periodic system of ordinary differential equations with a small parameter, which is quasilinear in a neighborhood of infinity, under the assumption that the right-hand side of the system has a critical linear approximation. In terms of the properties of the first homogeneous nonlinear approximation of the monodromy operator, we obtain conditions for the existence of a periodic solution whose initial value is infinitely large for an infinitesimal value of the parameter.

Parameter Stability Under Permanent Perturbations

For a normal periodic system of ordinary differential equations with a small parameter, we examine the stability property of the origin under the assumption that the right-hand side of the system has a critical linear approximation. The stability conditions are formulated in terms of estimates of the monodromy operator.

Continuous spectrum of periodically stationary pulses in a stretched-pulse laser.

A spectral method for determining the stability of periodically stationary pulses in fiber lasers is introduced. Pulse stability is characterized in terms of the spectrum (eigenvalues) of the monodromy operator, which is the linearization of the round trip operator about a periodically stationary pulse. A formula for the continuous (essential) spectrum of the monodromy operator is presented, which quantifies the growth and decay of continuous waves far from the pulse. The formula is verified by comparison with a fully numeric method for an experimental fiber laser. Finally, the effect of a saturable absorber on pulse stability is demonstrated.

In-process impulse response of milling to identify stability properties by signal processing

In this work we present a quantitative measurement-based method to describe the stability behaviour of a periodic dynamical system. In-process response for impulse during milling is analysed to provide operational stability prediction. The proposed method combines chatter detection techniques with the theory of time-periodic delay differential equations applied for general milling operations. Signal processing based on dynamic modal decomposition approach not only presents qualitative properties (like stability/instability) but also quantifies a measure of stability through the determination of the Floquet multipliers. The corresponding monodromy operator of the milling process is approximated from the measured system’s response during machining operation. By changing the technological parameters, the variation of the modulus of the Floquet multipliers can be monitored. The stability limit can precisely be interpolated with the unitary multiplier; furthermore, the stability limit can be extrapolated while the manufacturing parameters remain in the chatter-free region. The presented approach is validated by numerical results and laboratory tests.

Piecewise discretization of monodromy operators of delay equations on adapted meshes

<p style='text-indent:20px;'>Periodic solutions of delay equations are usually approximated as continuous piecewise polynomials on meshes adapted to the solutions' profile. In practical computations this affects the regularity of the (coefficients of the) linearized system and, in turn, the effectiveness of assessing local stability by approximating the Floquet multipliers. To overcome this problem when computing multipliers by collocation, the discretization grid should include the piecewise adapted mesh of the computed periodic solution. By introducing a piecewise version of existing pseudospectral techniques, we explain why and show experimentally that this choice is essential in presence of either strong mesh adaptation or nontrivial multipliers whose eigenfunctions' profile is unrelated to that of the periodic solution.</p>

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.

Asymptotic behavior of solutions of periodic linear partial functional differential equations on the half line

We study conditions for the abstract linear functional differential equation x ˙ = A x + F ( t ) x t + f ( t ) , t ≥ 0 to have asymptotic almost periodic solutions, where F ( ⋅ ) is periodic, f is asymptotic almost periodic. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the circular spectrum of the forcing term f . The obtained results extend recent results on the subject.

Automorphic vector bundles on the stack ofG-zips

AbstractFor a connected reductive groupGover a finite field, we study automorphic vector bundles on the stack ofG-zips. In particular, we give a formula in the general case for the space of global sections of an automorphic vector bundle in terms of the Brylinski-Kostant filtration. Moreover, we give an equivalence of categories between the category of automorphic vector bundles on the stack ofG-zips and a category of admissible modules with actions of a 0-dimensional algebraic subgroup a Levi subgroup and monodromy operators.

On the fixed-time stabilization of input delay systems using act-and-wait control

Abstract We address the state feedback stabilization of linear systems with input delay using the so-called act-and-wait control strategy. The latter approach induces an analysis of the closed-loop stability based on a finite-dimensional monodromy operator, and rendering this matrix nilpotent results in fixed-time stability. We show that any controllable planar system can be stabilized in a fixed time, and the constructive proof reveals that always two isolated solutions for the corresponding controller gain co-exist. Next, for the fixed-time stabilization of general linear systems, we present both a novel numerical approach for the computation of the controller gain, and an analytic approach relying on incorporating a predictor and an appropriately defined feedback transformation in the control scheme. Several illustrations complete the presentation.

The essential spectrum of periodically stationary solutions of the complex Ginzburg–Landau equation

We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic–quintic complex Ginzburg–Landau equation about a periodically stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg–Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically stationary pulses in ultrafast fiber lasers.

Eigenvalues and delay differential equations: periodic coefficients, impulses and rigorous numerics

We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We consider systems with a single delay where the period is at most equal to the delay, and the latter two are commensurate. We first represent the monodromy operator (period map) as an operator acting on a product of sequence spaces that represent the Chebyshev coefficients of the state-space vectors. Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation error in addition to some other technical operator norms, this leads to the method being suitable to computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs on two example problems. We also test our discretization scheme in floating point arithmetic on a gamut of randomly-generated high-dimensional examples with both periodic and constant coefficients to inspect the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check for periodic systems) for increasing numbers of Chebyshev modes.