Monodromy Operator Research Articles

Almost periodic solutions of periodic second order linear evolution equations

The paper is concerned with periodic linear evolution equations of the form $x''(t)=A(t)x(t)+f(t)$, where $A(t)$ is a family of (unbounded) linear operators in a Banach space $X$, strongly and periodically depending on $t$, $f$ is an almost (or asymptotic) almost periodic function. We study conditions for this equation to have almost periodic solutions on ${\mathbb R}$ as well as to have asymptotic almost periodic solutions on ${\mathbb R}^+$. We convert the second order equation under consideration into a first order equation to use the spectral theory of functions as well as recent methods of study. We obtain new conditions that are stated in terms of the spectrum of the monodromy operator associated with the first order equation and the frequencies of the forcing term $f$.

On Stability of Small Periodic Solutions

In this paper, we consider normal time-periodic systems of ordinary differential equations whose right-hand sides smoothly depend on phase variables and small parameters. Branching conditions for small periodic solution of the system are found. Lyapunov stability tests for small solutions with respect to parameters or variables are established. Our reasoning is based on the analysis of the first nonlinear approximation of the monodromy operator.

A combinatorial description of the monodromy of log curves

Let $$k$$ be an algebraically closed field of characteristic $$0$$ . For a log curve $$X/k^{\times }$$ over the standard log point (Kato in Int J Math 11(2):215–232, 2000), we define (algebraically) a combinatorial monodromy operator on its log-de Rham cohomology group. The invariant part of this action has a cohomological description, it is the Du Bois cohomology of $$X$$ (Du Bois in Bull Soc Math Fr 109(1):41–81, 1981). This can be seen as an analogue of the invariant cycles exact sequence for a semistable family (as in the complex, etale and $$p$$ -adic settings). In the specific case in which $$k={\mathbb {C}}$$ and $$X$$ is the central fiber of a semistable degeneration over the complex disc, our construction recovers the topological monodromy and the classical local invariant cycles theorem. In particular, our description allows an explicit computation of the monodromy operator in this setting.

Pseudospectral method for assessing stability robustness for linear time‐periodic delayed dynamical systems

SummaryThe article presents a pseudospectral approach to assess the stability robustness of linear time‐periodic delay systems, where periodic functions potentially present discontinuities and the delays may also periodically vary in time. The considered systems are subject to linear real‐valued time‐periodic uncertainties affecting the coefficient matrices, and the presented method is able to fully exploit structure and potential interdependencies among the uncertainties. The assessment of robustness relies on the computation of the pseudospectral radius of the monodromy operator, namely, the largest Floquet multiplier that the system can attain within a given range of perturbations. Instrumental to the adopted novel approach, a solver for the computation of Floquet multipliers is introduced, which results into the solution of a generalized eigenvalue problem which is linear w.r.t. (samples of) the original system matrices. We provide numerical simulations for popular applications modeled by time‐periodic delay systems, such as the inverted pendulum subject to an act‐and‐wait controller, a single‐degree‐of‐freedom milling model and a turning operation with spindle speed variation.

Spectrum-based Stability Analysis and Stabilization of Time-periodic Time-delay Systems

We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet multipliers, where global approximation is combined with local corrections. Correspondingly, we also propose two novel characterizations of left eigenvectors. Finally, from the nonlinear eigenvalue problem formulation, we derive computationally tractable expressions for derivatives of Floquet multipliers with respect to parameters, which are beneficial in the context of stability optimization. Several numerical examples show the efficacy and applicability of the presented results.

Robust stability of milling operations based on pseudospectral approach

The prediction of chatter-free machining parameters is not reliable for industrial applications without guaranteed robustness against modeling uncertainties. Measurement inaccuracies, fitting uncertainties, simplifications and modeling errors typically lead to a mismatch between the mathematical model and the real physical system. This paper presents the robust stability analysis of milling operations based on a pseudospectral approach to take into account the effect of bounded parametric uncertainties both in the cutting coefficients and in the modal parameters. In order to make the predictions more accurate, the operational modal analysis of the spindle during rotation was conducted. The natural frequencies and damping ratios of the dominant vibration modes were identified from the impact tests of the rotating tool at different spindle speeds. The uncertainties of the fitted parameters were included in the computation of the pseudospectral radius of the monodromy operator of the time-periodic system. The solver is tested in a case study and experimental chatter tests are also included for demonstration purposes.

Global structure of subharmonics in a class of periodic predator-prey models

This paper ascertains the global topological structure of the set of subharmonics of arbitrary order of the periodic predator-prey model introduced in López-Gómez et al (1996 Adv. Differ. Equ. 1 403–23). By constructing the iterates of the monodromy operator of the system, it is shown that the system admits subharmonics of all orders for the appropriate ranges of values of the parameters. Then, some sharp results of topological nature in the context of global bifurcation theory provide us with the fine topological structure of the components of subharmonics emanating from the T-periodic coexistence state.

Solution Space Monodromy of a Special Double Confluent Heun Equation and Its Applications

We consider three linear operators determining automorphisms of the solution space of a special double confluent Heun equation of positive integer order (L-operators). We propose a new method for describing properties of the solution space of this equation based on using eigenfunctions of one of the L-operators, called the universal L-operator. We construct composition laws for L-operators and establish their relation to the monodromy transformation of the solution space of the special double confluent Heun equation. We find four functionals quadratic in eigenfunctions of the universal automorphism; they have a property with respect to the considered equation analogous to the property of the first integral. Based on them, we construct matrix representations of the L-operators and also the monodromy operator. We give a method for extending solutions of the special double confluent Heun equation from the subset Re z > 0 of a complex plane to a maximum domain on which the solution exists. As an example of its application to the RSJ model theory of overdamped Josephson junctions, we give the explicit form of the transformation of the phase difference function induced by the monodromy of the solution space of the special double confluent Heun equation and propose a way to continue this function from a half-period interval to any given interval in the domain of the function using only algebraic transformations.

Harmonic stability of standing water waves

A numerical method is developed to study the stability of standing water waves and other time-periodic solutions of the free-surface Euler equations using Floquet theory. A Fourier truncation of the monodromy operator is computed by solving the linearized Euler equations about the standing wave with initial conditions ranging over all Fourier modes up to a given wave number. The eigenvalues of the truncated monodromy operator are computed and ordered by the mean wave number of the corresponding eigenfunctions, which we introduce as a method of retaining only accurately computed Floquet multipliers. The mean wave number matches up with analytical results for the zero-amplitude standing wave and is helpful in identifying which Floquet multipliers collide and leave the unit circle to form unstable eigenmodes or rejoin the unit circle to regain stability. For standing waves in deep water, most waves with crest acceleration below $A_c=0.889$ are found to be linearly stable to harmonic perturbations; however, we find several bubbles of instability at lower values of $A_c$ that have not been reported previously in the literature. We also study the stability of several new or recently discovered time-periodic gravity-capillary or gravity waves in deep or shallow water, finding several examples of large-amplitude waves that are stable to harmonic perturbations and others that are not. A new method of matching the Floquet multipliers of two nearby standing waves by solving a linear assignment problem is also proposed to track individual eigenvalues via homotopy from the zero-amplitude state to large-amplitude standing waves.

Analysis of a SIR model with pulse vaccination and temporary immunity: Stability, bifurcation and a cylindrical attractor

A time-delayed SIR model with general nonlinear incidence rate, pulse vaccination and temporary immunity is developed. The basic reproduction number is derived and it is shown that the disease-free periodic solution generically undergoes a transcritical bifurcation to an endemic periodic solution as the vaccination coverage drops below a critical level. Using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution as the vaccination coverage is decreased and a Hopf point is detected. This leads to a bifurcation to an attracting, invariant cylinder. As the vaccination coverage is further decreased, the geometry of the cylinder contracts along its length until it finally collapses to a periodic orbit when the vaccination coverage goes to zero. In the intermediate regime, phase locking on the cylinder is observed.

Almost Periodic Solutions of Periodic Linear Partial Functional Differential Equations

We study conditions for the abstract periodic linear functional differential equation $\dot{x}=Ax+F(t)x_t+f(t)$ to have almost periodic with the same structure of frequencies as $f$. The main conditions are stated in terms of the spectrum of the monodromy operator associated with the equation and the frequencies of the forcing term $f$. The obtained results extend recent results on the subject. A discussion on how the results could be extended to the case when $A$ depends on $t$ is given.

A Kulikov-type classification theorem for a one parameter family of K3-surfaces over a p-adic field and a good reduction criterion

In this paper, we prove a $p$-adic analogous of the Kulikov-Persson-Pinkham classification theorem [Persson:1981wp] for the central fiber of a degeneration of $K3$-surfaces in terms of the nilpotency degree of the monodromy of the family. Namely, let $X_K$ be a be a smooth, projective $K3$-surface which has a minimal semi-stable model $X$ over $\mathcal O_K$. If we let $N_{st}$ be the monodromy operator on $D_{st}(H^2_{et}}(X_{\overline K},\mathbb Q_p))$, then we prove that the degree of nilpotency of $N_{st}$ determines the type of the special fiber of $X$. As a consequence we give a criterion for the good reduction of the semi-stable $K3$-surface $X_K$ over the $p$-adic field $K$ in terms of its $p$-adic representation $H^2_{\text{et}}(X_{\overline K},\mathbb Q_p)$, which is similar to the criterion of good reduction for $p$-adic abelian varieties and curves given by Coleman-Iovita and Andreatta-Iovita-Kim.

Asymptotics for periodic systems

This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class of dissipative systems arising naturally in applications. For this class of systems we analyse in detail the spectral properties of the associated monodromy operator, showing in particular that it is a so-called Ritt operator under a natural ‘resonance’ condition. This allows us to deduce from our general result a precise description of the asymptotic behaviour of the corresponding solutions. In particular, we present conditions for rational rates of convergence to periodic solutions in the case where the convergence fails to be uniformly exponential. We illustrate our general results by applying them to concrete problems including the one-dimensional wave equation with periodic damping.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

Stokes Phenomenon and Confluence in Non-autonomous Hamiltonian Systems

This article studies a confluence of a pair of regular singular points to an irregular one in a generic family of time-dependent Hamiltonian systems in dimension 2. This is a general setting for the understanding of the degeneration of the sixth Painlevé equation to the fifth one. The main result is a theorem of sectoral normalization of the family to an integrable formal normal form, through which is explained the relation between the local monodromy operators at the two regular singularities and the non-linear Stokes phenomenon at the irregular singularity of the limit system. The problem of analytic classification is also addressed.

On the Milnor Monodromy of the Exceptional Reflection Arrangement of Type $G_{31}$

Combining recent results by A. D. Mačinic et al. [Doc. Math. (Bielefeld) 22, 135–150 (2017; Zbl 1358.14020)] with a spectral sequence and computer aided computations, we determine the monodromy action on H^1(F,\Bbb C) , where F denotes the Milnor fiber of the hyperplane arrangement associated to the exceptional irreducible complex reflection group G_{31} . This completes the description given by the first author of such monodromy operators for all the other irreducible complex reflection groups.

Finite Dimensional Approximation of the Monodromy Operator of a Periodic Delay Differential Equation with Piecewise Constant Orthonormal Functions

Using piecewise constant orthonormal functions, an approximation of the monodromy operator of a Linear Periodic Delay Differential Equation (PDDE) is obtained by approximating the integral equation corresponding to the PDDE as a linear operator over the space of initial conditions. This approximation allows us to consider the state space as finite dimensional resulting in a finite matrix approximation whose spectrum converges to the spectrum of the monodromy operator.

Superconformal index, BPS monodromy and chiral algebras

We show that specializations of the 4d mathcal{N}=2 superconformal index labeled by an integer N is given by Tr ℳN where ℳ is the Kontsevich-Soibelman monodromy operator for BPS states on the Coulomb branch. We provide evidence that the states enumerated by these limits of the index lead to a family of 2d chiral algebras {mathcal{A}}_N . This generalizes the recent results for the N = −1 case which corresponds to the Schur limit of the superconformal index. We show that this specialization of the index leads to the same integrand as that of the elliptic genus of compactification of the superconformal theory on S2× T2 where we turn on frac{1}{2}N units of U(1)r flux on S2.

Orbifold Milnor lattice and orbifold intersection form

For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define orbifold versions of the monodromy operator on the quantum (co)homology group, of the Milnor lattice, of the Seifert form and of the intersection form. We also describe some symmetry properties of invariants of invertible polynomials refining the known ones.

Homology graph of real arrangements and monodromy of Milnor fiber

We study the first homology group H1(F,C) of the Milnor fiber F of sharp arrangements A‾ in PR2. Our work relies on the minimal complex C⁎(S(A)) of the deconing arrangement A and its boundary map. We describe an algorithm which computes possible eigenvalues of the monodromy operator h1 of H1(F,C). We prove that, if a condition on some intersection points of lines in A is satisfied, then the only possible nontrivial eigenvalues of h1 are cubic roots of the unity. Moreover we give sufficient conditions for just eigenvalues of order 3 or 4 to appear in cases in which this condition is not satisfied.