Monodromy Operator Research Articles

Seasonal migration dynamics: periodicity, transition delay and finite-dimensional reduction

We study a model of bird migration between the summer breeding ground and the winter refuge site. The model involves time lags for the transition time between the patches, and the model parameters are periodic in time as the biological activities related to the migration and reproduction are seasonal. It has been shown in previous studies that the model system exhibits threshold dynamics: either all solutions converge to the trivial solution, or the system has a positive and globally attractive periodic solution. Two issues remain and will be addressed in this paper: how to express the threshold condition in terms of model parameters explicitly (rather than the abstract spectral radius of a certain monodromy operator) and how to describe the temporal pattern of the positive periodic solution. We make an interesting and surprising observation that the delay differential system is completely characterized by a finite-dimensional ordinary differential system and then a finite-dimensional map in the sense that the bird population at the initial time of spring migration determines the future status of the system. As a consequence, we derive the threshold condition, explicitly in terms of the model parameters, for the extinction and persistence of the considered bird species.

A unified Floquet theory for discrete, continuous, and hybrid periodic linear systems

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) systemʼs poles in the complex plane. We include several examples to show the utility of this theory.

Concatenated solutions of delay-differential equations and their representation with time-delay feedback systems

Stimulated by the monodromy operator approach to time-delay systems (TDSs) developed recently, this article studies the conversion problems of delay-differential equations (DDEs) into the representation as time-delay feedback systems. We give two conversion methods including the conversion of initial conditions, where we show that each of the two methods corresponds, in general, to one of the two different definitions for the solutions of DDEs, called pseudo concatenated solutions and continuous concatenated solution. The study is actually closely related to the subtle behaviours of the solutions of DDEs under discontinuous initial conditions, and simple examples illustrating such subtleties as well as the validity of the conversion methods are also provided. The results of this article suggest that the ability of representing TDSs is higher in the representation as time-delay feedback systems than in the representation as DDEs.

Quantum Monodromy and nonconcentration near a closed semi-hyperbolic orbit

For a large class of semiclassical operators P(h) ― z which includes Schrodinger operators on manifolds with boundary, we construct the Quantum Monodromy operator M(z) associated to a periodic orbit γ of the classical flow. Using estimates relating M(z) and P(h) — z, we prove semiclassical estimates for small complex perturbations of P(h) — z in the case γ is semi-hyperbolic. As our main application, we give logarithmic lower bounds on the mass of eigenfunctions away from semi-hyperbolic orbits of the associated classical flow. As a second application of the Monodromy Operator construction, we prove if γ is an elliptic orbit, then P(h) admits quasimodes which are well-localized near γ.

A Massera type criterion for almost automorphy of nonautonomous boundary differential equations

For abstract linear nonautonomous boundary differential equations with an almost automorphic forcing term, a Massera type criterion is established for the existence of an almost automorphic solution with the help of the spectrum of monodromy operator, which extends the classical theorem due to Massera on the existence of periodic solutions for linear periodic ordinary differential equations.

Instability of Slender Liquid Jet in AC Electric Field of Arbitrary Frequency

In the present work stability of capillary micro-jet of electrolyte solution in alternating longitudinal electric field is investigated theoretically. The gravity effects are neglected. The problem is described by strongly coupled nonlinear system of PDEs for ion transport, electric field and fluid flow under assumption of a viscous Newtonian liquid. The Debye layer thickness is supposed to be small compared with initial jet radius. The Peclet number based on the Debye layer thickness is assumed to be small. These assumptions lead to substantial simplification of the problem. Slender-body theory is used to further simplification of initial statement. Used asymptotic method allows to reduce initially infinite system to three-dimensional ODE with time-periodic coefficients. It is shown that monodromy operator has the only real unstable multiplier. In the case of high-frequency alternating electric field the results showed good agreement with the ones provided by averaging theory.

A CLASSIFICATION OF PERIODIC TIME-DEPENDENT GENERALIZED HARMONIC OSCILLATORS USING A HAMILTONIAN ACTION OF THE SCHRÖDINGER–VIRASORO GROUP

In the wake of a preceding article [31] introducing the Schrodinger–Virasoro group, we study its affine action on a space of (1+1)-dimensional Schrodinger operators with time- and space-dependent potential V periodic in time. We focus on the subspace corresponding to potentials that are at most quadratic in the space coordinate, which is in some sense the natural quantization of the space of Hill (Sturm–Liouville) operators on the one-dimensional torus. The orbits in this subspace have finite codimension, and their classification by studying the stabilizers can be obtained by extending Kirillov's results on the orbits of the space of Hill operators under the Virasoro group. We then explain the connection to the theory of Ermakov–Lewis invariants for time-dependent harmonic oscillators. These exact adiabatic invariants behave covariantly under the action of the Schrodinger–Virasoro group, which allows a natural classification of the orbits in terms of a monodromy operator on L2(ℝ) which is closely related to the monodromy matrix for the corresponding Hill operator.

INTEGRAL CONSTRAINTS ON THE MONODROMY GROUP OF THE HYPERKÄHLER RESOLUTION OF A SYMMETRIC PRODUCT OF A K3 SURFACE

Let S[n]be the Hilbert scheme of length n subschemes of a K3 surface S. H2(S[n],ℤ) is endowed with the Beauville–Bogomolov bilinear form. Denote by Mon the subgroup of GL [H*(S[n],ℤ)] generated by monodromy operators, and let Mon2be its image in OH2(S[n],ℤ). We prove that Mon2is the subgroup generated by reflections with respect to +2 and -2 classes (Theorem 1.2). Thus Mon2does not surject onto OH2(S[n],ℤ)/(±1), when n - 1 is not a prime power.As a consequence, we get counterexamples to a version of the weight 2 Torelli question for hyperKähler varieties X deformation equivalent to S[n]. The weight 2 Hodge structure on H2(X,ℤ) does not determine the bimeromorphic class of X, whenever n - 1 is not a prime power (the first case being n = 7). There are at least 2ρ(n - 1) - 1distinct bimeromorphic classes of X with a given generic weight 2 Hodge structure, where ρ(n - 1) is the Euler number of n - 1.The second main result states, that if a monodromy operator acts as the identity on H2(S[n],ℤ), then it acts as the identity on Hk(S[n],ℤ), 0 ≤ k ≤ n + 2 (Theorem 1.5). We conclude the injectivity of the restriction homomorphism Mon → Mon2, if n ≡ 0 or n ≡ 1 modulo 4 (Corollary 1.6).

On the $p$-adic cohomology of some $p$-adically uniformized varieties

Let $K$ be a finite extension of ${\mathbb Q}_p$ and let $X$ be Drinfel'd's symmetric space of dimension $d$ over $K$. Let $\Gamma\subset {\rm SL}_{d+1}(K)$ be a cocompact discrete (torsionfree) subgroup and let ${X}_{\Gamma}=\Gamma\backslash {X}$, a smooth projective ${K}$-variety. In this paper we investigate the de Rham and log crystalline (log convergent) cohomology of local systems on $X_{\Gamma}$ arising from $K[\Gamma]$-modules. (I) We prove the monodromy weight conjecture in this context. To do so we work out, for a general strictly semistable proper scheme of pure relative dimension $d$ over a cdvr of mixed characteristic, a rigid analytic description of the $d$-fold iterate of the monodromy operator acting on de Rham cohomology. (II) In cases of arithmetical interest we prove the (weak) admissibility of this cohomology (as a filtered $(\phi,N)$-module) and the degeneration of the relevant Hodge spectral sequence.

Fast-Lifting Approach to the Computation of the Spectral Radius of Neutral Time-Delay Systems

This paper discusses a further development of the lifting-based method for continuous-time time-delay systems (TDSs), in which the state transition is viewed in discrete-time and described by the monodromy operator. The method deals with only infinite-dimensional bounded operators without taking any process that directly reduces the infinite-dimensionality to finite-dimensionality, and only uses “pseudo-discretization” induced by the fast-lifting technique. A key role of fast-lifting lies in facilitating an appropriate approximation of the monodromy operator with a tractable one that is still infinite-dimensional, where the latter eventually leads to the reduction to finite-dimensionality. The method is thus purely operator-theoretic and also system-theoretic. Under such a framework, a finite-dimensional computation method of the spectrum of the monodromy operator was derived in a preceding paper, which is ensured to be asymptotically exact as the fast-lifting parameter N tends to infinity but was restricted to retarded time-delay systems. This paper shows that a similar method can be established also for neutral TDSs as far as the spectral radius computation is concerned.

Galois representations attached to Hilbert–Siegel modular forms

This article is a spinoff of the book of Harris and Taylor [HT], in which they prove the local Langlands conjecture for \operatorname{GL}(n) , and its companion paper by Taylor and Yoshida [TY] on local-global compatibility. We record some consequences in the case of genus two Hilbert–Siegel modular forms. In other words, we are concerned with cusp forms \pi on \operatorname{GSp}(4) over a totally real field, such that \pi_{\infty} is regular algebraic (that is, \pi is cohomological). When \pi is globally generic (that is, has a non-vanishing Fourier coefficient), and \pi has a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for \operatorname{GSp}(4) defined by Gan and Takeda in a recent preprint [GT]. Over \mathbb Q , for \pi as above, this leads to a new realization of the Galois representations studied previously by Laumon, Taylor and Weissauer. We are hopeful that our approach should apply more generally, once the functorial lift to \operatorname{GL}(4) is understood, and once the so-called book project is completed. An application of the above compatibility is the following special case of a conjecture stated in [SU]: If \pi has nonzero vectors fixed by a non-special maximal compact subgroup at v , the corresponding monodromy operator at v has rank at most one.

Level-raising for Saito–Kurokawa forms

AbstractThis paper provides congruences between unstable and stable automorphic forms for the symplectic similitude group GSp(4). More precisely, we raise the level of certain CAP representations Π arising from classical modular forms. We first transfer Π toπon a suitable inner formG; this is achieved byθ-lifting. Forπ, we prove a precise level-raising result that is inspired by the work of Bellaiche and Clozel and which relies on computations of Schmidt. We thus obtain a$\tilde {\pi }$congruent toπ, with a local component that is irreducibly induced from an unramified twist of the Steinberg representation of the Klingen parabolic. To transfer$\tilde {\pi }$back to GSp(4), we use Arthur’s stable trace formula. Since$\tilde {\pi }$has a local component of the above type, all endoscopic error terms vanish. Indeed, by results due to Weissauer, we only need to show that such a component does not participate in theθ-correspondence with any GO(4); this is an exercise in using Kudla’s filtration of the Jacquet modules of the Weil representation. We therefore obtain a cuspidal automorphic representation$\tilde {\Pi }$of GSp(4), congruent to Π, which is neither CAP nor endoscopic. It is crucial for our application that we can arrange for$\tilde {\Pi }$to have vectors fixed by the non-special maximal compact subgroups at all primes dividingN. SinceGis necessarily ramified at some primer, we have to show a non-special analogue of the fundamental lemma atr. Finally, we give an application of our main result to the Bloch–Kato conjecture, assuming a conjecture of Skinner and Urban on the rank of the monodromy operators at the primes dividingN.

Circular spectrum and bounded solutions of periodic evolution equations

We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u ′ = A ( t ) u + ϵ H ( t , u ) + f ( t ) , where A ( t ) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ϵ is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of “circular spectrum” and then apply it to study the linear equations u ′ = A ( t ) u + f ( t ) with general conditions on f. For small ϵ we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f, then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f.

Analytic methods in quantum computing

Here we consider a model of quantum computation, based on the monodromy representation of a Fuchsian system. The rôle of local and entangling operators in monodromic quantum computing is played by monodromy matrices of connections with logarithmic singularities acting on the fiber of a holomorphic vector bundle as on the space of qubits. The leading theme is the problem of constructing a set of universal gates as monodromy operators induced from a connection with logarithmic singularity. In the formal scheme developed by us, already known models — topological and holonomic — can be incorporated.

ON THE STABILITY OF PERIODICALLY TIME-DEPENDENT QUANTUM SYSTEMS

The main motivation of this article is to derive sufficient conditions for dynamical stability of periodically driven quantum systems described by a Hamiltonian H(t), i.e. conditions under which it holds true sup t ∈ ℝ|〈ψt, H(t)ψt〉| < ∞ where ψt denotes a trajectory at time t of the quantum system under consideration. We start from an analysis of the domain of the quasi-energy operator. Next, we show, under certain assumptions, that if the spectrum of the monodromy (Floquet) operator U(T, 0) is pure point then there exists a dense subspace of initial conditions for which the mean value of the energy is uniformly bounded in the course of time. Further, we show that if the propagator admits a differentiable Floquet decomposition then ‖H(t)ψt‖ is bounded in time for any initial condition ψ0, and one employs the quantum KAM algorithm to prove the existence of this type of decomposition for a fairly large class of H(t). In addition, we derive bounds uniform in time on transition probabilities between different energy levels, and we also propose an extension of this approach to the case of a higher order of differentiability of the Floquet decomposition. The procedure is demonstrated on a solvable example of the periodically time-dependent harmonic oscillator.

Picard–Lefschetz monodromy of products

Let f and g be reduced homogeneous polynomials in separate sets of variables. We establish a simple formula that relates the eigenspace decomposition of the monodromy operator on the Milnor fiber cohomology of f g to that of f and g separately. We use a relation between local systems and Milnor fiber cohomology that has been established by D. Cohen and A. Suciu.

The Stokes phenomenon in the confluence of the hypergeometric equation using Riccati equation

In this paper we study the confluence of two regular singular points of the hypergeometric equation into an irregular one. We study the consequence of the divergence of solutions at the irregular singular point for the unfolded system. Our study covers a full neighborhood of the origin in the confluence parameter space. In particular, we show how the divergence of solutions at the irregular singular point explains the presence of logarithmic terms in the solutions at a regular singular point of the unfolded system. For this study, we consider values of the confluence parameter taken in two sectors covering the complex plane. In each sector, we study the monodromy of a first integral of a Riccati system related to the hypergeometric equation. Then, on each sector, we include the presence of logarithmic terms into a continuous phenomenon and view a Stokes multiplier related to a 1-summable solution as the limit of an obstruction that prevents a pair of eigenvectors of the monodromy operators, one at each singular point, to coincide.

Isotopies of Legendrian 1-knots and Legendrian 2-tori

We construct a Legendrian 2-torus in the 1-jet space of $S^1 x $\Bbb R$ (or of $\Bbb R^2$) from a loop of Legendrian knots in the 1-jet space of $\Bbb R$. The differential graded algebra (DGA) for the Legendrian contact homology of the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation (linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.

Singular numbers of the monodromy operator and sufficient conditions of the asymptotic stability of periodic system of differential equations with fixed delay

To obtain sufficient conditions for the asymptotic stability of linear periodic systems with fixed delay commensurable with the period of coefficients, singular numbers of the monodromy operator are used. To find these numbers, a self-adjoint boundary value problem for ordinary differential equations is applied. We study the motion of eigenvalues of this boundary value problem under a variation of a parameter. Obtaining sufficient conditions for the asymptotic stability is reduced to finding the bifurcation value of the parameter for the boundary value problem.

Symmetries of the hypergeometric function \phantom{}_{𝑚}𝐹_{𝑚-1}

In this paper, we show that the generalized hypergeometric function m F m − 1 \phantom {}_mF_{m-1} has a one parameter group of local symmetries, which is a conjugation of a flow of a rational Calogero-Mozer system. We use the symmetry to construct fermionic fields on a complex torus, which have linear-algebraic properties similar to those of the local solutions of the generalized hypergeometric equation. The fields admit a nontrivial action of the quaternions based on the above symmetry. We use the similarity between the linear-algebraic structures to introduce the quaternionic action on the direct sum of the space of solutions of the generalized hypergeometric equation and its dual. As a side product, we construct a “good” basis for the monodromy operators of the generalized hypergeometric equation inspired by the study of multiple flag varieties with finitely many orbits of the diagonal action of the general linear group by Magyar, Weyman, and Zelevinsky. As an example of computational effectiveness of the basis, we give a proof of the existence of the monodromy invariant hermitian form on the space of solutions of the generalized hypergeometric equation (in the case of real local exponents) different from the proofs of Beukers and Heckman and of Haraoka. As another side product, we prove an elliptic generalization of Cauchy identity.