Monodromy Operator Research Articles

The Deligne–Simpson Problem and the Connectedness of some Varieties

The Deligne---Simpson problem is formulated as follows: give necessary and sufficient conditions for the choice of the conjugacy classes $C_{j} \subset {\text{GL}}{\left( {n,\mathbb{C}} \right)}$ or $c_{j} \subset {\text{gl}}{\left( {n,\mathbb{C}} \right)}$ so that there exist irreducible (p+1)-tuples of matrices M j ? C j or A j ? c j satisfying the equality M 1 ? M p+1 = I or A 1 + ? + A p+1 = 0. The matrices M j and A j are interpreted as monodromy operators of regular linear systems and as matrices-residues of Fuchsian ones on the Riemann sphere. We prove that in the so-called simple case the subset $\mathcal{W}^{\prime }$ or $\mathcal{V}^{\prime }$ of the variety $\mathcal{W}: = {\left\{ {{\left( {M_{1} , \ldots ,M_{{p + 1}} } \right)}\left| {M_{j} \in C_{j} ,M_{1} \cdots M_{{p + 1}} = I} \right.} \right\}}$ or $\mathcal{V}: = {\left\{ {{\left( {A_{1} , \ldots ,A_{{p + 1}} } \right)}\left| {A_{j} \in c_{j} ,A_{1} + } \right. \cdots + A_{{p + 1}} = 0} \right\}}$ consisting of all irreducible (p+1)-tuples (if nonempty) is connected. "Simple" means that the greatest common divisor of all quantities of Jordan blocks of a given size, of a given matrix M j or A j , and with a given eigenvalue is 1.

Quasi-energy spectral series for a nonlocal Gross-Pitaevskii equation

For a nonlocal Gross-Pitaevskii operator, quasi-energy spectral series have been constructed accurate to O(ħ3/2). These series correspond to the closed phase trajectories of the Hamilton-Ehrenfest system, which are stable in a linear approximation. The corresponding monodromy operator has been constructed accurate to O(ħ3/2) in the class of trajectory-coherent functions.

Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation

A technique for dimensional reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. The DDEs considered here have a canonical form with at most cubic nonlinearities and periodic coefficients. The nonlinear terms are multiplied by a perturbation parameter. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogeneous linear ordinary differential equations (ODEs) with time-periodic coefficients. One set of linear nonhomogeneous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The linear part of the map is the monodromy operator whose eigenvalues characterize stability. Dimensional reduction on the map is then carried out. In the case of critical eigenvalues, this corresponds to center manifold reduction, while for the noncritical case resonance conditions are derived. The accuracy of the nonlinear Chebyshev collocation map is demonstrated by finding the solution of a nonlinear delayed Mathieu equation and then a milling model via the method of steps. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation. The method of this paper is shown by example to be applicable to systems with strong parametric excitations.

On monodromies of a degeneration of irreducible symplectic Kähler manifolds

We study the monodromy operators on the Betti cohomologies associated to a good degeneration of irreducible symplectic manifold and we show that the unipotency of the monodromy operator on the middle cohomology is at least the half of the dimension. This implies that the “mildest” singular fiber of a good degeneration with non-trivial monodromy of irreducible symplectic manifolds is quite different from the generic degeneration of abelian varieties or Calabi–Yau manifolds.

Hyperbolicity of periodic solutions of functional differential equations with several delays

We study conditions for the hyperbolicity of periodic solutions to nonlinear functional differential equations in terms of the eigenvalues of the monodromy operator. The eigenvalue problem for the monodromy operator is reduced to a boundary value problem for a system of ordinary differential equations with a spectral parameter. This makes it possible to construct a characteristic function. We prove that the zeros of this function coincide with the eigenvalues of the monodromy operator and, under certain additional conditions, the multiplicity of a zero of the characteristic function coincides with the algebraic multiplicity of the corresponding eigenvalue.

Gates for quantum computing induced from monodromy operators

The monodromy approach to quantum computing is based on a holomorphic vector bundle with a meromorphic connection. The gates for computation are obtained from the monodromy matrices of the connection.

Error Bounds for Approximate Eigenvalues of Periodic-Coefficient Linear Delay Differential Equations

We describe a new Chebyshev spectral collocation method for systems of variable-coefficient linear delay differential equations with a single fixed delay. Computable uniform a posteriori bounds are given for this method. When the coefficients are periodic, the system has a unique compact nonnormal monodromy operator whose spectrum determines the stability of the system. The spectral method approximates this operator by a dense matrix of modest size. In cases where the coefficients are smooth we observe spectral convergence of the eigenvalues of that matrix to those of the operator. Our main result is a computable a posteriori bound on the eigenvalue approximation error in the case that the coefficients are analytic.

Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems

The problem on determining conditions for the asymptotic stability of linear periodic delay systems is considered. Solving this problem, we use the function space of states. Conditions for the asymptotic stability are determined in terms of the spectrum of the monodromy operator. To find the spectrum, we construct a special boundary value problem for ordinary differential equations. The motion of eigenvalues of this problem is studied as the parameter changes. Conditions of the stability of the linear periodic delay system change when an eigenvalue of the boundary value problem intersects the circumference of the unit disk. We assume that, at this moment, the boundary value problem is self-adjoint. Sufficient coefficient conditions for the asymptotic stability of linear periodic delay systems are given.

Mirror symmetry, Kobayashi's duality, and Saito's duality

M. Kobayashi introduced a notion of duality of weight systems. We tone this notion slightly down to a notion called coupling. We show that coupling induces a relation between the reduced zeta functions of the monodromy operators of the corresponding singularities generalizing an observation of K. Saito concerning Arnold's strange duality. We show that the weight systems of the mirror symmetric pairs of M. Reid's list of 95 families of Gorenstein K3 surfaces in weighted projective 3-spaces are strongly coupled. This includes Arnold's strange duality where the corresponding weight systems are strongly dual in Kobayashi's original sense. We show that the same is true for the extension of Arnold's strange duality found by the author and C. T. C. Wall.

Integral Operators and Delay Differential Equations

The monodromy operator of a linear delay differential equation with periodic coefficients is formulated as an integral operator. The kernel of this operator includes a factor formed from the fundamental solution of the linear delay differential equation. Although the properties of the fundamental solutions are known, in general there is no closed form for the fundamental solution. This paper describes a collocation procedure to approximate the fundamental solution before the integral operator is discretized. Using arguments on collectively compact operators, the eigenvalues of the discretized monodromy operator are shown to converge to the eigenvalues of the monodromy operator in integral form. The eigenvalues of the monodromy operator tell the stability of the linear delay differential equation. An application to several cases of the Van der Pol oscillator with delay will be given. AMS (MOS) Subject Classification. 34K11, 34K13, 34K28, 45P05, 47A75, 47B38, 47G10.

A spectral countability condition for almost automorphy of solutions of differential equations

We consider the almost automorphy of bounded mild solutions to equations of the form (*) dx/dt = A(t)x+f(t) with (generally unbounded) r-periodic A(·) and almost automorphic f(·) in a Banach space X. Under the assumption that X does not contain c 0 , the part of the spectrum of the monodromy operator associated with the evolutionary process generated by A(·) on the unit circle is countable. We prove that every bounded mild solution of (*) on the real line is almost automorphic.

On the Floquet Multipliers of Periodic Solutions to Non-linear Functional Differential Equations

For periodic solutions to the autonomous delay differential equation $$x^{\prime}(t) =-\mu x(t) + f(x(t-1))$$ with rational periods we derive a characteristic equation for the Floquet multipliers. This generalizes a result from an earlier paper where only periods larger than 2 were considered. As an application we obtain a criterion for hyperbolicity of certain periodic solutions, which are rapidly oscillating in the sense that the delay 1 is larger than the distance between consecutive zeros. The criterion is used to find periodic orbits which are unstable and hyperbolic. An example of a non-autonomous periodic linear delay differential equation with a monodromy operator which is not hyperbolic shows how subtle the conditions of the hyperbolicity criteria in the present paper and in its predecessor are. We also derive first results on Floquet multipliers in case of irrational periods. These are based on approximations by periodic solutions with rational periods.

Intersection homology and Alexander modules of hypersurface complements

Let V be a degree d , reduced hypersurface in \mathbb{CP}^{n+1} , n \geq 1 , and fix a generic hyperplane, H . Denote by \mathcal{U} the (affine) hypersurface complement, \mathbb{CP}^{n+1}-V \cup H , and let \mathcal{U}^c be the infinite cyclic covering of \mathcal{U} corresponding to the kernel of the total linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules H_i(\mathcal{U}^c;\mathbb{Q}) of the hypersurface complement and show that, if i \leq n , these are torsion over the ring of rational Laurent polynomials. We also obtain obstructions on the associated global polynomials: their zeros are roots of unity of order d and are entirely determined by the local topological information encoded by the link pairs of singular strata of a stratification of the pair (\mathbb{CP}^{n+1},V) . As an application, we give obstructions on the eigenvalues of monodromy operators associated to

Continuation of Bifurcations in Periodic Delay‐Differential Equations Using Characteristic Matrices

In this paper we describe a method for continuing periodic solution bifurcations in periodic delay-differential equations. First, the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is demonstrated. An alternative formulation of the characteristic matrix is given, which can be computed efficiently. Defining systems of bifurcations are constructed in a standard way, including the characteristic matrix and its derivatives. For following bifurcation curves in two parameters, the pseudo-arclength method is used combined with Newton iteration. Two test examples (an interrupted machining model and a traffic model with driver reaction time) conclude the paper. The algorithm has been implemented in the software tool PDDE-cont.

NUMERICAL COMPUTATION OF CHARACTERISTIC MULTIPLIERS FOR LINEAR TIME PERIODIC COEFFICIENTS DELAY DIFFERENTIAL EQUATIONS

in this work we address the question of asymptotic stability of linear delay differential equations (DDEs) with time periodic coefficients, a class which is recognized to be fundamental in machining tool.Since the dynamics of such a class of delay systems is governed by the dominant eigenvalues (multipliers) of the monodromy operator associated to the system of DDEs, i.e. the solution operator over the period of the coefficients, we discretize it by using pseudospectral differencing techniques based on collocation and approximate the dominant multipliers by the eigenvalues of the resulting matrix. The use of pseudospectral methods has already been proposed in the context of simpler DDEs. Here we fully generalize the method to the class of linear time periodic coefficients DDEs with arbitrary period and multiple discrete and distributed delays.The scheme is shown to have spectral accuracy by means of several numerical examples.

Global Strichartz estimates for the wave equation with time-periodic potentials

We obtain global Strichartz estimates for the solutions u of the wave equation ( ∂ t 2 − Δ x + V ( t , x ) ) u = F ( t , x ) for time-periodic potentials V ( t , x ) with compact support with respect to x. Our analysis is based on the analytic properties of the cut-off resolvent R χ ( z ) = χ ( U ( T ) − z I ) −1 ψ 1 , where U ( T ) = U ( T , 0 ) is the monodromy operator and T > 0 the period of V ( t , x ) . We show that if R χ ( z ) has no poles z ∈ C , | z | ⩾ 1 , then for n ⩾ 3 , odd, we have a exponential decal of local energy. For n ⩾ 2 , even, we obtain also an uniform decay of local energy assuming that R χ ( z ) has no poles z ∈ C , | z | ⩾ 1 , and R χ ( z ) remains bounded for z in a small neighborhood of 0.

Frobenius and monodromy operators in rigid analysis, and Drinfel’d’s symmetric space

We define Frobenius and monodromy operators on the de Rham cohomology of K K -dagger spaces (rigid spaces with overconvergent structure sheaves) with strictly semistable reduction Y Y , over a complete discrete valuation ring K K of mixed characteristic. For this we introduce log rigid cohomology and generalize the so-called Hyodo-Kato isomorphism to versions for non-proper Y Y , for non-perfect residue fields, for non-integrally defined coefficients, and for the various strata of Y Y . We apply this to define and investigate crystalline structure elements on the de Rham cohomology of Drinfel’d’s symmetric space X X and its quotients. Our results are used in a critical way in the recent proof of the monodromy-weight conjecture for quotients of X X given by de Shalit (2005).

Maximal unipotent monodromy for complete intersection CY manifolds

The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the moduli space is called a large radius limit (or large complex structure limit). In this paper we are going to construct one parameter families of n dimensional Calabi-Yau manifolds, which are complete intersections in toric varieties and which have a monodromy operator T such that (T N - id ) n +1 = 0 but (T N - id ) n ≠ 0, i.e., the monodromy operator is maximal unipotent.

On the hyperbolicity of rapidly oscillating periodic solutions of functional differential equations

We consider periodic solutions of nonlinear functional differential equations with rational periods less than 2. We study the spectral properties of monodromy operators and state a hyperbolicity criterion for such solutions.

Zero-curvature formalism of supersymmetric principal chiral model

We investigate one-parameter family of transformation on superfields of super principal chiral model and obtain different zero-curvature representations of the model. The parametric transformation is related to the super Riccati equations and an infinite set of local and non-local conservation laws is derived. A Lax representation of the model is presented which gives rise to a superspace monodromy operator.