Monodromy Matrix Research Articles

Actions of the monodromy matrix elements onto -invariant Bethe vectors

Multiple actions of the monodromy matrix elements onto off-shell Bethe vectors in the -invariant quantum integrable models are calculated. These actions are used to describe recursions for the highest coefficients in the sum formula for the scalar product. For simplicity, detailed proofs are given for the case. The results for the supersymmetric case can be obtained similarly and are formulated without proofs.

Discrete Hirota reductions associated with the lattice KdV equation

We study the integrability of a family of birational maps obtained as reductions of the discrete Hirota equation, which are related to travelling wave solutions of the lattice KdV equation. In particular, for reductions corresponding to waves moving with rational speed N/M on the lattice, where N, M are coprime integers, we prove the Liouville integrability of the maps when N + M is odd, and prove various properties of the general case. There are two main ingredients to our construction: the cluster algebra associated with each of the Hirota bilinear equations, which provides invariant (pre)symplectic and Poisson structures; and the connection of the monodromy matrices of the dressing chain with those of the KdV travelling wave reductions.

Correlation functions of charged free boson and fermion systems

Using the idea of the quantum inverse scattering method, we introduce the operators B(x), C(x) and corresponding to the off-diagonal entries of the monodromy matrix T for the phase model and i-boson model in terms of bc fermions and neutral fermions respectively, thus giving alternative treatment of the KP and BKP hierarchies. We also introduce analogous operators B*(x) and C*(x) for the charged free boson system and show that they are in complete analogy to those of bc fermionic fields. It is proved that the correlation function ⟨0|C(xN)⋯C(x1)B(y1)⋯B(yN)|0⟩ in the bc fermionic fields is the inverse of the correlation function ⟨0|C*(xN)⋯C*(x1)B*(y1)⋯B*(yN)|0⟩ in the charged free bosons.

Application of the Filippov Method to PV‐fed DC‐DC converters modeled as hybrid‐DAEs

SummaryIn this article, a method to study the nonlinear dynamics of a PV‐fed boost converter is developed. The model for the solar panel introduces an algebraic constraint into the system and the resulting mathematical model is a set of hybrid differential algebraic equations (DAEs). The behavior of the system is observed to exhibit undesired operation as parameter values vary. Conventional mathematical tools developed to analyze DC‐DC converters are not directly applicable to systems modeled as a set of hybrid DAEs. In order to find the monodromy matrix to assess the eigenvalues of the system, we modify the Filippov method to calculate the saltation matrix. The derived formula is general and versatile such that it can be applied to any PV‐fed DC‐DC converter irrespective of the topology or control algorithm employed. Two case studies are investigated: current mode control and maximum power point tracking. Each case study serves to demonstrate different considerations that must be taken into account when conducting stability analysis of hybrid‐DAEs.

Floquet spectrum and electronic transitions of tilted anisotropic Dirac materials under electromagnetic radiation: Monodromy matrix approach

We analyze the quasienergy-spectrum and the valence to conduction-band transition probabilities of a tilted anisotropic Dirac material subject to linearly and circularly polarized electromagnetic fields. The quasienergy-spectrum is numerically calculated from the monodromy matrix of the Schr\"odinger equation via the Floquet theorem for arbitrarily intense electromagnetic fields. To asses the valence to conduction-band transition times we deduced a Rabi-like formula in the rotating wave approximation. In the strong-field regime the spectrum as a function of the momentum components divides into two very distinctive regions. In the first, located around the Dirac point, the quasi-spectrum is significantly distorted by the field as the electronic parameters are renormalized by electronic-dressing. In the second, all the characteristics of the free carrier spectrum are retained. Linearly polarized light anisotropically deforms the spectrum according to the field polarization direction. Dirac-like points form around the original Dirac point. The quasi spectrum of circularly polarized light, instead, exhibits a gap formation in the Dirac point and has elliptical symmetry. We show that, in contrast to the single-photon resonant transitions that characterize the weak-field regime, the strong-field regime is dominated by multiphoton resonances.

Designing a hysteresis band in a boost flyback converter

In this paper, we present a method to control a boost flyback converter using a hysteresis band, which is designed using information of the magnetization current and a proportional integral control action of the error. The stability of the periodic orbit is proven via the monodromy matrix, and the robustness, after the first tests of the disturbance rejection, is proven using bifurcation diagrams. When the period-1 orbit losses the stability, it disappears and gives rise to other period-1 orbit with different topological sequence. In all cases that we tested, the system always presented a good performance in a period-1 orbit with very low error.

STABILITY ANALYSIS OF PARALLEL CONNECTED DC-DC CONVERTERS

Parallel controlled DC-DC converters are nonlinear and non-smooth systems, they show various nonlinear behaviour including smooth, non-smooth bifurcation, and chaos when they work outer their design conditions. Usually, the Poincaré map approach is the most common method for studying the stability of those nonlinear systems. Stability is indicated using the eigenvalues of the Jacobian of the map computed at the fixed point. The other method is the monodromy matrix approach, where the stability can be concluded by computed the eigenvalues of the matrix. In this paper, the nonlinear dynamics of parallel connected DC-DC converters are investigated. It is shown that the concept of the monodromy matrix can be applied to determine the stability of the system as well as the Poincare map approach.

Stability and bifurcation analysis of a bevel gear system supported by finite-length squeeze film dampers

To analyze the dynamic characteristics of gear systems supported by squeeze film dampers (SFD), the nonlinear oil-film force of SFD is usually obtained by the short bearing approximation (SBA), long bearing approximation (LBA) or finite difference method (FDM). However, the SBA and LBA methods only hold for the cases of infinitely short and infinitely long SFD, which may be not true in practice. Additionally, the FDM method is generally applied to the case of the regular film boundary. Hence, the present work proposes a finite element method to achieve the film pressure of finite-length SFDs (FLSFD) based on the variational principle. The proposed method is not plagued with the boundary conditions and is verified by the comparison with the classic methods. Then, a seven-degree-of-freedom dynamic model of a bevel gear system with FLSFD is developed incorporating the nonlinear film force. Based on Gram–Schmidt QR-decomposition, a strategy to calculate the Lyapunov spectrum of the high-dimensional gear system is presented, and the characteristic multipliers of the system are obtained by solving the eigenvalues of the monodromy matrix. The Lyapunov exponents, characteristic multipliers, and bifurcation diagrams, as well as phase portraits and Poincare sections, are utilized to qualify the nonlinear behaviors of the bevel gear system with and without FLSFD. The results show that the application of FLSFD can effectively reduce the occurrences of saddle-node bifurcation, Hopf bifurcation, and period-doubling, and suppress nonlinear characteristics like the bistable response and jump phenomenon.

Simulation and stability analysis of periodic flexible multibody systems

The dynamic response of many flexible multibody systems of practical interest is periodic. The investigation of such problems involves two intertwined tasks: first, the determination of the periodic response of the system and second, the analysis of the stability of this periodic solution. Starting from Hamilton’s principle, a unified solution procedure for continuous and discontinuous Galerkin methods is developed for these two tasks. In the proposed finite element formulation, the unknowns consist of the displacement and rotation components at the nodes, which are interpolated via the dual spherical linear interpolation technique. Periodic solutions are obtained by solving the discrete nonlinear equations resulting from continuous and discontinuous Galerkin methods. The monodromy matrix required for stability analysis is constructed directly from the Jacobian matrix of the solution process. Numerical examples are presented to validate the proposed approaches.

The monodromy matrix construction for executive object of a nonlinear system.

The article reveals one of a monodromy matrix constructing methods, reveals the essence of the simplest construction and calculation such matrix. This method is used to build a actuator mathematical model, which can be used to study transients and steady-state processes.

The Lyapunov exponents and the neighbourhood of periodic orbits

ABSTRACT We show that the Lyapunov exponents of a periodic orbit can be easily obtained from the eigenvalues of the monodromy matrix. It turns out that the Lyapunov exponents of simply stable periodic orbits are all zero, simply unstable periodic orbits have only one positive Lyapunov exponent, doubly unstable periodic orbits have two different positive Lyapunov exponents, and the two positive Lyapunov exponents of complex unstable periodic orbits are equal. We present a numerical example for periodic orbits in a realistic galactic potential. Moreover, the centre manifold theorem allowed us to show that stable, simply unstable, and doubly unstable periodic orbits are the mothers of families of, respectively, regular, partially, and fully chaotic orbits in their neighbourhood.

Weyl metrics and Wiener-Hopf factorization

We consider the Riemann-Hilbert factorization approach to the construction of Weyl metrics in four space-time dimensions. We present, for the first time, a rigorous proof of the remarkable fact that the canonical Wiener-Hopf factorization of a matrix obtained from a general (possibly unbounded) monodromy matrix, with respect to an appropriately chosen contour, yields a solution to the non-linear gravitational field equations. This holds regardless of whether the dimensionally reduced metric in two dimensions has Minkowski or Euclidean signature. We show moreover that, by taking advantage of a certain degree of freedom in the choice of the contour, the same monodromy matrix generally yields various distinct solutions to the field equations. Our proof, which fills various gaps in the existing literature, is based on the solution of a second Riemann-Hilbert problem and highlights the deep role of the spectral curve, the normalization condition in the factorization and the choice of the contour. This approach allows us to construct explicit solutions, including new ones, to the non-linear gravitational field equations, using simple complex analytic results.

On the Calculation of the Spectral Bands of One-Dimensional Photonic Crystals

In this work we consider the propagation of transverse electromagnetic waves in one-dimensional photonic crystals consisting of periodic arrays of slabs. On the basis of the Floquet theory we obtain the characteristic function of the periodic medium, which defines the photonic bands of the crystal. The characteristic function is constructed from the monodromy matrices of the slabs. Entries of monodromy matrices are explicitly given as power series of the spectral parameter. The present analysis can be applied not only to homogeneous slabs, but also to slabs with varying refractive indexes. The power series representation of the entries of monodromy matrices leads to an effective numerical method for the calculation of the spectral bands of one-dimensional photonic crystals.

Stability Analysis of Voltage Controlled Buck Converter Feed From a Periodic Input

The ripples of dc input voltage belong to cascaded systems are usually close to the sinusoidal voltage, and its stability is quite different from that of dc constant voltage converters. Under the premise of no loss of generality, this article takes the output voltage of front-end converter in a cascaded system as sine wave, discusses the principle of closed-loop switching converters driven by sinusoidal input voltage, and calculates the average value of fixed-point state variables by using a discrete-time mathematical model. In addition, Filippov method is adopted to predict bifurcation by calculating the eigenvalues of Monodromy matrix belonging to the whole system, and the mechanism of nonlinear phenomena is also addressed. The results show that bifurcation points change obviously under different input conditions. Moreover, the influence of sinusoidal ripples amplitude and phase shift on bifurcation behavior is discussed, the compensation technique is used to prevent bifurcation and obtain the extended stability region of the circuit parameters. The experimental results prove the validity of the analysis.

Lax pairs for string Newton Cartan geometry

In this paper, based on a systematic formulation of Lax pairs, we show classical integrability for nonrelativistic strings propagating over stringy Newton-Cartan (NC) geometry. We further construct the corresponding monodromy matrix which leads to an infinite tower of non-local conserved charges over stringy NC background.

Stabilization of chaotic systems via act-and-wait delayed feedback control using a high-precision direct integration method

A technique for the numerical analysis of the stability problem in chaotic systems via act-and-wait delayed feedback control is developed and justified. Recently, act-and-wait modification of a delayed feedback control method is proposed to stabilize unstable periodic orbits in dynamical systems. To overcome the difficulties in obtaining conditions under which the state of closed-loop system converges toward a periodic solution via the act-and-wait scheme, this paper presents a high-precision direct integration method for calculating the monodromy matrix corresponding to the closed-loop system based on the theory of Peano–Baker series. The capabilities of the proposed schemes in stabilization of unstable periodic orbits of chaotic systems are illustrated by numerical examples.

T, Q and periods in SU(3) mathcal{N} = 2 SYM

We consider the third order differential equation derived from the deformed Seiberg-Witten differential for pure mathcal{N} = 2 SYM with gauge group SU(3) in Nekrasov- Shatashvili limit of Ω-background. We show that this is the same differential equation that emerges in the context of Ordinary Differential Equation/Integrable Models (ODE/IM) correspondence for 2d A2 Toda CFT with central charge c = 98. We derive the corresponding QQ and related T Q functional relations and establish the asymptotic behaviour of Q and T functions at small instanton parameter q → 0. Moreover, numerical integration of the Floquet monodromy matrix of the differential equation leads to evaluation of the A-cycles a1,2,3 at any point of the moduli space of vacua parametrized by the vector multiplet scalar VEVs (tr \U0001d7192) and (tr \U0001d7193) even for large values of q which are well beyond the reach of instanton calculus. The numerical results at small q are in excellent agreement with instanton calculation. We conjecture a very simple relation between Baxter’s T -function and A-cycle periods a1,2,3, which is an extension of Alexei Zamolodchikov’s conjecture about Mathieu equation.

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.

Analysis of a distant retrograde orbit in the Hill three-body problem

In this study, spacecraft orbital motion in the Hill three-body problem is considered. When the secondary body moves in a circular orbit whose center is the primary body, a distant retrograde orbit (DRO) exists on the plane of revolution of the secondary body. In this study, a three-dimensional DRO including an out-of-plane motion is investigated. First, the motion in each direction is assumed as a simple trigonometric motion, and relations between the ratios of the amplitudes and the ratios of the angular velocities are derived. Then, a stability criterion of the three-dimensional DRO is obtained from the maximum eigenvalues of the monodromy matrix. A comparative study is executed between the present analytical results and the numerical simulations, and it validates the analytical results.

New boundary monodromy matrices for classical sigma models

The 2d principal models without boundaries have G×G symmetry. The already known integrable boundaries have either H×H or GD symmetries, where H is such a subgroup of G for which G/H is a symmetric space while GD is the diagonal subgroup of G×G. These boundary conditions have a common feature: they do not contain free parameters. We have found new integrable boundary conditions for which the remaining symmetry groups are either G×H or H×G and they contain one free parameter. The related boundary monodromy matrices are also described.