Monodromy Matrix Research Articles

Floquet theory and stability analysis for Hamiltonian PDEs

Abstract We analyze Floquet theory as it applies to the stability and instability of periodic traveling waves in Hamiltonian PDEs. Our investigation focuses on several examples of such PDEs, including the generalized KdV and BBM equations (third order), the nonlinear Schrödinger and Boussinesq equations (fourth order), and the Kawahara equation (fifth order). Our analysis reveals that the characteristic polynomial of the monodromy matrix inherits symmetry from the underlying PDE, enabling us to determine the essential spectrum along the imaginary axis and bifurcations of the spectrum away from the axis, employing the Floquet discriminant. We present numerical evidence to support our analytical findings.

Dimers for type D relativistic Toda model

We construct dimer graphs for type D relativistic Toda models by introducing impurities into the Y2N,0 square dimer graphs. By properly placing the impurities and changing the canonical variables assigned to the 1-loops on the dimer graph, we perform a “folding” of the graphs, which yields the type D relativistic Toda lattice Hamiltonian and monodromy matrix.

Koopman–Hill stability computation of periodic orbits in polynomial dynamical systems using a real-valued quadratic harmonic balance formulation

In this paper, we generalize the Koopman–Hill projection method, which was recently introduced for the numerical stability analysis of periodic solutions, to be included immediately in classical real-valued harmonic balance (HBM) formulations. We incorporate it into the Asymptotic Numerical Method (ANM) continuation framework, providing a numerically efficient stability analysis tool for frequency response curves obtained through HBM. The Hill matrix, which carries stability information and follows as a by-product of the HBM solution procedure, is often computationally challenging to analyze with traditional methods. To address this issue, we generalize the Koopman–Hill projection stability method, which extracts the monodromy matrix from the Hill matrix using a matrix exponential, from complex-valued to real-valued formulations. In addition, we propose a differential recast procedure, which makes this real-valued Hill matrix immediately available within the ANM continuation framework. Using as an example a nonlinear von Kármán beam, we demonstrate that these modifications improve computational efficiency in the stability analysis of frequency response curves.

Bifurcation analysis of piecewise-smooth engineering systems with delays through numeric continuation of periodic orbits

This paper presents a numeric continuation framework for periodic orbits of piecewise-smooth and hybrid dynamical systems with fixed point delays. For the numeric solution of the corresponding infinite dimensional multi-point boundary value problem, a novel discretization and interpolation scheme is developed employing Chebyshev polynomial based spectral collocation techniques. The same approach is employed for the formulation of the corresponding monodromy matrix enabling stability analysis on the found periodic orbits. Special care is attributed to the accurate detection of discontinuity induced bifurcations such as grazing and sliding, and the implemented pseudo-arclength framework is adapted to allow two parameter continuation of these critical points. The capabilities of the developed algorithms are demonstrated on a set of delayed piecewise-smooth and hybrid dynamical systems, showcasing potential engineering applications from the fields of control theory, traffic dynamics modelling, and machine tool vibrations. Finally, a detailed tutorial is attached in the appendix to accompany the open-source release of the developed codebase.

On Monodromy Matrices for a Difference Schrödinger Equation on the Real Line with a Small Periodic Potential

On Monodromy Matrices for a Difference Schrödinger Equation on the Real Line with a Small Periodic Potential

Transient modes for the coupled modified Korteweg-de Vries equations with negative cubic nonlinearity: Stability and applications of breathers.

Dynamics and properties of breathers for the modified Korteweg-de Vries equations with negative cubic nonlinearities are studied. While breathers and rogue waves are absent in a single component waveguide for the negative nonlinearity case, coupling can induce regimes of modulation instabilities. Such instabilities are correlated with the existence of rogue waves and breathers. Similar scenarios have been demonstrated previously for coupled systems of nonlinear Schrödinger and Hirota equations. Both real- and complex-valued modified Korteweg-de Vries equations will be treated, which are applicable to stratified fluids and optical waveguides, respectively. One special family of breathers for coupled, complex-valued equations is derived analytically. Robustness and stability of breathers are studied computationally. Knowledge of the growth rates of modulation instability of plane waves provides an instructive prelude on the robustness of breathers to deterministic perturbations. A theoretical formulation of the linear instability of breathers will involve differential equations with periodic coefficient, i.e., a Floquet analysis. Breathers associated with larger eigenvalues of the monodromy matrix tend to suffer greater instability and increased tendency of distortion. Predictions based on modulation instability and Floquet analysis show excellent agreements. The same trend is obtained for simulations conducted with random noise disturbances. Linear approaches like modulation instabilities and Floquet analysis, thus, generate a very illuminating picture of the nonlinear dynamics.

Method for determining the Lyapunov exponent of a continuous model using the monodrome matrix

The Lyapunov exponent is a measure of the sensitivity of a dynamic system to any changes and disturbances. It is therefore applicable in many fields of science, including physics, chemistry, economics, psychology, biology, medicine, and technology. Therefore, there is a need to have effective methods for determining the value of this exponent. In particular, it concerns the definition of its sign. The positive value of the Lyapunov exponent confirms the sensitivity of the system, especially when the system is chaotic. A negative value indicates the stability of the dynamic system being tested. The numerical value of the Lyapunov exponent indicates the degree of sensitivity of the dynamic system under study. Although the mathematical definition of the Lyapunov exponent is clear and simple, in practice determining its value is not a trivial matter, especially for continuous models. This paper presents a method for determining the Lyapunov exponent for continuous models. This method is based on the monodromy matrix. The work, for example, presents research on the following models of physical systems: the Van der Pol model with external forcing, the nonlinear mathematical pendulum model with external forcing and the Lorenz weather model.

Functional Bethe Ansatz for a sinh-Gordon Model with Real q

Recently, Bazhanov and Sergeev have described an Ising-type integrable model which can be identified as a sinh-Gordon-type model with an infinite number of states but with a real parameter q. This model is the subject of Sklyanin’s Functional Bethe Ansatz. We develop in this paper the whole technique of the FBA which includes: (1) Construction of eigenstates of an off-diagonal element of a monodromy matrix. The most important ingredients of these eigenstates are the Clebsh-Gordan coefficients of the corresponding representation. (2) Separately, we discuss the Clebsh-Gordan coefficients, as well as the Wigner’s 6j symbols, in details. The later are rather well known in the theory of 3D indices. Thus, the Sklyanin basis of the quantum separation of variables is constructed. The matrix elements of an eigenstate of the auxiliary transfer matrix in this basis are products of functions satisfying the Baxter equation. Such functions are called usually the Q-operators. We investigate the Baxter equation and Q-operators from two points of view. (3) In the model considered the most convenient Bethe-type variables are the zeros of a Wronskian of two well defined particular solutions of the Baxter equation. This approach works perfectly in the thermodynamic limit. We calculate the distribution of these roots in the thermodynamic limit, and so we reproduce in this way the partition function of the model. (4) The real parameter q, which is the standard quantum group parameter, plays the role of the absolute temperature in the model considered. Expansion with respect to q (tropical expansion) gives an alternative way to establish the structure of the eigenstates. In this way we classify the elementary excitations over the ground state.

Stability Analysis of Controlled DC Motor

Controlled dc motors are nonlinear systems, thatshow a nonlinear action in their operation including, sub-harmonics and chaos when they work outside their designspecifications. This nonlinearity forces the motor changingits normal operation to a random-like behaviour; In thispaper, the nonlinear dynamics of DC motors areinvestigated. It is shown that the concept of the Poincarémap approach and the monodromy matrix method can besuccessfully applied to determine the stability of DC motors

Riemann-Hilbert problems, Toeplitz operators and ergosurfaces

The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix M depending on the Weyl coordinates ρ and v, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the (ρ, v) plane on which some elements of M(ρ, v) tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces.

U folds from geodesics in moduli space

We exploit the presence of moduli fields in the AdS3×S3×CY2, where CY2=T4 or K3, solution to type IIB superstring theory, to construct a U-fold solution with geometry AdS2×S1×S3×CY2. This is achieved by giving a nontrivial dependence of the moduli fields in SO(4,n)/SO(4)×SO(n) (n=4 for CY2=T4 and n=20 for CY2=K3), on the coordinate η of a compact direction S1 along the boundary of AdS3, so that these scalars, as functions of η, describe a geodesic on the corresponding moduli space. The backreaction of these evolving scalars on spacetime amounts to a splitting of AdS3 into AdS2×S1 with a nontrivial monodromy along S1 defined by the geodesic. Choosing the monodromy matrix in SO(4,n;Z), this supergravity solution is conjectured to be a consistent superstring background. We generalize this construction starting from an ungauged theory in D=2d, d odd, describing scalar fields nonminimally coupled to (d−1) forms and featuring solutions with topology AdSd×Sd, and moduli scalar fields. We show, in this general setting, that giving the moduli fields a geodesic dependence on the η coordinate of an S1 at the boundary of AdSd is sufficient to split this space into AdSd−1×S1, with a monodromy along S1 defined by the starting and ending points of the geodesic. This mechanism seems to be at work in the known J-fold solutions in D=10 type IIB theory and hints toward the existence of similar solutions in the type IIB theory compactified on CY2. We argue that the holographic dual theory on these backgrounds is a 1+0 CFT on an interface in the 1+1 theory at the boundary of the original AdS3. Published by the American Physical Society 2024

Hamiltonian Monodromy via spectral Lax pairs

Hamiltonian Monodromy is the simplest topological obstruction to the existence of global action-angle coordinates in a completely integrable system. We show that this property can be studied in a neighborhood of a focus-focus singularity by a spectral Lax pair approach. From the Lax pair, we derive a Riemann surface which allows us to compute in a straightforward way the corresponding Monodromy matrix. The general results are applied to the Jaynes–Cummings model and the spherical pendulum.

Generating new gravitational solutions by matrix multiplication

Explicit solutions to the nonlinear field equations of some gravitational theories can be obtained, by means of a Riemann–Hilbert approach, from a canonical Wiener–Hopf factorization of certain matrix functions called monodromy matrices. In this paper, we describe other types of factorization from which solutions can be constructed in a similar way. Our approach is based on an invariance problem, which does not constitute a Riemann–Hilbert problem and allows to construct solutions that could not have been obtained by Wiener–Hopf factorization of a monodromy matrix. It gives rise to a novel solution generating method based on matrix multiplications. We show, in particular, that new solutions can be obtained by multiplicative deformation of the canonical Wiener–Hopf factorization, provided the latter exists, and that one can superpose such solutions. Examples of applications include Kasner, Einstein–Rosen wave and gravitational pulse wave solutions.

Towards a quadratic Poisson algebra for the subtracted classical monodromy of symmetric space sine-Gordon theories

Symmetric space sine-Gordon theories are two-dimensional massive integrable field theories, generalising the sine-Gordon and complex sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution.

REDUCIBILITY OF MULTIPERIODIC LINEAR SYSTEMS WITH A DIAGONAL DIFFERENTIATION OPERATOR AND ITS APPLICATION TO CONDITIONALLY PERIODIC SYSTEMS

Abstract. The main theorem is proven by substantiating the reducibility of an equivalent multiperiodic linear system, with a differentiation operator in the direction of diagonal of independent variable space. It is shown that helical lines with an elevation angle of  / 4, when applied to a P-circular cylindrical surface, are  -periodic characteristics of the differentiation operator.The reducibility of a multiperiodic linear system is investigated near the helix line which starts from the initial point of the phase circle, following the scheme of substantiating the reducibility of linear periodic systems. The concept of a monodromy matrix is introduced, which is constant along the first integrals of the characteristic equations of the differentiation operator and has the properties of multiperiodicity and smoothness. The existence of localised positive eigenvalues with all the properties of the monodromy matrix is proved. It is assumed that the monodromy matrix accepts the maximum number of different eigenvalues at the initial point of the phase circle compared to other points. In the neighbourhood of these eigenvalues, the localisation of eigenvalues of the monodromy matrix, defined on a cylindrical surface, is realized in its entirety. The Gershgorin method is employed for this purpose. The non-singularity of the monodromy matrix allows the existence of a curve of the complex plane that does not cover zero and surrounds the spectrum of the monodromy matrix. This proves the existence of the logarithm of the monodromy matrix, and therefore the reducibility of a multiperiodic linear system. Hence, a proof of the main theorem for a conditionally periodic system generating a multiperiodic system is obtained as a consequence.

Robustness and stability of doubly periodic patterns of the focusing nonlinear Schrödinger equation.

The nonlinear Schrödinger equation possesses doubly periodic solutions expressible in terms of the Jacobi elliptic functions. Such solutions can be realized through doubly periodic patterns observed in experiments in fluid mechanics and optics. Stability and robustness of these doubly periodic wave profiles in the focusing regime are studied computationally by using two approaches. First, linear stability is considered by Floquet theory. Growth will occur if the eigenvalues of the monodromy matrix are of a modulus larger than unity. This is verified by numerical simulations with input patterns of different periods. Initial patterns associated with larger eigenvalues will disintegrate faster due to instability. Second, formation of these doubly periodic patterns from a tranquil background is scrutinized. Doubly periodic profiles are generated by perturbing a continuous wave with one Fourier mode, with or without the additional presence of random noise. Effects of varying phase difference, perturbation amplitude, and randomness are studied. Varying the phase angle has a dramatic influence. Periodic patterns will only emerge if the perturbation amplitude is not too weak. The growth of higher-order harmonics, as well as the formation of breathers and repeating patterns, serve as a manifestation of the classical problem of Fermi-Pasta-Ulam-Tsingou recurrence.

Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz

В рамках обобщенного алгебраического анзаца Бете рассматривается спиновая цепочка $XYZ$. Вычислено действие элементов матрицы монодромии на векторы Бете, результат которого представляется в виде линейной комбинации новых векторов Бете. Также вычислено кратное действие калибровочно-преобразованных элементов матрицы монодромии на превекторы Бете. Результаты выражены через статистическую сумму восьмивершинной модели.

Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz

Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz

Effects of coupling coefficient dispersion on the Fermi-Pasta-Ulam-Tsingou recurrence in two-core optical fibers

In a two-core optical fiber (TCF), the linear coupling coefficient between the two cores in general varies with the optical wavelength, a phenomenon known as coupling coefficient dispersion (CCD). Here we study the effects of CCD on the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena in a TCF. Modulation instability tends to be a precursor in the formation of breathers. The subsequent decay of breathers and re-emergence of modulation instability lead to FPUT type periodic patterns. For the symmetric or antisymmetric continuous-wave (CW) state of a TCF, CCD does not affect the modulation instability gain spectra. However, if asymmetric perturbations are applied to a symmetric CW state, CCD can drastically modify FPUT. In the anomalous dispersion regime, without CCD, only FPUT dynamics without energy transfer between the two cores occurs. With CCD effects, FPUT dynamics with strong energy transfer between the two cores can arise. As CCD increases, FPUT dynamics without energy transfer never vanishes, but the corresponding number of FPUT cycles can be drastically reduced to only one. In the normal dispersion regime, no FPUT dynamics with energy transfer between the two cores occurs. The major CCD effect is to significantly decrease the FPUT cycles, and only one FPUT cycle can be observed at relatively large magnitude of CCD values. As the magnitude of CCD increases further, more FPUT cycles arise again. Moreover, larger CCD will lead to higher (lower) propagation speed of the wave patterns in the anomalous (normal) dispersion regime respectively. To enhance theoretical insight, we perform a Floquet analysis for the linearized stability equations with periodic coefficients based on the FPUT dynamics in the normal dispersion regime. The monodromy matrix and eigenvalues can be computed explicitly. Larger eigenvalues correlate with stronger instability and more rapid disintegration of the periodic patterns, agreeing well with the trends of the numerical simulations.

Augmented Legendrian cobordism in $J^1S^1$

We consider Legendrian links and tangles in J^1S^1 and J^1[0,1] equipped with Morse complex families over a field \mathbb{F} and classify them up to Legendrian cobordism. When the coefficient field is \mathbb{F}_2 , this provides a cobordism classification for Legendrians equipped with augmentations of the Legendrian contact homology DG-algebras. A complete set of invariants, for which arbitrary values may be obtained, is provided by the fiber cohomology, a graded monodromy matrix, and a mod 2 spin number. We apply the classification to construct augmented Legendrian surfaces in J^1M with \mathrm{dim} M = 2 realizing any prescribed monodromy representation, \Phi:\pi_1(M,x_0) \to \mathrm{GL}(\mathbf{n}, \mathbb{F}) .