Linear Homogeneous System Research Articles

Integral reduction with Kira 2.0 and finite field methods

We present the new version 2.0 of the Feynman integral reduction program Kira and describe the new features. The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly. This procedure can be parallelized on computer clusters with MPI. Furthermore, the support for user-provided systems of equations has been significantly improved. This mode provides the flexibility to integrate Kira into projects that employ specialized reduction formulas, direct reduction of amplitudes, or to problems involving linear system of equations not limited to relations among standard Feynman integrals. We show examples from state-of-the-art Feynman integral reduction problems and provide benchmarks of the new features, demonstrating significantly reduced main memory usage and improved performance w.r.t. previous versions of Kira. New version program summaryProgram title:KiraCPC Library link to program files:https://doi.org/10.17632/v3cmsnfrnn.2Developer's repository link:https://gitlab.com/kira-pyred/kiraLicensing provisions: GNU General Public License 3 (GPL)Programming language:C++Journal Reference of previous version: P. Maierhöfer, J. Usovitsch and P. Uwer, Kira—A Feynman integral reduction program, Comput. Phys. Commun.230 (2018) 99 [1705.05610].Does the new version supersede the previous version?: Yes.Reasons for the new version: Implementation of new features, some of which improve the performance significantly for many problems.Summary of revisions: The primary new feature is the reconstruction of the final coefficients in integration-by-parts reductions by means of finite field methods with the help of FireFly [1,2]. This procedure can be parallelized on computer clusters with MPI. Further improvements include the expanded support for user-provided systems of equations as well as a new feature to reduce the main memory usage when generating the system of equations.Nature of problem: The reduction of Feynman integrals to a smaller set of master integrals is a central strategy for high precision calculations in theoretical particle physics, e.g. for cross sections. Furthermore, the reduction is a key ingredient in many methods to calculate the master integrals themselves.Solution method:Kira generates a system of equations employing integration-by-parts [3,4] and Lorentz-invariance identities [5] as well as symmetry relations. Linearly dependent equations are removed and master integrals are identified by solving the system over a finite field. The resulting system can be solved with two methods: Since version 1.0, Kira can solve the system algebraically with the help of Fermat [6]. New in this version is the strategy of solving the system many times over finite fields and reconstructing the master integral coefficients with the help of FireFly [1,2]. Both strategies can also be applied to arbitrary homogeneous linear systems of equations.

A L2‐gain robust PID‐like protocol for time‐varying output formation‐containment of multi‐agent systems with external disturbance and communication delays

This paper addresses the time-varying output formation-containment control problem for homogeneous high-order linear Multi-Agent Systems (MASs). In this context, a PID-like output-feedback control strategy is proposed capable to guarantee that all followers reach and maintain a time-varying formation reference behaviour, as imposed by the multiple leaders, while keeping a prescribed offset with respect to it. Moreover, the proposed approach allows dealing with both communication impairments and external disturbances acting on agents dynamics. The asymptotic stability of the closed-loop MAS is analytically proved by means of a Lyapunov–Krasovskii approach. Stability conditions are expressed as a feasibility problem with Linear Matrix Inequalities constraints, whose solution provides the proper tuning of the robust PID control gains. Extensive numerical analysis confirms the theoretical framework and discloses the effectiveness of the proposed approach in ensuring the output time-varying formation-containment.

The absence of residual property for strong exponents of oscillation of linear systems

In this paper, we study various types of exponents of oscillation (upper or lower, strong or weak) of zeros, roots, hyperroots, strict and non-strict signs of non-zero solutions of linear homogeneous differential systems on the positive semi-axis. On the set of non-zero solutions of autonomous systems the relations between these exponents of oscillation are established. It is proved that all strong exponents of oscillations (unlike Sergeev's frequencies of sign changes, zeros and roots, as well as all the weak exponents of oscillations) considered as functions on the set of solutions to linear homogeneous $n$-dimensional differential systems with continuous coefficients on the semi-line are not residual (i.e. can be changed when changing solution on a finite interval). Besides, at any beforehand given natural $n\ge2$ we give the example of $n$-dimensional differential system, for some solution of which all strong oscillation exponents differ from corresponding weak exponents. In this case, all weak and all strong exponents on the chosen solution coincide with each other, respectively. When proving the results of this work, the case of parity and odd $n$ are considered separately.

Design of Distributed Event-Triggered Average Tracking Algorithms for Homogeneous and Heterogeneous Multiagent Systems

This article addresses the design problem of distributed event-triggered average tracking (DETAT) algorithms for homogeneous and heterogeneous multiagent systems. The objective of the DETAT problem is to develop a group of distributed cooperative control algorithms with event-triggered strategies for agents to track the average of multiple time-varying reference signals. First, for homogeneous linear multiagent systems, based on sampling measurements and model-relied holding techniques, a class of static-gain DETAT algorithms is proposed with a couple of local event-triggered functions for estimators and controllers, respectively. Compared with the existing distributed average tracking (DAT) algorithms, the static-gain DETAT algorithms greatly reduce the cost over communication networks and the frequency of control protocol updates. Second, to reduce the chattering phenomenon caused by nonsmooth items in static-gain algorithms and requirements of the global information of networks, smooth dynamic-gain DETAT algorithms are introduced based on boundary layer approximation methods and self-adaptive principles. Third, for heterogeneous linear multiagent systems, a new algorithm is established by using the output regulation techniques for the heterogeneous DETAT problem. The outputs of heterogeneous agents can ultimately track the average of multiple time-varying reference signals. To the best of our knowledge, it is the first time to study the DETAT problem for heterogeneous multiagent systems. Finally, some examples are presented to show the validity of theoretical results.

Light propagation in anisotropic and metamaterial media by a Finslerian vector eikonal method

A systematic study of light propagation in anisotropic and metamaterial media is presented by using a Finslerian vector eikonal method, that is, by considering the approximation of very short wavelengths of light in Maxwell’s equations and using particular metric tensors of the general Finsler metric tensor induced by an anisotropic medium. Green’s vector functions, and therefore general solutions, are obtained as a product of a complex scalar function with a local polarization unitary vector. The phase (wave surface or eikonal function) of such a complex scalar function is calculated using a Fermat integral derived from the Maxwell equations. This integral can be considered as the integral form of the eikonal equation and provides a Finslerian metric tensor which allows, together with energy conservation in anisotropic media, the derivation of a general expression depending on the eikonal function for obtaining the scalar amplitude of such a complex function. Likewise, the local polarization unitary vector u ( r ) is calculated by solving an approximated homogeneous linear equations system also depending on the eikonal function. Exact solutions are obtained for uniaxial materials and hyperbolic metamaterials, and approximate solutions are proposed for biaxial materials. Finally, an approximated vector Fresnel–Kirchhoff diffraction formula is presented by using vector Green’s functions in order to study optical propagation. For the sake of completeness, interference patterns produced by anisotropic materials are presented using the results obtained. This systematic study of anisotropic and metamaterial media by a Finslerian vector eikonal method is intended in particular for university physics teachers and as an advanced topic for undergraduate and first year postgraduate students.

Stability index of linear random dynamical systems

Given a homogeneous linear discrete or continuous dynamical system, its stability index is given by the dimension of the stable manifold of the zero solution. In particular, for the $n$ dimensional case, the zero solution is globally asymptotically stable if and only if this stability index is $n.$ Fixed $n,$ let $X$ be the random variable that assigns to each linear random dynamical system its stability index, and let $p_k$ with $k=0,1,\ldots,n,$ denote the probabilities that $P(X=k)$. In this paper we obtain either the exact values $p_k,$ or their estimations by combining the Monte Carlo method with a least square approach that uses some affine relations among the values $p_k,k=0,1,\ldots,n.$ The particular case of $n$-order homogeneous linear random differential or difference equations is also studied in detail.

Global existence and decay rates of the solutions for a chemotaxis system with Lotka-Volterra type model for chemoattractant and repellent

<p style='text-indent:20px;'>We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish <inline-formula><tex-math id="M1">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> time-decay for the linear homogeneous system by using a Fourier transform and finding Green's matrix. Then, we find <inline-formula><tex-math id="M2">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> time-decay for the nonlinear system using solution representation by Duhamel's principle and time-weighted estimates.</p>

On the Reduction of the Linear System of the Differential Equations with coefficients of oscillating type to the Triangular Kind in the Resonant Case

For the linear homogeneous differential system, whose coefficients are represented as an absolutely and uniformly convergent Fourier-series with slowly varying coefficients and frequency, the conditions of the existence of the transformation which leads it to triangular kind, are obtained in the resonant cases.

Green's function for impulsive periodic solutions in alternately advanced and delayed differential systems and applications

In this paper we investigate the existence of the periodic solutions of a nonlinear impulsive differential system with piecewise alternately advanced and retarded arguments, in short IDEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such system are obtained. In the process we use the Green's function for impulsive periodic solutions and convert the given the IDEPCAG into an equivalent integral equation system. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this type of nonlinear impulsive differential systems. We also use the contraction mapping principle to show the existence of a unique impulsive periodic solution. Appropriate examples are given to show the feasibility of our results.

Generalised surface waves at the boundary of piezo-poroelastic medium with arbitrary anisotropy.

This study considers the propagation of surface waves along all directions on the plane boundary of piezo-poroelastic half-space with arbitrary anisotropy. This generalised propagation is characterized through an anisotropic phase velocity, which should ensure the decay of wave-field with depth into the medium. A linear homogeneous system of six equations with complex coefficients governs the existence and propagation of surface waves in the considered medium. The real phase velocity of surface waves lies implicit in a complex determinantal equation, which ensures a non-trivial solution to the system of equations. Through a specific transformation, the system of complex equations is modified to yield a real secular equation, with phase velocity being the only unknown. This equation can always be solved numerically for phase velocity of surface wave along any direction on the plane boundary of anisotropic piezo-poroelastic medium. The phase velocity has been used further to calculate the components of energy flux at the boundary. Horizontal components of energy flux define the group velocity and ray direction for the surface wave. A numerical example is solved to analyse the phase/group velocity curves at the boundary of the medium.

Fundamental fractional exponential matrix: New computational formulae and electrical applications

It is well known that the fundamental fractional exponential matrix (FFEM) is closely related to the formal solution of the homogeneous and non-homogeneous linear time-conformable fractional dynamical differential system (LT-CFDDS) with delay in control and it also plays a central role in the solution of any other (matrix) fractional differential system (FDS). In this paper, we present several new theoretical results and computational formulae of the FFEM eAtαα for any arbitrary square matrix A. Moreover, some attractive and interesting special cases for FFEM are also derived and discussed. In addition, three important and interesting physical applications related to linear fractional electrical circuits (ECs) in 2 and 3-dimentions are considered and solved by means of FFEM. These applications are the fractional RC, RL and RLC-ECs. The method exists by converting the data of a given linear fractional EC into homogeneous or non-homogeneous LT-CFDDS for which solutions can be easily computed. For further and simplicity analysis, we provide an illustrative example for each problem and present the general exact solution (GES) in a simple form based on our new approach. Moreover, graphical results are created and discussed in order to ensure that the solutions of specific problems are stable at different values of α and assure that our suggestion technique is a simple, efficient, accurate, powerful analytic tool and can be applied successfully to solve many other conformable fractional differential problems in various fields. Finally, the classical behaviors of physical problems are recovered when the fractional order α is equal to 1.

Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators

Beam lattice materials are characterized by a periodic microstructure realizing a geometrically regular pattern of elementary cells. The dispersion properties governing the free dynamic propagation of elastic waves can be studied by formulating parametric discrete models of the cellular microstructure and applying the Floquet-Bloch theory. Within this framework, governing the wave propagation by means of spectral design techniques and/or energy dissipation mechanisms is a major issue of theoretical and applied interest. Specifically, the wave propagation can be inhibited by purposely designing the microstructural parameters to open stop bands in the material spectrum at target center frequencies. Based on these motivations, a general dynamic formulation is presented for determining the dispersion properties of mechanical metamaterials, modeled as locally resonant beam lattices with generic coordination number. The mechanism of local resonance is realized by tuning periodic auxiliary oscillators, viscoelastically coupled with the beam lattice microstructure. As peculiar aspect, the viscoelastic coupling is derived by a mechanical formulation based on the Boltzmann superposition integral, whose kernel is approximated by a Prony series. Consequently, the free propagation of damped waves is governed by a linear homogeneous system of integro-differential equations of motion. Therefore, differential equations of motion with frequency-dependent viscoelastic coefficients are obtained by applying the in-space Z-transform and in-time bilateral Laplace transform. The complex-valued branches characterizing the dispersion spectrum are determined and parametrically analyzed for the beam lattice characterized by quadrilateral periodic cell. The spectral branches may exceed the model dimension, due to the occurrence of pure-damping spectral components. Particularly, the spectra corresponding to Laurent series approximations of the viscoelastic coefficients are investigated and the solution admissibility and convergence for increasing order series is analyzed. The standard dynamic equations with linear viscous damping are recovered at the first-order approximation. Low-order approximations are found to underestimate the real and imaginary parts of the spectrum, as well as the low-frequency stop bandwidth. Finally, the forced response to a harmonic mono-frequent external point excitation is investigated. The metamaterial responses to non-resonant, resonant and quasi-resonant external forces are compared and discussed from a qualitative and quantitative viewpoint.

A STUDY ABOUT STABILITY OF TWO AND THREE SPECIES POPULATION MODELS

In this article, we have discussed the stability of second order linear and non-linear systems by characteristic roots. In the case of non-linear system, we linearize the nonlinear system under certain specified conditions and study the stability of critical points of the linearized systems. Necessary theories have been presented, applied, and illustrated with examples. A self-contained theory for a homogeneous linear system of third order is built by using the basic concept of the differential equation.

Green’s Function for Periodic Solutions in Alternately Advanced and Delayed Differential Systems

In this paper we investigate the existence of the periodic solutions of a nonlinear differential equation with a general piecewise constant argument, in short DEPCAG, that is, the argument is a general step function. We consider the critical case, when associated linear homogeneous system admits nontrivial periodic solutions. Criteria of existence of periodic solutions of such equations are obtained. In the process we use the Green’s function for periodic solutions and convert the given DEPCAG into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii’s fixed point theorem to show the existence of a periodic solution of this type of nonlinear differential equations. We also use the contraction mapping principle to show the existence of a unique periodic solution. Appropriate examples are given to show the feasibility of our results.

Loop transfer recovery design for synchronization of homogeneous linear multi-agent systems

In this paper we propose the use of loop transfer recovery procedure in the context of cooperative control to aid the design of output feedback synchronization controllers with desirable performance characteristics. It is shown that, for a certain class of linear multi-agent systems, an upper bound on the singular values of the sensitivity matrix for the global networked system can be related to the dynamics of a classical single-system Kalman filter. A numerical example is presented to illustrate the main conclusions of this paper.

A suboptimality approach to distributed [formula omitted] control by dynamic output feedback

This paper deals with suboptimal distributed H2 control by dynamic output feedback for homogeneous linear multi-agent systems. Given a linear multi-agent system, together with an associated H2 cost functional, the objective is to design dynamic output feedback protocols that guarantee the associated cost to be smaller than an a priori given upper bound while synchronizing the controlled network. A design method is provided to compute such protocols. The computation of the two local gains in these protocols involves two Riccati inequalities, each of dimension equal to the dimension of the state space of the agents. The largest and smallest nonzero eigenvalue of the Laplacian matrix of the network graph are also used in the computation of one of the two local gains. A simulation example is provided to illustrate the performance of the proposed protocols.

Global synchronization of coupled reaction–diffusion neural networks with general couplings via an iterative approach

Abstract We establish a framework to investigate the global synchronization of coupled reaction–diffusion neural networks with time delays. The coupled networks under consideration can incorporate both the internal delays in each individual network and the transmission delays across different networks. The coupling scheme for the coupled networks is rather general, and its performance is not adversely affected by the restrictions commonly imposed by existing relevant investigations. Based on the proposed iterative approach, the problem of global synchronization is transformed into that of solving the corresponding homogeneous linear system of algebraic equations. The synchronization criterion is subsequently derived and can be verified with simple computations. Three numerical examples are presented to illustrate the effectiveness of the synchronization theory presented in this paper.

Checkpoints in searching for rational solutions of linear ordinary difference and differential systems

It is quite common that search algorithms for those solutions of difference and differential equations and systems that belong to a fixed class of functions are designed so that nonexistence of solutions of the desired type is detected only in the last stages of the algorithm. However, performing additional tests on the intermediate results makes it possible to stop the algorithm as soon as these tests imply that no solutions of the desired type exist. This gives an opportunity to save time and other computing resources. So, it makes sense to equip algorithms with checkpoints and some tests. We consider these questions in connection with the search for rational solutions of linear homogeneous difference and differential systems with polynomial coefficients, and propose a scheme equipped with such checkpoints and tests, and also report results of experiments with our implementation of the scheme in Maple.

Quasi-canonicalization for linear homogeneous nonholonomic systems**Project supported by the National Natural Science Foundation of China (Grant Nos. 11972177, 11972122, 11802103, 11772144, 11872030, and 11572034) and the Scientific Research Starting Foundation for Scholars with Doctoral Degree of Guangdong Medical University (Grant Nos. B2019042 and B2019021).

For conservative linear homogeneous nonholonomic systems, there exists a cotangent bundle with the symplectic structure dπμ ∧ dξμ, in which the motion equations of the system can be written into the form of the canonical equations by the set of quasi-coordinates πμ and quasi-momenta ξμ. The key to construct this cotangent bundle is to define a set of suitable quasi-coordinates πμ by a first-order linear mapping, so that the reduced configuration space of the system is a Riemann space with no torsion. The Hamilton–Jacobi method for linear homogeneous nonholonomic systems is studied as an application of the quasi-canonicalization. The Hamilton–Jacobi method can be applied not only to Chaplygin nonholonomic systems, but also to non-Chaplygin nonholonomic systems. Two examples are given to illustrate the effectiveness of the quasi-canonicalization and the Hamilton–Jacobi method.

State Consensus for Discrete-Time Multiagent Systems Over Time-Varying Graphs

We study the state consensus problem for linear shift-invariant discrete-time homogeneous multiagent systems (MASs) over time-varying graphs. A novel approach based on the small gain theorem is proposed to design the consensus control protocols for both neutrally stable and neutrally unstable MASs. It is shown that the state consensus can be achieved for neutrally stable MASs under the uniform observability condition; for neutrally unstable MASs, the state consensus entails more than the uniform observability. Two numerical examples are worked out to illustrate our consensus results.