Quadratic Systems Research Articles

Lumped Element Method Based Conductivity Reconstruction Algorithm for Localization Using Symmetric Discrete Operators on Coarse Meshes

The inverse conductivity problem in electrical impedance tomography involves the solving of a nonlinear and under-determined system of equations. This paper presents a new approach, which leads to a quadratic and overdetermined system of equations. The aim of the paper is to establish new research directions in handling of the inverse conductivity problem. The basis of the proposed method is that the material, which can be considered as an isotropic continuum, is modeled as a linear network with concentrated parameters. The weights of the obtained graph represent the properties of the discretized continuum. Further, the application of the developed procedure allows for the dielectric constant to be used in the multi-frequency approach, as a result of which the optimized system of equations always remains overdetermined. Through case studies, the efficacy of the reconstruction method by changing the mesh resolution applied for discretizing is presented and evaluated. The presented results show, that, due to the application of discrete, symmetric mathematical structures, the new approach even at coarse mesh resolution is capable of localizing the inhomogeneities of the material.

Symmetry classes of dissipative topological insulators with edge dark states

We classify the dissipative topological insulators (TIs) with edge dark states (EDSs) by using the 38-fold way of non-Hermitian systems in this paper. The dissipative dynamics of these quadratic open fermionic systems is captured by a non-Hermitian single-particle matrix which contains both the internal dynamics and the dissipation, referred to as the damping matrix $X$. The dark states in these systems are the eigenmodes of $X$ in which the eigenvalues' imaginary part vanishes. However, there is a constraint on $X$, namely, that the modes in which the eigenvalues' imaginary parts are positive are forbidden. In other words, the imaginary line-gap of $X$ is ill-defined, so the topological band theory classifying the dark states cannot be applied to $X$. To reveal the topological protection of the EDSs, we propose the double damping matrix $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}=\text{diag}(X,{X}^{*})$, where the imaginary line-gap is well-defined. Thus, the 38-fold way can be applied to $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}$, and the topological protection of the EDS is uncovered. Different from previous studies of EDSs in purely dissipative dynamics, the EDSs in the dissipative TIs are robust against the inclusion of Hamiltonians. Furthermore, the topological classification of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{X}$ not only reflects the topological protection of EDSs in dissipative TIs but also provides a paradigm to predict the appearance of EDSs in other open free fermionic systems.

Enhancement of the group delay in quadratic coupling optomechanical systems subjected to an external force

We theoretically study the effect of the quadratic coupling strength on optomechanical systems subjected to a continuous external force. Quadratic coupling strength originates from strong coupling between the optical and the mechanical degrees of freedom. We show that the quadratic coupling strength reduces the amplitude of the dispersion spectra at the resonance in both blue- and red-sideband regimes. However, it increases (decreases) the amplitude of the absorption spectrum in the blue- (red-)sideband regime. Furthermore, in both sideband regimes, the effective detuning between the pump and the cavity deviates with the quadratic coupling strength. Thereby, appropriate selection of the quadratic coupling strength results in an important magnification (in absolute value) of the group delay for both slow and fast light exiting from the optomechanical cavity.

On the security of multivariate-based ring signature and other related primitives

The multivariate public-key cryptography (MPKC) provides a promising class of post-quantum signature schemes. Its theoretical security comes from the intractability of the multivariate quadratic problem (MQ problem), which is an NP-hard problem [1]. Nevertheless, not every instance of MQ-problem is hard [2–6] while the problem itself is NP-hard. Thus, it is important to understand and analyze the nature of the quadratic systems used in existing MPKC so that the security of the multivariate schemes can be properly assessed. Herein, we present a detailed cryptanalysis of the multivariate based ‘ring’ signature scheme presented by Wang et al. [7]. We utilize the attacks on the underdetermined system of multivariate quadratic equations. We prove both theoretically and experimentally that the scheme (Wang et al. [7]) can be broken in polynomial time even for rings of small size. Additionally, we observed that the attack strategy we proposed in this work is not only limited to [7], but can also be applied to break the unforgeability of other multivariate based schemes such as verifiable ring signature scheme [8], multivariate group signature scheme [9], and Gui and GeMSS based ring signature scheme [10,11].

Hopf bifurcations by perturbing a class of reversible quadratic systems

This paper first investigates the dynamical behavior of a class of reversible quadratic systems, providing all possible phase portraits on the plane. Then, we use generalized Melnikov function method to study the Hopf bifurcation of reversible quadratic systems under the perturbation of piecewise quadratic systems, finding 4 more limit cycles than the smooth case.

One-Dimensional Quadratic Chaotic System and Splicing Model for Image Encryption

Digital image transmission plays a very significant role in information transmission, so it is very important to protect the security of image transmission. Based on the analysis of existing image encryption algorithms, this article proposes a new digital image encryption algorithm based on the splicing model and 1D secondary chaotic system. Step one is the algorithm of this article divides the plain image into four sub-parts by using quaternary coding, and these four sub-parts can be coded separately. Only by acquiring all the sub-parts at one time can the attacker recover the useful plain image. Therefore, the algorithm has high security. Additionally, the image encryption scheme in this article used a 1D quadratic chaotic system, which makes the key space big enough to resist exhaustive attacks. The experimental data show that the image encryption algorithm has high security and a good encryption effect.

Combustion synthesis, characterization, and photodegradation performance of La2-xBixCuO4

Quaternary metal oxide system of the type, La2-xBixCuO4, where x = 0, 0.01, 0.03, 0.06, 0.08 have been synthesized by the combustion method. The crystalline phases and purity of nanostructures were confirmed by X-ray diffraction, which showed that they crystallized in the orthorhombic system of Bamb space group. Furthermore, they tend to adopt the quadratic system upon the incorporation of Bi atom into the lattices of the compound. A vibrational study shows the disappearance of absorption band around 680 cm−1 due to crystallographic system transformation. Morphological characterization by scanning electron microscopy (SEM) indicated that the micrographs are composed of nanometric crystallite particles with porous microstructure. The photocatalytic property of these materials was evaluated by monitoring the degradation of Malachite green under sunlight and the disappearance of the band characteristic of malachite green around 620 nm occurred after 240 min. The efficiency of La2-xBixCuO4 is 64, 68, 75, 89, and 83% for the nanostructures for x = 0, 0.01, 0.03, 0.06, 0.08, respectively.

The quantitative analysis of homoclinic orbits from quadratic isochronous systems

In this paper, we investigate a pair of symmetric homoclinic orbits given by a Perturbation-Incremental (PI) method for the quadratic isochronous systems, which can bifurcate from the periodic orbits of the quadratic isochronous centers. The existing researches on the isochronous system are mostly from a qualitative perspective, while we have studied the quadratic isochronous system through the quantitative analysis method. As a quantitative analysis method, the PI method can not only determine the position of the limit cycle and the homoclinic (heteroclinic) orbit, but also provide an analytical expression of the asymptotic solution. At the same time, this method combines the significant characteristics of the perturbation method and the parameter incremental method. By utilizing the perturbation method, we obtain the analytic expressions of the zero-order perturbation solutions of homoclinic orbits when the parameters of systems are small. Then, the approximate analytical expressions of homoclinic orbits can be obtained by using the parameter incremental method when they are iterated from small parameters to large parameters. The numerical results will be compared with those from the Runge–Kutta method. The theoretical results agree very well with numerical simulation.

A New Analysis of the Kipnis-Shamir Method Solving the MinRank Problem

The MinRank problem is investigated as a problem related to rank attacks in multivariate cryptography and the decoding of rank codes in coding theory. The Kipnis-Shamir method is one of the methods to solve the problem, and recently, significant progress has been made in its complexity estimation by Verbel et al. As this method reduces the problem to an MQ problem, which asks for a solution to a system of quadratic equations, its complexity depends on the solving degree of a quadratic system deduced from the method. A theoretical value introduced by Verbel et al. approximates the minimal solving degree of the quadratic systems in the method although their value is defined under a certain limit for the system considered. A quadratic system outside their limitation often has a larger solving degree, but the solving complexity is not always higher because it has a smaller number of variables and equations. Thus, in order to discuss the best complexity of the Kipnis-Shamir method, a theoretical value is needed to approximate the solving degree of each quadratic system deduced from the method. A quadratic system deduced from the Kipnis-Shamir method always has a multi-degree, and the solving complexity is influenced by this property. In this study, we introduce a theoretical value defined by such a multi-degree and show that it approximates the solving degree of each quadratic system. Thus, the systems deduced from the method are compared, and the best complexity is discussed. As an application, for the MinRank attack using the Kipnis-Shamir method against the multivariate signature scheme Rainbow, we show a case in which a deduced quadratic system outside Verbel et al.'s limitation is the best. In particular, the complexity estimation of the MinRank attack using the KS method against the Rainbow parameter sets I, III and V is reduced by about 172, 140 and 212 bits, respectively, from Verbel et al.'s estimation.

The Legendre pseudorandom function as a multivariate quadratic cryptosystem: security and applications

AbstractSequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs.

Bifurcations and Saddle-Sink-Source Networks in Variable-Independent Quadratic Systems

Bifurcations and Saddle-Sink-Source Networks in Variable-Independent Quadratic Systems

State-Estimation-Based Control Strategy Design for Connected Cruise Control With Delays

The research on connected and automated vehicles attracts much attention recently. In connected vehicle systems, connected cruise control (CCC) is a wildly concerned technique to improve traffic safety and mitigate congestion. In this article, an optimal state-estimation-based (SE-based) control algorithm is proposed in a backward recursion manner to regulate CCC vehicle's longitudinal motion under stochastic communication delays, state noises, and channel interferences. Based on the analysis of error dynamics of vehicles and optimal state estimation strategy, the SE-based system dynamics is first formulated by using the Kalman filtering method. Then, a two-step SE-based control algorithm aiming at minimizing the quadratic system cost is proposed to enhance the vehicular platoon stability. In particular, the optimal control strategy can be generated online as a linear function of the estimated state information and previous control signals, and the corresponding coefficient can be iteratively calculated in an offline step. Finally, the results are successfully extended to the case of arbitrary communication delays, and the condition for the asymptotically mean square stability of the proposed optimal SE-based CCC algorithm is derived. Simulation results indicate that the proposed SE-based control algorithm has better system performance and control stability compared with existing works.

The reverse Hölder inequality for matrix-valued stochastic exponentials and applications to quadratic BSDE systems

In this paper, we study the connections between three concepts — the reverse Hölder inequality for matrix-valued martingales, the well-posedness of linear BSDEs with unbounded coefficients, and the well-posedness of quadratic BSDE systems. In particular, we show that a linear BSDE with bmo coefficients is well-posed if and only if the stochastic exponential of a related matrix-valued martingale satisfies a reverse Hölder inequality. Furthermore, we give structural conditions under which these equivalent conditions are satisfied. Finally, we apply our results on linear equations to obtain global well-posedness results for two new classes of non-Markovian quadratic BSDE systems with special structure.

About the Jacobi Stability of a Generalized Hopf–Langford System through the Kosambi–Cartan–Chern Geometric Theory

In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi–Cartan–Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point.

Limit cycles near a homoclinic loop connecting a tangent saddle in a perturbed quadratic Hamiltonian system

In this paper, we study bifurcation of limit cycles from a homoclinic loop connecting a saddle of tangent type for a quadratic Hamiltonian system perturbed by nth degree polynomials, n=1,2,…,13. The main tool is the asymptotic expansion of the related Abelian integral near the homoclinic loop, and the maximal number of independent coefficients gives exact number of limit cycles. Our aim is to obtain more limit cycles by exploring more coefficients in the asymptotic expansion. However, it is usually very difficult to obtain the coefficients of the terms with degree greater than or equal to 2 in the asymptotic expansion. To overcome the difficulty, we derive two auxiliary systems and investigate the expansions for the related Abelian integral. The coefficients of lower degree terms in the new asymptotic expansions are equivalent to those of higher degree terms in the original asymptotic expansion. We obtain n−1−n−24 limit cycles near the non-regular homoclinic loop and n−24 limit cycles near the center, and it totally has at least n−1 limit cycles, when n∈{1,2,…,13}. The cyclicity of period annulus is also estimated by the first order Melnikov functions for n=3.

Cyclicity of rigid centres on centre manifolds of three-dimensional systems

We work with polynomial three-dimensional rigid differential systems. Using the Lyapunov constants, we obtain lower bounds for the cyclicity of the known rigid centres on their centre manifolds. Moreover, we obtain an example of a quadratic rigid centre from which is possible to bifurcate 13 limit cycles, which is a new lower bound for three-dimensional quadratic systems.

Number of Limit Cycles for Planar Systems with Invariant Algebraic Curves

AbstractFor planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non-existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non-existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Liénard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.

Existence, nonexistence and uniqueness for Lane–Emden type fully nonlinear systems

We study existence, nonexistence, and uniqueness of positive radial solutions for a class of nonlinear systems driven by Pucci extremal operators under a Lane–Emden coupling configuration. Our results are based on the analysis of the associated quadratic dynamical system and energy methods. For both regular and exterior domain radial solutions we obtain new regions of existence and nonexistence. Besides, we show an exclusion principle for regular solutions, either in or in a ball, by exploiting the uniqueness of trajectories produced by the flow. In particular, for the standard Lane–Emden system involving the Laplacian operator, we prove that the critical hyperbola of regular radial positive solutions is also the threshold for existence and nonexistence of radial exterior domain solutions with Neumann boundary condition. As a byproduct, singular solutions with fast decay at infinity are also found.

Exactly Solvable Quadratic Differential Equation Systems Through Generalized Inversion

We study the autonomous systems of quadratic differential equations of the form {dot{x}}_i(t)={textbf{x}}(t)^T {textbf{A}}_i {textbf{x}}(t) + {textbf{v}}_i^T {textbf{x}}(t) with {textbf{x}}(t) = (x_1(t),x_2(t), ldots ,x_i(t),dots ) which, in general, cannot be solved exactly. In the present paper, we introduce a subclass of analytically solvable quadratic systems, whose solution is realized through a multi-dimensional generalization of the inversion which transforms a quadratic system into a linear one. We provide a constructive algorithm which, on one hand, decides whether the system of differential equations is analytically solvable with the inversion transformation and, on the other hand, provides the solution. The presented results apply for arbitrary, finite number of variables.

Quadratic closed G2$\mathrm{G}_2$‐structures

This article studies closed G 2 $\mathrm{G}_2$ -structures satisfying the quadratic condition, a second-order PDE system introduced by Bryant involving a parameter λ $\lambda$ . For certain special values of λ $\lambda$ the quadratic condition is equivalent to the Einstein condition for the metric induced by the closed G 2 $\mathrm{G}_2$ -structure (for λ = 1 / 2 $\lambda = 1/2$ ), the extremally Ricci-pinched (ERP) condition (for λ = 1 / 6 $\lambda =1/6$ ), and the condition that the closed G 2 $\mathrm{G}_2$ -structure be an eigenform for the Laplace operator (for λ = 0 $\lambda = 0$ ). Prior to the work in this article, solutions to the quadratic system were known only for λ = 1 / 6 $\lambda = 1/6$ , − 1 / 8 $-1/8$ , and 2 / 5 $2/5$ , and for these values only a handful of solutions were known. In this article, we produce infinitely many new examples of ERP G 2 $\mathrm{G}_2$ -structures, including the first example of a complete inhomogeneous ERP G 2 $\mathrm{G}_2$ -structure and a new example of a compact ERP G 2 $\mathrm{G}_2$ -structure. We also give a classification of homogeneous ERP G 2 $\mathrm{G}_2$ -structures. We provide the first examples of quadratic closed G 2 $\mathrm{G}_2$ -structures for λ = − 1 $\lambda = -1$ , 1 / 3 $1/3$ and 3 / 4 $3/4$ , as well as infinitely many new examples for λ = − 1 / 8 $\lambda = -1/8$ and 2 / 5 $2/5$ . Our constructions involve the notion of special torsion for closed G 2 $\mathrm{G}_2$ -structures, a new concept that is likely to have wider applicability. In the final section of the article, we provide two large families of inhomogeneous complete steady gradient solitons for the Laplacian flow, the first known such examples.