Simple Zero Eigenvalue Research Articles

Active Synchronization for Double-Integrator Network Systems Without Velocity Information

Of particular interest in this paper is to study the active synchronization (act-synchronization) problem for double-integrator network systems without velocity information. To this end, scaling parameters are suggested to quantify the degree of act-synchronization for different agents. And weighted gains are employed to assure that there is a simple zero eigenvalue with the other ones sharing positive real parts. An auxiliary variable is introduced to mitigate the unavailability on velocity information. Thus only the directed spanning tree requirement in the content of algebraic graph theory is required. This exhibits a clear distinction from signed-graph or scaled consensus as non-unitary signs for scaling parameter and weighted gain are allowed. It is shown that the condition for act-synchronization is related to weighted gain, scaling parameter, the real and imaginary parts of the involved eigenvalues. With the constraint on identical weighted gain and scaling parameter for position and auxiliary variables, it is proven that act-synchronization can be exponentially guaranteed using the Lyapunov-based technique. Finally, the multiple unmanned ground vehicles system and the 10-generator in IEEE 39-bus system are given to facilitate the proposed configuration and the theoretical findings.

Double-Integrator Dynamics for Multiagent Systems With Antagonistic Reciprocity.

This article is dedicated to the consensus problem for interacting agents of the double-integrator dynamics subject to antagonistic reciprocity, described by negative scalar parameters. To this end, we first show the existence of the weighted gains which play an essential role for solving the consensus problem. Then, we establish the relationship between the weighted gains and scalar parameters to guarantee that the underlying "Laplacian" matrix contains a simple zero eigenvalue and the remaining eigenvalues enjoy positive real parts. Based on the above analysis, we further proceed to solve the considered problem. A major difficulty is that the Laplacian matrices, associated with the position and velocity information, are entirely distinct from each other, leading to the failure of the conventional consensus method for the second-order dynamics. We derive some criteria involving the weighted gains, the scaling parameters, and the real/image parts of the Laplacian matrix of the interaction graph. Moreover, some special frameworks, which have been extensively studied in the literature, are also elaborated on. Compared with the Altafini's model, we do not need to redefine a new Laplacian matrix, and more important, the restriction on the digon sign-symmetry property is removed. It is worth mentioning that the proposed consensus algorithm cannot be deduced by the Altafini's model or its variants. Finally, a wheeled multirobot system is formulated to validate the efficiency of the theoretical results.

Bipartite Synchronization and Convergence Analysis for Network of Harmonic Oscillator Systems With Signed Graph and Time Delay

This paper considers the distributed control problem for a network of harmonic oscillator systems with delayed velocity coupling and signed graph topology containing a directed spanning tree. In sharp contrast to the dynamical behaviors of traditional coupled harmonic oscillators over unsigned graphs, it is found that the considered signed network can exhibit some more interesting behaviors, including bipartite synchronization, stabilization, and convergent behavior. First, when the underlying graph is structurally balanced, some necessary and sufficient conditions in terms of coupling strength and time delay are established to achieve bipartite synchronization in the signed network. Then, under the assumption that the signed graph is structurally unbalanced, it is shown that the network state converges either to the origin or to a nontrivial trajectory. In particular, a novel approach is developed to explicitly determine the converged trajectory based on the left and right eigenvectors associated with the simple zero eigenvalue of the signed graph’s Laplacian matrix. Finally, the theoretical analysis is illustrated by some numerical examples.

Sign-changing diagonal perturbations of Laplacian matrices of graphs

In this paper we provide sufficient conditions for positive (semi)definiteness of sign-changing diagonal perturbations of positive semidefinite difference operators and their matrix representations, the Laplacian matrices of graphs. Our estimates arise from the discrete version of the Poincaré inequality and essentially depend on the algebraic connectivity of the underlying graph, i.e., the second smallest eigenvalue of the graph Laplacian matrix. We generalize our results to positive semidefinite matrices with simple zero eigenvalue and illustrate our results by numerical experiments and discuss the optimality of our assumptions.

On Semidefiniteness of Signed Laplacians with Application to Microgrids

Abstract: The paper investigates the positive semidefiniteness of signed Laplacians. It is noted that a symmetric signed Laplacian defines a unique resistive electrical network, wherein the negative weights correspond to negative resistances. As such, the positive semidefiniteness of the signed Laplacians is equivalent to the passivity of the associated resistive networks. By utilizing n -port circuit theory, we obtain several equivalent conditions for signed Laplacians to be positive semidefinite with a simple zero eigenvalue. These conditions characterize a set of negative weights that maintain the semidefiniteness of the Laplacian. The results are used to analyze the small-disturbance angle stability of microgrids as an application.

Asymptotic analysis of linear parabolic problems with singularities

Asymptotic methods are used to investigate a second-order linear parabolic problem with lower order coefficients rapidly oscillating in time. The coefficient multiplying the principal stationary operator is assumed to be singular, i.e., it has a simple zero eigenvalue. Under certain additional conditions, the problem is proved to have a unique time-periodic solution and its complete asymptotic expansion is constructed and justified.

Stability and Sensitivity Analysis of a Plant Disease Model with Continuous Cultural Control Strategy

In this paper, a plant disease model with continuous cultural control strategy and time delay is formulated. Then, how the time delay affects the overall disease progression and, mathematically, how the delay affects the dynamics of the model are investigated. By analyzing the transendental characteristic equation, stability conditions related to the time delay are derived for the disease-free equilibrium. Specially, whenR0=1, the Jacobi matrix of the model at the disease-free equilibrium always has a simple zero eigenvalue for allτ≥0. The center manifold reduction and the normal form theory are used to discuss the stability and the steady-state bifurcations of the model near the nonhyperbolic disease-free equilibrium. Then, the sensitivity analysis of the threshold parameterR0and the positive equilibriumE*is carried out in order to determine the relative importance of different factors responsible for disease transmission. Finally, numerical simulations are employed to support the qualitative results.

Bifurcation and chaos of an airfoil with cubic nonlinearity in incompressible flow

Using a combination of analytical and numerical methods, the paper studies bifurcations and chaotic motions of a two-dimensional airfoil with cubic nonlinearity in incompressible flow. One type of critical points (characterized by a negative eigenvalue, a simple zero eigenvalue and a pair of purely imaginary eigenvalues) for the bifurcation response equations is considered. With the aid of the normal form theory, the explicit expressions of the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. The stability of the bifurcation solutions is also investigated. By using the undetermined coefficient method, the homoclinic orbit is found, and the uniform convergence of the homoclinic orbit series expansion is proved. It analytically demonstrates that there exists a homoclinic orbit joining the initial equilibrium point to itself, therefore Smale horseshoe chaos occurs for this system via Si’lnikov criterion. The system evolves into chaotic motion through period-doubling bifurcation, and is periodic again as the dimensionless airflow speed increases. Numerical simulations are also given, which confirm the analytical results.

Stationary Bifurcations Control with Applications

Given a family of nonlinear control systems, where the Jacobian of the driver vector field at one equilibrium has a simple zero eigenvalue, with no other eigenvalues on the imaginary axis, we split it into two parts, one of them being a generic family, where it is possible to control the stationary bifurcations: saddle-node, transcritical and pitchfork bifurcations, and the other one being a non-generic family, where it is possible to control the transcritical and pitchfork bifurcations. The polynomial control laws designed are given in terms of the original control system. The center manifold theory is used to simplify the analysis to dimension one. Finally, the results obtained are applied to two underactuated mechanical systems: the pendubot and the pendulum of Furuta.

Stability and multiple bifurcations of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity

We investigate the dynamics of a damped harmonic oscillator with delayed feedback near zero eigenvalue singularity. We perform a linearized stability analysis and multiple bifurcations of the zero solution of the system near zero eigenvalue singularity. Taking the time delay as the bifurcation parameter, the presence of steady-state bifurcation, Bogdanov-Takens bifurcation, triple zero, and Hopf-zero singularities is demonstrated. In the case when the system has a simple zero eigenvalue, center manifold reduction and normal form theory are used to investigate the stability and the types of steady-state bifurcation. The stability of the zero solution of the system near the simple zero eigenvalue singularity is completely solved.

Numerical analysis of higher order singularities in chemical process models

In this contribution, a tool is presented that allows the continuation of singularities of higher codimension for complex chemical process models. The tool is an extension of the process-modelling tool ProMoT. It allows creating augmented systems for simple zero eigenvalue bifurcation points. Parameter continuation of such points allows to produce different varieties for hysteresis, isolas, pitchforks or winged cusps singularities. Required higher order directional derivatives are obtained analytically via an interface to the computer algebra system Maxima.

Spectral Properties in Modon Stability Theory

Spectral properties of an invariant functional, denoted by H, for the linear stability equation associated with the modon, or solitary drift vortex, solutions of the quasi‐geostrophic equivalent barotropic potential vorticity, or Charney–Hasegawa–Mima (CHM), equation are investigated. It is shown that H, which is the only known quadratic invariant in modon stability theory, is identical in form to the second variation of a “Benjamin‐like” variational principle for solitary vortices. However, such a principle does not exist for the modon. The discrete spectrum of the “form operator” in H contains two simple negative eigenvalues and the simple zero eigenvalue. For the leftward‐traveling solution there are only a finite number of positive eigenvalues. For the rightward‐traveling solution, there are a countable infinity of positive eigenvalues. A sharp lower bound on the spectrum, for both the rightward‐ and leftward‐traveling solutions, and a sharp upper bound for the leftward traveling solution, is determined. For the leftward‐traveling solutions, the eigenfunctions span a finite‐dimensional vector space and are orthogonal with respect to an inner product which is valid for all of L2. For the rightward‐traveling solutions, the eigenfunctions span an infinite‐dimensional Hilbert space, but are orthogonal with respect to an inner product, which is not valid for all of L2.

Lyapunov exponents for two nonlinear systems driven by real noises

For a van der Pol–Duffing oscillator and a codimension–two bifurcation system excited parametrically by a small real noise, a model of enhanced generality is developed by assuming that the real noise is the first component of an output vector of a linear filter system, which conforms with the detailed balance condition. The strong mixing condition and the hypothesis (i.e. the infinitesimal generator of the noise has only one isolated simple zero eigenvalue) employed by Arnold and co–workers in 1986 are not considered. To tackle the complexity encountered in the present work, the asymptotic analysis approach and the eigenfunction expansion for the solution to the Fokker–Planck equations are applied to calculate the asymptotic expansions of the invariant measures and the maximal Lyapunov exponents for the relevant systems.

ALMOST-SURE STABILITY OF LINEAR GYROSCOPIC SYSTEMS

This paper studies the stability behaviour of a linear gyroscopic system parametrically perturbed by a (multiplicative) real noise of small intensity. To this end, its maximal Lyapunov exponent is calculated using the method of Sri Namachchivaya et al.[1]. The results derived are suitable for cases where the response frequenciesω1 , ω2are non-commensurable and the infinitesimal generator associated with the noise process, ξ (t) has a simple zero eigenvalue. These results are then employed to determine the almost-sure stability boundaries of a rotating shaft subjected to random axial loading.

Bifurcation Stabilization with Local Output Feedback

Local output feedback stabilization with smooth nonlinear controllers is studied for parameterized nonlinear systems for which the linearized system possesses either a simple zero eigenvalue or a pair of imaginary eigenvalues and the bifurcated solution is unstable at the critical value of the parameter. It is assumed that the unstable mode corresponding to the critical eigenvalue of the linearized system is not linearly controllable. Results are established for bifurcation stabilization using output feedback where the critical mode can be either linearly observable or linearly unobservable. The stabilizability conditions are characterized in explicit forms that can be used to synthesize stabilizing controllers. The results obtained in this paper are applied to rotating stall control for axial flow compressors as an application example.

Application of center manifold reduction to nonlinear system stabilization

The Center Manifold Theorem is applied to the local feedback stabilization of nonlinear systems in critical cases. The paper addresses two particular critical cases of which the system linearization at the equilibrium point of interest is assumed to possess either a simple zero eigenvalue or a complex conjugate pair of simple and pure imaginary eigenvalues. In either case, the noncritical eigenvalues are taken to be stable. The results on stabilizability and stabilization are given explicitly in terms of the nonlinear model of interest in its original form, i.e., before reduction to the center manifold. Moreover, the formulation given in this paper uncovers connections between results obtained using the center manifold reduction and those of an alternative approach.

New Asymptotic Stability Criterions for Nonlinear Systems in Critical Cases

Abstract In this paper, asymptiotic stability of a nonlinear system in critical cases (NSICC) is studied. Two asymptotic stability criterions for NSICC with either a pair of pure imaginary eigenvalues or a simple zero eigenvalue are obtained by means of Lyapunov stability analysis and center manifold reduction technology . The two criterions proposed here mainly have two advantages: (l)the computation concerns only the second order derivatives instead of tedious manipulation of matrix; (2)the representation of the criterions reveals the relationship between the asymptotic stability of a NSICC and the modes of its linearized system.

Asymptotic stability of structural systems based on Lyapunov exponents and moment Lyapunov exponents

An asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the moment Lyapunov exponent, the asymptotic growth rate of the moments of the response, have been obtained for systems driven by a small intensity real noise process. The systems under consideration are general four-dimensional dynamical systems with two critical modes. Almost-sure and moment stability conditions are obtained provided the natural frequencies of these critical modes are non-commensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. In this paper, the results obtained are applied to a thin rectangular beam under the action of a stochastic follower force and a model of a vehicle traveling over a rough road. The stability regions predicted by the two different criteria are then compared.

The problem of an elastic-plastic beam dynamics and an incomplete co-dimension two bifurcation

In this paper, a continuous fourth-order ordinary differential equation Shanley-type model is suggested for analytical analysis of the problem of elastic-plastic beam dynamics. A codimension three bifurcation problem and its simplified case, an incomplete co-dimension two bifurcation of a pair of pure imaginary eigenvalues and a simple zero eigenvalue are presented and analyzed, and the normal form analysis and the unfoldings of 2-jet and 4-jet cases of the incomplete normal forms are provided. Since elastic-plastic beam dynamics are of great non-linear complexity and the vector fields are multiple degeneracies, small differences of physical parameters cause dramatic essential changes of behavior of the motion. Through these results the rich dynamical behaviors of the elastic-plastic beam dynamics, including the counterintuitive behavior and its sensitivity to small parameters of this problem, can be illustrated.

Almost-sure asymptotic stability of a general four-dimensional system driven by real noise

In the first part of this paper, we construct an asymptotic expansion for the maximal Lyapunov exponent, the exponential growth rate of solutions to a linear stochastic system, and the rotation numbers for a general four-dimensional dynamical system driven by a small-intensity real noise process. Stability boundaries are obtained provided the natural frequencies are noncommensurable and the infinitesimal generator associated with the noise process has an isolated simple zero eigenvalue. This work is an extension of the work of Sri Namachchivaya and Van Roessel and is general in the sense that general stochastic perturbations of nonautonomous systems with two noncommensurable natural frequencies are considered.