Homogeneous Nonlinear Systems Research Articles

Rapid computation of Hopf bifurcation points of continuous and discrete systems through minimization

For an autonomous nonlinear system, the Hopf bifurcation point along the equilibrium path is a critical feature that indicates whether the values of the parameters change from exhibiting fixed-point behavior to having a periodic orbit. To solve these problems, we developed a method of transforming an eigenvalue problem based on the Jacobian matrix at equilibrium into a minimization problem, enabling the rapid identification of a solution. Specifically, this generalized eigenvalue problem is solved by identifying the vector variable after reducing the number of eigenequations by one in the nonhomogeneous linear system. This can be achieved by normalizing the value of a selected nonzero component of the eigenvector and then moving the column containing this component to the other side of the equation. An appropriate merit function was established in terms of the Euclidean norm of the eigenequation, and this merit function was minimized using the golden section search algorithm to determine the eigenparameters of the bifurcation point. The accuracy of the method for identifying the parameter values and the corresponding imaginary eigenvalues at the Hopf bifurcation points was evaluated for numerous examples for both the continuous and discrete systems. The method was both fast and accurate. Moreover, its stability in the presence of noise was investigated, and the method was robust.

Practical exponential stability and applications of switched homogeneous nonlinear systems with external inputs and impulses

The practical exponential stability of switched homogeneous nonlinear systems (SHNS) with external inputs and impulses is studied in this paper, which includes two types of continuous-time systems and discrete-time systems. Specifically, we add a kind of nonnegative impulses which is consistent with the switching behaviours for the systems. We first study switched homogeneous positive nonlinear systems (SHPNS) case, then we extend the SHPNS to a time-varying SHNS case. The primary difficulty of this research is the estimation of Lyapunov function value at switching-impulse points under the coupling effects of synchronous switches and state jumps. By employing the max-separable Lyapunov function approach, we present new sufficient stability criteria and design new switching-impulse signals, and successfully establish the new relation between the average dwell time parameter and impulse strength. Then, we apply the obtained results to the time-varying vehicle formation model with impulses. At last, four examples are given to demonstrate the feasibility of our results, which includes the specific algorithm verification process.

A mathematical model with uncertainty quantification for allelopathy with applications to real-world data

We revisit a deterministic model for studying the dynamics of allelopathy. The model is formulated in terms of a non-homogeneous linear system of differential equations whose forcing or source term is a piecewise constant function (square wave). To account for the inherent uncertainties present in this natural phenomenon, we reformulate the model as a system of random differential equations where all model parameters and the initial condition are assumed to be random variables, while the forcing term is a stochastic process. Taking extensive advantage of the so-called Random Variable Transformation (RVT) method, we obtain the solution of the randomized model by providing explicit expressions of the first probability density function of the solution under very general assumptions on the model data. We also determine the joint probability density function of the non-trivial equilibrium point, which is a random vector. If the source term is a time-dependent stochastic process, the RVT method might not be applicable since no explicit solution of the model is available. We then show an alternative approach to overcome this drawback by applying the Liouville–Gibbs partial differential equation. All the theoretical findings are illustrated through several examples, including the application of the randomized model to real-world data on alkaloid contents from leaching thornapple seed.

Stability Analysis of Nonlinear Dynamic Systems Using the Lyapunov Method Approach

A new approach to the stability analysis of homogeneous nonlinear systems is proposed, based on the concept of candidate Lyapunovfunctions, where the focus is not on the positive definiteness of the candidate Lyapunovfunctions, but on the negative definiteness of their derivatives. After having ensured the negative definiteness of the derivative function, on the basis of the sign assignment of the primitive function, the stability of the equilibrium is analyzed, where the necessary and sufficient conditions are declared at the same time. The selection of the tendency of the Lyapunov candidate function is primarily performed in the form of a linear combination of some simple functions. The unknown coefficients in the structure of the candidate function are computed based on the negative definiteness of the derivative function. Then, using these coefficients in Lyapunov function, sign of primitive function in state space is argued. Thus, the stability/instability of the equilibrium point can be deduced from the triple sign settings of the candidate function. Furthermore, in the process of negative definiteness of the derivative function, the coefficients are obtained using two independent methods. The proposed theoretical results are supported and their effectiveness is demonstrated by numerical simulations.

Three-dimensional Analysis of Electromagnetic Nanomaterial Flow and Thermal Variations for Forced Convection

This paper investigates the three-dimensional motion of electromagnetic nanofluid under the influence of heat source/sink, nonlinear heat radiation, magnetic field, and altered Arrhenius equation. Nonlinear stretching in the velocity is considered in the x-direction. Thermophoresis (Nt) and Brownian motion (Nb) are also considered in nanoparticle concentration profiles and temperature analysis. The boundary layer equations are transformed into nonlinear ODEs using suitable similarity transformations. The coupled nonlinear homogeneous system of ordinary differential equations is tackled by the MAPLE software. Non-dimensional system of the equation contains fourteen physical parameters Fr, Nb, M, γ, λ, Rd, δ, Pr, Nt, S, E, Sc, Bi and power index, which are governed by the physical model. Graphs are presented to show the impact of the abovementioned parameters on temperature, concentration and velocity profile. The present study contributes by observing how the aforementioned parameters influence the heat dissipation rate of nanofluids. This study has broad applications in the field of nanofluids like oil production, metal extrusion, heat exchangers, catalytic reactors etc. Also, results for a particular case found good concurrence with earlier work.

Caputo-fabrizio fractional-order systems: periodic solution and stabilization of non-periodic solution with application to gunn diode oscillator

Fractional-order autonomous systems do not possess any non-constant periodic solutions, and to the best of our knowledge, there are no existing results regarding the existence of the periodic solution for fractional-order non-autonomous systems. The main objective of this work is to fill the above gap by studying the existence of a periodic solution of the Caputo-Fabrizio fractional-order system and also to find ways to stabilize a non-periodic solution. First, by using the concepts of an equilibrium point, it is proved that an autonomous Caputo-Fabrizio system cannot admit a non-constant periodic solution. Under a similar assumption as the one for an integer-order differential system, and by using the properties of the Caputo-Fabrizio derivative, the existence of a periodic solution of a non-autonomous Caputo-Fabrizio fractional-order differential system is established. The main result is utilized in constructing and finding the periodic solution of the linear non-homogeneous Caputo-Fabrizio system. By using the result on linear systems, we derive a periodic solution of a fractional-order Gunn diode oscillator under a periodic input voltage, and observe that the diameter of the periodic orbit keeps reducing as the fractional-order continuously increases. In the end, by using the result on a linear non-homogeneous system, and by constructing a suitable linear feedback control, the solution of the linear non-homogeneous fractional-order system is stabilized to a periodic solution. An example is presented to support the obtained result. The main advantage of the proposed method over others is the simple considerations like the concept of equilibrium point and the utilization of the property of the Caputo-Fabrizio derivatives instead of other types of fractional derivatives.

An Algorithm for the Solution of Second Order Linear Fuzzy System With Mechanical Applications

In this paper, we consider homogeneous and non-homogeneous second order linear fuzzy systems under granular differentiability. The concept of continuous n-dimensional fuzzy functions on the space of n-dimensional fuzzy numbers are introduced. Developed an algorithm for the solution of a non-homogeneous second order linear fuzzy system under granular differentiability. The proposed algorithm is applied to solve some well-known mechanical problems with fuzzy uncertainty.

Practical exponential stability of switched homogeneous positive nonlinear systems with stable and unstable modes

In this article, we primarily investigate practical exponential stability of switched homogeneous positive nonlinear systems (SHPNSs) that have partial unstable modes and perturbation. First of all, the max-separable Lyapunov function (MSLF) technique and a special pre-setting switching sequence are used to define a number of significant stability conditions that ensure the state trajectories of the system converge to a confined region in continuous-time and discrete-time domains. In addition, the key results include several earlier findings as special situations and can be directly applied to general switched systems, not necessarily positive. Finally, two important instances are provided to further illustrate the validity of the theoretical findings.

P Components of Cluster-Lag Consensus for Second-Order Multiagent Systems With Adaptive Controller on Cooperative-Competitive Networks.

The consensus tracking problem means that a group of followers tracks the desired trajectory with local communication. In this article, partial components of cluster consensus have been considered. In this scenario, the p components of the followers in different clusters track the leader at different lag times, while p components of each agent in the same cluster reach a consensus, which is called p components of cluster-lag (PCCL) consensus. By using a seminorm ||xi||2,p and a Lyapunov-Krasovskii functional, PCCL consensus for second-order multiagent systems with homogeneous nonlinear systems on cooperative-competitive networks has been considered. For the case that the communication network graph is undirected, a decentralized adaptive controller, which is based on the exchanged neighbors' information from the same cluster, is designed such that all the agents reach PCCL consensus. For the directed graph case, an adaptive protocol based on the intracoupling strength is constructed for each cluster to achieve PCCL consensus. Finally, two simulation examples are illustrated to show the effectiveness of the proposed control protocols.

Stability of homogeneous positive systems with time-varying delays

Stability analysis of positive linear systems has been well-developed, while attention to positive homogeneous nonlinear systems is notably less paid. This note deepens and extends some previous results on stability of homogeneous systems with bounded time-varying delays. Based on positivity property, exponential stability of a class of delayed homogeneous system with degree confined to an interval is studied, and its state boundary is further investigated in any fixed time, which is more precise than the former result in some cases. Additionally, finite-time stability is obtained for a delay-free homogeneous system and an explicit boundary is presented.

Continuous–discrete time observers for homogeneous nonlinear systems with sampled-data outputs

In this article, we propose a new observer design for systems whose output is sampled in an aperiodic and asynchronous way and for which a continuous-time homogeneous observer of negative degree is available. The proposed method consists in adapting the existing continuous-time observer in order to cope with the fact that only sampled measurements are available instead of continuous ones. The obtained observer error is shown to be globally uniformly ultimately bounded for any upper bound on the sampling periods. Furthermore, the ultimate bound decreases as the upper bound on the sampling periods decreases. The stability analysis is based on a Lyapunov approach and the performances of the proposed observer are illustrated with simulations.

Backstepping leaderless dynamic consensus among nonlinear systems over directed networks: application to attitude synchronisation control*

We present a distributed consensus controller for multi-agent homogeneous nonlinear systems over directed networks. The systems are assumed to be of second order and the control design relies on a standard backstepping approach. In that light, the control design also hinges on the ability to construct a strict Lyapunov function for the multi-agent nonlinear system interconnected over a directed graph for which the only assumption is that it is connected. That is, that there exists a directed spanning tree, but there is no requirement of conservative conditions such as strong or balanced connectivity. To construct a strict Lyapunov function, we use a generalised Lyapunov equation for the directed-graph Laplacian matrix, which characterises the spanning-tree-existence condition. Then, we establish exponential stability of the consensus manifold. In addition, we implement our dynamic consensus controller on a multi-agent satellite system in the context of attitude synchronisation and demonstrate its efficacy in numerical simulations.

Free Vibrations of Multi-Degree Structures: Solving Quadratic Eigenvalue Problems with an Excitation and Fast Iterative Detection Method

For the free vibrations of multi-degree mechanical structures appeared in structural dynamics, we solve the quadratic eigenvalue problem either by linearizing it to a generalized eigenvalue problem or directly treating it by developing the iterative detection methods for the real and complex eigenvalues. To solve the generalized eigenvalue problem, we impose a nonzero exciting vector into the eigen-equation, and solve a nonhomogeneous linear system to obtain a response curve, which consists of the magnitudes of the n-vectors with respect to the eigen-parameters in a range. The n-dimensional eigenvector is supposed to be a superposition of a constant exciting vector and an m-vector, which can be obtained in terms of eigen-parameter by solving the projected eigen-equation. In doing so, we can save computational cost because the response curve is generated from the data acquired in a lower dimensional subspace. We develop a fast iterative detection method by maximizing the magnitude to locate the eigenvalue, which appears as a peak in the response curve. Through zoom-in sequentially, very accurate eigenvalue can be obtained. We reduce the number of eigen-equation to n−1 to find the eigen-mode with its certain component being normalized to the unit. The real and complex eigenvalues and eigen-modes can be determined simultaneously, quickly and accurately by the proposed methods.

Advancing contact of a 2D elastic curved beam indented by a rigid pin with clearance

A two-dimensional analytical solution is presented for stresses and displacements in an elastic curved beam forming an incomplete ring in frictionless and unbonded contact with a rigid pin loaded by a point force and in the presence of clearance. The circular beam is modeled as an incomplete elastic thick ring, constrained at both ends and in a plane stress state. The stress and displacement fields within the beam are derived from a biharmonic Airy stress function, according to the Michell solution in polar coordinates. The mixed boundary value problem is reduced to a set of dual series equations and then to a non-homogeneous linear system of infinite equations, which is then solved by truncation. The non-linear relations between the applied load and the contact angle or the pressure distribution are obtained by using an inverse method. The analytical results are compared with finite element predictions for a pin-lug connection and a reasonable agreement is observed for several typical geometries. The peaks of contact pressure and von Mises equivalent stress and their location within the curved beam are evidenced.

Blood Flow through an Elliptical Stenosed Artery with the Heat Source and Chemical Reaction

The research article scrutinized blood circulation in the arteries in abnormal (elliptical) form. In the evaluation, the mathematical model originated as coupled partial differential equations solved employing the parameter variation method through the boundary conditions by solving the non-homogeneous linear system of matrices in the Bessel function. To understand the influence of blood flow characterized by velocity, temperature, wall shear stress, concentration, frequency rate and volumetric flow rate for different values of dimensionless parameters, the current investigation is helpful in the biomedical field for treating various cardiovascular diseases.

Practical stability of switched homogeneous positive nonlinear systems: Max-separable Lyapunov function method

The practical stability problem of switched homogeneous positive nonlinear systems (SHPNS) is addressed in this study, which includes two instances in terms of continuous-time and discrete-time. Sufficient conditions are presented by using the max-separable Lyapunov function (MSLF) approach, such that each solution of SHPNS is practically stable. The distinction between the existing results and the obtained results is that ours are not only relatively concise but also easily verifiable, and the theoretical results are also extended to a more general case without restricting the systems to be positive. Eventually, a pair of examples are proposed to explain the approach’s validity.

Finite-time state bounding for homogeneous nonlinear positive systems with time-varying delay and bounded disturbance

In this paper, the problem of finite-time state bounding for homogeneous nonlinear positive systems with time-varying delay and bounded disturbance is studied for the first time. By exploiting an approach presented in positive systems which is different from the Lyapunov–Krasovskii functional technique, an explicit criterion is formulated for the existence of a special ball such that all the state trajectories of homogeneous nonlinear positive systems with degree 0<α<1 converge within it in finite time. We then extend the main result to the general nonlinear time variant systems. Finally, a numerical example is presented to demonstrate the effectiveness of the theoretical results.

Data Fitting with Rational Functions: Scaled Null Space Method with Applications of Fitting Large Scale Shocks on Economic Variables and S-Parameters

Curve fitting discrete data (x, y) with a smooth function is a complex problem when faced with sharply oscillating data or when the data are very large in size. We propose a straightforward method, one that is often overlooked, to fit discrete data (s, ys) with rational functions. This method serves as a solid data fitting choice that proves to be fast and highly accurate. Its novelty lies on scaling positive explanatory data to the interval [0, 1], before solving the associated linear problem Ax=0. A rescaling is performed once the fitting function is derived. Each solution in the null space of A provides a rational fitting function. Amongst them, the best is chosen based on a pointwise error check. This avoids solving an overdetermined nonhomogeneous linear system Ax=b with a badly conditioned and scaled matrix A. With large data, the latter can lack accuracy and be computationally expensive. Furthermore, any linear combination of at least one solution in the basis of the null space produces a new fitting function, which gives the flexibility to choose the best rational function that fits the constraints of specific problems. We tested our method with many economic variables that experienced sharp oscillations owing to the effects of COVID-19-related shocks to the economy. Such data are intrinsically difficult to fit with a smooth function. Deriving such continuous model functions over a desired period is important in the analysis and prediction algorithms of such economic variables. The method can be expanded to model behaviors of interest in other applied sciences, such as electrical engineering, where the method was successfully fitted into network scattering parameter measurements with high accuracy.

Wave processes in the constructions of mechanisms and buildings

In the article a methodology for researching wave processes in weakly nonlinear homogeneous one-dimensional systems – elements of machine and building structures is proposed. It is based on D'Alembert's method of constructing solutions of wave equations and the asymptotic method of nonlinear mechanics. Resonant and non-resonant cases are considered. In the non-resonant case, the influence of viscoelastic forces and small periodic disturbances on the oscillatory process is taken into account. In the resonant case, the influence of the disturbing force and the phase difference of natural oscillations is considered. Numerical methods for linear differential equations were used to analyze the obtained differential equations.

Oscillation and non-oscillation criteria for linear nonhomogeneous systems of two first-order ordinary differential equations

The Riccati equation method is used to establish oscillation and non-oscillation criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillation criterion is a generalization of J.S.W. Wong's oscillation criterion.