Homogeneous Nonlinear Systems Research Articles

Convergence of Solutions of Nonhomogeneous Linear Difference Systems with Delays

Sufficient conditions are given for the asymptotic constancy of the solutions of a linear system of difference equations with delays. Moreover, it is shown that the limits of the solutions, as t→∞, can be computed in terms of the initial function and a special matrix solution of the corresponding adjoint equation.

Elliptic balance solution to two degree of freedom, undamped, homogeneous systems having cubic nonlinearities

The application of the elliptic balance method to the solution of undamped, two degree of freedom homogeneous nonlinear systems is described. This method uses Jacobian elliptic functions in the balance and is based on the concept of averaging with respect to complete elliptic integrals of the first kind. To assess the accuracy of the approximate solution thus obtained, we consider the motion of a linear vibration absorber attached to a rigid body that is supported symmetrically by incompressible, homogeneous and isotropic hyperelastic shear blocks. It is shown that the amplitude–time response of the model system is well predicted by the elliptic balance method solution even for relatively large parameter values.

On stability of switched homogeneous nonlinear systems

In this paper, the problem of stability of switched homogeneous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogeneous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogeneous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogeneous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planar switched homogeneous systems, an LMI approach to stability of planar switched homogeneous systems is presented. Similar to the condition for linear systems, the LMI-type condition is easily verifiable. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.

A Finite-time State Feedback Stabilization Method for a Class of Second Order Nonlinear Systems

The finite-time feedback stabilization problem of a class of second-order nonlinear systems is discussed. Three kinds of global finite-time feedback stabilization approaches based on continuous state feedback are given. First, using homogeneous properties of nonlinear homogeneous systems, a state feedback controller is designed to guarantee that the closed loop system is asymptotically stable and satisfies negative homogeneity. Sec-ond, using backstepping constructive approach based on finite-time Lyapunov function, a finite-time controller is developed. Finally, using nonsingular terminal sliding mode technology, a feedback stabilizing controller which can make the closed loop system converge to equilibrium in finite time is given. Simu-lation results are provided to demonstrate the effectiveness of these approaches.

Generalizations of the notion of hyperbolicity

We describe several generalizations of the classical notion of hyperbolicity for a sequence of linear mappings. It is shown that the following three statements are equivalent: (i) the corresponding linear non-homogeneous system has a bounded solution for any bounded nonhomogeneity, (ii) the sequence has a (C, λ)-structure, (iii) the sequence is piecewise hyperbolic with long enough intervals of hyperbolicity.

Soliton ratchets in homogeneous nonlinear Klein-Gordon systems

We study in detail the ratchetlike dynamics of topological solitons in homogeneous nonlinear Klein-Gordon systems driven by a biharmonic force. By using a collective coordinate approach with two degrees of freedom, namely the center of the soliton, X(t), and its width, l(t), we show, first, that energy is inhomogeneously pumped into the system, generating as result a directed motion; and, second, that the breaking of the time shift symmetry gives rise to a resonance mechanism that takes place whenever the width l(t) oscillates with at least one frequency of the external ac force. In addition, we show that for the appearance of soliton ratchets, it is also necessary to break the time-reversal symmetry. We analyze in detail the effects of dissipation in the system, calculating the average velocity of the soliton as a function of the ac force and the damping. We find current reversal phenomena depending on the parameter choice and discuss the important role played by the phases of the ac force. Our analytical calculations are confirmed by numerical simulations of the full partial differential equations of the sine-Gordon and phi4 systems, which are seen to exhibit the same qualitative behavior. Our results show features similar to those obtained in recent experimental work on dissipation induced symmetry breaking.

Identification of closed Michaelis‐Menten system by a linear approach

PurposeThis paper aims at identifying the closed Michaelis‐Menten (M‐M) system. The system governs many phenomena that are based on “enzyme reactions”.Design/methodology/approachA linearization method is used. The system, however is not homogeneous and consequently earlier identification results obtained by the author cannot be applied. The approach is to show that the closed Michaelis‐Mentan system can be approximated by a sequence of open M‐M systems. The M‐M parameters are then obtained by continuity from the Cauchy's problem solution.FindingsThe main results – the identification of a closed M‐M system is described. The linearization of the closed M‐M compartmental system led to a linear nonhomogeneous system. Earlier classical methods could not applied but the “Reasonable Biological Hypothesis” of the author allowed further study.Research limitations/implicationsThis study takes a preferred theoretical aspect. A generalization to n, n≥3 is considered to be interesting because we can consider an open compartmental system having n compartments as a closed system with n+1 compartments. Further work on this generalization is contemplated.Practical implicationsThis theoretical study will further increase the use compartmental analysis to modelling any phenomena depending only on time.Originality/valueProvides a methodology for identification of M‐M system using a linear strategy.

Internal Mode Mechanism for Collective Energy Transport in Extended Systems

We study directed energy transport in homogeneous nonlinear extended systems in the presence of homogeneous ac forces and dissipation. We show that the mechanism responsible for unidirectional motion of topological excitations is the coupling of their internal and translation degrees of freedom. Our results lead to a selection rule for the existence of such motion based on resonances that explain earlier symmetry analysis of this phenomenon. The direction of motion is found to depend both on the initial and the relative phases of the two harmonic drivings, even in the presence of noise.

Identifiability of nonlinear homogeneous polynomial systems

New results are presented concerning the state isomorphism approach to global identifiability analysis of parameterized classes of nonlinear homogeneous systems with specified initial states. For such systems, the local state isomorphism for a pair of indistinguishable parameter vectors is homogeneous of degree one. Under certain conditions, which may only be satisfied for homogeneous polynomial systems, the local state isomorphism is linear. Here, the key issue is whether or not the observability rank condition holds at the origin. The controllability rank condition is shown to play a truly secondary role. The results are generalized to the multivariable case and a worked example demonstrates how identifiability analysis may be simplified along these lines.

Chebyshev rational matrix approximation with a priori error bounds for linear and Riccati matrix equations

This paper deals with initial value problems for Lipschitz continuous coefficient matrix Riccati equations. Using Chebyshev polynomial matrix approximations the coefficients of the Riccati equation are approximated by matrix polynomials in a constructive way. Then using the Fröbenius method developed in [1], given an admissible error ϵ > 0 and the previously guaranteed existence domain, a rational matrix polynomial approximation is constructed so that the error is less than ϵ in all the existence domain. The approach is also considered for the construction of matrix polynomial approximations of nonhomogeneous linear differential systems avoiding the integration of the transition matrix of the associated homogeneous problem.

Asymptotic Behavior of a Nonhomogeneous Linear Recurrence System

Consider the nonhomogeneous linear recurrence systemxn+1=(A+Bn)xn+gn,where A and Bn (n=0,1,…) are square matrices and gn (n=0,1,…) are column vectors. In this paper, we describe, in terms of the initial condition, the asymptotic behavior of the solutions of this equation in the case when A has a simple dominant eigenvalue λ0,∑n=0∞⫫Bn⫫<∞ and ∑n=0∞|λ0|−n⫫gn⫫<∞. The proof is based on the duality between the solutions of the above equation and the solutions of the associated adjoint equation. As a consequence, we obtain a similar result for higher order scalar equations.

Estimates of Exponents of Linear Nonhomogeneous Systems via the Grobman Improperness Coefficient

Estimates of Exponents of Linear Nonhomogeneous Systems via the Grobman Improperness Coefficient

Extension of the Perron--Frobenius Theorem to Homogeneous Systems

This paper deals with a particular class of positive systems. The state components of a positive system are positive or zero for all positive times. These systems are often encountered in applied areas such as chemical engineering or biology. It is shown that for this particular class the first orthant contains an invariant ray in its interior. An invariant ray generalizes the concept of an eigenvector of linear systems to nonlinear homogeneous systems. Then sufficient conditions for uniqueness of this ray are given. The main result states that the vector field on an invariant ray determines the stability properties of the zero solution with respect to initial conditions in the first orthant. The asymptotic behavior of the solutions is examined. Finally, we compare our results to the Perron--Frobenius theorem, which gives a detailed picture of the dynamical behavior of positive linear systems.

Periodic boundary value problem of quasilinear system

The existence and uniqueness results about quasilinear periodic boundary value problems are established by using the global inverse function theorem and the result about the existence and uniquencess of periodic solutions for the periodic boundary value problem of nonhomogeneous linear periodic system.

Numerical simulation of new particle formation over the northwest Atlantic using the MM5 mesoscale model coupled with sulfur chemistry

The production of sulfuric acid vapor and new particle formation are studied over the northwest Atlantic for two synoptic cases: a “wet” case in April with a large amount of cloud and precipitation in the simulation domain and a relatively “dry” one in March. The Fifth‐Generation Pennsylvania State University/National Center for Atmospheric Research Mesoscale Model (MM5) is utilized coupled on‐line with a simple sulfur chemistry mechanism. Our numerical simulations show that there are three regions where significant binary homogeneous nucleation occurs. One region is in convective outflow where high anthropogenic SO2 concentration transported upward and low air temperature in the free troposphere (FT) favor the binary homogeneous nucleation of sulfuric acid gas and water vapor. The second region is near the top of marine boundary layer (MBL) clouds. The third region is in the clear FT at 500–600 hPa. In the three regions, significant nucleation events are all connected with relative humidity (RH) higher than 60%. For the northwest Atlantic we found that homogeneous nucleation generally occurs from the late morning through the afternoon on the northwest side of our simulation domain where anthropogenic SO2 concentration is high enough to produce sufficient sulfuric acid gas through the oxidation by OH radical. Significant nucleation does not necessarily happen in regions with very low preexisting particle surface area in our simulated cases. Our simulated total aerosol number concentrations in regions with significant new particle formation are generally consistent with field measurements. Nucleation rates are enhanced by NH3 especially in the MBL. Sensitivity tests show that a grid of 10 km (and time step of 30 s) yielded significantly higher nucleation as compared with a grid size of 30 km (and time step of 90 s). The latter grid may be too coarse to resolve the concentration gradients, which likely strongly impact the highly nonlinear homogeneous nucleation system.

Second order optimality conditions for differentiable multiobjective problems

A second order optimality condition for multiobjective optimization with a set constraint is developed; this condition is expressed as the impossibility of nonhomogeneous linear systems. When the constraint is given in terms of inequalities and equalities, it can be turned into a John type multipliers rule, using a nonhomogeneous Motzkin Theorem of the Alternative. Adding weak second order regularity assumptions, Karush, Kuhn-Tucker type conditions are therefore deduced.

H ∞ control, stabilization and input output stability of nonlinear systems based on homogeneous techniques 1

Abstract In this paper, new relationships are studied between H∞ control and stabilization of (homogeneous) nonlinear control systems, and between Lp stability and Lyapunov stability of (homogeneous) nonlinear systems. In the first part, by virtue of Hamilton-Jacobi-Isaacs (HJI) inequality, nonlinear H∞ control problems of stabilizable homogeneous systems and the systems that can be approximated by homogeneous ones are discussed for their designs. One of the main results is that, if the considered system, without exogenous input signals, can be stabilized via homogeneous feedback, then its H^a control problem can be solved with continuous (but maybe noiismooth) feedback laws. In the second part, the problem of input-output stability (IOS) is considered with relations among the lowest homogeneity degrees of the terms of considered systems, and both global and local results are obtained in different cases

A symmetry group and controllability of nonlinear homogeneous systems 1

Abstract A 1-dimensional Lie group and its left action on convex cones are chosen, with respect to which nonlinear homogeneous control systems possess symmetries. Controllability of this class of systems is equivalently simplified, via this symmetry Lie group and its action, into two lower dimensional problems.

Simultaneous Stabilization of Linear and Nonlinear Systems by Means of Nonlinear State Feedback

The simultaneous stabilization (resp., asymptotic stabilization) of a countable family of control systems consists of finding a control which stabilizes (resp., asymptotically stabilizes) all the systems in the family. In this paper, we introduce a new method which enables us to show that, given any countable family of stabilizable nonlinear systems, there exists a continuous state feedback law which simultaneously stabilizes (not asymptotically) the family. Then, by enriching this method, we prove that any finite family of stabilizable linear time invariant (LTI) systems can be simultaneously exponentially stabilized by means of nonlinear time-varying state feedback. We also derive sufficient conditions for the simultaneous asymptotic stabilizability of countably infinite families of LTI systems. Finally, sufficient conditions for the simultaneous asymptotic stabilizability of finite families of nonlinear systems are provided and used for the simultaneous asymptotic stabilization of certain pairs of nonlinear homogeneous systems.

Existence and structure of complete formal solutions of non-linear meromorphic systems of ordinary differential equations

For general non-linear n-dimensional homogeneous systems of meromorphic ODE, we show existence of a complete formal solution, formally depending upon n free parameters. We investigate the structure of such formal solutions and, in particular, obtain a factorization, analogous to results of J.-P. Ramis and (independently) the author for linear systems.