Stability criteria for a class of fractional order systems

This paper deals with the stability problem in LTI fractional order systems having fractional orders between 1 and 1.5. Some sufficient algebraic conditions to guarantee the BIBO stability in such systems are obtained. The obtained conditions directly depend on the coefficients of the system equations, and consequently using them is easier than the use of conditions constructed based on solving the characteristic equation of the system. Some illustrations are presented to show the applicability of the obtained conditions. For example, it is shown that these conditions may be useful in stabilization of unstable fractional order systems or in taming fractional order chaotic systems.     

Existence and uniqueness of periodic solutions for fourth-order nonlinear periodic systems

In this paper, we study the existence, uniqueness and stability of the periodic solutions for fourth-order nonlinear nonhomogeneous periodic systems with slowly changing coefficients by using the method of Liapunor Function. We obtain some sufficient conditions which guarantee the existence, uniqueness and asymptotic stability of the periodic solutions of these systems and estimate the extent to which the coefficients are allowed to change.     

Stability of motion of quasiperiodic systems in critical cases

A method of setting up a matrix-valued Lyapunov function for a system of differential equations with quasiperiodic coefficients is proposed. This function is used to establish asymptotic-stability conditions for some class of mechanical systems described by nonlinear systems of equations. The stability of motion of these systems in critical cases is analyzed     

Stability of Solutions of a Degenerate Linear System of Differential Equations

We introduce the concept of stability of solutions of a system of linear differential equations with an identically degenerate matrix as the coefficient of the derivative. We find necessary and sufficient conditions for the stability of such systems. We generalize the Floquet–Lyapunov theory to systems of this type with periodic coefficients.     

Two dimensional analysis of transient flow and heat transfer in a natural circulation loop

The transient heat transfer, fluid flow and pressure in a natural circulation loop have been studied under laminar flow conditions. Most studies of these systems have utilized a onedimensional approach which requires a priori specifications of the friction and the heat-transfer coefficients. In the present work the variation of the friction and heat-transfer coefficients are determined. Detailed pressure, temperature and velocity distributions are presented.     

RESEARCH OF THE PERIODIC SOLUTION FOR A CLASS OF NONLINEAR DIFFERENTIAL EQUATIONS

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Stability of the motion of linear large-scale oscillating systems

A method for setting up a time-periodic Lyapunov function for a linear system with periodic coefficients based on an auxiliary matrix-valued function is proposed and developed. New sufficient conditions for the stability of large-scale periodic systems decomposed into an even number of subsystems are formulated     

On the stability of multiple parameter time-varying dynamic systems

A technique is presented for determaining the stability of lumped-pararaeter, time-varying, dynamic systems with aperiodic coefficients. An “energy like” function is used to develop stability conditions which are direct in terms of the coefficient matrices. The significance of what is presented here is twofold. First, it gives stability conditions applicable to systems which are not necessarily periodic. Second, is allows for a systematic categorization of the effects of the parameter changes on system response and stability, in order to provide a better understanding of the behavior of this class of dynamic systems as they arise in various areas of engineering.     

Analysis of periodic-quasiperiodic nonlinear systems via Lyapunov-Floquet transformation and normal forms

In this paper a general technique for the analysis of nonlinear dynamical systems with periodic-quasiperiodic coefficients is developed. For such systems the coefficients of the linear terms are periodic with frequency ω while the coefficients of the nonlinear terms contain frequencies that are incommensurate with ω. No restrictions are placed on the size of the periodic terms appearing in the linear part of system equation. Application of Lyapunov-Floquet transformation produces a dynamically equivalent system in which the linear part is time-invariant and the time varying coefficients of the nonlinear terms are quasiperiodic. Then a series of quasiperiodic near-identity transformations are applied to reduce the system equation to a normal form. In the process a quasiperiodic homological equation and the corresponding ‘solvability condition’ are obtained. Various resonance conditions are discussed and examples are included to show practical significance of the method. Results obtained from the quasiperiodic time-dependent normal form theory are compared with the numerical solutions. A close agreement is found.     

The complete infinite series solution of systems governed by the wave equation with boundary damping

A common method of solving initial boundary value problems is separation of variables, denoted as modal analysis in the field of flexible structures. For systems with undamped boundary conditions the method is well-established, but for systems with boundary damping it does not provide closed form solutions. In this paper the exact modal series solution for second order systems with damped boundaries is derived with explicit expressions for the series coefficients. Knowledge of these coefficients enables practical applications of the solution, such as finite dimension approximation. The key element of the derivation is a new orthogonality condition for the damped eigenfunctions. The modal series is also transformed into a traveling wave form. The solution, which is the extension of the classical D’Alembert formula, is represented by a single equivalent propagating wave. A component of the solution, denoted by “end waves”, is identified to provide the continuity of the systems displacement response.     

Crack-tip field on mode Ⅱ interface crack of double dissimilar orthotropic composite materials

Two systems of non-homogeneous linear equations with 8 unknowns are obtained.This is done by introducing two stress functions containing 16 undetermined coefficients and two real stress singularity exponents with the help of boundary conditions.By solving the above systems of non-homogeneous linear equations,the two real stress singularity exponents can be determined when the double material parameters meet certain conditions.The expression of the stress function and all coefficients are obtained based on the uniqueness theorem of limit.By substituting these parameters into the corresponding mechanics equations,theoretical solutions to the stress intensity factor,the stress field and the displacement field near the crack tip of each material can be obtained when both discriminants of the characteristic equations are less than zero.Stress and displacement near the crack tip show mixed crack characteristics without stress oscillation and crack surface overlapping.As an example,when the two orthotropic materials are the same,the stress singularity exponent,the stress intensity factor,and expressions for the stress and the displacement fields of the orthotropic single materials can be derived.     

Technical Stability of Forced Automatic-Control Variable-Structure Systems

Sufficient conditions for technical stability of functional states of controlled systems with a variable structure are derived. Allowance is made for external perturbations acting on the given process for all possible initial distributions from the set of the process' initial states predetermined in a quadratic measure. The differential equations characterizing the system under consideration include coefficients that vary stepwise with stepwise change in the parameters of the discontinuous control function. It is shown that the conditions of technical stability obtained do not necessarily depend on the existence of sliding modes in variable-structure systems. The eigenvalues of the quadratic forms of the corresponding Lyapunov functions are found to relate to the criteria of technical stability of automatic-control variable-structure processes     

The Maslov Index and Spectral Counts for Linear Hamiltonian Systems on [0, 1]

Working with general linear Hamiltonian systems on [0, 1], and with a wide range of self-adjoint boundary conditions, including both separated and coupled, we develop a general framework for relating the Maslov index to spectral counts. Our approach is illustrated with applications to Schrödinger systems on \({\mathbb {R}}\) with periodic coefficients, and to Euler–Bernoulli systems in the same context.     

On the strong ellipticity for orthotropic micropolar elastic bodies in a plane strain state

In the present paper we consider an orthotropic micropolar elastic material subject to a state of plane strain. In this context, we establish necessary and sufficient conditions for the strong ellipticity of constitutive coefficients. Furthermore, we study existence of progressive plane waves under the strong ellipticity conditions previously determined. Finally, we detail the results obtained for a specific class of materials related to tetragonal systems.     

Setting up Lyapunov functions for the class of systems with quasiperiodic coefficients

The paper proposes a method to set up a matrix-valued Lyapunov function for a system of differential equations with quasiperiodic coefficients. This function is used to establish asymptotic stability conditions for a class of linear systems Translated from Prikladnaya Mekhanika, Vol. 44, No. 12, pp. 121–130, December 2008.     

Revisiting the theory of stability on time scales of the class of linear systemswith structural perturbations

Linear systems of dynamic equations with periodic coefficients and structural perturbations on time scale are analyzed for Lyapunov stability. Sufficient conditions for the asymptotic stability of the equations are established based on the matrix-value concept of Lyapunov’s direct method for all values of the structural matrix from the structural set. A system of two dynamic equations on time scale is considered as an example of applying the theoretical results obtained     

A direct analysis of nonlinear systems with external periodic excitations via normal forms

This study presents a direct methodology for a quantitative analysis of nonlinear dynamic systems with external periodic forcing via an application of the theory of normal forms. Rather than introducing a new state variable to reduce the problem to a homogeneous one, a set of time-dependant near-identity transformations is applied to construct the normal forms. In the process, the total response of the system is expressed as superposition of a steady state solution and a transient solution. A steady state solution of the system is obtained by the method of harmonic balance and the transient solution is obtained by solving a set of time periodic homological equations. The proposed method can be applied to time-invariant as well as time varying systems. After discussing the time-invariant case, the methodology is extended to systems with time-periodic coefficients. The case of time periodic systems is handled through an application of the Lyapunov–Floquet (L–F) transformation. Application of the L–F transformation produces a dynamically equivalent system in which the linear part of the system is time-invariant, making the system amenable to near-identity transformations. An example for each type of system, namely, constant coefficients and time-varying coefficients, is included to demonstrate effectiveness of the method. Various resonance conditions are discussed. It is observed that the linear parametric excitation term need not be small as generally assumed in perturbation and averaging techniques. Results obtained by proposed methods are compared with numerical solutions. Close agreements are found in some typical applications.     

Complex Attractors and Patterns in Reaction–Diffusion Systems

We consider semiflows generated by initial boundary value problems for reaction–diffusion systems. In these systems, reaction terms satisfy general conditions, which admit a transparent chemical interpretation. It is shown that the semiflows generated by these initial boundary value problems exhibit a complicated large time behavior. Any structurally stable finite dimensional dynamics (up to an orbital topological equivalence) can be realized by these semiflows by a choice of appropriate external sources and diffusion coefficients (nonlinear terms are fixed). Results can be applied to the morphogenesis and pattern formation problems.     

The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity

In this article we make a full study of the class of non-degenerate real planar quadratic differential systems having all points at infinity (in the Poincaré compactification) as singularities. We prove that all such systems have invariant affine lines of total multiplicity 3, give all their configurations of invariant lines and show that all these systems are integrable via the method of Darboux having cubic polynomials as inverse integrating factors. After constructing the topologically distinct phase portraits in this class we give invariant necessary and sufficient conditions in terms of the 12 coefficients of the systems for the realization of each one of them and give representatives of the orbits under the action of the affine group and time rescaling. We construct the moduli space of this class for this action and give the corresponding bifurcation diagram. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday