An algebraic criterion of the Darboux integrability of differential-difference equations and systems

The article investigates systems of differential-difference equations of hyperbolic type, integrable in sense of Darboux. The concept of a complete set of independent characteristic integrals underlying Darboux integrability is discussed. A close connection is found between integrals and characteristic Lie-Rinehart algebras of the system. It is proved that a system of equations is Darboux integrable if and only if its characteristic algebras in both directions are finite-dimensional.

Sharp Bound of the Number of Zeros for a Liénard System with a Heteroclinic Loop

In the presented paper, the Abelian integral I h of a Liénard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to find the least upper bound of the number of limit cycles bifurcating from periodic annulus.

New formulations of algebraic criteria for controllability and observability of a linear MIMO-system

New formulations of algebraic criteria for controllability and observability of a linear dynamical system with multiple inputs and outputs (MIMO-systems) are given, the corresponding theorems are formulated. The criteria are based on algebraic relations between linear combinations of the control matrix columns and own vectors of the free dynamics matrix. Keywords algebraic criterion; controllability; observability; linear MIMO-system; own value; own vector; Krylov vector and matrix; kernel; cokernel

Globally projective synchronization for Caputo fractional quaternion-valued neural networks with discrete and distributed delays

<abstract><p>This paper is devoted to discussing the globally projective synchronization of Caputo fractional-order quaternion-valued neural networks (FOQVNNs) with discrete and distributed delays. Without decomposing the FOQVNNs into several subsystems, by employing the Lyapunov direct method and inequality techniques, the algebraic criterion for the globally projective synchronization is derived. The effectiveness of the proposed result is illustrated by the MATLAB toolboxes and numerical simulation.</p></abstract>

Determination of Nonchaotic Behavior for Some Classes of Polynomial Jerk Equations

In this work, by using an algebraic criterion presented by us in an earlier paper, we determine the conditions on the parameters in order to guarantee the nonchaotic behavior for some classes of nonlinear third-order ordinary differential equations of the form [Formula: see text] called jerk equations, where [Formula: see text] is a polynomial of degree [Formula: see text]. This kind of equation is often used in literature to study chaotic dynamics, due to its simple form and because it appears as mathematical model in several applied problems. Hence, it is an important matter to determine when it is chaotic and also nonchaotic. The results stated here, which are proved using the mentioned algebraic criterion, corroborate and extend some results already presented in literature, providing simpler proofs for the nonchaotic behavior of certain jerk equations. The algebraic criterion proved by us is quite general and can be used to study nonchaotic behavior of other types of ordinary differential equations.

Hopf Bifurcation of Heated Panels Flutter in Supersonic Flow

A differential equation of panel vibration in supersonic flow is established on the basis of the thin-plate large deflection theory under the assumption of a quasi-steady temperature field. The equation is dimensionless, and the derivation of its second-order Galerkin discretization yields a four-dimensional system. The algebraic criterion of the Hopf bifurcation is applied to study the motion stability of heated panels in supersonic flow. We provide a supplementary explanation for the proof process of a theorem, and analytical expressions of flutter dynamic pressure and panel vibration frequencies are derived. The conclusion is that the algebraic criterion of Hopf bifurcation can be applied in high-dimensional problems with many parameters. Moreover, the computational intensity of the method established in this work is less than that of conventional eigenvalue computation methods using parameter variation.

Analysis of Snapback Repellers Using Methods of Symbolic Computation

This paper presents an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multidimensional discrete dynamical systems based on a modified Marotto’s theorem developed by Li and Chen (called “Marotto–Li–Chen theorem”). Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and propose an algorithmic approach to solve this algebraic problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.