Quadratic Right-hand Sides Research Articles

Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases.

New algebraically solvable systems of two autonomous first-order ordinary differential equations with purely quadratic right-hand sides

We identify many new solvable subcases of the general dynamical system characterized by two autonomous first-order ordinary differential equations with purely quadratic right-hand sides and the solvable character of these dynamical systems amounting to the possibility to obtain the solution of their initial value problem via algebraic operations. Equivalently, by considering the analytic continuation of these systems to complex time, their algebraically solvable character corresponds to the fact that their general solution either is single-valued or features only a finite number of algebraic branch points as functions of complex time (the independent variable). Thus, our results provide a major enlargement of the class of solvable systems beyond those with a single-valued general solution identified by Garnier about 60 years ago. An interesting property of several of these new dynamical systems is the elementary character of their general solution, identifiable as the roots of a polynomial with explicitly obtainable time-dependent coefficients. We also mention that, using a time-dependent change of (dependent and independent) variables involving the imaginary parameter iω, isochronous variants of each of the algebraically solvable models we identify, can be explicitly exhibited: by this we mean that these variants all feature the remarkable property that their generic solution is periodic with a period that is a fixed integer multiple of the basic period T = 2π/ω.

Homogeneous systems with quadratic integrals, Lie-Poisson quasibrackets, and Kovalevskaya's method

We consider differential equations with quadratic right-hand sides that admit two quadratic first integrals, one of which is a positive- definite quadratic form. We indicate conditions of general nature under which a linear change of variables reduces this system to a certain ` canonical' form. Under these conditions, the system turns out to be divergenceless and can be reduced to a Hamiltonian form, but the corresponding linear Lie-Poisson bracket does not always satisfy the Jacobi identity. In the three-dimensional case, the equations can be reduced to the classical equations of the Euler top, and in four-dimensional space, the system turns out to be superintegrable and coincides with the Euler-Poincaré equations on some Lie algebra. In the five-dimensional case we find a reducing multiplier after multiplying by which the Poisson bracket satisfies the Jacobi identity. In the general case for we prove the absence of a reducing multiplier. As an example we consider a system of Lotka-Volterra type with quadratic right-hand sides that was studied by Kovalevskaya from the viewpoint of conditions of uniqueness of its solutions as functions of complex time. Bibliography: 38 titles.

The dynamics of systems with servoconstraints. II

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.

Динамика систем с сервосвязями. I

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.

Improved Duality Estimates and Applications to Reaction-Diffusion Equations

We present a refined duality estimate for parabolic equations. This estimate entails new results for systems of reaction-diffusion equations, including smoothness and exponential convergence towards equilibrium for equations with quadratic right-hand sides in two dimensions. For general systems in any space dimension, we obtain smooth solutions of reaction-diffusion systems coming out of reversible chemistry under an assumption that the diffusion coefficients are sufficiently close one to another.

Stability of Nonlinear Autonomous Quadratic Discrete Systems in the Critical Case

Many processes are mathematically simulated by systems of discrete equations with quadratic right‐hand sides. Their stability is thought of as a very important characterization of the process. In this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete autonomous systems in a critical case in the presence of a simple eigenvalue λ = 1 of the matrix of linear terms. In addition to the stability investigation, we also estimate stability domains.

Stability investigation of nonlinear quadratic discrete dynamics systems in the critical case

Many important processes (e.g. in industry and biology) are mathematically simulated with systems of discrete equations with quadratic right-hand sides. The paper presents stability results of quadratic discrete systems in the critical case (in the presence of a simple eigenvalue of the matrix of linear terms that is equal to unity and the others are smaller than unity in absolute value). In addition to the stability investigation of a zero solution, we also estimate stability domains.

Stability investigation of quadratic systems with delay

Systems of differential equations with quadratic right‐hand sides with delay are considered in the paper. Compact matrix notation form is proposed for the systems of such type. Stability investigations are performed by Lyapunov′s second method with functions of quadratic form. Stability conditions of quadratic systems with delay, uniformly by argument deviation, and with delay depending on the system′s parameters are derived. A guaranteed radius of the ball of asymptotic stability region for zero solution is obtained.

The stability of uniformly accelerated motions

Differential equations with quadratic right-hand sides and additional constant terms are considered. Important examples are self-driven gyroscopes and the problem of the motion of a rigid body in an unbounded volume of ideal fluid subject to a force and a torque which are constant in an attached frame of reference. Under certain simple conditions, these equations have solutions that increase linearly with time. In problems of dynamics they describe uniformly accelerated motions of mechanical systems. The stability of such motions is investigated in the first approximation and using bundles of integrals. The general results are used to investigate the stability of uniformly accelerated screw motions of a rigid body in a fluid.

Majorant estimates for solutions of quadratic difference systems with lag

Systems of differential equations with lag with quadratic right-hand sides are considered. A compact matrix form for representing these systems in the general form is suggested. Upper and lower bounds for solutions of systems of quadratic differential equations with arbitrary lag are obtained by using quadratic Lyapunov functions.