Qualitative analysis of nonlinear quasi-monotone dynamical systems described by functional-differential equations

This paper discusses properties related to the stability of a nonlinear quasi-monotone dynamical system described by a functionaldifferential equationẋ =F(x_t, t) + u(t). Specially, mathematical conditions which guarantee the same qualitative behavior inherent in a nonlinear off-diagonally monotone dynamical systemẋ=f(x(t),t)+u(t)are discussed. We first consider the basic properties of solutions: lower and upper bound preservation and ordering preservation of solutions. By using these properties, we estimate the trajectory. behavior by means of a partial ordering relation, and derive the following results: IfFis independent oft, anduis a constant input, then every bounded solution converges to a unique equilibrium pointx^{}under some natural conditions. In addition, ifFis a nonlinear functional with separate variables, then every solution converges toxᵃˢᵗunder the same conditions; IfF(x_t,{·})andu(·)are periodic and have the same periodω, then, under certain natural conditions, there is aω-periodic solutionx^{}(·), and every solution converges to it if it is a uniquew-periodic solution.