The existence results of positiveω-periodic solutions are obtained for the third-order ordinary differential equation with delaysu′′′(t)+a(t)u(t)=f(t,u(t-τ0),u′(t-τ1),u′′(t-τ2)),t∈ℝ,wherea∈C(ℝ,(0,∞))isω-periodic function andf:ℝ×[0,∞)×ℝ2→[0,∞)is a continuous function which isω-periodic int,and τ0,τ1,τ2are positive constants. The discussion is based on the fixed-point index theory in cones.
In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x‴+ft,x,x′,x″=0,t∈ℝ≔−∞,∞, where f:ℝ4⟶ℝ is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.
The paper considers the existence of quasi-periodic solutions in three-dimensional systems. Since these solutions commonly arise as a consequence of a Neimark–Sacker bifurcation of a limit cycle, a fairly general relation connected to this phenomenon is pointed out as the main result of the paper. Then, the application of harmonic balance techniques makes possible to exploit such a relation. In particular, a simplified condition denoting the quasi-periodicity onset can be derived, in making evident the main elements for this transition in terms of structure and parameters, and hence some remarks on the features of the interested systems. Several examples show the application of the above condition to detect "tori" in the state space in a qualitative (not simply numerical) way. They consider classical systems — Rössler, where such behavior seems to be unknown, Chua, forced Van der Pol — and new quadratic systems.
By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence,
nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.
Several numerical approaches have been developed to capture nonlinear effects of dynamical systems. In this paper we present a mixed shooting-harmonic balance method to solve large mechanical systems with local nonlinearities efficiently. The Harmonic Balance Method as well as the shooting method have both their pros and cons. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore advantages of both.
Abstract
A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.
Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays
