Time-Delay Two-Dimension Mathieu Equation in Synchrotron Dynamics
نویسنده
چکیده
Two dimensions Mathieu equation containing periodic terms as well as the delayed parameters has been investigated in the present work. The present system represents to a generalized form of the one-dimension delay Mathieu equation. The mathematical difficulty for delay the coupled Mathieu equation has been overcome by using the matrices method. Properties of inverse complex matrices enable us to transform the vector form of the solvability conditions to the scalar form. Small oscillation about a marginal state is introduced by using the method of multiple scales. Stability criteria for the complex matrices have been established and lead to obtain resonance curves. The analysis has been extended so that the delay 2-dimensions Mathieu equation containing weak complex damping part. Stability conditions and the transition curves that included the influence of both the delayed as well the complex damping terms has been obtained. The transition curves are analyzed using the method of harmonic balance. We note that the delayed higher dimension of the parametric excitation has a great interest and application to the design of nuclear accelerators.
منابع مشابه
Delay-Coupled Mathieu Equations in Synchrotron Dynamics Revisited: Delay Terms in the Slow Flow
In a previous work, we applied perturbation methods to a system of two delay-coupled Mathieu equations, resulting in a slow flow which contains delayed variables. This previous treatment involved a convenient approximation which involved replacing delay terms in the slow flow by non-delay terms. The current paper explores the effect of keeping delay terms in the slow flow with the hope of illus...
متن کاملDelay-Coupled Mathieu Equations in Synchrotron Dynamics
This paper investigates the dynamics of two couplecd Mathieu equations. The coupling functions involve both delayed and nondelay terms. We use a perturbation method to obtain a slow flow which is then studied using Routh-Hurwitz stability criterion. Analytic results are shown to compare favorably with numerical integration. The numerical integrator, DDE23, is shown to inadvertently add damping....
متن کامل2:1 Resonance in the delayed nonlinear Mathieu equation
We investigate the dynamics of a delayed nonlinear Mathieu equation: ẍ + (δ + εα cos t)x + εγ x = εβx(t − T ) in the neighborhood of δ = 1/4. Three different phenomena are combined in this system: 2:1 parametric resonance, cubic nonlinearity, and delay. The method of averaging (valid for small ε) is used to obtain a slow flow that is analyzed for stability and bifurcations. We show that the 2:1...
متن کاملStability of the Damped Mathieu Equation With Time Delay
In the space of the system parameters, the stability charts are determined for the delayed and damped Mathieu equation defined as ẍ~t!1k ẋ~t!1~d1« cos t!x~t!5bx~t22p! . This stability chart makes the connection between the Strutt-Ince chart of the damped Mathieu equation and the Hsu-Bhatt-Vyshnegradskii chart of the autonomous second order delay-differential equation. The combined charts descri...
متن کاملLyapunov Spectrum of Linear Delay Differential Equations with Time-Varying Delay
To understand the dynamics of systems with fluctuations in the retarded argument, the Lyapunov spectra of a scalar, linear delay differential equation (DDE) are studied. Depending on the structure of the deviated argument some Lyapunov exponents can be equal to minus infinity, which indicate the difference between the dimension of the state space and the asymptotic solution space. In order to v...
متن کامل