scholarly journals Computation of double Hopf points for delay differential equations

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Yingxiang Xu ◽  
Tingting Shi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Delay Systems ◽  
System Of Equations ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Defining System ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Branch Switching

AbstractRelating to the crucial problem of branch switching, the calculation of codimension 2 bifurcation points is one of the major issues in numerical bifurcation analysis. In this paper, we focus on the double Hopf points for delay differential equations and analyze in detail the corresponding eigenspace, which enable us to obtain the finite dimensional defining system of equations of such points, instead of an infinite dimensional one that happens naturally for delay systems. We show that the double Hopf point, together with the corresponding eigenvalues, eigenvectors and the critical values of the bifurcation parameters, is a regular solution of the finite dimensional defining system of equations, and thus can be obtained numerically through applying the classical iterative methods. We show our theoretical findings by a numerical example.

NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

2001 ◽  
Vol 11 (03) ◽  
pp. 737-753 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
KOEN ENGELBORGHS ◽  
DIRK ROOSE
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Numerical Bifurcation Analysis

In this paper we apply existing numerical methods for bifurcation analysis of delay differential equations with constant delay to equations with state-dependent delay. In particular, we study the computation, continuation and stability analysis of steady state solutions and periodic solutions. We collect the relevant theory and describe open theoretical problems in the context of bifurcation analysis. We present computational results for two examples and compare with analytical results whenever possible.

Galerkin Projections for Delay Differential Equations

2004 ◽  
Vol 127 (1) ◽  
pp. 80-87 ◽  
Author(s):  
Pankaj Wahi ◽  
Anindya Chatterjee
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Galerkin Projection ◽  
Shape Functions ◽  
Projection Technique ◽  
Finite Dimensional ◽  
Systems Of Odes ◽  
Qualitative Dynamics ◽  
Delay Differential ◽  
Galerkin Projections

We present a Galerkin projection technique by which finite-dimensional ordinary differential equation (ODE) approximations for delay differential equations (DDEs) can be obtained in a straightforward fashion. The technique requires neither the system to be near a bifurcation point, nor the delayed terms to have any specific restrictive form, or even the delay, nonlinearities, and/or forcing to be small. We show through several numerical examples that the systems of ODEs obtained using this procedure can accurately capture the dynamics of the DDEs under study, and that the accuracy of solutions increases with increasing numbers of shape functions used in the Galerkin projection. Examples studied here include a linear constant coefficient DDE as well as forced nonlinear DDEs with one or more delays and possibly nonlinear delayed terms. Parameter studies, with associated bifurcation diagrams, show that the qualitative dynamics of the DDEs can be captured satisfactorily with a modest number of shape functions in the Galerkin projection.

scholarly journals Periodic Solutions of a System of Delay Differential Equations for a Small Delay

2002 ◽  
Vol 7 (2) ◽  
pp. 295
Author(s):  
Adu A.M. Wasike ◽  
Wandera Ogana
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Original System ◽  
System Of Equations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Stable Periodic ◽  
Delay Differential

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.   

Estimation of Time-Delays Using the Characteristic Roots of Delay Differential Equations

2011 ◽  
Author(s):  
Sun Yi ◽  
Sangseok Yu
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Time Delays ◽  
Delay Systems ◽  
Lambert W Function ◽  
Short Paper ◽  
Characteristic Roots ◽  
General Systems ◽  
Delay Differential ◽  
Estimation Of Time

In this short paper, the preliminary result of a new method for estimation of time-delays of time-delay systems is presented. The presented method makes use of the Lambert W function, and is for scalar first-order delay differential equations (DDEs). Possible extension to general systems of DDEs and application to physical systems are also discussed.

scholarly journals Dependence of Dynamics of a System of Two Coupled Generators with Delayed Feedback on the Sign of Coupling

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1790
Author(s):  
Alexandra Kashchenko
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Coupling Parameter ◽  
Delayed Feedback ◽  
Structure Of Solutions ◽  
Asymptotics Of Solutions ◽  
Finite Dimensional ◽  
Method Of Steps ◽  
Delay Differential ◽  
Negative Coupling

In this paper, we study the nonlocal dynamics of a system of delay differential equations with large parameters. This system simulates coupled generators with delayed feedback. Using the method of steps, we construct asymptotics of solutions. By these asymptotics, we construct a special finite-dimensional map. This map helps us to determine the structure of solutions. We study the dependence of solutions on the coupling parameter and show that the dynamics of the system is significantly different in the case of positive coupling and in the case of negative coupling.

Pole Placement for Delay Differential Equations with Time-Periodic Delays Using Galerkin Approximations (CND-20-1378)

2021 ◽  
Author(s):  
Shanti Swaroop Kandala ◽  
Thomas K. Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Second Order ◽  
Delay Systems ◽  
First Order ◽  
The Galerkin Method ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Abstract Many practical systems have inherent time delays that cannot be ignored; thus, their dynamics are described using delay differential equations (DDEs). The Galerkin approximation method is one strategy for studying the stability of time-delay systems. In this work, we consider delays that are time-varying and, specifically, time-periodic. The Galerkin method can be used to obtain a system of ordinary differential equations (ODEs) from a second-order time-periodic DDE in two ways: either by converting the DDE into a second-order time-periodic partial differential equation (PDE) and then into a system of second-order ODEs, or by first expressing the original DDE as two first-order time-periodic DDEs, then converting into a system of first-order time-periodic PDEs, and finally converting into a first-order time-periodic ODE system. The difference between these two formulations in the context of control is presented in this paper. Specifically, we show that the former produces spurious Floquet multipliers at a spectral radius of 1. We also propose an optimization-based framework to obtain feedback gains that stabilize closed-loop control systems with time-periodic delays. The proposed optimization-based framework employs the Galerkin method and Floquet theory, and is shown to be capable of stabilizing systems considered in the literature. Finally, we present experimental validation of our theoretical results using a rotary inverted pendulum apparatus with inherent sensing delays as well as additional time-periodic state-feedback delays that are introduced deliberately.

scholarly journals Numerical Solution of Piecewise Constant Delay Systems Based on a Hybrid Framework

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
H. R. Marzban ◽  
S. Hajiabdolrahmani
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Operational Matrix ◽  
Delay Systems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Constant Delay ◽  
Piecewise Constant ◽  
Delay Function ◽  
Delay Differential

An efficient numerical scheme for solving delay differential equations with a piecewise constant delay function is developed in this paper. The proposed approach is based on a hybrid of block-pulse functions and Taylor’s polynomials. The operational matrix of delay corresponding to the proposed hybrid functions is introduced. The sparsity of this matrix significantly reduces the computation time and memory requirement. The operational matrices of integration, delay, and product are employed to transform the problem under consideration into a system of algebraic equations. It is shown that the developed approach is also applicable to a special class of nonlinear piecewise constant delay differential equations. Several numerical experiments are examined to verify the validity and applicability of the presented technique.

scholarly journals Numerical Solution of Fuzzy Multiple Hybrid Single Retarded Delay Differential Equations

2019 ◽  
Vol 8 (3) ◽  
pp. 1946-1949
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Exact Solutions ◽  
Numerical Solution ◽  
Hybrid Systems ◽  
Delay Differential Equations ◽  
Fuzzy Systems ◽  
Numerical Solutions ◽  
Delay Systems ◽  
Delay Differential

In this Paper, We are combining so many mathematical-cum-engineering topics such as Fuzzy systems, Delay systems and Hybrid Systems under one roof called Numerical Solutions. The fuzzy valued problem was solved numerically and that approximate solution was compared with that of exact solutions. The non fuzzy and fuzzy valued numerical solutions and their graphical illustrations are also provided for the better understanding of the multiple hybrid single retarded delay problems.

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