scholarly journals THE HOMOTOPY ANALYSIS METHOD IN BIFURCATION ANALYSIS OF DELAY DIFFERENTIAL EQUATIONS

2012 ◽  
Vol 22 (08) ◽  
pp. 1230024 ◽  
Author(s):  
ANDREA BEL ◽  
WALTER REARTES
Keyword(s):  
Periodic Solutions ◽  
Bifurcation Analysis ◽  
Homotopy Analysis Method ◽  
Delay Differential Equations ◽  
Analysis Method ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential ◽  
Bifurcation And Stability

In this paper we apply the homotopy analysis method (HAM) to study the van der Pol equation with a linear delayed feedback. The paper focuses on the calculation of periodic solutions and associated bifurcations, Hopf, double Hopf, Neimark–Sacker, etc. In particular, we discuss the behavior of the systems in the neighborhoods of double Hopf points. We demonstrate the applicability of HAM to the analysis of bifurcation and stability.

Search for Periodic Orbits in Delay Differential Equations

2014 ◽  
Vol 24 (06) ◽  
pp. 1450084 ◽  
Author(s):  
Romina Cobiaga ◽  
Walter Reartes
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Heuristic Search ◽  
Delay Differential Equations ◽  
Anharmonic Oscillator ◽  
Analysis Method ◽  
Van Der Pol ◽  
Homotopy Analysis ◽  
Delay Differential

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

scholarly journals A Natural Homotopy Analysis Method for Nonlinear Delay Differential Equations

2019 ◽  
Vol 1 (2) ◽  
pp. 86-90
Author(s):  
Aminu Barde
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Homotopy Analysis Method ◽  
Delay Differential Equations ◽  
Analysis Method ◽  
Linear Delay ◽  
Homotopy Analysis ◽  
Nonlinear Delay Differential Equations ◽  
Functional Differential ◽  
Delay Differential

Delay differential equation (DDEs) is a type of functional differential equation arising in numerous applications from different areas of studies, for example biology, engineering population dynamics, medicine, physics, control theory, and many others. However, determining the solution of delay differential equations has become a difficult task more especially the nonlinear type. Therefore, this work proposes a new analytical method for solving non-linear delay differential equations. The new method is combination of Natural transform and Homotopy analysis method. The approach gives solutions inform of rapid convergence series where the nonlinear terms are simply computed using He's polynomial. Some examples are given, and the results obtained indicate that the approach is efficient in solving different form of nonlinear DDEs which reduces the computational sizes and avoid round-off of errors.

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