Journal ArticleDOI
Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations
S. Das,Anindya Chatterjee +1 more
Reads0
Chats0
TLDR
In this article, small perturbations of three linear Delay Differential Equations (DDEs) close to Hopf bifurcation points are studied, and it is shown that the method of multiple scales, on simply discarding the infinitely many exponentially decaying components of the complementary solutions obtained at each stage of the approximation, can bypass the explicit center manifold calculation.Abstract:
We study small perturbations of three linear Delay Differential Equations (DDEs) close to Hopf bifurcation points. In analytical treatments of such equations, many authors recommend a center manifold reduction as a first step. We demonstrate that the method of multiple scales, on simply discarding the infinitely many exponentially decaying components of the complementary solutions obtained at each stage of the approximation, can bypass the explicit center manifold calculation. Analytical approximations obtained for the DDEs studied closely match numerical solutions.read more
Citations
More filters
Journal ArticleDOI
An Efficient Method for Studying Weak Resonant Double Hopf Bifurcation in Nonlinear Systems with Delayed Feedbacks
TL;DR: The perturbation‐incremental scheme is employed to investigate the delay‐induced weak resonant double Hopf bifurcation and dynamics in the van der Pol–Duffing and the Stuart–Landau systems with delayed feedback.
Journal ArticleDOI
Subcritical bifurcation and bistability in thermoacoustic systems
TL;DR: In this article, an example of a Rijke tube model with an explicit time delay is presented, and a linear stability analysis of the model is performed to identify parameter values at the onset of linear instability via a Hopf bifurcation.
Journal ArticleDOI
Galerkin Projections for Delay Differential Equations
Pankaj Wahi,Anindya Chatterjee +1 more
TL;DR: In this article, a Galerkin projection technique was proposed to obtain finite-dimensional ODE approximations for delay differential equations (DDEs) in a straightforward fashion, requiring 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.
Regenerative tool chatter near a codimension-2 hopf point
Pankaj Wahi,Anindya Chatterjee +1 more
TL;DR: In this paper, a delay differential equation (DDE) near a codimension 2 Hopf bifurcation point is studied, and analytical expressions for the double Hopf points are obtained.
Journal ArticleDOI
Regenerative Tool Chatter Near a Codimension 2 Hopf Point Using Multiple Scales
Pankaj Wahi,Anindya Chatterjee +1 more
TL;DR: In this paper, a delay differential equation (DDE) near a codimension 2 Hopf bifurcation point is studied, and analytical expressions for the double Hopf points are obtained.
References
More filters
Book
Introduction to Functional Differential Equations
TL;DR: The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977 and attempts to maintain the spirit of that book and have retained approximately one-third of the material intact.
Book
Perturbation Methods
Ali H. Nayfeh,Vimal Singh +1 more
TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
Book
Introduction to perturbation techniques
TL;DR: In this paper, the authors introduce the notion of forced Oscillations of the Duffing Equation and the Mathieu Equation for weakly nonlinear systems with quadratic and cubic nonlinearities.
Journal ArticleDOI
Normal Forms for Retarded Functional Differential Equations with Parameters and Applications to Hopf Bifurcation
Teresa Faria,Luis T. Magalhães +1 more
TL;DR: In this article, the authors considered the problem of normal forms associated with the flow on a finite-dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity.
Journal ArticleDOI
Normal Forms for Retarded Functional Differential Equations and Applications to Bogdanov-Takens Singularity
Teresa Faria,Luis T. Magalhães +1 more
TL;DR: In this paper, a phase space appropriate to the computation of normal forms associated with the flow on a finite-dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity is introduced, and adequate nonresonance conditions for the normal forms are derived.