DELAYED BIFURCATION IN FIRST-ORDER SINGULARLY PERTURBED PROBLEMS WITH A NONGENERIC TURNING POINT

2012 ◽  
Vol 22 (12) ◽  
pp. 1250302 ◽  
Author(s):  
JIANHE SHEN ◽  
MAOAN HAN
Keyword(s):  
Asymptotic Behavior ◽  
Hopf Bifurcation ◽  
Turning Point ◽  
Upper And Lower Solutions ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
First Order ◽  
Delayed Bifurcation ◽  
Dynamical Properties ◽  
Theoretical Results

In this paper, based on the method of upper and lower solutions, delayed bifurcation in first-order singularly perturbed problems with a nongeneric turning point is studied. The asymptotic behavior of the solutions is understood by constructing the upper and lower solutions with the desired dynamical properties. As an application of the obtained results, delayed phenomenon of degenerate Hopf bifurcation in a planar polynomial differential system with a slowly varying parameter is discussed in detail and the maximal delay is calculated. Numerical simulations are carried out to verify the theoretical results.

Stability and Hopf bifurcation analysis for Nicholson's blowflies equation with non-local delay

2012 ◽  
Vol 23 (6) ◽  
pp. 777-796 ◽  
Author(s):  
RUI HU ◽  
YUAN YUAN
Keyword(s):  
Hopf Bifurcation ◽  
Upper And Lower Solutions ◽  
Laboratory Data ◽  
Steady State Solution ◽  
Steady States ◽  
Nicholson’S Blowflies Equation ◽  
Non Local ◽  
The Stability ◽  
Manifold Theory ◽  
Theoretical Results

We consider a diffusive Nicholson's blowflies equation with non-local delay and study the stability of the uniform steady states and the possible Hopf bifurcation. By using the upper- and lower solutions method, the global stability of constant steady states is obtained. We also discuss the local stability via analysis of the characteristic equation. Moreover, for a special kernel, the occurrence of Hopf bifurcation near the steady state solution and the stability of bifurcated periodic solutions are given via the centre manifold theory. Based on laboratory data and our theoretical results, we address the influence of various types of vaccinations in controlling the outbreak of blowflies.

scholarly journals SCHEMES CONVERGENT Ε-UNIFORMLY FOR PARABOLIC SINGULARLY PERTURBED PROBLEMS WITH A DEGENERATING CONVECTIVE TERM AND A DISCONTINUOUS SOURCE

2015 ◽  
Vol 20 (5) ◽  
pp. 641-657 ◽  
Author(s):  
Carmelo Clavero ◽  
Jose Luis Gracia ◽  
Grigorii I. Shishkin ◽  
Lidia P. Shishkina
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Euler Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Interior Layer ◽  
Convective Term ◽  
Singularly Perturbed Problems ◽  
First Order

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed.

scholarly journals Uniform Hybrid Difference Scheme for Singularly Perturbed Differential-Difference Turning Point Problems Exhibiting Boundary Layers

2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Wondwosen Gebeyaw Melesse ◽  
Awoke Andargie Tiruneh ◽  
Getachew Adamu Derese
Keyword(s):  
Difference Scheme ◽  
Boundary Layers ◽  
Turning Point ◽  
Original Problem ◽  
Second Order ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problem ◽  
Uniformly Convergent ◽  
Theoretical Results ◽  
Taylor’S Series

In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.

Singularly perturbed problems with a turning point: The non-compatible case

2014 ◽  
Vol 12 (03) ◽  
pp. 293-321 ◽  
Author(s):  
Chang-Yeol Jung ◽  
Roger Temam
Keyword(s):  
Asymptotic Expansions ◽  
Turning Point ◽  
Limit Problem ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems ◽  
Compatibility Conditions ◽  
Compactly Supported ◽  
The Given

The singularly perturbed problems with a turning point were discussed in [21]. The case where the limit problem is compatible with the given data was fully resolved. However, with limited compatibility conditions on the data, the asymptotic expansions were constructed only up to the order of the level of compatibilities. In this paper, using a smooth cut-off function compactly supported around the turning point we resolve the difficulties incurred from the non-compatible data and finally provide the full asymptotic expansions up to any order.

Regularization of Integral Operators in Singularly Perturbed Problems Using Normal Forms

Vestnik MEI ◽  
2019 ◽  
Vol 6 ◽  
pp. 131-137
Author(s):  
Abdukhafiz A. Bobodzhanova ◽  
◽  
Valeriy F. Safonov ◽  
Keyword(s):  
Normal Forms ◽  
Integral Operators ◽  
Singularly Perturbed ◽  
Singularly Perturbed Problems
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