scholarly journals Interaction of homoclinic solutions and Hopf points in functional differential equations of mixed type

2009 ◽  
Vol 139 (1) ◽  
pp. 123-155 ◽  
Author(s):  
Marc Georgi
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Homoclinic Orbit ◽  
Mixed Type ◽  
Functional Differential Equations ◽  
Homoclinic Orbits ◽  
Homoclinic Solution ◽  
Equations Of Mixed Type ◽  
Functional Differential ◽  
Equation Of Mixed Type

We study a homoclinic bifurcation in a general functional differential equation of mixed type. More precisely, we investigate the case when the asymptotic steady state of a homoclinic solution undergoes a Hopf bifurcation. Bifurcations of this kind are diffcult to analyse due to the lack of Fredholm properties. In particular, a straightforward application of a Lyapunov–Schmidt reduction is not possible.As one of the main results we prove the existence of centre-stable and centre-unstable manifolds of steady states near homoclinic orbits. With their help, we can analyse the bifurcation scenario similar to the case for ordinary differential equations and can show the existence of solutions which bifurcate near the homoclinic orbit, are decaying in one direction and oscillatory in the other direction. These solutions can be visualized as an interaction of the homoclinic orbit and small periodic solutions that exist on account of the Hopf bifurcation, for exactly one asymptotic direction t→8 or t→−∞.

scholarly journals Oscillation Criteria for Certain Second-Order Nonlinear Neutral Differential Equations of Mixed Type

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Zhenlai Han ◽  
Tongxing Li ◽  
Chenghui Zhang ◽  
Ying Sun
Keyword(s):  
Differential Equations ◽  
Mixed Type ◽  
Second Order ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Equations Of Mixed Type ◽  
Positive Integers

Some oscillation criteria are established for the second-order nonlinear neutral differential equations of mixed type[(x(t)+p1x(t−τ1)+p2x(t+τ2))γ]′​′=q1(t)xγ(t−σ1)+q2(t)xγ(t+σ2),t≥t0, whereγ≥1is a quotient of odd positive integers. Our results generalize the results given in the literature.

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