Smoothness with respect to a parameter of an exponentially dichotomous invariant torus of a quasilinear system of differential equations

1989 ◽  
Vol 41 (10) ◽  
pp. 1228-1230
Author(s):  
O. V. Shpakovich
Keyword(s):  
Differential Equations ◽  
Invariant Torus ◽  
Quasilinear System ◽  
System Of Differential Equations

scholarly journals Showcase of Blue Sky Catastrophes

2014 ◽  
Vol 24 (08) ◽  
pp. 1440003 ◽  
Author(s):  
Leonid Pavlovich Shilnikov ◽  
Andrey L. Shilnikov ◽  
Dmitry V. Turaev
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Invariant Torus ◽  
Complex Dynamics ◽  
Klein Bottle ◽  
System Of Differential Equations ◽  
Homoclinic Connection ◽  
Global Stable Manifold

Let a system of differential equations possess a saddle periodic orbit such that every orbit in its unstable manifold is homoclinic, i.e. the unstable manifold is a subset of the (global) stable manifold. We study several bifurcation cases of the breakdown of such a homoclinic connection that causes the blue sky catastrophe, as well as the onset of complex dynamics. The birth of an invariant torus and a Klein bottle is also described.

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