Exponential stability of an invariant torus of a nonlinear countable system of differential equations

1990 ◽  
Vol 42 (3) ◽  
pp. 357-360
Author(s):  
Yu. V. Teplinskii ◽  
P. I. Avdeyuk
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Invariant Torus ◽  
Countable 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.

scholarly journals Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

2019 ◽  
Vol 3 (4) ◽  
pp. 50 ◽  
Author(s):  
Gani Stamov ◽  
Anatoliy Martynyuk ◽  
Ivanka Stamova
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Practical Stability ◽  
Volterra Equations ◽  
System Of Differential Equations ◽  
First Time ◽  
New Criteria

In this paper, an impulsive fractional-like system of differential equations is introduced. The notions of practical stability and boundedness with respect to h-manifolds for fractional-like differential equations are generalized to the impulsive case. For the first time in the literature, Lyapunov-like functions and their derivatives with respect to impulsive fractional-like systems are defined. As an application, an impulsive fractional-like system of Lotka–Volterra equations is considered and new criteria for practical exponential stability are proposed. In addition, the uncertain case is also investigated.

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