Splitting of a system of differential equations with slowly varying phase in the neighborhood of an asymptotically stable invariant torus

1986 ◽  
Vol 37 (6) ◽  
pp. 617-621 ◽  
Author(s):  
A. M. Samoilenko ◽  
M. Ya. Svishchuk
Keyword(s):  
Differential Equations ◽  
Invariant Torus ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Stable Invariant ◽  
Stable Invariant Torus

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.

PERIODIC ORBITS IN THE NEWTON–LEIPNIK SYSTEM

2002 ◽  
Vol 12 (03) ◽  
pp. 511-523 ◽  
Author(s):  
BENJAMIN A. MARLIN
Keyword(s):  
Periodic Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Periodic Orbits ◽  
Closed Orbit ◽  
Numerical Results ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Promising Candidate ◽  
Closed Orbits

This paper considers an autonomous nonlinear system of differential equations derived in [Leipnik, 1979]. A criterion for the existence of closed orbits in similar systems is presented. Numerical results are made rigorous by the use of interval analytic techniques in establishing the existence of a periodic solution which is not asymptotically stable. The limitations of the method of locating orbits are considered when a promising candidate for a closed orbit is shown not to intersect itself.

scholarly journals Asymptotic Properties of Solutions to Delay Differential Equations Describing Plankton—Fish Interaction

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3064
Author(s):  
Maria A. Skvortsova
Keyword(s):  
Differential Equations ◽  
Equilibrium Point ◽  
Delay Differential Equations ◽  
Asymptotic Properties ◽  
Asymptotic Behavior Of Solutions ◽  
Asymptotically Stable ◽  
System Of Differential Equations ◽  
Estimates Of Solutions ◽  
Two Delays ◽  
Delay Differential

We consider a system of differential equations with two delays describing plankton–fish interaction. We analyze the case when the equilibrium point of this system corresponding to the presence of only phytoplankton and the absence of zooplankton and fish is asymptotically stable. In this case, the asymptotic behavior of solutions to the system is studied. We establish estimates of solutions characterizing the stabilization rate at infinity to the considered equilibrium point. The results are obtained using Lyapunov–Krasovskii functionals.

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