A Stable Manifold Theorem for a System of Volterra Integro-Differential Equations

1975 ◽  
Vol 6 (3) ◽  
pp. 506-522 ◽  
Author(s):  
R. K. Miller ◽  
J. A. Nohel
Keyword(s):  
Differential Equations ◽  
Stable Manifold ◽  
Stable Manifold Theorem

Local stable manifold theorem for fractional systems

2015 ◽  
Vol 83 (4) ◽  
pp. 2435-2452 ◽  
Author(s):  
Amey Deshpande ◽  
Varsha Daftardar-Gejji
Keyword(s):  
Stable Manifold ◽  
Fractional Systems ◽  
Stable Manifold Theorem ◽  
Local Stable Manifold

scholarly journals Integrability of continuous bundles

2019 ◽  
Vol 2019 (752) ◽  
pp. 229-264 ◽  
Author(s):  
Stefano Luzzatto ◽  
Sina Tureli ◽  
Khadim War
Keyword(s):  
Dynamical Systems ◽  
Stable Manifold ◽  
Arbitrary Dimension ◽  
Sufficient Conditions ◽  
Classical Theorem ◽  
Uniqueness Of Solutions ◽  
Stable Manifold Theorem ◽  
New Criteria

Abstract We give new sufficient conditions for the integrability and unique integrability of continuous tangent subbundles on manifolds of arbitrary dimension, generalizing Frobenius’ classical theorem for {C^{1}} subbundles. Using these conditions, we derive new criteria for uniqueness of solutions to ODEs and PDEs and for the integrability of invariant bundles in dynamical systems. In particular, we give a novel proof of the Stable Manifold Theorem and prove some integrability results for dynamically defined dominated splittings.

On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter

1991 ◽  
Vol 2 (2) ◽  
pp. 133-158 ◽  
Author(s):  
John G. Byatt-Smith
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Unstable Manifold ◽  
Stable Manifold ◽  
Phase Plane ◽  
Careful Analysis ◽  
Heteroclinic Orbits ◽  
Perturbed System ◽  
Heteroclinic Connection ◽  
Small Dispersion

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

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