Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points

Palmer, Kenneth J.

doi:10.1007/978-3-322-96656-8_5

Cited by 165 publications

(164 citation statements)

References 3 publications

Supporting

Mentioning

164

Contrasting

Order By: Relevance

“…possesses by (Palmer 1988, Lemma 2.7) for each sequence h ∈ X a unique bounded solution on . Thus D x Γ(ξ ,λ ) is also surjective and therefore it is a homeomorphism.…”

Section: A3mentioning

confidence: 98%

See 1 more Smart Citation

Homoclinic trajectories of non-autonomous maps

Hüls

2011

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

For non-autonomous difference equations of the form x n+1 = f (x n , λ n ), n ∈ we consider homoclinic trajectories. These are pairs of trajectories that converge in both time directions towards each other. Assuming hyperbolicity, we derive a numerical method to compute homoclinic trajectories in two steps. In the first step one trajectory is approximated by the solution of a boundary value problem and precise error estimates are given. In particular, influences of parameters λ n with |n| large are discussed in detail. A second trajectory that is homoclinic to the first one is computed in a subsequent step as follows. We transform the original system into a topologically equivalent form having zero as an n-independent fixed point. Applying the boundary value ansatz to the transformed system, we obtain a non-autonomous homoclinic orbit, converging towards the origin, cf. Hüls (2006). Transforming back to the original coordinates leads to the desired homoclinic trajectories. The numerical method and the validity of the error estimates are illustrated by examples.

show abstract

“…possesses by (Palmer 1988, Lemma 2.7) for each sequence h ∈ X a unique bounded solution on . Thus D x Γ(ξ ,λ ) is also surjective and therefore it is a homeomorphism.…”

Section: A3mentioning

confidence: 98%

“…Palmer (1988), where G is Green's function, introduced in (16). Note that due to our assumptions, r n is bounded from above by some constant R for all n ∈ and r n = 0 for n ∈ J = [n − , n + ], since λ n = µ n for n ∈ J.…”

Section: Influence Ofλ N With Large |N|mentioning

confidence: 99%

Homoclinic trajectories of non-autonomous maps

Hüls

2011

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

show abstract

“…It is well known (cf. [12]) that this orbit is transversal according to Definition 1, if these manifolds intersect transversally. In this section, we generalize this result to our non-autonomous context.…”

Section: Transversalitymentioning

confidence: 99%

“…Φ(m, n)u < ∞ for all m ≥ n, cf. [12,Proposition 5.4] for the autonomous counterpart. First, we prove a technical but important lemma, generalizing the ideas of [12,Section 5].…”

Section: Transversally Intersecting Fiber Bundlesmentioning

confidence: 99%

See 1 more Smart Citation

Homoclinic orbits of non-autonomous maps and their approximation

Hüls

2006

Journal of Difference Equations and Applications

View full text Add to dashboard Cite

We consider homoclinic orbits in non-autonomous discrete time dynamical systems of the formwhere it is assumed that an n independent fixed point exists. A numerical method for computing finite approximations of transversal homoclinic orbits is introduced and a detailed error analysis is presented. The non-autonomous setup requires special tools. We prove that the analytic condition of transversality of the orbit corresponds to a transversal intersection of the corresponding invariant fiber bundles. The approximation method and the validity of the error estimate is illustrated by an example.

show abstract

References

2021

Generalized Ordinary Differential Equations in Abstract Spaces and Applications

View full text Add to dashboard Cite

Exponential Dichotomies, the Shadowing Lemma and Transversal Homoclinic Points

Cited by 165 publications

References 3 publications

Homoclinic trajectories of non-autonomous maps

Homoclinic trajectories of non-autonomous maps

Homoclinic orbits of non-autonomous maps and their approximation

References

Contact Info

Product

Resources

About