A spectral theory for linear differential systems

Sacker, Robert J.; Sell, George R.

doi:10.1016/0022-0396(78)90057-8

Cited by 451 publications

(349 citation statements)

References 8 publications

Supporting

Mentioning

343

Contrasting

Unclassified

Order By: Relevance

“…To classify the strength of attractivity and repulsivity of linear systems, the concept of the dichotomy spectrum is essential. For linear skew product flows with compact base sets, the so-called Sacker-Sell spectrum (see SACKER & SELL [153]) has become widely accepted. In SIEGMUND [171] and AULBACH & SIEGMUND [19], this spectrum has been adapted for arbitrary classes of linear differential and difference equations, respectively (for the noninvertible case, see AULBACH & SIEGMUND [20]).…”

Section: Dichotomy Spectramentioning

confidence: 99%

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Rasmussen

2007

Lecture Notes in Mathematics

View full text Add to dashboard Cite

Section: Dichotomy Spectramentioning

confidence: 99%

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Rasmussen

2007

Lecture Notes in Mathematics

View full text Add to dashboard Cite

“…Moreover, the decomposition of R 2 into the direct sum of W s ͑ ͒ and W u ͑ ͒, for ⌰, implies that every initial condition ͑v , ͒ not lying on the stable subbundle is attracted to the unstable subbundle and grows exponentially in norm under the evolution given by ͑4͒. Looking at directions ͑which is what the polar Harper map ͑6͒ retains͒, forward orbits with initial condition ͑ , ͒ ͑other than ͑ s ͑ ͒ , ͒͒ are exponentially attracted to ⌽ u , that is 28,29 In such a case, the projectivizations ⌽ s , ⌽ u are continuous invariant curves of the Harper map.…”

Section: B Nonuniform Hyperbolicity and Snamentioning

confidence: 99%

Strange nonchaotic attractors in Harper maps

Haro

Puig

2006

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

We study the existence of strange nonchaotic attractors ͑SNA͒ in the family of Harper maps. We prove that for a set of parameters of positive measure, the map possesses a SNA. However, the set is nowhere dense. By changing the parameter arbitrarily small amounts, the attractor is a smooth curve and not a SNA. © 2006 American Institute of Physics. ͓DOI: 10.1063/1.2259821͔In the last two decades there has been an enormous interest in the so-called strange nonchaotic attractors (SNA). These attracting invariant objects of dynamical systems capture the evolution of a large subset of the phase space and are very relevant for their description. In particular, SNA are geometrically complicated (they are strange) and their dynamics is regular (in most of the examples, quasiperiodic). SNA are typically observed in systems where there is a coupling between different dynamics. In contrast to the vast amount of numerical and experimental work in the area, there are only few rigorous proofs. The goal of this paper is to prove the existence and abundance of SNA in the family of Harper maps. This family arises from a model in mathematical physics and it is a paradigm of a 1D quasiperiodically forced map. Our approach connects the spectral theory of Schrödinger operators and the theory of nonuniformly hyperbolic systems. So, even if the proof is made for a concrete family, many of the arguments apply to other families.

show abstract

“…The theory of linear skew-product flows with finite dimensional fibers has come in our days into widespread usage in the area in asymptotics theory of differential equations (see e.g. [9,10,22,23,26]). Also, it is well known the approach of J.…”

Section: U (T τ )F (τ )Dτ T ≥ S T S ∈ Jmentioning

confidence: 99%

“…In the finite dimensional context the Sacker-Sell spectrum offered an interesting way to describe all of these properties (see [9,23,24,25]). This kind of approach was extended in the last decade to the case of normcontinuous cocycles on infinite dimensional Banach spaces by Latushkin and Stepin.…”

Section: U (T τ )F (τ )Dτ T ≥ S T S ∈ Jmentioning

confidence: 99%

On the uniform exponential stability of linear skew‐product semiflows

Preda

2006

Journal of Function Spaces

View full text Add to dashboard Cite

Abstract. The problem of uniform exponential stability of linear skew-product semiflows on locally compact metric space with Banach fibers, is discussed. It is established a connection between the uniform exponential stability of linear skewproduct semiflows and some admissibility-type condition. This approach is based on the method of "test functions", using a very large class of function spaces, the so-called Orlicz spaces.

show abstract

A spectral theory for linear differential systems

Cited by 451 publications

References 8 publications

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Attractivity and Bifurcation for Nonautonomous Dynamical Systems

Strange nonchaotic attractors in Harper maps

On the uniform exponential stability of linear skew‐product semiflows

Contact Info

Product

Resources

About