By Neil Fenichel (auth.), Christopher K. R. T. Jones, Urs Kirchgraber, Hans-Otto Walther (eds.)
This booklet comprises 4 first-class contributions on subject matters in dynamical platforms by way of authors with a world attractiveness: "Hyperbolic and Exponential Dichotomy for Dynamical Systems", "Feedback Stabilizability of Time-periodic Parabolic Equations", "Homoclinic Bifurcations with Weakly increasing middle Manifolds" and "Homoclinic Orbits in a 4-dimensional version of a Perturbed NLS Equation: a geometrical Singular Perturbation Study". all of the authors supply a cautious and readable presentation of modern study effects, addressed not just to experts but additionally to a broader diversity of readers together with graduate students.
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Extra info for Dynamics Reported: Expositions in Dynamical Systems
The condition above ensures that we can find a branch of the logarithm defined on a neighbourhood of uu. This allows us to define a logarithm of the operator Vu(O) E ~(Xu(O» by using the well-known functional calculus given by the Dunford-integral (cf. ). 1. There exists an operator B E ~(Xu(O», namely 1 B:= -T"logVu(O), satisfying The operator B is not necessarily unique since, as a rule, different branches of the logarithm will produce different B's. We now set for each t E IR Q(t):= Uu(t, O)e IB E ~(Xu(O), Xu(t».
The references , , , ,  and  are to more recent work and have been added at the suggestion of one of the referees. See, in particular, the nonstationary Hadamard-Perron Theorem in Hasselblatt and Katok  for a result which is closely related to Lemma L The bootstrapping method for proving smoothness of the stable and unstable manifolds is found also in the book of Hirsch, Pugh and Shub . References  D. V. Anosov, Geodesic flows on closed Riemanian manifolds with negative curvature, Proceedings of the Steklov Institute, 90 (1967) AMS, Providence, RI.
We introduce a so called feedback mechanism which shall enable us to stabilize the trivial solution of (I). L(t}(~} , u=O, u(· ,O} = UO in nx on an x (O,oo), in (O,oo), (4) n. (O} = C(O}', and represents an observation or sensor. The function f : 0 x JR -+ JR is also smooth and T periodic in t E JR, and represents a so called controller. The correct interpretation is the following: The sensor observes the state of the system at time t > 0 and this information is used to modify the law governing the evolution of the population to be controlled via the controller f (for instance by the judicious use of some pesticide which is sprayed over the habitat).
Dynamics Reported: Expositions in Dynamical Systems by Neil Fenichel (auth.), Christopher K. R. T. Jones, Urs Kirchgraber, Hans-Otto Walther (eds.)