# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A variational theory of hyperbolic Lagrangian coherent structures. (English) Zbl 1214.37056

This article deals with the development of a self-consistent theory of coherent trajectory patterns in dynamical systems defined over a finite time-interval. It is used the term Lagrangian coherent structures (LCSs) [G. Haller and G. Yuan, Physica D 147, No. 3–4, 352–370 (2000; Zbl 0970.76043)] to describe the core surfaces around which such trajectory patterns form. As proposed there repelling LCSs are the core structures generating stretching, attracting LCSs act as centerpieces of folding, and shear LCS delineate swirling and jet-type tracer patterns. In order to act as organizing center for Lagrangian patterns, LCSs are expected to have two key properties:

$\left(1\right)$ An LCs should be a material surface, i.e., a codimension-one invariant surface in the extended phase of a dynamical system. This is because (a) an LCS must have sufficiently high dimension to have visible impact and act as a transport barrier and (b) an LCS must move with the flow to act as observable core of evolving Lagrangian patterns.

$\left(2\right)$ An LCS should exhibit locally the strongest attraction, repulsion or shearing in the flow. This is essential to distinguish the LCS from all nearby material surfaces that will have the same stability type, as implied by the continuous dependence of the flow on initial conditions over finite times.

The authors develop a mathematical theory that clarifies the relationship between observable LCSs and invariants of the Cauchy-Green strain tensor field. Motivated by physical observations of trajectory patterns hyperbolic LCSs are defined as material surfaces (i.e., codimension-one invariant manifolds in the extendend phase space) that extremize an appropriate finite-time normal repulsion or attraction measure over all nearby material surfaces. Weak LCss (WLCSs) are defined as stationary solutions of the above variational problem. Solving these variational problems computable sufficient and necessary criteria are obtained for WLCSs and LCSs that link them rigorously to the Cauchy-Green strain tensor field. The notion of a Constrained LCS (CLCS) is introduced that extremizes normal repulsion or attractor under constraints. This construction allows for the extraction of a unique observed LCS from linear systems, and for the identification of the most influential weak unstable manifold of an unstable node.

##### MSC:
 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76F25 Turbulent transport, mixing 37D35 Thermodynamic formalism, variational principles, equilibrium states