Finding finite-time invariant manifolds in two-dimensional velocity fields. (English) Zbl 0979.37012
Summary: For two-dimensional velocity fields defined on finite time intervals, we derive an analytic condition that can be used to determine numerically the location of uniformly hyperbolic trajectories. The conditions of our main theorem will be satisfied for typical velocity fields in fluid dynamics where the deformation rate of coherent structures is slower than individual particle speeds. We also propose and test a simple numerical algorithm that isolates uniformly finite-time hyperbolic sets in such velocity fields. Uniformly hyperbolic sets serve as the key building blocks of Lagrangian mixing geometry in applications.
MSC:
37D10 | Invariant manifold theory for dynamical systems |
37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |
76F20 | Dynamical systems approach to turbulence |
37N99 | Applications of dynamical systems |
References:
[1] | DOI: 10.1017/S0022112093003192 · Zbl 0800.76206 · doi:10.1017/S0022112093003192 |
[2] | DOI: 10.1016/S0167-2789(98)00091-8 · Zbl 1194.76089 · doi:10.1016/S0167-2789(98)00091-8 |
[3] | DOI: 10.1063/1.166399 · Zbl 0987.37080 · doi:10.1063/1.166399 |
[4] | DOI: 10.1016/S0167-2789(97)00115-2 · Zbl 0925.76260 · doi:10.1016/S0167-2789(97)00115-2 |
[5] | DOI: 10.1175/1520-0485(1999)029<1649:GOCSMI>2.0.CO;2 · doi:10.1175/1520-0485(1999)029<1649:GOCSMI>2.0.CO;2 |
[6] | DOI: 10.1063/1.870155 · Zbl 1149.76515 · doi:10.1063/1.870155 |
[7] | DOI: 10.1016/0167-2789(91)90088-Q · Zbl 0716.76025 · doi:10.1016/0167-2789(91)90088-Q |
[8] | DOI: 10.1142/S0218127491000440 · Zbl 0874.58053 · doi:10.1142/S0218127491000440 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.