Discrete inertial manifolds. (English) Zbl 1146.39030
Summary: This work is devoted to attractive invariant manifolds for nonautonomous difference equations, occurring in the discretization theory for evolution equations. Such invariant sets provide a discrete counterpart to inertial manifolds of dissipative FDEs and evolutionary PDEs. We discuss their essential properties, like smoothness, the existence of an asymptotic phase, normal hyperbolicity and attractivity in a nonautonomous framework of pullback attraction.
As application we show that inertial manifolds of the Allen-Cahn and complex Ginzburg-Landau equation persist under discretization. For the Ginzburg-Landau equation we can also estimate the dimension of the inertial manifold.
As application we show that inertial manifolds of the Allen-Cahn and complex Ginzburg-Landau equation persist under discretization. For the Ginzburg-Landau equation we can also estimate the dimension of the inertial manifold.
MSC:
| 39A12 | Discrete version of topics in analysis |
| 39A70 | Difference operators |
| 34K19 | Invariant manifolds of functional-differential equations |
| 35B42 | Inertial manifolds |
| 34C40 | Ordinary differential equations and systems on manifolds |
Keywords:
invariant fiber bundles; inertial manifold; normal hyperbolicity; discretization; discrete Ginzburg-Landau equation; pullback attraction; nonautonomous difference equationsReferences:
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