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Local stable and unstable manifolds and their control in nonautonomous finite-time flows. (English) Zbl 1364.37067

The author studies a problem related to nonautonomous fluids whose dynamics can be modelled by a system of differential equations of the form \[ \dot x = F(x,t,\varepsilon), \quad x\in \mathbb R^2,\tag{1} \] where \(\varepsilon\) is a non-negative small parameter, and \(F(x,t,0)=f(x)\) is independent of \(t\). It is also assumed that there exists a hyperbolic equilibrium \(a\in\mathbb R^2\) for (1), and that the eigenvalues of \((Df)(a)\) have opposite sign. Under these conditions, the equilibrium \(a\) presents stable and unstable manifolds with respect to the vector field \(f(x)\).
When \(\varepsilon>0\), under certains conditions, the equilibrium \(a\) is replaced by a hyperbolic trajectory \(a(t)\). The author’s purpose is to characterize how the stable and unstable manifolds of \(a(t)\) for \(\varepsilon>0\) are related to the stable and unstable manifold of \(a\) for \(\varepsilon=0\). In particular, the rotation angles of the tangent spaces to these manifolds are investigated.
The case for which (1) is defined for \(t\in(-\infty,+\infty)\) and the case for which \(t\in(-T,+T)\) require different treatment.
Then the author considers an inverse problem determining a function \(c(x,t)\) such that the equation \[ \dot x = f(x) +c(x,t)\tag{2} \] has a hyperbolic trajectory \(a(t)\) approximating a given \(\tilde a(t)\), in order to control the rotation angles of the stable and unstable manifolds.
The paper contains also simulations and examples.

MSC:

37D10 Invariant manifold theory for dynamical systems
37D05 Dynamical systems with hyperbolic orbits and sets
76R05 Forced convection
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations

Software:

DG-FTLE
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References:

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