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

DG-FTLE
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