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Partial linearization for planar nonautonomous differential equations. (English) Zbl 1316.34041

This interesting paper contains a partial linearization result for Carathéodory differential equations in the plane. The linear part is supposed to have one center- and one hyperbolic direction. More precisely, near a reference solution \(\mu:{\mathbb R}\to{\mathbb R}^2\) the planar differential equation \(\dot x=f(t,x)\) is shown to be \(C^k\)-equivalent to the simplified system \[ \dot y_1=a_1(t)y_1+p(t,y_1),\quad \dot y_2=a_2(t)y_2+q(t,y_1)y_2, \] provided an appropriate gap condition is fulfilled in terms of the Sacker-Sell spectrum for the variational equation \(\dot y=D_2f(t,\mu(t))y\). The proof is based on a flattening of the unstable manifold combined with a normal form reduction.
As an application the authors consider a Duffing-van der Pol oscillator, whose linear part is perturbed by a bounded measurable function.

MSC:

34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37C60 Nonautonomous smooth dynamical systems
37G05 Normal forms for dynamical systems
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C41 Equivalence and asymptotic equivalence of ordinary differential equations
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References:

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