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Nonautonomous gradient-like vector fields on the circle: Classification, structural stability and autonomization

To Jürgen, with the gratitude and best wishes

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  • We study a class of scalar differential equations on the circle $ S^1 $. This class is characterized mainly by the property that any solution of such an equation possesses an exponential dichotomy both on the semi-axes $ \mathbb R_+ $ and $ \mathbb R_+ $. Also we impose some other assumptions on the structure of the foliation into integral curves for such the equation. Differential equations of this class are called gradient-like ones. As a result, we describe the global behavior of a foliation, introduce a complete invariant of the uniform equivalency, give standard models for the equations of this distinguished class. The case of almost periodic gradient-like equations is also studied, their classification is presented.

    Mathematics Subject Classification: Primary: 34C27, 34C40, 34D30; Secondary: 70G60.


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  • Figure 1.  Bifurcation from infinities, $ \mu = 0, $ $ \mu > 0 $

    Figure 2.  Bifurcation at the violation of Assumption 3

    Figure 3.  Foliation inside the rectangle

    Figure 4.  Neighborhoods: crosses correspond to u-solutions, bold points correspond to s-solutions

    Figure 5.  Construction of asymptotically autonomous ODE: crosses correspond to u-solutions, bold points correspond to s-solutions

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