On classical results for discontinuous and constrained differential systems
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2019-08-27
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Universidade Federal de Goiás
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The present work concerns the study of classes of discontinuous differential systems addressing the following topics: global attractors, linearization, and codimension--one singularities for constrained differential systems. The Markus--Yamabe conjecture deals with global stability and it states that if a differentiable system x’=f(x) has a singularity and the Jacobian matrix Df(x) has everywhere eigenvalues with negative real part, then the singularity is a global attractor. This conjecture was proved for planar vector fields of class C^1 and counterexamples were presented in higher dimension. Let Z=(X,Y) be a piecewise linear differential systems separated by one straight line ∑, an extension of the Markus--Yamabe conjecture for Z affirm that if 0 є ∑, Y(0)=0, X(0)≠0, and the Jacobian matrices DX(x) and DY(x) have eigenvalues with negative real part for any xє R^2 then the origin is a global attractor. In this work we prove that about these conditions Z can has one crossing limit cycle. This means that under similar hypotheses to that of the Markus--Yamabe conjecture the origin is not necessarily a global attractor of Z. The Grobman-Hartman Theorem is a classical result on linearization that provide a linear differential system that is topologically equivalent to x’=X(x) around a hyperbolic singularity.
Let Z=(X,Y) a discontinuous differential systems defined in R^n, the generic singularities of Z consist of the hyperbolic singularities of X and Y, the hyperbolic singularities of the sliding vector fields, and the tangency--regular points of Z. On linearization for discontinuous differential systems we provide a piecewise linear differential system that is ∑-equivalent to Z, around of the generic singularities, so that the sliding vector fields is also linear.
Let A(x) be a nxn matrix valued function, n≥2, and F(x) a vector field defined on R^n. Assuming that A and F are smooth, we define a constrained differential system as a differential system of the form A(x)x’=F(x), where xєR^n. In this thesis we classify the codimension-one singularities of a constrained system defined on R^3. Moreover we provide the respective normal forms in the one parameter space.
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MENEZES, L. A. S. On classical results for discontinuous and constrained differential systems. 2019. 91 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2019.