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The Riccati equation: pinching of forcing and solutions. (English) Zbl 1059.34004

Consider the Riccati equation \(x'+x^2=k(t)\) with \(k(t)>0\), studied from the point of view of dynamical systems and differential geometry. A matrix form of this equation is important for the asymptotic spectral information on solutions. One obtains asymptotic bounds of the solutions (absolute bunching). Theorem 1.1 gives “absolute bunching” information from the absolute pinching of the sectional curvature. Instead of absolute bunching, it suffices to get relative bunching, to obtain regularity of the invariant foliations. Absolute bunching gives control of some ratio of bounds on contraction and expansion rates. Relative bunching corresponds to bounding a ratio of solutions of the Riccati equation. Some explicit solutions are found and asymptotic ratios are obtained. Some numerical work and open questions finish the paper.

MSC:

34A34 Nonlinear ordinary differential equations and systems
34H05 Control problems involving ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
37C10 Dynamics induced by flows and semiflows
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
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