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.


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.)
Full Text: DOI Euclid EuDML Link


[1] DOI: 10.1070/RM1967v022n05ABEH001228 · Zbl 0177.42002
[2] Bellman Richard, Stability Theory of Differential Equations. (1953)
[3] Bittanti Sergio, The Riccati equation. (1991) · Zbl 0734.34004
[4] Harold Davis T., Introduction to Nonlinear Differential and Integral Equations. (1990)
[5] DOI: 10.1512/iumj.1971.21.21017 · Zbl 0246.58015
[6] Jaap Geluk L., Regular Variation, Extensions and Tauberian Theorems (1987) · Zbl 0624.26003
[7] DOI: 10.1017/S0143385700008105 · Zbl 0821.58032
[8] Hasselblatt Boris, Journal of Differential Geometry 39 (1) pp 57– (1994) · Zbl 0795.53026
[9] DOI: 10.1017/S0143385799133868 · Zbl 1069.37031
[10] Hirsch Morris, Journal of Differential Geometry 10 pp 225– (1975) · Zbl 0312.58008
[11] Hirsch Morris, Invariant Manifolds. (1977)
[12] Hopf Eberhard, Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physikalische Klasse 91 pp 261– (1939)
[13] Katok Anatole, Introduction to the Modem Theory of Dynamical Systems. (1995) · Zbl 0878.58020
[14] Klingenberg Wilhelm, Riemannian Geometry. (1982)
[15] DOI: 10.1007/BFb0103952 · Zbl 0946.34001
[16] Omey, Edward. ”Rapidly Varying Behaviour of the Solutions of a Second Order Linear Differential Equation.”. Proceedings of the Seventh International Colloquium on Differential Equations. pp.295–303. Utrecht: VSP. [Omey 97] · Zbl 0956.34023
[17] DOI: 10.1007/978-1-4612-1600-1
[18] Watson, George Neville. 1922.A Treatise on the Theory of Bessel FunctionsVol. 1944, 1966Cambridge, UK: Cambridge University Press. [Watson 95], 1995
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.