A modulus of stability for the Sil’nikov theorem. (English) Zbl 0812.58075

Bamon, R. (ed.) et al., Dynamical systems. Proceedings of the 3rd international school of dynamical systems, Santiago de Chile, 1990. Harlow, Essex: Longman Scientific & Technical. Pitman Res. Notes Math. Ser. 285, 10-23 (1993).
In this note a result of I. M. Ovsyannikov and L. P. Shil’nikov [Math. USSR, Sb. 58, 557-574 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 4(8), 552-570 (1986; Zbl 0628.58044)] on dynamical systems with a saddle-focus homoclinic curve is generalized to infinite dimensional Banach spaces. Under a number of conditions which are too involved to repeat here the equation \(\dot z + Az = g(z)\) is studied. It is shown that the following two sets of systems are dense in the space of all systems satisfying the conditions:
1) those having a structurally unstable periodic orbit;
2) those having a structurally unstable Poincaré homoclinic curve.
For the entire collection see [Zbl 0773.00013].


37-XX Dynamical systems and ergodic theory
34D30 Structural stability and analogous concepts of solutions to ordinary differential equations
58B99 Infinite-dimensional manifolds


Zbl 0628.58044