## 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].

### MSC:

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

### Keywords:

structural stability; dynamical systems

Zbl 0628.58044