## Exponential dichotomy and trichotomy of nonlinear differential equations.(English)Zbl 0722.34053

The main contribution of this paper is to extend the notions of dichotomy and trichotomy to nonlinear differential equations. For linear systems $$(1)\quad \dot x=A(t)x$$ the concept of dichotomy is now classical. In a previous paper, the authors introduced the notion of trichotomy for (1). This is somewhat intermediate between dichotomy on $${\mathbb{R}}$$ and dichotomies on both $${\mathbb{R}}^+$$ and $${\mathbb{R}}^-$$. It turns out to characterize the equations $$\dot x=A(t)x+b(t),$$ which have a bounded solution for all integrally bounded forcing b(t). In the first section of the paper the authors study trichotomy of linear systems. Considering two systems, one being a perturbation of the other, it is natural to inquire whether for any given solution of the unperturbed one, there exists a solution of the perturbed one that tracks the first one, i.e. that stays within a distance $$\rho$$. For linear systems a positive answer is given provided the homogeneous part (1) has a trichotomy. An important feature is that the tracking constant depends in some simple way on the perturbation but not on the given solution. For nonlinear systems, this tracking property is a key to an extension of trichotomy and dichotomy. Elementary consequences of nonlinear dichotomy are investigated, suggesting relations with structural stability. An application to control systems which hold approximatively the state to a point is given. - The last section presents dichotomy in variation which is an alternative way to investigate nonlinear dichotomy.

### MSC:

 34G10 Linear differential equations in abstract spaces 34D05 Asymptotic properties of solutions to ordinary differential equations 34D30 Structural stability and analogous concepts of solutions to ordinary differential equations 34D10 Perturbations of ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 34A30 Linear ordinary differential equations and systems 34A34 Nonlinear ordinary differential equations and systems 93C73 Perturbations in control/observation systems