Abstract
An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations.
Highlights
Exponential dichotomy of a linear system of differential equations is a type of conditional stability that goes back to an idea in Perron [1]
It has been used to show the existence of chaotic behaviour in non autonomous perturbations of autonomous nonlinear equations having a homoclinic solution, since transverse intersection of stable and unstable manifolds along a homoclinic solution corresponds to the fact that the linearization of the nonlinear system along it has an exponential dichotomy on R [2]
Starting from a linear system with an exponential dichotomy, shifting the coefficient matrix by α− β νI, ν = 2, we can assume that the exponents are the same
Summary
Exponential dichotomy of a linear system of differential equations is a type of conditional stability that goes back to an idea in Perron [1]. From Gronwall inequality it follows that, on a compact interval, the linear system (1) has an exponential dichotomy with any projection P and exponents α and β (but the constant may change). Sometimes it becomes important to determine this rate of convergence, and the exponents of the dichotomy, for example when studying chaotic behaviour of discontinuous systems [8] or developing Melnikov theory for implicit nonlinear differential equations [9]. As a matter of fact we work directly in the space of continuous functions decaying to zero as t → ∞ at a certain given rate. This approach leads us to derive the first of the two exponential estimates given in (2).
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