ON SMOOTH SOLUTIONS OF NON LINEAR DYNAMICAL SYSTEMS, f_n+1 = u(f_n), PART I

Abstract

We consider the dynamical system, f_n+1 = u(f_n), (1) (where usually n, is time) defined by a continuous map u. Our target is to find a flow of the system for each initial state f₀, i.e., we seek continuous solutions of (1), with the same smoothness degree as u. We start with the introduction of continued forms which are a generalization of continued fractions. With the use of continued forms and a modulator function (i.e., weight function) m, we construct a sequence of smooth functions, which come arbitrarily close to a smooth flow of (1). The limit of this sequence is a functional transform, K_m[u], of u, with respect to m. The functional transform is a solution of (1), in the sense that, K_m[u] (y + c), is a flow of (1) for each translation constant c. Here we present the first part of our work where we consider a subclass of dissipative dynamical systems in the sence that they have wandering sets of positive measure. In particular we consider strictly increasing real univariate maps, u: D⟶D, D = (a+∞), where, a≤0, or, a=-∞, with the property, u(x)-x≥ε>0, which implies that u, has no real fixed points. We briefly give some mathematical and physical applications and we discuss some open problems. We demonstrate the method on the simple non-linear dynamical system, f_n+1 = (f_n)²+1.