ON SMOOTH SOLUTIONS OF NON LINEAR DYNAMICAL SYSTEMS, fn+1 = u(fn), PART I
- 1 National and Kapodistrian University of Athens, Greece
Copyright: © 2021 Kyriakos Kefalas. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
We consider the dynamical system, fn+1 = u(fn), (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, Km[u], of u, with respect to m. The functional transform is a solution of (1), in the sense that, Km[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, fn+1 = (fn)2+1.
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- Non Linear Dynamical Systems
- Smooth Flows
- Functional Transform
- Continuous Iterates
- Continued Forms
- Abel Functional Equation
- Iterative Functional Equations
- Iterative Roots