Abstract
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.
Highlights
With the use of continued forms and a modulator function m, we construct a sequence of smooth functions, which come arbitrarily close to a smooth flow of (1)
In theorem T5 we show that any functional transform, K m[u], is a solution of (1) and that, K m[u]( y + c), interpolates the complete orbit, O( f0), of the dynamical system (1)
We show a smooth solution of the system using the logistic function as a modulator function, along the way pointing out some computational difficulties
Summary
Let Z, be any totally ordered set of integers. We denote index sets for the rest of this article as, Z[k, n] , Z[k, ±∞) , Z We say that a function is Ck smooth if it has continuous derivatives of kth order. We denote the non-negative integer iterates of an univariate function, f : D → , where, f (D) ⊂ D ⊂ , as:. We adopt the bracket notation for the iteration exponent to avoid any confusion with powers. D ⊂ f (D) , we define the negative integer iterates of f, as:. For the successor function, S(x) = x +1 , we define real continuous ‘principal’ iterates as: S[a](x) ≡ x + a, a ∈. Where, the meaning of ‘principal’ iterates is explained in section 4.5 on homologous and principal functions
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