Wrapping effect and wrapping function (Q1276122)

Let \(X^0\in I(\mathbb{R})^n\) (= set of interval vectors, i.e., vectors with \(n\) real compact intervals as components), let \(D\subseteq \mathbb{R}^n\) be an open set and let \(f: [t_0,\overline t]\times D\to \mathbb{R}^n\) with \(t_0<\overline t\). Assume that the initial value problem \[ \dot x= f(t,x),\quad x(t_0)= x^0\tag{1} \] has a unique solution on \([t_0,\overline t]\) for any \(x^0\in X^0\). Denote by \(x(t_0, X^0;t)\) the set of all values which the solutions \(x\) of (1) can assume at \(t\) when \(x^0\) varies over \(X^0\). Let \([x(t_0, X^0; t)]\in I(\mathbb{R})^n\) be the tightest interval enclosures of \(x(t_0, X^0;t)\). Any validating interval method for (1) produces an interval function \(S(h; t)\) such that \([x(t_0, X^0; t)]\subseteq S(h;t)\). The idealized enclosure method \({\mathcal S}(h; t)\) is defined by \({\mathcal S}(h; t_0)= X^0\), \({\mathcal S}(h,t)= [x(t_k,{\mathcal S}(h; t_k); t)]\), \(t\in [t_k, t_{k+1}]\), \(k= 0,1,\dots, p-1\), where we assume that \({\mathcal S}\) is based on mesh \(\{t_0,\dots, t_p\}\) with \(t_k= t_0+ kh\), \(k= 0,1,\dots, p\), \(t_p=\overline t\). The set \({\mathcal S}(h; t_{k+1}\setminus x(t_k,{\mathcal S}(h; t_k); t_{k+ 1})\) is called wrapping excess at \(t_{k+1}\). It is decisively influenced by the wrapping excesses at the previous mesh points \(t_i\), \(i= 1,\dots,k\) and may cause an increasing blowing up of \({\mathcal S}(h; t_k)\) over \(x(t_0, X^0; t_k)\) when \(k\) increases. This phenomenon of accumulated wrapping excess is called wrapping effect. The authors introduce a function \(\widehat X: [t_0,\overline t]\to I(\mathbb{R})^n\) which they call wrapping function to zero. This property is proved for any interval enclosure \(S(h; t)\) of \(x(t_0, X^0; t)\) which satisfies some additional assumptions. The wrapping function can be used to quantify the wrapping effect associated with (1). Examples illustrate this topic. Finally, a class of functions \(f\) for (1) is characterized such that no wrapping effect occurs at all.