Treffer: Self-stochasticity in deterministic boundary value problems

It is considered the simplest nonlinear problem (1) \(\frac{\partial u}{\partial t}= \frac{\partial u}{\partial x}\), \(x\in [0,1]\), \(t\in \mathbb{R}^+\); (2) \(u|_{x=1}= f(u)|_{x=0}\); (3) \(u|_{t=0}= \varphi(x)\) with \(f\in C^1(I,I)\) be an irreversible map of a closed interval \(I\) into itself and \(\varphi\in C^1(I,I)\) complying with the consistency relations \(\varphi(1)= f((\varphi(0))\) and \(\varphi'(1)= f'(\varphi(0)) \varphi'(0)\). The simplest nonlinear problem is reduced to a difference equation \(w(\tau+1)= f(w(\tau))\), \(\tau\in \mathbb{R}^+\), where \(u(x,t)= w(x+t)\). Thus (2) and (3) generate the dynamical system (4) \(\{C^k ([0,1], I),T,S^t\}\) with (5) \(S^t [\varphi](x)= (f^{[t+x]} \circ \varphi)(t+x)\). It is proved that for every nonsingular \(\varphi\in C^1\), the \(\omega\)-limit set of the trajectory \(S^t[\varphi]\) consist of random functions that combine into a cycle of (4) and (5) with period \(p\), more precisely \(\omega[\varphi]= \bigcup_{t\in [0,p)} S^t [f^\# \circ \varphi]\), where \(f^\#\) is a purely random process specified by the distribution function \(F_{f^\#}(u,z):= p\mu (J_i\cap (-\infty,z))\) for \(u\in J_i\) with \(J_i= \bigcup_{j\geq 0} f^{-jp(J_i)}\), \(1\leq i\leq p\), and superposition \(f^\#\circ \varphi\) means a random process specified by the distributions \(F_{f^\#}\circ \varphi(x_1,\dots, x_r,\dots; z_1,\dots, z_r):= F_{f^\#} (\varphi (x_1,\dots, \varphi(x_r); z_1,\dots, z_r)\), where (4) and (5) being uniformly continuous with respect to some metric \(\rho^\#\), induces a dynamical system on the extended space \(C^\#\). There exist classes of boundary value problems (among which is (1)--(3)) such that, under certain conditions, the corresponding dynamical system possesses a massive set of trajectories that are compact in \(C^\#\), and \(C^\#\) is the completion of the phase space \(C^k(D, E)\) via the metric \(\rho^\#\) with random and deterministic functions having the property \(F_\psi^\varepsilon (x_1,\dots, z_r)\to F_\psi(x_1\dots z_r)\) as \(\varepsilon\to 0\) for all points \((x_1,\dots, z_r)\in X^r\times Z^r\) outside of a set of Lebesgue measure zero.