Simulation of spatial-temporal chaos: The simplest mathematical patterns and computer graphics
The authors discuss the solution of a linear system of two (uncoupled) first-order hyperbolic equations in two space variables on a strip in \(\mathbb{R}^ 2\) with nonlinear boundary conditions and \(C^ 1\) initial conditions which agree with the boundary data. They use the reduction method and study the types of oscillatory solutions for large time and the dependence of these solutions on the initial data. The space-time evolution of the vector field \((u, v)\) solving this system of equations is related in an intimate way to the iterations of the map \[ f_ A(w):= (1- A)^{- 1}(1- w^ 2)- 1, \] where \(A\in (0, 1/2]\) is a parameter and \(w\) is the left-hand side of the difference problem corresponding to the original system by the reduction method. The authors study the space-time behavior of \(u\) and \(v\) for different values of the parameter \(A\) and are particularly interested in the onset of chaos. The map \(f_ A\) has two fixed points if \(A\in (0, 1/3]\) and it has one fixed point and a cycle of period 2 if \(A\in (1/3, ((\sqrt 6- 1)/(\sqrt 6+ 1))]\). Results: For \(A\in (((\sqrt 8- 1)/(\sqrt 8+1)), 0.4794182\dots)\) the map \(f_ A\) has an attracting cycle of period 3 and the limiting vector field is asymptotically periodic. If \(B\) is the value of the parameter \(A\) such that, for any \(i= 0,1,\dots\), the map \(f_ B\) has a cycle of period \(2^ i\) and has no cycles of other periods \((B= 0,43976738\dots)\) and the vector field is asymptotically almost periodic. For \(A= 1/2\), the set \(\{f^{- n}_ A(0), n= 0,1,\dots\}\) is dense in \([- 1,1]\) and the vector field is asymptotically stochastic. These results are found by computer experiments.
scientific article; zbMATH DE number 726947