There are several reasons to investigate nonlinear maps. Maps dynamics is very complicated and PC makes quickly enough amazing fractal pictures (integration of differential equations is much more boring). One can study flows dynamics by Poincare maps too. Surprisingly, very simple maps turn out to yield rather good qualitative models for behavior in ordinary and partial differential equations.
The blue square mixing by the standard map. N is number of iterations. Press "-" button to trace the mixing process. |
Often it is possible to partition the phase space of a system in such a way that its orbits are determined uniquely by the sequence of passing of these partitioned regions (symbolic sequence). For example symbolic dynamics is constructed if dynamical system is reduced to the Smale horseshoe map. By means of symbolic dynamics it is possible to show that system have unstable periodic orbit with any period and continuum of non-periodic chaotic orbits. Horseshoe map exists if there is a homoclinic point i.e. an intersection of stable and unstable manifolds of a saddle point. |
For the quadratic map parameter C regions with regular dynamics are dense in [-2, 1/4]. Therefore arbitrary close to C_{ch} with chaotic dynamics there is a region with regular attracting cycle. But close to C_{ch} period of this orbit grows, chaotic transient duration tend to infinity and it is impossible practically to destinguish regular and chaotic dynamics (see intricate entangling of regular and chaotic dynamics regions on the bifurcation diagram to the left). |