Piecewise smooth maps appear as models of various physical, economical and other systems.In such maps bifurcations can occur when a fixed point or periodic orbit crosses or collides with the border between two regions of smooth behaviour as a system parameter is varied.These bifurcations have little analogue in standard bifurcation theory for smooth maps and are often more complex.They are now known as "border collision bifurcations".The classification of border collision bifurcations is only available for one-dimensional maps.For two and higher dimensional piecewise smooth maps the study of border collision bifurcations is far from complete.In this thesis we investigate some of the bifurcation phenomena in two-dimensional continuous piecewise smooth discrete-time systems.There are a lot of studies and observations already done for piecewise smooth maps where the determinant of the Jacobian of the system has modulus less than 1, but relatively few consider models which allow area expansions.We show that the dynamics of systems with determinant greater than 1 is not necessarily trivial.Although instability of the systems often gives less useful numerical results, we show that snap-back repellers can exist in such unstable systems for appropriate parameter values, which makes it possible to predict the existence of chaotic solutions.This chaos is unstable because of the area expansion near the repeller, but it is in fact possible that this chaos can be part of a strange attractor.We use the idea of Markov partitions and a generalization of the affine locally eventually onto property to show that chaotic attractors can exist and are fully two-dimensional regions, rather than the usual fractal attractors with dimension less than two.We also study some of the local and global bifurcations of these attracting sets and attractors.Some observations are made, and we show that these sets are destroyed in boundary crises and some conditions are given.Finally we give an application to a coupled map system.
|Date of Award||1 Aug 2012|
- The University of Manchester
|Supervisor||Paul Glendinning (Supervisor) & Jeremy Huke (Supervisor)|
- Piecewise smooth maps
- Dynamical systems
- Border collision bifurcations