Fixed and variable time-stepping numerical methods for dynamical systems.
This thesis is concerned with the numerical solution of dynamical systems by fixed and variable
time-stepping methods. Chapter I, reports on current work in the field and states principal results.
Chapter 2, briefly reviews dynamical systems theory for ordinary differential equations.
In Chapter 3, standard numerical methods for continuation of solution branches are summarised.
Chapter 4, continues the discussion on bifurcations and spurious solutions for numerical methods.
The mechanism by which the presence of spurious numerical solutions degrades the numerical
approximation of an attractor of the underlying system is studied. Further, an investigation into
how well real bifurcations in the family of dynamical systems are approximated as the step-size
varies is carried out. In general, the preservation of bifurcation structures and stability under numerical
simulations is discussed. In addition, the behaviour of numerical solutions generated by a
Runge-Kutta method applied to a dynamical system whose analytical solution undergoes a Hopf
bifurcation is investigated. Hopf bifurcation results for the numerical solution are presented and
analysed. In Chapter 5, the stability step-size constraints are discussed further. In particular, it
is proved that for any dynamical system with locally Lipschitz I, trajectories of solutions neither
cross nor merge in phase space. A necessary condition to stop merging or crossing of trajectories
in numerical simulations is derived using linear theory. Finally, in Chapter 6, a phase space error
control "PS8 error control" is introduced which bounds the truncation error at each step by a
fraction of the solution arc length over the corresponding time interval. It is shown that this error
control can be incorporated within a standard algorithm as an additional constraint at negligible
additional computational cost. Numerical results are given to demonstrate that the new error control
has positive effects on the linear stability properties around true fixed points and moreover,
prevents spurious fixed points that might otherwise be allowed by the adaptive algorithm. Also,prevents spurious fixed points that might otherwise be allowed by the adaptive algorithm. Also,
since step-size selection is non-trivial for phase space error controls as they are not based on a
simple error estimate, a new step-size selection scheme is introduced which leads to stable stepsizes
(with fast linear convergence to a constant value) near fixed points. Numerical simulations
that illustrate and confirm the analysis, as regards the dynamics of the numerical solution and the
step-size sequences near to stable and saddle points, are also presented.