Bifurcations of maps: numerical algorithms and applications

Dynamical systems theory provides mathematical models for systems which evolve in time according to a rule, originally expressed in analytical form as a system of equations. Discrete-time dynamical systems defined by an iterated map depending on control parameters, \begin{equation} \label{Map:g} g(x,\alpha) := f^{(J)}(x,\alpha)= \underbrace{f(f(f(\cdots f}_{J \mbox{~times}}(x,\alpha),\alpha),\alpha),\alpha), \end{equation} appear naturally in, e.g., ecology and economics, where $x\in \R^n$ and $\alpha \in \R^k$ are vectors of state variables and parameters, respectively. %The system dynamics describe a sequence of points \left\{x_k{\right\} \subset \R^n (orbit), provided an initial $x_0 \in \R^n$ is given. The main goal in the study of a dynamical system is to find a complete characterization of the geometry of the orbit structure and the change in orbit structure under parameter variation. An aspect of this study is to identify the invariant objects and the local behaviour around them. This local information then needs to be assembled in a consistent way by means of geometric and topological arguments, to obtain a global picture of the system. At local bifurcations, the number of steady states can change, or the stability properties of a steady state may change. The computational analysis of local bifurcations usually begins with an attempt to compute the coefficients that appear in the normal form after coordinate transformation. These coefficients, called critical normal form coefficients, determine the direction of branching of new objects and their stability near the bifurcation point. After locating a codim 1 bifurcation point, the logical next step is to consider the variation of a second parameter to enhance our knowledge about the system and its dynamical behaviour. % % In codim 2 bifurcation points branches of various codim 1 bifurcation curves are rooted. % These curve can be computed by a combination of parameter-dependent center manifold reduction and asymptotic expressions for the new emanating curves. We implemented new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits of maps in the {\sc Matlab} software package {\sc Cl\_MatcontM}. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points, and their continuation in two control parameters, as well as detection and location of all codim 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed with finite directional differences, automatic differentiation and symbolic derivatives of the original map. Asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period to continue the double and quadruple period bifurcations. In the case $n=2$ we compute one-dimensional invariant manifolds and their transversal intersections to obtain initial connections of homoclinic and heteroclinic orbits orbits to fixed points of (\ref{Map:g}). We continue connecting orbits, using an algorithm based on the continuation of invariant subspaces, and compute their fold bifurcation curves, corresponding to the tangencies of the invariant manifolds. {\sc Cl\_MatcontM} is freely available at {\bf www.matcont.ugent.be} and {\bf www. sourceforge.net}

Zombies: a simple discrete model of the apocalypse

A simple discrete-time two-dimensional dynamical system is constructed and analyzed numerically, with modelling motivations drawn from the zombie virus of popular horror fiction, and with suggestions for further exercises or extensions suitable for an introductory undergraduate course

Zombies: a simple discrete model of the apocalypse

A simple discrete-time two-dimensional dynamical system is constructed and analyzed numerically, with modelling motivations drawn from the zombie virus of popular horror fiction, and with suggestions for further exercises or extensions suitable for an introductory undergraduate course

Switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in ODEs

The paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n-dimensional ODEs. We discuss the implementation and the performance of the algorithm in several examples, including an extended Lorenz-84 model and a laser system.Comment: 17 pages, 7 figures, submitted to Physica

Numerical treatment of nonlinear optimal control problems

In this paper, we present iterative and non-iterative methods for the solution of nonlinear optimal con- trol problems (NOCPs) and address the sufficient conditions for uniqueness of solution. We also study convergence properties of the given techniques. The approximate solutions are calculated in the form of a convergent series with easily computable components. The efficiency and simplicity of the methods are tested on a numerical example