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}

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

The Reproducing Kernel Method for Some Variational Problems Depending on Indefinite Integrals

In this paper we introduce the reproducing kernel method to solve a class of variational problems (VPs) depending on indefinite integrals. We discuss an analysis of convergence and error for the proposed method. Some test examples are presented to demonstrate the validity and applicability of method. The results of numerical examples indicate that the proposed method is computationally very simple and attractive

Numerical bifurcation and stability analysis of an predator-prey system with generalized Holling type III functional response

We perform a bifurcation analysis of a predator-prey model with Holling functional response. The analysis is carried out both analytically and numerically. We use dynamical toolbox MATCONT to perform numerical bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous types of bifurcation phenomena, including fold, subcritical Hopf, cusp, Bogdanov-Takens. By starting from a Hopf bifurcation point, we approximate limit cycles which are obtained, step by step, using numerical continuation method and compute orbitally asymptotically stable periodic orbits

A new application of the homotopy analysis method in solving the fractional Volterra's population system

summary:This paper considers a Volterra's population system of fractional order and describes a bi-parametric homotopy analysis method for solving this system. The homotopy method offers a possibility to increase the convergence region of the series solution. Two examples are presented to illustrate the convergence and accuracy of the method to the solution. Further, we define the averaged residual error to show that the obtained results have reasonable accuracy