Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps

Abstract

We discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits (cycles) of maps and their implementation in matcont, a MATLAB toolbox for continuation and bifurcation analysis of dynamical systems. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points (i.e., limit point, period-doubling, and Neimark-Sacker) and their continuation in two control parameters, as well as detection and location of all codimension 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed, both numerically with finite directional differences and using symbolic derivatives of the original map. Using a parameter-dependent center manifold reduction, explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period. These asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations. We provide two examples illustrating the developed techniques: a generalized Hénon map and a juvenile/adult competition model from mathematical biology. Introduction.Discrete-time dynamical systems generated by iterated maps appear in many scientific areas, such as economics, engineering, and ecology. The usual method for getting an insight into their behavior is to compute many orbits, starting from various initial points (i.e., simulation). Another tool for gaining a better understanding of these systems is bifurcation analysis. It provides theoretic results on the classification of possible modes of behavior, which may explain results of the simulations at different values of control parameters. The application of such theoretical results requires robust numerical computational tools, as a purely symbolic approach is often futile. In the present paper we describe how cl matcont, which supports only ODEs [7,8], has been complemented by the addition of the new package cl matcontm, which supports local bifurcation analysis of generic finite-dimensional smooth maps. The software and a manual are freely available at http://www.matcont.ugent.be/. Before discussing and comparing existing computer tools for maps, we fix some notation and discuss some events which any standard *

Resonance and bifurcation in a discrete-time predator–prey system with Holling functional response

“…For the LP P D bifurcation it follows from the output be < 0 that no new local bifurcation curves are rooted in this point, cf. [8]. Similarly, for the R2 bifurcation the output c > 0 has the same implication.…”

“…This is based on the numerical computation of the normal form coefficients of these bifurcations (see [19], Ch. 8 and [8]). However, a detailed study in [15] (Theorem 4.1 and Theorem 5.1) suggests that other cases might be possible for certain combinations of parameter values.…”

“…In Section 3 we numerically compute curves of codim-1 bifurcations and the critical normal form coefficients of codim-2 bifurcation points, using the Matlab toolbox MatContM [8,9]. These tools enable us to compute stability boundaries of different cycles.…”

“…So the discriminant of (8) is ∆ = c 2 12 − c 11 c 22 = c 2 12 = 1 2 > 0. If Y (s) is any branch of fixed points of (2) with Y (s 0 ) = (0, 0, 1), then its derivative Y s (s 0 ) can be written as Y s (s 0 ) = αφ 1 + βφ 2 where α and β are scaled roots of (8). We find…”

“…In this section we perform a numerical bifurcation analysis by using the MAT-LAB package MatContM, see [8] and [9]. The bifurcation analysis is based on continuation methods, tracing out the solution manifolds of fixed points while some of the parameters of the map vary, see [2].Testruns for these computations willl be made available via [9].…”