Advances in numerical bifurcation software : MatCont

The mathematical background of MatCont, a freely available toolbox, is bifurcation theory which is a field of hard analysis. Bifurcation theory treats dynamical systems from a high-level point of view. In the case of continuous dynamical systems this means that it considers nonlinear differential equations without any special form and without restrictions except for differentiability up to a sufficiently high order (in the present state of MatCont never higher than five.) The number of equations is not fixed in advance and neither is the number of variables or the number of parameters, some of which can be active and others not. The aim of bifurcation theory is to understand and classify the qualitative changes of the solutions to the differential equations under variation of the parameters. This knowledge cannot be applied to practical situations without numerical software, except in some artificially constructed situations. Matcont is a toolbox that computes bifurcation diagrams through numerical methods, namely continuation. This dissertation describes the advances and innovations that were made including the detection and continuation of new bifurcations in discrete-time systems

Numerical bifurcation analysis of homoclinic orbits embedded in one-dimensional manifolds of maps

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in MatContM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. MatContM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features

Chaos and periodic solutions in a dynamic monopoly model

T. Puu [2,3] proposed a 2D monopoly model with cubic price and quadratic marginal cost functions. He provides incomplete information on the existence of cycles of period 4 and the chaotic behavior in his model [3]. Though the recent literature still deals with simplified versions of the monopoly model of T. Puu, and none of them analyzes the dynamic behavior of the T. Puu model in detail. In a recent paper [1], we reconsider the dynamic monopoly model. We present fundamental corrections to the fixed point stability analysis presented in [3]. By simulations, the existence of solutions of period 4, 5, 10, 13, 17 and the chaotic behavior are investigated. Continuation and bifurcation analysis is used to get information about the stability of 5,10,13,17-cycles under parameter variation. In all regions, further period-doubling bifurcations are found which implies the existence of orbits with higher periods as well. A general formula for solutions of period 4 is derived. Among other things, we discuss the symmetry properties of these solutions. The analytical stability analysis for the 4-cycles proves that they are never linearly asymptotically stable. Therefore, the stability of the 4-cycle is investigated by studying the effect of small displacements applied to the eigenvector corresponding to the eigenvalue located at the stability boundary. This work, combined with simulation and the basin of attraction analysis for the 4-cycle allows us to determine the stability region of the 4-cycle. This region is larger than the one obtained in [3] which is based on an incorrect linear stability analysis. We analyze the chaotic behavior of the monopoly model by computing the largest Lyapunov exponents. This analysis confirms our results. References [1] Bashir Al-Hdaibat, Willy Govaerts, , Niels Neirynck. On periodic and chaotic behavior in a two-dimensional monopoly model, Chaos, Solitons & Fractals, 70 (2015), pp. 27–37. [2] T. Puu, The chaotic monopolist, Chaos, Solitons & Fractals, 5 (1) (1995), pp. 35–44. [3] T. Puu, Attractors, bifurcations, and chaos: nonlinear phenomena in economics, Springer-Verlag, Berlin (2000)

Using MatContM in the study of a nonlinear map in economics

MatContM is a MATLAB interactive toolbox for the numerical study of iterated smooth maps, their Lyapunov exponents, fixed points, and periodic, honioclinic and heteroclinic orbits as well as their stable and unstable invariant iwniifolds. The bifurcation;1.t aysis is based on continuation methods, tracing out solution iiianifolds of various types of objects while some of the parameters of the map vary. In particular, MatContM computes codimension 1 bifurcation curves of cycles and supports the computation of the normal form coefficients of their codimension two bifurcations, and allows branch switching from codimension 2 points to secondary curves. MatContM builds on an earlier command-line MATLAB package Cl_MatContM but provides new coinputational routines and functionalities, as well as a graphical user interface, enabling internctive, control of all computations, data handling and archi Ong. We, apply MatContM in our study of the monopoly model of T. Puu with cubic price and quadratic marginal cost functions. Using MatContM, we analyze the fixed points and their stability and we compute branches of solutions of period 5, 10, 13 17. The chaotic and periodic behavior of the monopoly model is further analyzed by computing the largest Lyapunov exponents

Homoclinic orbits embedded in one-dimensional invariant manifolds of maps

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing onedimensional stable and unstable manifolds. The methods are implemented in MatcontM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows to detect all known codimension 1 and 2 bifurcations in 3D maps, including tangencies and generalized tangencies. MatcontM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing to compute all homoclinic orbits to saddle points in three-dimensional maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features

Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing one-dimensional stable and unstable manifolds. The methods are implemented in MatContM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits, and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows us to detect all known codimension 1 and 2 bifurcations in three-dimensional (3D) maps, including tangencies and generalized tangencies. MatContM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature, we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing us to compute all homoclinic orbits to saddle points in 3D maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis, and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features

Homoclinic orbits embedded in one-dimensional invariant manifolds of maps

We describe new methods for initializing the computation of homoclinic orbits for maps in a state space with arbitrary dimension and for detecting their bifurcations. The initialization methods build on known and improved methods for computing onedimensional stable and unstable manifolds. The methods are implemented in MatcontM, a freely available toolbox in Matlab for numerical analysis of bifurcations of fixed points, periodic orbits and connecting orbits of smooth nonlinear maps. The bifurcation analysis of homoclinic connections under variation of one parameter is based on continuation methods and allows to detect all known codimension 1 and 2 bifurcations in 3D maps, including tangencies and generalized tangencies. MatcontM provides a graphical user interface, enabling interactive control for all computations. As the prime new feature we discuss an algorithm for initializing connecting orbits in the important special case where either the stable or unstable manifold is one-dimensional, allowing to compute all homoclinic orbits to saddle points in three-dimensional maps. We illustrate this algorithm in the study of the adaptive control map, a 3D map introduced in 1991 by Frouzakis, Adomaitis and Kevrekidis, to obtain a rather complete bifurcation diagram of the resonance horn in a 1:5 Neimark-Sacker bifurcation point, revealing new features

Bistability and Stabilization of Human Visual Perception under Ambiguous Stimulation

We discuss a computational model that describes stabilization of percept choices under intermittent viewing of an ambiguous visual stimulus at long stimulus intervals. Let Toff and Ton be the time that the stimulus is off and on, respectively. The behavior was studied by direct numerical simulation in a grid of (Toff, Ton) values in a 2007 paper of Noest, van Ee, Nijs, and van Wezel. They found that both alternating and repetitive sequences of percepts can appear stably, sometimes even for the same values of Toff and Ton. Longer Toff, however, always leads to a situation where, after transients, only repetitive sequences of percepts exist. We incorporate Toff and Ton explicitly as bifurcation parameters of an extended mathematical model of the perceptual choices. We elucidate the bifurcations of periodic orbits responsible for switching between alternating and repetitive sequences. We show that the stability borders of the alternating and repeating sequences in the (Toff, Ton) -parameter plane consist of curves of limit point and period-doubling bifurcations of periodic orbits. The stability regions overlap, resulting in a wedge with bistability of both sequences. We conclude by comparing our modeling results with the experimental results obtained by Noest, van Ee, Nijs, and van Wezel

Bistability and stabilization of human visual perception under ambiguous stimulation

We discuss a computational model that describes stabilization of percept choices under intermittent viewing of an ambiguous visual stimulus at long stimulus intervals. Let Toff and Ton be the time that the stimulus is off and on, respectively. The behavior was studied by direct numerical simulation in a grid of (Toff, Ton) values in a 2007 paper of Noest, van Ee, Nijs, and van Wezel. They found that both alternating and repetitive sequences of percepts can appear stably, sometimes even for the same values of Toff and Ton. Longer Toff, however, always leads to a situation where, after transients, only repetitive sequences of percepts exist. We incorporate Toff and Ton explicitly as bifurcation parameters of an extended mathematical model of the perceptual choices. We elucidate the bifurcations of periodic orbits responsible for switching between alternating and repetitive sequences. We show that the stability borders of the alternating and repeating sequences in the (Toff, Ton)-parameter plane consist of curves of limit point and period-doubling bifurcations of periodic orbits. The stability regions overlap, resulting in a wedge with bistability of both sequences. We conclude by comparing our modeling results with the experimental results obtained by Noest, van Ee, Nijs, and van Wezel