Numerical Bifurcation Analysis of Maps (original) (raw)
Numerical Methods for Two-Parameter Local Bifurcation Analysis of Maps
SIAM Journal on Scientific Computing, 2007
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. LP, PD and NS), 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.
Numerical analysis of the flip bifurcation of maps
Applied Mathematics and Computation, 1991
A numerical procedure for the analysis of the flip bifurcation of maps in R" is described. The procedure is based on normal forms and the center-manifold approach and can be applied to study period doubling of limit cycles in autonomous systems as well as period doubling of periodic solutions of time-periodic systems. The procedure has been programmed in FORTRAN-77, and the code is available on request from the second author. 1. *This work has been supported by the Italian Ministry of Scientific Research and Technology, contract MURST 40% Teoria dei sistemi e de1 controllo.
Calculation of Bifurcation Curves by Map Replacement
International Journal of Bifurcation and Chaos, 2010
The complex bifurcation structure in the parameter space of the general piecewise-linear scalar map with a single discontinuity — nowadays known as nested period adding structure — was completely studied analytically by N. N. Leonov already 50 years ago. He used an elegant and very efficient recursive technique, which allows the analytical calculation of the border-collision bifurcation curves, causing the nested period adding structure to occur. In this work, we have demonstrated that the application of Leonov's technique is not resticted to that particular bifurcation structure. On the contrary, the presented map replacement approach, which is an extension of Leonov's technique, allows the analytical calculation of border-collision bifurcation curves for periodic orbits with high periods and complex symbolic sequences using appropriate composite maps and the bifurcation curves for periodic orbits with much lower periods.
Bifurcation Analysis of Periodic Orbits of Maps in Matlab
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., LP, PD and NS), 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 i...
Numerical Bifurcation Analysis of Homoclinic Orbits Embedded in One-Dimensional Manifolds of Maps
ACM Transactions on Mathematical Software, 2018
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.
Physica D: Nonlinear …, 1999
Bifurcation diagrams of periodic windows of scalar maps are often found to be not only topologically equivalent, but in fact to be related by a nearly linear change of parameter coordinates. This effect has been observed numerically for one-parameter families of maps, and we offer an analytical explanation for this phenomenon. We further present numerical evidence of the same phenomenon for two-parameter families, and give a mathematical explanation like that for the one-parameter case.
Continuation techniques and interactive software for bifurcation analysis of ODEs and iterated maps
Physica D: Nonlinear Phenomena, 1993
We present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase-parameter space. The invariant sets involved in our study are equilibrium points and limit cycles of autonomous ODEs, periodic solutions of time-periodic nonautonomous ODEs, fixed points and periodic orbits of iterated maps. The more general concept of a continuation strategy is also discussed. It allows the analysis of various singularities of generic systems and of their mutual relationships. The approach is extended to codimension three singularities. We introduce several bifurcation functions and show how to use them to construct well-posed continuation problems. The described continuation technique is supported by an interactive graphical program called LOCBIF. We discuss briefly the concepts of the LOCBIF interface and give some examples of typical applications.
Numerical Bifurcation Analysis of Large Scale Systems
Preface These lecture notes are, like many websites, " under construction ". Some parts are based on material obtained from Dr. (Utrecht University). I am grateful to these colleagues for providing me their material.
An algebraic analysis for bifurcation problems
Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 1991
An algorithm for computing the codimensions of tangent space for bifurcation equations using the Grobner basis is presented. When a bifurcation equation with perturbation added becomes equivalent to the original bifurcation problem, the perturbation term belongs to the tangent space; this contains an ideal of a certain equation set on real functions. Therefore, perturbation terms that prevent the equivalence relationship belong to the codimensional space of the tangent space. On the other hand, the residue system of ideals on polynomial rings is found as a set of terms that are not multiples of the highest order of the Grobner basis. languages seems preferable. Furthermore, application of the Grobner basis of a polynomial ideal simplifies the algorithm. We can expect that the higher-dimensional case can be dealt with similarly.
Bifurcations of compactifying maps
Reports on Mathematical Physics, 1994
The concept'of bifurcation belongs to one of Poincart's pioneering studies. In 1885, Poincare introduced the notion of bifurcation, discussed the elements of the theory and described the phenomenon to an extremely detailed extent. The basic elements of the theory may be grouped into three areas: critical solutions, dynamic stability and structural stability. Our work concerns the area of structural stability, namely we use the topological degree theory methods to get local and global bifurcation results.
Topological Degree Approach to Bifurcation Problems
Topological Degree Approach to Bifurcation Problems, 2008
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GLOBAL BIFURCATIONS OF CLOSED INVARIANT CURVES IN TWO-DIMENSIONAL MAPS: A COMPUTER ASSISTED STUDY
International Journal of Bifurcation and Chaos, 2005
In this paper we describe some sequences of global bifurcations involving attracting and repelling closed invariant curves of two-dimensional maps that have a fixed point which may lose stability both via a supercritical Neimark bifurcation and a supercritical pitchfork or flip bifurcation. These bifurcations, characterized by the creation of heteroclinic and homoclinic connections or homoclinic tangles, are first described through qualitative phase diagrams and then by several numerical examples. Similar bifurcation phenomena can also be observed when the parameters in a two-dimensional parameter plane cross through many overlapping Arnold's tongues.