Numerical Bifurcation Analysis of Maps: From Theory to Software

This book combines a comprehensive state-of-the-art analysis of bifurcations of discrete-time dynamical systems with concrete instruction on implementations (and example applications) in the free MATLAB® software MatContM developed by the authors. While self-contained and suitable for independent study, the book is also written with users in mind and is an invaluable reference for practitioners. Part I focuses on theory, providing a systematic presentation of bifurcations of fixed points and cycles of finite-dimensional maps, up to and including cases with two control parameters. Several complementary methods, including Lyapunov exponents, invariant manifolds and homoclinic structures, and parts of chaos theory, are presented. Part II introduces MatContM through step-by-step tutorials on how to use the general numerical methods described in Part I for simple dynamical models defined by one- and two-dimensional maps. Further examples in Part III show how MatContM can be used to analyze more complicated models from modern engineering, ecology, and economics. Provides state-of-the-art analysis of bifurcations of discrete-time dynamical systems Theory is connected with practical applications, as well as step-by-step tutorials on how to analyze particular bifurcations using the free MATLAB® software MatContM This book is an ideal reference volume for professionals searching for results for a particular bifurcatio

Homoclinic saddle to saddle-focus transitions in 4D systems

A saddle to saddle-focus homoclinic transition when the stable leading eigenspace is three-dimensional (called the 3DL bifurcation) is analyzed. Here a pair of complex eigenvalues and a real eigenvalue exchange their position relative to the imaginary axis, giving rise to a 3D stable leading eigenspace at the critical parameter values. This transition is different from the standard Belyakov bifurcation, where a double real eigenvalue splits either into a pair of complex-conjugate eigenvalues or two distinct real eigenvalues. In the wild case, we obtain sets of codimension 1 and 2 bifurcation curves and points that asymptotically approach the 3DL bifurcation point and have a structure that differs from that of the standard Belyakov case. We give an example of this bifurcation in a perturbed Lorenz–Stenflo 4D ordinary differential equation model

Improved homoclinic predictor for Bogdanov-Takens bifurcation

An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical "blow-up" technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit rst- and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its eciency are discussed

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

On local bifurcations in neural field models with transmission delays

Neural fieldmodels with transmission delays may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results

Initialization of Homoclinic Solutions near Bogdanov-Takens Points: Lindstedt-Poincaré Compared with Regular Perturbation Method

To continue a branch of homoclinic solutions starting from a Bogdanov-Takens (BT) point in parameter and state space, one needs a predictor based on asymptotics for the bifurcation parameter values and the corresponding small homoclinic orbits in the phase space. We derive two explicit asymptotics for the homoclinic orbits near a generic BT point. A recent generalization of the Lindstedt-Poincaré (L-P) method is applied to approximate a homoclinic solution of a strongly nonlinear autonomous system that results from blowing up the BT normal form. This solution allows us to derive an accurate second-order homoclinic predictor to the homoclinic branch rooted at a generic BT point of an $n$-dimensional ordinary differential equation (ODE). We prove that the method leads to the same homoclinicity conditions as the classical Melnikov technique, the branching method, and the regular perturbation (R-P) method. However, it is known that the R-P method leads to a “parasitic turn‿ near the saddle point. The new asymptotics based on the L-P method do not have this turn, making them more suitable for numerical implementation. We show how to use these asymptotics to calculate the initial data to continue homoclinic orbits in two free parameters. The new homoclinic predictors are implemented in the MATLAB continuation package MatCont to initialize the continuation of homoclinic orbits from a BT point. Two examples with multidimensional state spaces are included

Analysis of bifurcations of limit cycles with Lyapunov exponents and numerical normal forms

In this paper we focus on the combination of normal form and Lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous ODEs. The normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary-value problems. The formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations, where 3-tori and 4-tori can be present. We apply our techniques to the study of a known laser model, a novel model from population biology, and a model of mechanical vibrations. These models exhibit Limit Point–Neimark–Sacker, Period-Doubling–Neimark–Sacker, and double Neimark–Sacker bifurcations. Lyapunov exponents are computed to numerically confirm the results of the normal form analysis, in particular with respect to the existence of stable invariant tori of various dimensions. Conversely, the normal forms are essential to understand the significance of the Lyapunov exponents

Numerical continuation of connecting orbits of maps in MATLAB

We present new or improved methods to continue heteroclinic and homoclinic orbits to fixed points in iterated maps and to compute their fold bifurcation curves, corresponding to the tangency of the invariant manifolds. The proposed methods are applicable to general n-dimensional maps and are implemented in matlab. They are based on the continuation of invariant subspaces (CIS) algorithm, which is presented in a novel way. The systems of defining equations include the Riccati equations appearing in CIS for bases of the generalized stable and unstable eigenspaces. We use the bordering techniques to continue the folds, and provide full algorithmic details on how to treat the Jacobian matrix of the defining system as a sparse matrix in matlab. For a special - but important in applications - case n = 2 we describe the first matlab implementation of known algorithms to grow one-dimensional stable and unstable manifolds of the fixed points of noninvertible maps. The methods are applied to study heteroclinic and homoclinic connections in the generalized Hénon map