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 dynamical properties of diffeomorphisms with homoclinic tangencies

We study bifurcations of a homoclinic tangency to a saddle fixed point without non-leading multipliers. We give criteria for the birth of an infinite set of stable periodic orbits, an infinite set of coexisting saddle periodic orbits with different instability indices, non-hyperbolic periodic orbits with more than one multiplier on the unit circle, and an infinite set of stable closed invariant curves (invariant tori). The results are based on the rescaling of the first-return map near the orbit of homoclinic tangency, which is shown to bring the map close to one of four standard quadratic maps, and on the analysis of the bifurcations in these maps

Dynamics of conservative Bykov cycles: tangencies, generalized Cocoon bifurcations and elliptic solutions

This paper presents a mechanism for the coexistence of hyperbolic and non-hyperbolic dynamics arising in a neighbourhood of a conservative Bykov cycle where trajectories turn in opposite directions near the two saddle-foci. We show that within the class of divergence-free vector fields that preserve the cycle, tangencies of the invariant manifolds of two hyperbolic saddle-foci densely occur. The global dynamics is persistently dominated by heteroclinic tangencies and by the existence of infinitely many elliptic points coexisting with non-uniformly hyperbolic suspended horseshoes. A generalized version of the Cocoon bifurcations for conservative systems is obtained.info:eu-repo/semantics/publishedVersio

Homoclinic orbits: Since Poincaré till today

The history and the contemporary results in homoclinic orbits are reported

Theory and computation of covariant Lyapunov vectors

Lyapunov exponents are well-known characteristic numbers that describe growth rates of perturbations applied to a trajectory of a dynamical system in different state space directions. Covariant (or characteristic) Lyapunov vectors indicate these directions. Though the concept of these vectors has been known for a long time, they became practically computable only recently due to algorithms suggested by Ginelli et al. [Phys. Rev. Lett. 99, 2007, 130601] and by Wolfe and Samelson [Tellus 59A, 2007, 355]. In view of the great interest in covariant Lyapunov vectors and their wide range of potential applications, in this article we summarize the available information related to Lyapunov vectors and provide a detailed explanation of both the theoretical basics and numerical algorithms. We introduce the notion of adjoint covariant Lyapunov vectors. The angles between these vectors and the original covariant vectors are norm-independent and can be considered as characteristic numbers. Moreover, we present and study in detail an improved approach for computing covariant Lyapunov vectors. Also we describe, how one can test for hyperbolicity of chaotic dynamics without explicitly computing covariant vectors.Comment: 21 pages, 5 figure

Computational dynamical systems analysis : Bogdanov-Takens points and an economic model

The subject of this thesis is the bifurcation analysis of dynamical systems (ordinary differential equations and iterated maps). A primary aim is to study the branch of homoclinic solutions that emerges from a Bogdanov-Takens point. The problem of approximating such branch has been studied intensively but neither an exact solution was ever found nor a higher-order approximation has been obtained. We use the classical ``blow-up'' technique to reduce an appropriate normal form near a Bogdanov-Takens bifurcation in a generic smooth autonomous ordinary differential equations to a perturbed Hamiltonian system. With a regular perturbation method and a generalization of the Lindstedt-Poincare' perturbation method, we derive two explicit third-order corrections of the unperturbed homoclinic orbit and parameter value. We prove that both methods lead to the same homoclinic parameter value as the classical Melnikov technique and the branching method. We show that the regular perturbation method leads to a ``parasitic turn'' near the saddle point while the Lindstedt-Poincare' solution does not have this turn, making it more suitable for numerical implementation. To obtain the normal form on the center manifold, we apply the standard parameter dependent center manifold reduction combined with the normalization, using the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we correct the parameter transformation existing in the literature. The generic homoclinic predictors are applied to explicitly compute the homoclinic solutions in the Gray-Scott kinetic model. The actual implementation of both predictors in the MATLAB continuation package MatCont and five numerical examples illustrating its efficiency are discussed. Besides, the thesis discusses the possibility to use the derived homoclinic predictor of generic ordinary differential equations to continue the branches of homoclinic tangencies in the Bogdanov-Takens map. The second part of this thesis is devoted to the application of bifurcation theory to analyze the dynamic and chaotic behaviors of a nonlinear economic model. The thesis studies the monopoly model with cubic price and quadratic marginal cost functions. We present fundamental corrections to the earlier studies of the model and a complete discussion of the existence of cycles of period 4. A numerical continuation method is used to compute branches of solutions of period 5, 10, 13 and 17 and to determine the stability regions of these solutions. General formulas for solutions of period 4 are derived analytically. We show that the solutions of period 4 are never linearly asymptotically stable. A nonlinear stability criterion is combined with basin of attraction analysis and simulation to determine the stability region of the 4-cycles. This corrects the erroneous linear stability analysis in previous studies of the model. The chaotic and periodic behavior of the monopoly model are further analyzed by computing the largest Lyapunov exponents, and this confirms the above mentioned results

Homoclinic puzzles and chaos in a nonlinear laser model

We present a case study elaborating on the multiplicity and self-similarity of homoclinic and heteroclinic bifurcation structures in the 2D and 3D parameter spaces of a nonlinear laser model with a Lorenz-like chaotic attractor. In a symbiotic approach combining the traditional parameter continuation methods using MatCont and a newly developed technique called the Deterministic Chaos Prospector (DCP) utilizing symbolic dynamics on fast parallel computing hardware with graphics processing units (GPUs), we exhibit how specific codimension-two bifurcations originate and pattern regions of chaotic and simple dynamics in this classical model. We show detailed computational reconstructions of key bifurcation structures such as Bykov T-point spirals and inclination flips in 2D parameter space, as well as the spatial organization and 3D embedding of bifurcation surfaces, parametric saddles, and isolated closed curves (isolas).Comment: 28 pages, 23 figure