A study of resonance tongues near a Chenciner bifurcation using MatcontM

MatcontM is a matlab toolbox for numerical analysis of bifurcations of fixed points and periodic orbits of maps. It computes codim 1 bifurcation curves and supports the computation of normal coefficients including branch switching from codim 2 points to secondary curves. Recently, the initialization and computation of connecting orbits was improved. Moreover, a graphical user interface was added enabling interactive control of all these computations. To further support these computations it allows to compute orbits of the map and its iterates and to represent them in 2D, 3D and numeric windows. We demonstrate the use of the toolbox in a study of Arnol'd tongues near a degenerate Neimark-Sacker (Chenciner) bifurcation. Here we illustrate the recent theory of [Baesens&Mackay,2007] how resonance tongues interact with a quasi-periodic saddle-node bifurcation of invariant curves in maps. Using normal form coefficients we find evidence for one of their cases, but not the other. Actually, we find another unfolding, i.e. a third possibility. We also find a structure that resembles a quasi-periodic cusp bifurcation of invariant curves

Bifurcations of maps: numerical algorithms and applications

Dynamical systems theory provides mathematical models for systems which evolve in time according to a rule, originally expressed in analytical form as a system of equations. Discrete-time dynamical systems defined by an iterated map depending on control parameters, \begin{equation} \label{Map:g} g(x,\alpha) := f^{(J)}(x,\alpha)= \underbrace{f(f(f(\cdots f}_{J \mbox{~times}}(x,\alpha),\alpha),\alpha),\alpha), \end{equation} appear naturally in, e.g., ecology and economics, where $x\in \R^n$ and $\alpha \in \R^k$ are vectors of state variables and parameters, respectively. %The system dynamics describe a sequence of points \left\{x_k{\right\} \subset \R^n (orbit), provided an initial $x_0 \in \R^n$ is given. The main goal in the study of a dynamical system is to find a complete characterization of the geometry of the orbit structure and the change in orbit structure under parameter variation. An aspect of this study is to identify the invariant objects and the local behaviour around them. This local information then needs to be assembled in a consistent way by means of geometric and topological arguments, to obtain a global picture of the system. At local bifurcations, the number of steady states can change, or the stability properties of a steady state may change. The computational analysis of local bifurcations usually begins with an attempt to compute the coefficients that appear in the normal form after coordinate transformation. These coefficients, called critical normal form coefficients, determine the direction of branching of new objects and their stability near the bifurcation point. After locating a codim 1 bifurcation point, the logical next step is to consider the variation of a second parameter to enhance our knowledge about the system and its dynamical behaviour. % % In codim 2 bifurcation points branches of various codim 1 bifurcation curves are rooted. % These curve can be computed by a combination of parameter-dependent center manifold reduction and asymptotic expressions for the new emanating curves. We implemented new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits of maps in the {\sc Matlab} software package {\sc Cl\_MatcontM}. This includes the numerical continuation of fixed points of iterates of the map with one control parameter, detecting and locating their bifurcation points, and their continuation in two control parameters, as well as detection and location of all codim 2 bifurcation points on the corresponding curves. For all bifurcations of codim 1 and 2, the critical normal form coefficients are computed with finite directional differences, automatic differentiation and symbolic derivatives of the original map. Asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period to continue the double and quadruple period bifurcations. In the case $n=2$ we compute one-dimensional invariant manifolds and their transversal intersections to obtain initial connections of homoclinic and heteroclinic orbits orbits to fixed points of (\ref{Map:g}). We continue connecting orbits, using an algorithm based on the continuation of invariant subspaces, and compute their fold bifurcation curves, corresponding to the tangencies of the invariant manifolds. {\sc Cl\_MatcontM} is freely available at {\bf www.matcont.ugent.be} and {\bf www. sourceforge.net}

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

A Taylor series-based continuation method for solutions of dynamical systems

International audienceThis paper describes a generic Taylor series based continuation method, the so-called Asymptotic Numerical Method, to compute the bifurcation diagrams of nonlinear systems. The key point of this approach is the quadratic recast of the equations as it allows to treat in the same way a wide range of dynamical systems and their solutions. Implicit Differential-Algebraic Equations, forced or autonomous, possibly with time-delay or fractional order derivatives are handled in the same framework. The static, periodic and quasi-periodic solutions can be continued as well as transient solutions

A computational and geometric approach to phase resetting curves and surfaces

"Vegeu el resum a l'inici del document del fitxer adjunt"

A mathematical framework for critical transitions: normal forms, variance and applications

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. For the Stommel-Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator-inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator-prey model explosive population growth near a codimension two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio