New features of the software MatContfor bifurcation analysis of dynamical systems

Abstract

Bifurcation software is an essential tool in the study of dynamical systems. From the beginning (the first packages were written in the 1970's) it was also used in the modelling process, in particular to determine the values of critical parameters. More recently, it is used in a systematic way in the design of dynamical models and to determine which parameters are relevant. MATCONT and CL_MATCONT are freely available MATLAB numerical continuation packages for the interactive study of dynamical systems and bifurcations. MATCONT is the GUI-version, CL_MATCONT is the command-line version. The work started in 2000 and the first publications appeared in 2003. Since that time many new functionalities were added. Some of these are fairly simple but were never before implemented in continuation codes, e.g. Poincare´maps. Others were only available as toolboxes that can be used by experts, e.g. continuation of homoclinic orbits. Several others were never implemented at all, such as periodic normal forms for codimension 1 bifurcations of limit cycles, normal forms for codimension 2 bifurcations of equilibria, detection of codimension 2 bifurcations of limit cycles, automatic computation of phase response curves and their derivatives, continuation of branch points of equilibria and limit cycles. New numerical algorithms for these computations have been published or will appear elsewhere; here we restrict to their software implementation. We further discuss software issues that are in practice important for many users, e.g. how to define a new system starting from an existing one, how to import and export data, system descriptions, and computed results.

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

“…They fit perfectly into a continuation context, where limit cycles and their bifurcations are computed using the BVP-approach [32], without numerical approximation of the Poincaré map or its derivatives. Being implemented into the matlab toolbox matcont [2,3], the methods developed are freely available to assist an advanced two-parameter bifurcation analysis of dynamical systems generated by ODEs and maps from various applications.…”

“…Solving M (0) = 0 and substituting p, q we obtain c 2 = θΘ(δ − 1) 3 − δ∆(1 − θ) 3 (δ − 1) 3 (2δθ − δ − θ) .…”

“…It is hard to use simulations to characterize such transitions correctly and efficiently. Numerical continuation software such as auto [1] or matcont [2,3] may be used to track bifurcations from a stable equilibrium to a periodic oscillation by a Hopf bifurcation and even the appearance of (un)stable invariant tori with multi-frequency oscillations by a secondary Hopf, or Neimark-Sacker bifurcation. Bifurcations of these invariant tori T m≥2 into other tori or chaos, however, are out of reach of the standard numerical analysis.…”

Emergence and Bifurcations of Lyapunov Manifolds in Nonlinear Wave Equations

“…The result is listed in the Appendix. We then performed numerical bifurcation analysis, using MATCONT (Dhooge et al 2008), to see how various modes interact. In the averaged system, nontrivial equilibria correspond to periodic solutions in the original system and limit cycles correspond to 2-tori, a Hopf bifurcation of a limit cycle in the averaged system corresponds with a Neimark-Sacker bifurcation in the original system yielding generically a 3-torus.…”

“…Our analysis shows that perturbation analysis does not fail, but that surprisingly enough, a complicated bifurcation structure destroys this picture for relatively small values of the small parameter ε. For this part of the analysis, we use higher order averaging in the case of near-resonance (Sanders et al 2007) and we use numerical bifurcation techniques as described in Kuznetsov (2004) and implemented in Kuznetsov andLevitin (1995-2001), Dhooge et al (2008).…”