Abstract
We consider generic parameterized autonomous ODEs of the form dx/dt ≡ ẋ = f(x, α), where x ∈ ℝn is the vector of state variables, α ∈ ℝm represents parameters, and f(x, α) ∈ ℝn. There are several interactive software packages for analysis of dynamical systems defined by ODEs. The most widely used are AUTO86/97[1], CONTENT[2] and XPPAUT.The Matlab software package MATCONT provides an interactive environment for the continuation and normal form analysis of dynamical systems. This analysis is complementary to the simulation of the systems which is also included in the package and can be used in their identification, control, and optimization. MATCONT is designed to exploit the power of Matlab. It is developed in parallel with the continuation toolbox CL_MATCONT, a package of Matlab routines that can be used from the command line.We consider the following model of an autonomous electronic circuit where x, y and z are state variables and β,γ,ν,r,a3,b3 are parameters: [see pdf for formula]We compute a branch of equilibria with free parameter ν stating from the trivial solution x = 0.00125, y = -0.001, z = 0.00052502 at β = 0.5, γ = -0.6, r = -0.6, a3 = 0.32858, b3 = 0.93358, ν = -0.9, ε = 0.001. We start a curve of periodic orbits from a Hopf point on this curve choosing ν as the free parameter. We detect a torus bifurcation point at ν = -0.59575. We continue the torus bifurcation in two parameters ν, ε and find that it shrinks to a single point for decreasing values of ν (Figure 2).
- E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede and X. J. Wang, AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), User's Guide, Concordia University, Montreal, Canada 1997. (http://indy.cs.concordia.ca).Google Scholar
- Yu. A. Kuznetsov and V. V. Levitin, CONTENT: A multiplatform environment for analyzing dynamical systems, Dynamical Systems Laboratory, CWI, Amsterdam 1995--1997 (ftp.cwi.nl/pub/CONTENT).Google Scholar
Index Terms
- MATCONT: a Matlab package for numerical bifurcation analysis of ODEs
Recommendations
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
MATCONT is a graphical MATLAB software package for the interactive numerical study of dynamical systems. It allows one to compute curves of equilibria, limit points, Hopf points, limit cycles, period doubling bifurcation points of limit cycles, and fold ...
Numerical continuation of fold bifurcations of limit cycles in MATCONT
ICCS'03: Proceedings of the 1st international conference on Computational science: PartIMATCONT is a Matlab continuation package with a GUI for the interactive numerical study of a range of parameterized nonlinear problems. In the case of ODEs it allows to compute curves of equilibria, limit points, Hopf points, limit cycles, period ...
Stability and Hopf bifurcation of a Lorenz-like system
Hopf bifurcation is one of the important dynamical behaviors. It could often cause some phenomena, such as quasiperiodicity and intermittency. Consequently, chaos will happen due to such dynamical behaviors. Since chaos appears in the Lorenz-like system,...
Comments