MATCONT: a Matlab package for numerical bifurcation analysis of ODEs

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 ∈ ℝ^m 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).