MATCONT: a Matlab package for numerical bifurcation analysis of ODEs
MATCONT: a Matlab package for numerical bifurcation analysis of ODEs
复制标题
DOI：
10.1145/980175.980184
复制
发表时间：
2004-03
期刊：
影响因子：
--
通讯作者：
Annick Dhooge;W. Govaerts;Y. Kuznetsov
中科院分区：
文献类型：
--
作者：
Annick Dhooge;W. Govaerts;Y. Kuznetsov
考虑一般的参数化自治常微分方程dx/dt ∞ = f（x，α），其中x ∈ ∞ n是状态变量的向量，α ∈ ∞ m表示参数，f（x，α）∈ ∞ n.有几个交互式软件包用于分析由常微分方程定义的动力系统。最常用的是AUTO 86/97[1]、CONTENT[2]和XPPAUT。Matlab软件包MATCONT为动力系统的连续性和规范形分析提供了一个交互式环境。这种分析是对系统仿真的补充，系统仿真也包含在软件包中，可用于系统的识别、控制和优化。MATCONT旨在利用Matlab的强大功能。它是与连续工具箱CL_MATCONT并行开发的，CL_MATCONT是一个可以从命令行使用的Matlab例程包。我们考虑以下自治电子电路的模型，其中x，y和z是状态变量，β，γ，ν，r，a3，b3是参数：[公式见pdf]我们计算一个带有自由参数ν的平衡点的分支，从平凡解x = 0.00125，y =-0.001开始，z = 0.00052502，β = 0.5，γ =-0.6，r =-0.6，a3 = 0.32858，b3 = 0.93358，ν =-0.9，ε = 0.001。我们从这条曲线上的一个霍普夫点开始，选择ν作为自由参数，得到一条周期轨道曲线。我们在ν =-0.59575处检测到环面分叉点。我们继续两个参数ν，ε的环面分叉，发现它收缩到一个单一的点，减少ν的值（图2）。
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).