生成2D系统的分叉图 [英] Generate bifurcation diagram for 2D system

问题描述

一维系统的分叉图很清楚，但是如果我有以下格式的二维系统 dx/dt = f(x，y，r)，dy/dt = g(x，y，r)

Drawing bifurcation diagram for 1D system is clear but if I have 2D system on the following form dx/dt=f(x,y,r), dy/dt=g(x,y,r)

我想在MATLAB中为x与r生成分叉图.这样做的主要目的是什么，或者有什么可能对我有帮助的提示?

And I want to generate a bifurcation diagram in MATLAB for x versus r. What is the main idea to do that or any hints which could help me?

推荐答案

您首先必须做一些数学运算:

将每个函数设置为零会为您提供两个函数y(x)(称为零线)，您可以在相图中进行绘制.两条线相交的位置是系统的固定点(平衡点).

Setting each of the functions to zero gives you two functions y(x) (called the nullclines), which you can plot in a phase diagram. Where the two lines intersect are the fixed-points (equilibria) of your system.

现在，您必须将系统的jacobian插入每个固定点，这将为您提供系统的线性稳定性分析.

Now, you have to take the jacobian of your system and plug each of those fixed-points in, which will give you the linear stability analysis of the system.

现在可以通过改变r(分叉参数)来计算固定点的位置和每个点的稳定性.

The location of the fixed points and the stability of each point can now be computed as a you vary r (the bifurcation parameter).

用于编程:

-使用牛顿方法(在MATLAB中为 fsolve )来查找方程为零的位置- eig 将帮助您找到系统的特征值.

-use newton's method (fsolve in MATLAB) to find where the equations are zero -eig will help you find the eigenvalues of the system.

但是

这取决于您的系统.

如果您要寻找极限环或混乱之类的东西，则必须使用其中一个极点求解器，然后分析变得更加棘手.我想您可以开发一个poincare-bendixson算法，但这会涉及到算法，并且细节将取决于您的系统.

If you're supposed to be looking for limit cycles or chaos or something, you'll have to use one of the ode solvers and then the analysis becomes more tricky. I suppose you could develop a poincare-bendixson algorithm, but that would be involved and details would depend on your system.

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