Today, the tutor of our dynamical system course said that in the exam one part will be to determine equilibria and to compute the local equations of stable and unstable manifold.

I do not know what is meant by giving the local equations for stable/ unstable manifold.

Can you give me a task/ example?


You need to be more specific, at least whether you have discrete or continuous time. But a (local) stable manifold is simply a graph over the stable space, so you can (with reasonable regularity assumptions) write it say with the first few terms of a Taylor series.

For example, take $x'=-x+xy$, $y'=y+x^2$. The stable manifold will be the graph of the function $\varphi(x)=ax^2+bx^3+\cdots$ (in this example you can go up to any order), that is, the set of points $(x,y)$ with $y=\varphi(x)$. Computing you get $$ y'=\varphi'(x)x'=(2ax+3bx^2+\cdots)(-x+x(ax^2+bx^3+\cdots)). $$ On the other hand, $$ y'=y+x^2=ax^2+bx^3+\cdots+x^2. $$ The two right-hand sides are equal (write the first few terms), which gives you equations to determine $a$, $b$, etc successively.


We are taking as definition of stable manifold any smooth invariant curve that is tangent to the stable space at the origin. That is, $$V^s=\{(x,\varphi(x)):x\in(-\delta,\delta)\}$$ for some $C^k$ function $\varphi\colon(-\delta,\delta)\to\mathbb R$ with $\varphi(0)=\varphi'(0)=0$ and with the property that any solution starting in $V^s$ remains in $V^s$ for all positive time. This means that if $(x(t),y(t))$ is a solution with $(x(t_0),y(t_0))\in V^s$ for some $t_0$, then $(x(t),y(t))\in V^s$ for all $t>t_0$. This is why we must have $y(t)=\varphi(x(t))$ (written above simply as $y=\varphi(x)$.

  • $\begingroup$ I am not sure what you mean with this function $\phi(x)$. To my computations, the equilibrium is $(0,0)$, the linearization matrix is $A=\begin{pmatrix}-1 & 0\\0 & 1\end{pmatrix}$ and $A$ has two eigenvalues $\lambda_1=+1$ and $\lambda_2=-1$. Hence, the solutions should be $x=x_0e^{t}, y=y_0e^{-t}$. So, $(0,0)$ is a saddle and locally, the $x$-axis should be the stable manifold where the $y$-axis is the unstable manifold? $\endgroup$
    – Rhjg
    Jan 15 '16 at 21:36
  • $\begingroup$ Fine if it is, the example applies to whatever nonlinear part you want, provided that you keep the linear part unchanged. $\endgroup$
    – John B
    Jan 15 '16 at 22:50
  • $\begingroup$ Sorry! I still do not understand you and what you are aiming at. $\endgroup$
    – Rhjg
    Jan 15 '16 at 22:54
  • $\begingroup$ You write "the $x$-axis should be the stable manifold where the $y$-axis is the unstable manifold". This might be true for some examples, but if fails in general. What then your definition of "stable manifold"? $\endgroup$
    – John B
    Jan 15 '16 at 22:56
  • $\begingroup$ Good question. We never had such situations. - Ok, so back to your idea. I cannot see where your function comes from. $\endgroup$
    – Rhjg
    Jan 15 '16 at 22:59

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