Reference Request: Parameter Dependent Center Manifold Theorem for ODEs

Question

Suppose we have an $n$-dimensional first order ODE of the form $\frac{dx}{dt}= f_{\mu}(x)$ with $\mu \in \mathbb{R}^k$ a parameter and with an equilibrium at $x=0$ $(f_{\mu}(0) =0)$.

For fixed $\mu$ I am able to apply the center manifold theorem which will give me information about the stable, unstable and center manifolds.

I am curious to understand how these manifolds depend on the parameter $\mu$.

Does anyone know a good reference for the corresponding theorem?

Thanks in advance.

Answer 1

So you need extend the system as pointed out by John: \begin{cases} x'=f_\mu(x),\\ \mu'=0 \end{cases} and then you apply the center-unstable, center-stable or center manifold theorem depending on which manifold's smoothness you want to show.

Kelley, Al. "Stability of the center-stable manifold." Journal of Mathematical Analysis and Applications 18.2 (1967): 336-344.

Answer 2

Actually you don't need a new theorem. The usual trick is to add the equation $\mu'=0$, and so to consider the extended vector field $F(x,\mu)=(f_\mu(x),0)$. Notice that the new equation $$ \begin{cases} x'=f_\mu(x),\\ \mu'=0 \end{cases} $$ has an additional central direction and so its center manifold has one additional dimension (applying the center manifold theorem that you already mentioned). The main advantage is that you get for free that the invariant manifolds are as smooth on $\mu$ as the map $(x,\mu)\mapsto f_\mu(x)$ is (as usual, you write them as a graph over the central directions, so including $\mu$).

Good references are (depending really on what you need):

• J. Carr (1981), Applications of Centre Manifold Theory, Springer-Verlag.

• J. Guckenheimer and P. Holmes (1983), Nonlinear Oscillations, Dynamical systems and Bifurcations of Vector Fields. Springer-Verlag.

• A. Vanderbauwhede (1989). Center Manifolds, Normal Forms and Elementary Bifurcations, In Dynamics Reported, Vol. 2. Wiley.

Answer 3

This book should be able to aide you. The second one is more readable and just better put together in my option.

1.)Normally Hyperbolic Invariant Manifolds in Dynamical Systems by Stephen Wiggins

2.)Nonautonomous Dynamical Systems (Mathematical Surveys and Monographs) by Peter E. Kloeden (Author), Martin Rasmussen (Author)

http://www.amazon.com/Hyperbolic-Invariant-Manifolds-Dynamical-Mathematical/dp/038794205X/ref=sr_1_1?ie=UTF8&qid=1462209154&sr=8-1&keywords=Normally+Hyperbolic+Invariant+Manifolds+in+Dynamical+Systems+by+Stephen+Wiggins

http://www.amazon.com/Nonautonomous-Dynamical-Systems-Mathematical-Monographs/dp/0821868713/ref=sr_1_1?s=books&ie=UTF8&qid=1462209321&sr=1-1&keywords=dynamical+systems+with+manifolds

Answer 4

You might look up "Shoshitaishvili reduction principle". The original paper is A. N. Šošitaĭšvili, Trudy Sem. Petrovsk. Vyp. 1 (1975), 279–309, MR0478239.