I try to verify example 20.4.10 from Wiggins - Introduction to Applied Nonlinear Dynamical Systems and Chaos and I am quite new to the topic so please be patient.

In the book is written that the family of maps \begin{equation*} f(x,\mu) = \mu + ax^2 \end{equation*} is a generic family. If I understand the theory correctly I have to show that the 2-jet of $f(x,\mu)$ is transversal to the submanifold $B := \{(x,f(x), Df(x),D^2f(x)) \in J^2(\mathbb{R},\mathbb{R})~|~ f(x) = Df(x) = 0\}$ and the assertion follows from Thoms transversality theorem.

To get the 2-jet of $f$ I apply the 2-jet extension and get: \begin{equation*} j^2f(x,\mu) = j^2(\mu+ax^2) = (x, \mu+ax^2, 2ax, 2a)\end{equation*}

To show that $j^2f \pitchfork B$ I have to show that \begin{equation} T_xj^2f(T_xJ^2(\mathbb{R},\mathbb{R})) + T_{j^2f}B = T_{j^2f}J^2(\mathbb{R},\mathbb{R}).\end{equation} (Are the spaces in the equation above correct?)

Here are the questions I have:

  1. What is the property that is meant to be generic for $f(x,\mu)$? To be a versal deformation or to have an non hyperbolic fixed point?
  2. At which point should I show that$j^2f \pitchfork B$? Should that be the rest point of $f(x,\mu)$?
  3. How do I show that $T_xj^2f(T_xJ^2(\mathbb{R},\mathbb{R})) + T_{j^2f}B = T_{j^2f}J^2(\mathbb{R},\mathbb{R}).$

Thank you for your help!


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