The background the ignition delay problem in combustion.
My state vector is $y=[T,Y_i]^T$, $T$ is temperature and $Y_i$ is mass fraction of species.
The evolution of the state vector can be described as an ODEs $\dot y =f(y)$
The quantity of interest is the iginition delay time $\tau$ defined as: $\ddot T(\tau) = 0$
I do not want to integrate along the time to guess where is the ignition delay. However, I cannot find math term to describe the problem. It is very similar to the minimum time control problem. But there is no control and the ODEs is autonomous.
Definition of ignintion delay looks like this: It is the inflection point of temperature with time.
You have the state equation
$$\dot{\boldsymbol{x}} =\boldsymbol{f}(\boldsymbol{x})$$
in which $\boldsymbol{x} = \begin{bmatrix} T &Y_i\end{bmatrix}^T$ and $\boldsymbol{f} = \begin{bmatrix}f_1 & f_2 \end{bmatrix}^T.$
What you are looking for is $$\ddot{T}=\dfrac{d}{dt}{\dot{T}}=\dfrac{d}{dt}f_1(\boldsymbol{x})=\dfrac{\partial f_1}{\partial\boldsymbol{x}}\dot{\boldsymbol{x}}=\begin{bmatrix}\dfrac{\partial f_1}{\partial x_1} & \dfrac{\partial f_1}{\partial x_2}\end{bmatrix}\begin{bmatrix}f_1\\f_2 \end{bmatrix}=f_1\dfrac{\partial f_1}{\partial x_1}+f_2\dfrac{\partial f_1}{\partial x_2}.$$
Evaluating the last expression at $\tau$ and setting it equal to $0$ will give you a nonlinear equation that you need to solve. Note, that you will also need to know at the same time what your $\boldsymbol{x}(\tau)$ is.
I think you first need to solve the given condition for $\boldsymbol{x}(\tau)$ and then find the $\tau$ that is corresponding to this $\boldsymbol{x}(\tau)$.