linear time-invariant differential equation
State-space representation is another model for systems and is suitable for non-linear systems.
Essentially, state-space model change nth-order differential equation into n simultaneous first-order equations. It seems to me that the state-space model is the mostly used ODE modeling methods in systems biology.
Test signals with different waveforms can be used to study systems.
The basic analysis of a system is to evaluate the time response of a system.
A sensitivity analysis can yield the percentage of change in a specification as a function of a change in a system parameter.
In biology, many ODEs has nonlinear terms with product of variables. So, transfer function cannot be applied, but state-space method can be used.
Controllability and Observability are well understood in continuous time-invariant linear state-space model, see https://en.wikipedia.org/wiki/State-space_representation#State_variables
Stability: a system is stable if every bounded input yields a bounded output. So, does aging changes a stable gene network into an unstable network?
Observability: If the initial state vector x(t0) can be found from input u(t) and output y(t) over a finite interval of time from t0, the system is observable; otherwise it is unobservable.
Observability is the ability to deduce state variables from knowledge of input u(t) and output y(t).