# Stability analysis for ODEs with non constant inputs

For a project, I have to deal with systems of ODE's with non constant input such as:

$$\begin{cases}\dot x =I(t)x+x^2\\ \dot y=x\end{cases}$$ where I(t) is a random input (for example). In any case, I don't have $I(t)$ as an explicit function. It may be random or it may be something that depends on other system of ODE's. I wonder if these systems have a name and there is some sort of theory behind them (any reference will be very appreciated).

In particular I would like to be able to do stability analysis of some of these systems, such as computing equilibria and their stability, finding periodic orbits and so on. I'm not sure if this is possible, but I would like to know how much can be known analytically of such systems.

Thanks a lot!

• Can you provide any additional constraints on $I$? For example, is it a noisy process? If so, this can be reformulated as a stochastic differential equation. Is it bounded? Is it positive? Oct 15 '14 at 0:30
• @MrSlunk $I$ will be a random process (Gaussian white noise for example) in one type of situation. In the second type that I'm interested $I$ will be the output of integrating another ODE for $\tau$ units of time. After that, this ode will be integrated for $\tau$ units of time and so on and so forth (up to four ODE's). However, the first case -the one you mention- is the one I'm more interested in. Thanks Oct 15 '14 at 7:23

I think they are called non-autonomous differential equations, or non-autonomous dynamic systems, or also time-varying (since "t" explicitly appears in the righthand side) differential equations. Lyapunov stability theory might apply to specific cases to the best of my knowledge.

• well, they are non-autonomous for sure. But the complication is not there, but in the fact that we don't have a explicit representation of the non-autonomous term. Sep 7 '14 at 15:43
• well, if $I(t)$ depends on another system of ODE's then the system is called a cascade system if it is in the following form, $\dot{x}_{1}=f_{1}(t,x_{1},x_{2})$, $\dot{x}_{2}=f_{2}(t,x_{2})$ (you can take $x_{2}=I(t)$ for instance). the stability of these type of systems can be studied using input-to-state stability. you can check the book of Hassan K. Khalil called "Nonlinear Systems" for further information. Sep 7 '14 at 16:32

In the case where $I(t)$ is a noisy process, this would be modelled as a stochastic differential equation. The best reference I've found is this on by Gardiner.

Some observations:

$x,y$ decouple; So there won't be any periodic orbits in $(x,y)$.

Naively looking at the fixed points of this problem; we have $\dot{x}=0$ which implies $x=0$ or $x=-I(t)$. If we look at a ball about the $x=0$ solution then your stability is tied to whether $I$ is positive or negative. When we linearise about the $x=-I(t)$ solution, we find that the stability of this is exactly the opposite of the $x=0$ solution.

Essentially, $x$ with jump from $x=0$ to $x=I(t)$ when $I>0$ and back $x=0$ when $I<0$. $y$ will act like a kind of accumulator and not have any fixed points unless $I<0$ for all time.