All Questions

I am looking at the discretization of an ODE: $$x_{n+1} = x_n + \alpha f(x_n),$$ where $x_n\in R^d$ and $f$ is continuously differentiable and such that $f(0)=0$ and $f'(0)$ is Hurwitz (i.e., the real ...

Let $A,B$ be $n\times n$ matrices. I am interested in the following ODE in $\mathbb{R}^n$ $$ \frac{dx_t}{dt}=Ax_t+Bx^+_t $$ where $x_t^+=(x^+_{1,t},...,x^+_{n,t})$ and $(\cdot)^+$ is the rectifier: $...

Consider the following autonomous system of differential equations: $$\frac{\mathrm d\mathbf x}{\mathrm dt} = \mathbf v(\mathbf x)$$ where $\mathbf x, \mathbf v \in \mathbb R^n$. Assume that $\...

Please, is there any reference for proposition below or does it perhaps follow from a standard fact? I've got it for some other problem but I actually do not know how to comment it in my article. ...

Consider a linear ODE system $$\dot x_k=\sum_{j=1}^ma_{kj}(t)x_j,\qquad k=1,\ldots, m,\quad a_{kj}(t)\in C[0,\infty).\quad (1)$$ Proposition. Suppose that $$\sup_{t\ge 0}\Big\{\int_0^t\Big(a_{kk}(...